ChemWiki - User contributions [en]

Rep:Mod:SA4213MgO

2016-02-25T14:53:01Z

Sa4213: /* Conclusion */

== Introduction ==

In the experiment the thermal expansion of magnesium oxide (MgO), its phonon dispersion and energy were studied using a quasi-harmonic approximation and a molecular dynamics approach. The energies and vibrations in the lattice were calculated to find the free energy of the crystal and thermal expansion.

The quasi-harmonic approximation (QHA) was used to compute volume-dependent thermal effects - such as the thermal expansion coefficient. This model is based on a harmonic oscillator, where each possible bond length is approximated by a quadratic function; but the QHA contains an additional anharmonic factor. This factor more allows the approximation to more closely mirror reality and as such we can account for thermal expansion, as the equilibrium bond length is no longer independent of temperature.

The molecular dynamics approach governs the motion of the atoms with Newtonian mechanics from interatomic forces. Therefore it is necessary to provide initial velocities and positions of the atoms, and then the computation propagates by iteratively repeating the algorithm with a set time step. New positions and velocities are set by calculation from the applied force and therefore acceleration (F = ma) that occurs between the atoms.

The software used in this experiment is RedHat Linux, DLVisualize (DLV) and General Utility Lattice Program (GULP). GULP is primarily used to perform simulations on materials using various boundary conditions, for example 0D (molecules), 1D (polymers), 2D (surfaces) or 3D (periodic solids), in our experiments we have an emphasis on 3D lattice dynamics. DLV is a general purpose graphical user interface for visualising the output of calculations.

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978 Å and internal angle α = 60°. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å <ref name=made>O. Madelung, U. Rössler, M. Schulz. Calcium oxide (CaO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-3. DOI: 10.1007/10681719_224</ref>).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90°. As before this also allies with literature (4.211 Å <ref name=ross>U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6</ref>. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell.<ref>G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.</ref> For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency.<ref>R. Hoffmann, Angew. Chem. Int. Ed. Engl., 1987, 56, pp 846-878; DOI: 10.1002/anie.198708461</ref> Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å<ref>M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109</ref>) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å.<ref>J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311</ref>, will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å<ref name=ross/> and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of figure 11 as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material<ref>J.R. Vinson, Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware; Springer; 2005</ref>, as the value is essentially 3 - within in the error caused by limitations in the theory - which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.<ref name=made/> When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 <ref>B. B. Karki, R. M. Wentzcovitch, S. de Gironcoli, S. Baroni, Phys. Rev. B, 2000, 61, p8793; DOI: 10.1103/PhysRevB.61.8793</ref>, compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figures 12 and 13 show that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treat the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math> - therefore there is a positive linear increase in E with T. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. From 0-400 K molecular dynamics doesn't follow the curved section of the QHA approach, in this region the anharmonicity isn't as prevalent as the higher temperature regime so QHA is more accurate. MD on the other hand doesn't accurately reproduce how MgO reacts in reality at lower temperatures. The shape of the QHA approach (figure 13) is more similar to literature graphs than MD, as such the equations for the thermal expansion coefficient are a more complex polynomial than the simplified linear graphs we've created between 300 to 3100 K - αV = 2.6025 10-5 + 1.3535 10-8 T + 6.5687 10-3 T-1 - 1.8281 T-2. <ref>L.S. Dubrovinsky, S.K. Saxena, Phys Chem Minerals, 1997, 24, pp 547–550</ref>

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==

In this experiment we completed our objective of studying the thermal expansion of MgO and phonon dispersion, and comparing the quasi-harmonic approximation and a molecular dynamics approach. Both approaches were able to find thermal expansion coefficients similar to literature within the range 400-1000 K. At lower temperatures QHA was more accurate as anharmonic effects were less relevant to the vibrations.

== References ==

Rep:Mod:SA4213MgO

2016-02-25T14:38:42Z

Sa4213: /* Molecular Dynamics */

Rep:Mod:SA4213MgO

2016-02-25T14:22:23Z

Sa4213: /* Molecular Dynamics */

== Introduction ==

In the experiment the thermal expansion of magnesium oxide (MgO), its phonon dispersion and energy were studied using a quasi-harmonic approximation and a molecular dynamics approach. The energies and vibrations in the lattice were calculated to find the free energy of the crystal and thermal expansion.

The quasi-harmonic approximation (QHA) was used to compute volume-dependent thermal effects - such as the thermal expansion coefficient. This model is based on a harmonic oscillator, where each possible bond length is approximated by a quadratic function; but the QHA contains an additional anharmonic factor. This factor more allows the approximation to more closely mirror reality and as such we can account for thermal expansion, as the equilibrium bond length is no longer independent of temperature.

The molecular dynamics approach governs the motion of the atoms with Newtonian mechanics from interatomic forces. Therefore it is necessary to provide initial velocities and positions of the atoms, and then the computation propagates by iteratively repeating the algorithm with a set time step. New positions and velocities are set by calculation from the applied force and therefore acceleration (F = ma) that occurs between the atoms.

The software used in this experiment is RedHat Linux, DLVisualize (DLV) and General Utility Lattice Program (GULP). GULP is primarily used to perform simulations on materials using various boundary conditions, for example 0D (molecules), 1D (polymers), 2D (surfaces) or 3D (periodic solids), in our experiments we have an emphasis on 3D lattice dynamics. DLV is a general purpose graphical user interface for visualising the output of calculations.

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978 Å and internal angle α = 60°. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å <ref name=made>O. Madelung, U. Rössler, M. Schulz. Calcium oxide (CaO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-3. DOI: 10.1007/10681719_224</ref>).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90°. As before this also allies with literature (4.211 Å <ref name=ross>U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6</ref>. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell.<ref>G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.</ref> For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency.<ref>R. Hoffmann, Angew. Chem. Int. Ed. Engl., 1987, 56, pp 846-878; DOI: 10.1002/anie.198708461</ref> Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å<ref>M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109</ref>) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å.<ref>J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311</ref>, will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å<ref name=ross/> and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of figure 11 as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material<ref>J.R. Vinson, Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware; Springer; 2005</ref>, as the value is essentially 3 - within in the error caused by limitations in the theory - which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.<ref name=made/> When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 <ref>B. B. Karki, R. M. Wentzcovitch, S. de Gironcoli, S. Baroni, Phys. Rev. B, 2000, 61, p8793; DOI: 10.1103/PhysRevB.61.8793</ref>, compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figures 12 and 13 show that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treat the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math> - therefore there is a positive linear increase in E with T. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. From 0-400 K molecular dynamics doesn't follow the curved section of the QHA approach. This could be due to the oversimplification of the method when at low temperatures, and so MD doesn't accurately reproduce how MgO reacts in reality at lower temperatures. The shape of the QHA approach (figure 13) is more similar to literature graphs than MD, as such the equations for the thermal expansion coefficient are a more complex polynomial than the simplified linear graphs we've created - αV = 2.6025 10-5 + 1.3535 10-8 T + 6.5687 10-3 T-1 - 1.8281 T-2. <ref>L.S. Dubrovinsky, S.K. Saxena, Phys Chem Minerals, 1997, 24, pp 547–550</ref>

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==
== References ==

Rep:Mod:SA4213MgO

2016-02-25T13:37:12Z

Sa4213: /* Molecular Dynamics */

== Introduction ==

In the experiment the thermal expansion of magnesium oxide (MgO), its phonon dispersion and energy were studied using a quasi-harmonic approximation and a molecular dynamics approach. The energies and vibrations in the lattice were calculated to find the free energy of the crystal and thermal expansion.

The quasi-harmonic approximation (QHA) was used to compute volume-dependent thermal effects - such as the thermal expansion coefficient. This model is based on a harmonic oscillator, where each possible bond length is approximated by a quadratic function; but the QHA contains an additional anharmonic factor. This factor more allows the approximation to more closely mirror reality and as such we can account for thermal expansion, as the equilibrium bond length is no longer independent of temperature.

The molecular dynamics approach governs the motion of the atoms with Newtonian mechanics from interatomic forces. Therefore it is necessary to provide initial velocities and positions of the atoms, and then the computation propagates by iteratively repeating the algorithm with a set time step. New positions and velocities are set by calculation from the applied force and therefore acceleration (F = ma) that occurs between the atoms.

The software used in this experiment is RedHat Linux, DLVisualize (DLV) and General Utility Lattice Program (GULP). GULP is primarily used to perform simulations on materials using various boundary conditions, for example 0D (molecules), 1D (polymers), 2D (surfaces) or 3D (periodic solids), in our experiments we have an emphasis on 3D lattice dynamics. DLV is a general purpose graphical user interface for visualising the output of calculations.

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978 Å and internal angle α = 60°. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å <ref name=made>O. Madelung, U. Rössler, M. Schulz. Calcium oxide (CaO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-3. DOI: 10.1007/10681719_224</ref>).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90°. As before this also allies with literature (4.211 Å <ref name=ross>U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6</ref>. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell.<ref>G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.</ref> For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency.<ref>R. Hoffmann, Angew. Chem. Int. Ed. Engl., 1987, 56, pp 846-878; DOI: 10.1002/anie.198708461</ref> Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å<ref>M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109</ref>) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å.<ref>J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311</ref>, will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å<ref name=ross/> and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of figure 11 as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material<ref>J.R. Vinson, Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware; Springer; 2005</ref>, as the value is essentially 3 - within in the error caused by limitations in the theory - which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.<ref name=made/> When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 <ref>B. B. Karki, R. M. Wentzcovitch, S. de Gironcoli, S. Baroni, Phys. Rev. B, 2000, 61, p8793; DOI: 10.1103/PhysRevB.61.8793</ref>, compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figures 12 and 13 show that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treat the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math> - therefore there is a positive linear increase in E with T. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. Between 0-400 K the two graphs differ due to the intricacies of the methods applied.

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==
== References ==

Rep:Mod:SA4213MgO

2016-02-25T13:06:29Z

Sa4213:

== Introduction ==

In the experiment the thermal expansion of magnesium oxide (MgO), its phonon dispersion and energy were studied using a quasi-harmonic approximation and a molecular dynamics approach. The energies and vibrations in the lattice were calculated to find the free energy of the crystal and thermal expansion.

The quasi-harmonic approximation (QHA) was used to compute volume-dependent thermal effects - such as the thermal expansion coefficient. This model is based on a harmonic oscillator, where each possible bond length is approximated by a quadratic function; but the QHA contains an additional anharmonic factor. This factor more allows the approximation to more closely mirror reality and as such we can account for thermal expansion, as the equilibrium bond length is no longer independent of temperature.

The molecular dynamics approach governs the motion of the atoms with Newtonian mechanics from interatomic forces. Therefore it is necessary to provide initial velocities and positions of the atoms, and then the computation propagates by iteratively repeating the algorithm with a set time step. New positions and velocities are set by calculation from the applied force and therefore acceleration (F = ma) that occurs between the atoms.

The software used in this experiment is RedHat Linux, DLVisualize (DLV) and General Utility Lattice Program (GULP). GULP is primarily used to perform simulations on materials using various boundary conditions, for example 0D (molecules), 1D (polymers), 2D (surfaces) or 3D (periodic solids), in our experiments we have an emphasis on 3D lattice dynamics. DLV is a general purpose graphical user interface for visualising the output of calculations.

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978 Å and internal angle α = 60°. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å <ref name=made>O. Madelung, U. Rössler, M. Schulz. Calcium oxide (CaO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-3. DOI: 10.1007/10681719_224</ref>).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90°. As before this also allies with literature (4.211 Å <ref name=ross>U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6</ref>. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell.<ref>G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.</ref> For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency.<ref>R. Hoffmann, Angew. Chem. Int. Ed. Engl., 1987, 56, pp 846-878; DOI: 10.1002/anie.198708461</ref> Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å<ref>M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109</ref>) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å.<ref>J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311</ref>, will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å<ref name=ross/> and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of figure 11 as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material<ref>J.R. Vinson, Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware; Springer; 2005</ref>, as the value is essentially 3 - within in the error caused by limitations in the theory - which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.<ref name=made/> When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 <ref>B. B. Karki, R. M. Wentzcovitch, S. de Gironcoli, S. Baroni, Phys. Rev. B, 2000, 61, p8793; DOI: 10.1103/PhysRevB.61.8793</ref>, compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. Between 0-400 K the two graphs differ.

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==
== References ==

Rep:Mod:SA4213MgO

2016-02-25T12:53:45Z

Sa4213: /* Calculating the Thermal Expansion Coefficients */

== Introduction ==

In the experiment the thermal expansion of magnesium oxide (MgO), its phonon dispersion and energy were studied using a quasi-harmonic approximation and a molecular dynamics approach. The energies and vibrations in the lattice were calculated to find the free energy of the crystal and thermal expansion.

The quasi-harmonic approximation (QHA) was used to compute volume-dependent thermal effects - such as the thermal expansion coefficient. This model is based on a harmonic oscillator, where each possible bond length is approximated by a quadratic function; but the QHA contains an additional anharmonic factor. This factor more allows the approximation to more closely mirror reality and as such we can account for thermal expansion, as the equilibrium bond length is no longer independent of temperature.

The molecular dynamics approach governs the motion of the atoms with Newtonian mechanics from interatomic forces. Therefore it is necessary to provide initial velocities and positions of the atoms, and then the computation propagates by iteratively repeating the algorithm with a set time step. New positions and velocities are set by calculation from the applied force and therefore acceleration (F = ma) that occurs between the atoms.

The software used in this experiment is RedHat Linux, DLVisualize (DLV) and General Utility Lattice Program (GULP). GULP is primarily used to perform simulations on materials using various boundary conditions, for example 0D (molecules), 1D (polymers), 2D (surfaces) or 3D (periodic solids), in our experiments we have an emphasis on 3D lattice dynamics. DLV is a general purpose graphical user interface for visualising the output of calculations.

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978 Å and internal angle α = 60°. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å <ref name=made>O. Madelung, U. Rössler, M. Schulz. Calcium oxide (CaO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-3. DOI: 10.1007/10681719_224</ref>).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90°. As before this also allies with literature (4.211 Å <ref name=ross>U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6</ref>. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell.<ref>G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.</ref> For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency.<ref>R. Hoffmann, Angew. Chem. Int. Ed. Engl., 1987, 56, pp 846-878; DOI: 10.1002/anie.198708461</ref> Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å<ref>M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109</ref>) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å.<ref>J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311</ref>, will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å<ref name=ross/> and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of figure 11 as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material<ref>J.R. Vinson, Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware; Springer; 2005</ref>, as the value is essentially 3 - within in the error caused by limitations in the theory - which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.<ref name=made/> When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 <ref>B. B. Karki, R. M. Wentzcovitch, S. de Gironcoli, S. Baroni, Phys. Rev. B, 2000, 61, p8793; DOI: 10.1103/PhysRevB.61.8793</ref>, compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. Between 0-400 K the two graphs differ.

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-25T12:39:10Z

Sa4213: /* Phonon Density of States (DOS) */

== Introduction ==

In the experiment the thermal expansion of magnesium oxide (MgO), its phonon dispersion and energy were studied using a quasi-harmonic approximation and a molecular dynamics approach. The energies and vibrations in the lattice were calculated to find the free energy of the crystal and thermal expansion.

The quasi-harmonic approximation (QHA) was used to compute volume-dependent thermal effects - such as the thermal expansion coefficient. This model is based on a harmonic oscillator, where each possible bond length is approximated by a quadratic function; but the QHA contains an additional anharmonic factor. This factor more allows the approximation to more closely mirror reality and as such we can account for thermal expansion, as the equilibrium bond length is no longer independent of temperature.

The molecular dynamics approach governs the motion of the atoms with Newtonian mechanics from interatomic forces. Therefore it is necessary to provide initial velocities and positions of the atoms, and then the computation propagates by iteratively repeating the algorithm with a set time step. New positions and velocities are set by calculation from the applied force and therefore acceleration (F = ma) that occurs between the atoms.

The software used in this experiment is RedHat Linux, DLVisualize (DLV) and General Utility Lattice Program (GULP). GULP is primarily used to perform simulations on materials using various boundary conditions, for example 0D (molecules), 1D (polymers), 2D (surfaces) or 3D (periodic solids), in our experiments we have an emphasis on 3D lattice dynamics. DLV is a general purpose graphical user interface for visualising the output of calculations.

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978 Å and internal angle α = 60°. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å <ref name=made>O. Madelung, U. Rössler, M. Schulz. Calcium oxide (CaO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-3. DOI: 10.1007/10681719_224</ref>).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90°. As before this also allies with literature (4.211 Å <ref name=ross>U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6</ref>. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell.<ref>G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.</ref> For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency.<ref>R. Hoffmann, Angew. Chem. Int. Ed. Engl., 1987, 56, pp 846-878; DOI: 10.1002/anie.198708461</ref> Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å<ref>M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109</ref>) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å.<ref>J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311</ref>, will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å<ref name=ross/> and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of figure 11 as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material<ref>J.R. Vinson, Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware; Springer; 2005</ref>, as the value is essentially 3 - within in the error caused by limitations in the theory - which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.] When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 [INSERT http://journals.aps.org/prb/pdf/10.1103/PhysRevB.61.8793], compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. Between 0-400 K the two graphs differ.

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-25T12:38:16Z

Sa4213: /* Internal Energy of an MgO Crystal */

== Introduction ==

In the experiment the thermal expansion of magnesium oxide (MgO), its phonon dispersion and energy were studied using a quasi-harmonic approximation and a molecular dynamics approach. The energies and vibrations in the lattice were calculated to find the free energy of the crystal and thermal expansion.

The quasi-harmonic approximation (QHA) was used to compute volume-dependent thermal effects - such as the thermal expansion coefficient. This model is based on a harmonic oscillator, where each possible bond length is approximated by a quadratic function; but the QHA contains an additional anharmonic factor. This factor more allows the approximation to more closely mirror reality and as such we can account for thermal expansion, as the equilibrium bond length is no longer independent of temperature.

The molecular dynamics approach governs the motion of the atoms with Newtonian mechanics from interatomic forces. Therefore it is necessary to provide initial velocities and positions of the atoms, and then the computation propagates by iteratively repeating the algorithm with a set time step. New positions and velocities are set by calculation from the applied force and therefore acceleration (F = ma) that occurs between the atoms.

The software used in this experiment is RedHat Linux, DLVisualize (DLV) and General Utility Lattice Program (GULP). GULP is primarily used to perform simulations on materials using various boundary conditions, for example 0D (molecules), 1D (polymers), 2D (surfaces) or 3D (periodic solids), in our experiments we have an emphasis on 3D lattice dynamics. DLV is a general purpose graphical user interface for visualising the output of calculations.

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978 Å and internal angle α = 60°. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å <ref name=made>O. Madelung, U. Rössler, M. Schulz. Calcium oxide (CaO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-3. DOI: 10.1007/10681719_224</ref>).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90°. As before this also allies with literature (4.211 Å <ref name=ross>U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6</ref>. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell.<ref>G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.</ref> For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency.<ref>R. Hoffmann, Angew. Chem. Int. Ed. Engl., 1987, 56, pp 846-878; DOI: 10.1002/anie.198708461</ref> Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å<ref>M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109</ref>) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å.<ref>J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311</ref>, will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å<ref>U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3</ref> and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of figure 11 as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material<ref>J.R. Vinson, Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware; Springer; 2005</ref>, as the value is essentially 3 - within in the error caused by limitations in the theory - which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.] When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 [INSERT http://journals.aps.org/prb/pdf/10.1103/PhysRevB.61.8793], compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. Between 0-400 K the two graphs differ.

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-25T12:36:29Z

Sa4213: /* Calculating the Thermal Expansion Coefficients */

== Introduction ==

In the experiment the thermal expansion of magnesium oxide (MgO), its phonon dispersion and energy were studied using a quasi-harmonic approximation and a molecular dynamics approach. The energies and vibrations in the lattice were calculated to find the free energy of the crystal and thermal expansion.

The quasi-harmonic approximation (QHA) was used to compute volume-dependent thermal effects - such as the thermal expansion coefficient. This model is based on a harmonic oscillator, where each possible bond length is approximated by a quadratic function; but the QHA contains an additional anharmonic factor. This factor more allows the approximation to more closely mirror reality and as such we can account for thermal expansion, as the equilibrium bond length is no longer independent of temperature.

The molecular dynamics approach governs the motion of the atoms with Newtonian mechanics from interatomic forces. Therefore it is necessary to provide initial velocities and positions of the atoms, and then the computation propagates by iteratively repeating the algorithm with a set time step. New positions and velocities are set by calculation from the applied force and therefore acceleration (F = ma) that occurs between the atoms.

The software used in this experiment is RedHat Linux, DLVisualize (DLV) and General Utility Lattice Program (GULP). GULP is primarily used to perform simulations on materials using various boundary conditions, for example 0D (molecules), 1D (polymers), 2D (surfaces) or 3D (periodic solids), in our experiments we have an emphasis on 3D lattice dynamics. DLV is a general purpose graphical user interface for visualising the output of calculations.

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978 Å and internal angle α = 60°. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å <ref>O. Madelung, U. Rössler, M. Schulz. Calcium oxide (CaO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-3. DOI: 10.1007/10681719_224</ref>).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90°. As before this also allies with literature (4.211 Å <ref>U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6</ref>. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell.<ref>G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.</ref> For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency.<ref>R. Hoffmann, Angew. Chem. Int. Ed. Engl., 1987, 56, pp 846-878; DOI: 10.1002/anie.198708461</ref> Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å<ref>M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109</ref>) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å.<ref>J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311</ref>, will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å<ref>U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3</ref> and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of figure 11 as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material<ref>J.R. Vinson, Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware; Springer; 2005</ref>, as the value is essentially 3 - within in the error caused by limitations in the theory - which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.] When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 [INSERT http://journals.aps.org/prb/pdf/10.1103/PhysRevB.61.8793], compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. Between 0-400 K the two graphs differ.

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-25T12:06:31Z

Sa4213: /* Computing the Phonon Dispersion Curves */

== Introduction ==

In the experiment the thermal expansion of magnesium oxide (MgO), its phonon dispersion and energy were studied using a quasi-harmonic approximation and a molecular dynamics approach. The energies and vibrations in the lattice were calculated to find the free energy of the crystal and thermal expansion.

The quasi-harmonic approximation (QHA) was used to compute volume-dependent thermal effects - such as the thermal expansion coefficient. This model is based on a harmonic oscillator, where each possible bond length is approximated by a quadratic function; but the QHA contains an additional anharmonic factor. This factor more allows the approximation to more closely mirror reality and as such we can account for thermal expansion, as the equilibrium bond length is no longer independent of temperature.

The molecular dynamics approach governs the motion of the atoms with Newtonian mechanics from interatomic forces. Therefore it is necessary to provide initial velocities and positions of the atoms, and then the computation propagates by iteratively repeating the algorithm with a set time step. New positions and velocities are set by calculation from the applied force and therefore acceleration (F = ma) that occurs between the atoms.

The software used in this experiment is RedHat Linux, DLVisualize (DLV) and General Utility Lattice Program (GULP). GULP is primarily used to perform simulations on materials using various boundary conditions, for example 0D (molecules), 1D (polymers), 2D (surfaces) or 3D (periodic solids), in our experiments we have an emphasis on 3D lattice dynamics. DLV is a general purpose graphical user interface for visualising the output of calculations.

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978 Å and internal angle α = 60°. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å <ref>O. Madelung, U. Rössler, M. Schulz. Calcium oxide (CaO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-3. DOI: 10.1007/10681719_224</ref>).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90°. As before this also allies with literature (4.211 Å <ref>U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6</ref>. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell.<ref>G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.</ref> For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency.<ref>R. Hoffmann, Angew. Chem. Int. Ed. Engl., 1987, 56, pp 846-878; DOI: 10.1002/anie.198708461</ref> Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å<ref>M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109</ref>) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å.<ref>J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311</ref>, will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å<ref>U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3</ref> and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of figure 11 as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.] When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 [INSERT http://journals.aps.org/prb/pdf/10.1103/PhysRevB.61.8793], compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. Between 0-400 K the two graphs differ.

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-25T11:48:54Z

Sa4213: /* Internal Energy of an MgO Crystal */

== Introduction ==

In the experiment the thermal expansion of magnesium oxide (MgO), its phonon dispersion and energy were studied using a quasi-harmonic approximation and a molecular dynamics approach. The energies and vibrations in the lattice were calculated to find the free energy of the crystal and thermal expansion.

The quasi-harmonic approximation (QHA) was used to compute volume-dependent thermal effects - such as the thermal expansion coefficient. This model is based on a harmonic oscillator, where each possible bond length is approximated by a quadratic function; but the QHA contains an additional anharmonic factor. This factor more allows the approximation to more closely mirror reality and as such we can account for thermal expansion, as the equilibrium bond length is no longer independent of temperature.

The molecular dynamics approach governs the motion of the atoms with Newtonian mechanics from interatomic forces. Therefore it is necessary to provide initial velocities and positions of the atoms, and then the computation propagates by iteratively repeating the algorithm with a set time step. New positions and velocities are set by calculation from the applied force and therefore acceleration (F = ma) that occurs between the atoms.

The software used in this experiment is RedHat Linux, DLVisualize (DLV) and General Utility Lattice Program (GULP). GULP is primarily used to perform simulations on materials using various boundary conditions, for example 0D (molecules), 1D (polymers), 2D (surfaces) or 3D (periodic solids), in our experiments we have an emphasis on 3D lattice dynamics. DLV is a general purpose graphical user interface for visualising the output of calculations.

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978 Å and internal angle α = 60°. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å <ref>O. Madelung, U. Rössler, M. Schulz. Calcium oxide (CaO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-3. DOI: 10.1007/10681719_224</ref>).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90°. As before this also allies with literature (4.211 Å <ref>U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6</ref>. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. <ref>G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.</ref> For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å <ref>M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109</ref>) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. <ref>J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311</ref>, will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å <ref>U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3</ref> and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of figure 11 as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.] When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 [INSERT http://journals.aps.org/prb/pdf/10.1103/PhysRevB.61.8793], compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. Between 0-400 K the two graphs differ.

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T17:57:03Z

Sa4213:

== Introduction ==

In the experiment the thermal expansion of magnesium oxide (MgO), its phonon dispersion and energy were studied using a quasi-harmonic approximation and a molecular dynamics approach. The energies and vibrations in the lattice were calculated to find the free energy of the crystal and thermal expansion.

The quasi-harmonic approximation (QHA) was used to compute volume-dependent thermal effects - such as the thermal expansion coefficient. This model is based on a harmonic oscillator, where each possible bond length is approximated by a quadratic function; but the QHA contains an additional anharmonic factor. This factor more allows the approximation to more closely mirror reality and as such we can account for thermal expansion, as the equilibrium bond length is no longer independent of temperature.

The molecular dynamics approach governs the motion of the atoms with Newtonian mechanics from interatomic forces. Therefore it is necessary to provide initial velocities and positions of the atoms, and then the computation propagates by iteratively repeating the algorithm with a set time step. New positions and velocities are set by calculation from the applied force and therefore acceleration (F = ma) that occurs between the atoms.

The software used in this experiment is RedHat Linux, DLVisualize (DLV) and General Utility Lattice Program (GULP). GULP is primarily used to perform simulations on materials using various boundary conditions, for example 0D (molecules), 1D (polymers), 2D (surfaces) or 3D (periodic solids), in our experiments we have an emphasis on 3D lattice dynamics. DLV is a general purpose graphical user interface for visualising the output of calculations.

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60°. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å <ref>O. Madelung, U. Rössler, M. Schulz. Calcium oxide (CaO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-3. DOI: 10.1007/10681719_224</ref>).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90°. As before this also allies with literature (4.211 Å – <ref>U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6</ref>. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. <ref>G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.</ref> For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å <ref>M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109</ref>) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. <ref>J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311</ref>, will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å <ref>U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3</ref> and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of figure 11 as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.] When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 [INSERT http://journals.aps.org/prb/pdf/10.1103/PhysRevB.61.8793], compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. Between 0-400 K the two graphs differ.

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T17:55:18Z

Sa4213:

== Introduction ==

In the experiment the thermal expansion of magnesium oxide (MgO), its phonon dispersion and energy were studied using a quasi-harmonic approximation and a molecular dynamics approach. The energies and vibrations in the lattice were calculated to find the free energy of the crystal and thermal expansion.

The quasi-harmonic approximation (QHA) was used to compute volume-dependent thermal effects - such as the thermal expansion coefficient. This model is based on a harmonic oscillator, where each possible bond length is approximated by a quadratic function; but the QHA contains an additional anharmonic factor. This factor more allows the approximation to more closely mirror reality and as such we can account for thermal expansion, as the equilibrium bond length is no longer independent of temperature.

The molecular dynamics approach governs the motion of the atoms with Newtonian mechanics from interatomic forces. Therefore it is necessary to provide initial velocities and positions of the atoms, and then the computation propagates by iteratively repeating the algorithm with a set time step. New positions and velocities are set by calculation from the applied force and therefore acceleration (F = ma) that occurs between the atoms.

The software used in this experiment is RedHat Linux, DLVisualize (DLV) and General Utility Lattice Program (GULP). GULP is primarily used to perform simulations on materials using various boundary conditions, for example 0D (molecules), 1D (polymers), 2D (surfaces) or 3D (periodic solids), in our experiments we have an emphasis on 3D lattice dynamics. DLV is a general purpose graphical user interface for visualising the output of calculations.

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60°. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å <ref>O. Madelung, U. Rössler, M. Schulz. Calcium oxide (CaO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-3. DOI: 10.1007/10681719_224</ref>).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90°. As before this also allies with literature (4.211 Å – <ref>U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6</ref>. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. <ref>G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.</ref> For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å <ref>M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109</ref>) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. <ref>J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311</ref>, will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å <ref>U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3<ref> and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of figure 11 as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.] When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 [INSERT http://journals.aps.org/prb/pdf/10.1103/PhysRevB.61.8793], compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. Between 0-400 K the two graphs differ.

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T17:48:36Z

Sa4213: /* Calculating the Thermal Expansion Coefficients */

== Introduction ==

In the experiment the thermal expansion of magnesium oxide (MgO), its phonon dispersion and energy were studied using a quasi-harmonic approximation and a molecular dynamics approach. The energies and vibrations in the lattice were calculated to find the free energy of the crystal and thermal expansion.

The quasi-harmonic approximation (QHA) was used to compute volume-dependent thermal effects - such as the thermal expansion coefficient. This model is based on a harmonic oscillator, where each possible bond length is approximated by a quadratic function; but the QHA contains an additional anharmonic factor. This factor more allows the approximation to more closely mirror reality and as such we can account for thermal expansion, as the equilibrium bond length is no longer independent of temperature.

The molecular dynamics approach governs the motion of the atoms with Newtonian mechanics from interatomic forces. Therefore it is necessary to provide initial velocities and positions of the atoms, and then the computation propagates by iteratively repeating the algorithm with a set time step. New positions and velocities are set by calculation from the applied force and therefore acceleration (F = ma) that occurs between the atoms.

The software used in this experiment is RedHat Linux, DLVisualize (DLV) and General Utility Lattice Program (GULP). GULP is primarily used to perform simulations on materials using various boundary conditions, for example 0D (molecules), 1D (polymers), 2D (surfaces) or 3D (periodic solids), in our experiments we have an emphasis on 3D lattice dynamics. DLV is a general purpose graphical user interface for visualising the output of calculations.

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of figure 11 as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.] When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 [INSERT http://journals.aps.org/prb/pdf/10.1103/PhysRevB.61.8793], compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. Between 0-400 K the two graphs differ.

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T17:37:27Z

Sa4213: /* Introduction */

== Introduction ==

In the experiment the thermal expansion of magnesium oxide (MgO), its phonon dispersion and energy were studied using a quasi-harmonic approximation and a molecular dynamics approach. The energies and vibrations in the lattice were calculated to find the free energy of the crystal and thermal expansion.

The quasi-harmonic approximation (QHA) was used to compute volume-dependent thermal effects - such as the thermal expansion coefficient. This model is based on a harmonic oscillator, where each possible bond length is approximated by a quadratic function; but the QHA contains an additional anharmonic factor. This factor more allows the approximation to more closely mirror reality and as such we can account for thermal expansion, as the equilibrium bond length is no longer independent of temperature.

The molecular dynamics approach governs the motion of the atoms with Newtonian mechanics from interatomic forces. Therefore it is necessary to provide initial velocities and positions of the atoms, and then the computation propagates by iteratively repeating the algorithm with a set time step. New positions and velocities are set by calculation from the applied force and therefore acceleration (F = ma) that occurs between the atoms.

The software used in this experiment is RedHat Linux, DLVisualize (DLV) and General Utility Lattice Program (GULP). GULP is primarily used to perform simulations on materials using various boundary conditions, for example 0D (molecules), 1D (polymers), 2D (surfaces) or 3D (periodic solids), in our experiments we have an emphasis on 3D lattice dynamics. DLV is a general purpose graphical user interface for visualising the output of calculations.

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of [FIGURE SORT IT OUT] as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.] When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 [INSERT http://journals.aps.org/prb/pdf/10.1103/PhysRevB.61.8793], compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. Between 0-400 K the two graphs differ.

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T17:13:41Z

Sa4213: /* Introduction */

== Introduction ==

In the experiment the thermal expansion of magnesium oxide (MgO), its phonon dispersion and energy were studied using a quasi-harmonic approximation and a molecular dynamics approach. The energies and vibrations in the lattice are calculated to find the compute the free energy of the crystal and thermal expansion.

By way of investigating the atomic interactions in the MgO crystals, one can calculate the internal energy and the vibrations (phonons) of the crystals. Then, having obtained the vibrational energy levels, thermodynamics can be used to calculate the free energy, in particular the Helmholtz free energy, of the crystal. The study of the vibrational energy levels and the computation of the free energy will allow us to predict how the MgO crystals expand when subjected to heating by use of the harmonic and quasi-harmonic approximations. Subsequently, the MgO crystal will be thermally expanded using the Molecular Dynamics approach. The final objective of this experiment is to calculate thermal expansion coefficients, αV, and compare the thermal expansion of MgO from both the Molecular Dynamics and the Quasi-Harmonic computational methods at a variety of temperature ranges.

The quasi-harmonic approximation (QHA) is an extension of the harmonic approximation but containing an additional anharmonic factor. In the harmonic phonon model all interatomic forces are purely harmonic. This does not account for thermal expansion since the equilibrium distance between the atoms is independent of temperature. Therefore, in the quasi-harmonic approximation the intermolecular distance changes with temperature and this change is represented by thermal expansion. The free energy is computed as a sum over the vibrational modes of the infinite crystal.

The software used in this experiment is RedHat Linux, DLVisualize (DLV) and General Utility Lattice Program (GULP). GULP is primarily used to perform simulations on materials using various boundary conditions, for example 0D (molecules), 1D (polymers), 2D (surfaces) or 3D (periodic solids), in our experiments we have an emphasis on 3D lattice dynamics. DLV is a general purpose graphical user interface for visualising the output of calculations.

With the molecular dynamics approach the system iteratively evolves according to Newton's Second Law - F = ma, where F is the force, m is the mass and a is the acceleration. The atoms are allowed to interact for a fixed period of time, then atom positions and energies are calculated and the

and the results are trajectories are determined by numerically solving Newton's Laws of motion for a system of interacting particles. Numerical methods bypasses the problem of it being impossible to determine the properties of complex systems containing many particles analytically. The ergodic hypothesis states that all accessible microstates are equally probable over a long period of time. Therefore, systems that follow this hypothesis can be used to determine the macroscopic thermodynamic properties of the system in which the time averages correspond to ensemble averages. Thus, this model computes the trajectories for the atoms and outputs time averages.

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of [FIGURE SORT IT OUT] as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.] When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 [INSERT http://journals.aps.org/prb/pdf/10.1103/PhysRevB.61.8793], compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. Between 0-400 K the two graphs differ.

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T16:43:31Z

Sa4213: /* Molecular Dynamics */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of [FIGURE SORT IT OUT] as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.] When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 [INSERT http://journals.aps.org/prb/pdf/10.1103/PhysRevB.61.8793], compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion: 
αV:QHA = 2.89 x10-5 K-1 
αV:MD = 3.00 x10-5 K-1 
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. Between 0-400 K the two graphs differ.

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T16:40:23Z

Sa4213: /* Molecular Dynamics */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of [FIGURE SORT IT OUT] as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.] When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 [INSERT http://journals.aps.org/prb/pdf/10.1103/PhysRevB.61.8793], compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion:
<math>\alpha_V:QH = 2.89\times 10^{-5}</math>K-1
<math>\alpha_V:MD = 3.00\times 10^{-5}</math>K-1
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range. Between 0-400 K the two graphs differ.

There are limitations for both models. Both describe atoms as hard, charged spheres that interact in a classical manner; therefore there is no consideration of atom overlap that would be considered in a quantum mechanical approach. This sets a ceiling to which the accuracy of both models can achieve. Additionally the models approximate long range interactions to be equal to zero, which wouldn't be the case for atoms just outside the closest neighbours of the atom under study.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T16:34:47Z

Sa4213: /* Calculating the Thermal Expansion Coefficients */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of [FIGURE SORT IT OUT] as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.] When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 [INSERT http://journals.aps.org/prb/pdf/10.1103/PhysRevB.61.8793], compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion:
<math>\alpha_V:QH = 2.89\times 10^{-5}</math>K-1
<math>\alpha_V:MD = 3.00\times 10^{-5}</math>K-1
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range.

Between 0-400 K the two graphs differ.

[INSERT explanation of more accurate and origin of effect]

The QH model describes the thermodynamic properties of the crystal lattice using the Grüneisen parameter (γ), describing the effect of a change in temperature on the volume of the lattice, and is a measure of the anharmonicity of the system. This method measures the properties using a primitive cell, which is comparable of the larger system used in the MD simulations. A larger size allows movements that are not in phase and an increase in temperature and hence volume would result in more random movement, and so the values obtained from this value differ from those using the QH model.

Both models assume that atoms can be described by hard, charged spheres that interact in a classical manner; no consideration of quantum mechanics is taken into account. This limits both models to the type of calculation that can be performed and prevents highly accurate calculations to be obtained. This was demonstrated with the Free energy calculations, which were found to not agree within the same order of magnitude.

One of the major problems with these models was that it only considers interactions between atoms that are relatively close. There is no consideration of the effect of interatomic forces from slightly outside its closest neighbours. Hence, a realistic description of the system is difficult to obtain.

From analysis of MgO at large temperatures, it was found that prediction of the motion of atoms after the melting temperature was not accurate. These calculations were performed on a unit cell, or a small multiple of unit cells, which were assumed to provide a description of the solid from repeating this unit cell along the axis to produce a crystal. However, as a liquid has no long range periodic order, it can not be described by a unit cell.

Vibrational motion of atoms, in the QHA, did not take into account interactions between atoms of different unit cells at large energy. These vibrations had such significant vibrations that the atom was almost found outside of its unit cell. Clearly, if this was to occur, there would be significant overlap with other atoms; thereby, increasing the energy profoundly and preventing it from happening.

One limitation of MD calculations is from the potential energy functions which are used to calculate forces experienced by atoms. If the potential energy can not be described realistically with simple functions, there will have to be a compromise; this compromise will either be in the accuracy of the potential or time taken to complete the calculation.

The main limitation for MD calculations is determining a reasonable time scale to perform calculations over. For a small number of atoms, which was the case for this report, it is relatively simple to perform accurate calculations on a 1 ps timescale with a 0.5 fs time step between calculations. These time parameters allowed visualisation of vibrations for MgO. However, for large systems, of 10,000 atoms, it is not trivial to model a system over a biological time frame, whilst keeping information about atomic vibrations.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T16:34:18Z

Sa4213: /* Calculating the Thermal Expansion Coefficients */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of [FIGURE SORT IT OUT] as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.] When comparing the value for αV at 300 K, the measured literature is 3.12 x10-5 [http://journals.aps.org/prb/pdf/10.1103/PhysRevB.61.8793], compared to our value of 2.37 x10-5. Whilst of the same magnitude, the literature value is 31.6% larger, which could be a manifestation of the phonon interaction or anharmonicity that is neglected by QHA becoming prominent.

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion:
<math>\alpha_V:QH = 2.89\times 10^{-5}</math>K-1
<math>\alpha_V:MD = 3.00\times 10^{-5}</math>K-1
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range.

Between 0-400 K the two graphs differ.

[INSERT explanation of more accurate and origin of effect]

The QH model describes the thermodynamic properties of the crystal lattice using the Grüneisen parameter (γ), describing the effect of a change in temperature on the volume of the lattice, and is a measure of the anharmonicity of the system. This method measures the properties using a primitive cell, which is comparable of the larger system used in the MD simulations. A larger size allows movements that are not in phase and an increase in temperature and hence volume would result in more random movement, and so the values obtained from this value differ from those using the QH model.

Both models assume that atoms can be described by hard, charged spheres that interact in a classical manner; no consideration of quantum mechanics is taken into account. This limits both models to the type of calculation that can be performed and prevents highly accurate calculations to be obtained. This was demonstrated with the Free energy calculations, which were found to not agree within the same order of magnitude.

One of the major problems with these models was that it only considers interactions between atoms that are relatively close. There is no consideration of the effect of interatomic forces from slightly outside its closest neighbours. Hence, a realistic description of the system is difficult to obtain.

From analysis of MgO at large temperatures, it was found that prediction of the motion of atoms after the melting temperature was not accurate. These calculations were performed on a unit cell, or a small multiple of unit cells, which were assumed to provide a description of the solid from repeating this unit cell along the axis to produce a crystal. However, as a liquid has no long range periodic order, it can not be described by a unit cell.

Vibrational motion of atoms, in the QHA, did not take into account interactions between atoms of different unit cells at large energy. These vibrations had such significant vibrations that the atom was almost found outside of its unit cell. Clearly, if this was to occur, there would be significant overlap with other atoms; thereby, increasing the energy profoundly and preventing it from happening.

One limitation of MD calculations is from the potential energy functions which are used to calculate forces experienced by atoms. If the potential energy can not be described realistically with simple functions, there will have to be a compromise; this compromise will either be in the accuracy of the potential or time taken to complete the calculation.

The main limitation for MD calculations is determining a reasonable time scale to perform calculations over. For a small number of atoms, which was the case for this report, it is relatively simple to perform accurate calculations on a 1 ps timescale with a 0.5 fs time step between calculations. These time parameters allowed visualisation of vibrations for MgO. However, for large systems, of 10,000 atoms, it is not trivial to model a system over a biological time frame, whilst keeping information about atomic vibrations.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T16:03:20Z

Sa4213: /* Molecular Dynamics */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of [FIGURE SORT IT OUT] as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.]

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion:
<math>\alpha_V:QH = 2.89\times 10^{-5}</math>K-1
<math>\alpha_V:MD = 3.00\times 10^{-5}</math>K-1
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range.

Between 0-400 K the two graphs differ.

[INSERT explanation of more accurate and origin of effect]

The QH model describes the thermodynamic properties of the crystal lattice using the Grüneisen parameter (γ), describing the effect of a change in temperature on the volume of the lattice, and is a measure of the anharmonicity of the system. This method measures the properties using a primitive cell, which is comparable of the larger system used in the MD simulations. A larger size allows movements that are not in phase and an increase in temperature and hence volume would result in more random movement, and so the values obtained from this value differ from those using the QH model.

Both models assume that atoms can be described by hard, charged spheres that interact in a classical manner; no consideration of quantum mechanics is taken into account. This limits both models to the type of calculation that can be performed and prevents highly accurate calculations to be obtained. This was demonstrated with the Free energy calculations, which were found to not agree within the same order of magnitude.

One of the major problems with these models was that it only considers interactions between atoms that are relatively close. There is no consideration of the effect of interatomic forces from slightly outside its closest neighbours. Hence, a realistic description of the system is difficult to obtain.

From analysis of MgO at large temperatures, it was found that prediction of the motion of atoms after the melting temperature was not accurate. These calculations were performed on a unit cell, or a small multiple of unit cells, which were assumed to provide a description of the solid from repeating this unit cell along the axis to produce a crystal. However, as a liquid has no long range periodic order, it can not be described by a unit cell.

Vibrational motion of atoms, in the QHA, did not take into account interactions between atoms of different unit cells at large energy. These vibrations had such significant vibrations that the atom was almost found outside of its unit cell. Clearly, if this was to occur, there would be significant overlap with other atoms; thereby, increasing the energy profoundly and preventing it from happening.

One limitation of MD calculations is from the potential energy functions which are used to calculate forces experienced by atoms. If the potential energy can not be described realistically with simple functions, there will have to be a compromise; this compromise will either be in the accuracy of the potential or time taken to complete the calculation.

The main limitation for MD calculations is determining a reasonable time scale to perform calculations over. For a small number of atoms, which was the case for this report, it is relatively simple to perform accurate calculations on a 1 ps timescale with a 0.5 fs time step between calculations. These time parameters allowed visualisation of vibrations for MgO. However, for large systems, of 10,000 atoms, it is not trivial to model a system over a biological time frame, whilst keeping information about atomic vibrations.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T15:20:10Z

Sa4213: /* Molecular Dynamics */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of [FIGURE SORT IT OUT] as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.]

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion:
<math>\alpha_V:QH = 2.89\times 10^{-5}</math>K-1
<math>\alpha_V:MD = 3.00\times 10^{-5}</math>K-1
With the difference coming from the difference in volume of the cell at 400 K rather than the step increase in volume per Kelvin. These two methods correlate in the 400-1000 K temperature range.

Between 0-400 K the two graphs differ.

The QH model describes the thermodynamic properties of the crystal lattice using the Grüneisen parameter (γ), describing the effect of a change in temperature on the volume of the lattice, and is a measure of the anharmonicity of the system. This method measures the properties using a primitive cell, which is comparable of the larger system used in the MD simulations. A larger size allows movements that are not in phase and an increase in temperature and hence volume would result in more random movement, and so the values obtained from this value differ from those using the QH model.

AO Pupto-5

When high temperature calculations were performed with the QHA, the ions were significantly displaced. The amplitudes were so large that ions were almost encroaching into the next cell. Therefore the QHA model can not provide an accurate description of MgO close to its melting point. Molecular dynamics provided a better description of atomic motion. Temperatures from 2500 through 4000 K were analysed, with incremental steps of 500 K. It was found that, as temperature increased, not only did the vibrational amplitudes increase, but so did the type of vibrations; this was not seen in QHA. At 2500 K, 625 K below melting, there was significant vibrational motion in the super cell. However, the amplitudes of ions remained relatively tame, with no substantial displacements from equilibrium. Vibrations appeared to be random and no long range order in these vibrations were observed. As the temperature increase to and beyond melting, these two observations changed. A temperature of 4000 K, which is significantly over melting, resulting in substantial vibrations; where an ion could be seen to approach the equilibrium position of another ion. Furthermore, long range vibrations had appeared. At some points in time, for short periods, all ions would move in the same direction. As a liquid can not be described by a unit cell, owing to the lack of long range periodic order, the calculations can not be performed for temperatures above, or near melting. For a more accurate description, significantly larger numbers of atoms would be required, which causes the expense of the calculation to increase drastically.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T15:03:41Z

Sa4213: /* Molecular Dynamics */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of [FIGURE SORT IT OUT] as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.]

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion. If you compare the 0-400 K region QHA takes on the shape of a quadratic curve whereas the MD approach continues along the line described in 400-1000 K.

AO Pupto-5

When high temperature calculations were performed with the QHA, the ions were significantly displaced. The amplitudes were so large that ions were almost encroaching into the next cell. Therefore the QHA model can not provide an accurate description of MgO close to its melting point. Molecular dynamics provided a better description of atomic motion. Temperatures from 2500 through 4000 K were analysed, with incremental steps of 500 K. It was found that, as temperature increased, not only did the vibrational amplitudes increase, but so did the type of vibrations; this was not seen in QHA. At 2500 K, 625 K below melting, there was significant vibrational motion in the super cell. However, the amplitudes of ions remained relatively tame, with no substantial displacements from equilibrium. Vibrations appeared to be random and no long range order in these vibrations were observed. As the temperature increase to and beyond melting, these two observations changed. A temperature of 4000 K, which is significantly over melting, resulting in substantial vibrations; where an ion could be seen to approach the equilibrium position of another ion. Furthermore, long range vibrations had appeared. At some points in time, for short periods, all ions would move in the same direction. As a liquid can not be described by a unit cell, owing to the lack of long range periodic order, the calculations can not be performed for temperatures above, or near melting. For a more accurate description, significantly larger numbers of atoms would be required, which causes the expense of the calculation to increase drastically.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T15:02:49Z

Sa4213: /* Thermal Expansion of MgO */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion. The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of [FIGURE SORT IT OUT] as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.]

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

In a diatomic molecule, assuming a perfect harmonic potential, increasing temperature wouldn't change the equilibrium bond length - as the harmonic oscillations are symmetrical. The amplitude of vibration would increase with temperature though it would still be vibrating about its mean bond length.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion. If you compare the 0-400 K region QHA takes on the shape of a quadratic curve whereas the MD approach continues along the line described in 400-1000 K.

AO Pupto-5

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T14:52:06Z

Sa4213: /* Calculating the Thermal Expansion Coefficients */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of [FIGURE SORT IT OUT] as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1
 

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>
Choosing L0 as 200 K we get 7.852 x10-6 K-1, which is similar to the literature value at 200 K of 7.39 x10-6 K-1.[INSERT REFERENCE: O. Madelung, U. Rössler, M. Schulz. Magnesium oxide (MgO) crystal structure, lattice parameters, thermal expansion. In: II-VI and I-VII Compounds; Semimagnetic Compounds. Landolt-Börnstein - Group III Condensed Matter(41B). Springer Berlin Heidelberg;1999: p1-6. DOI: 10.1007/10681719_206.]

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. To simplify computing anharmonicity, the phonon frequencies are volume dependent. This means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

____________________________________________________________________________________________________________________________________________________________________

Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion.

'''MELTING'''

When high temperature calculations were performed with the QHA, the ions were significantly displaced. The amplitudes were so large that ions were almost encroaching into the next cell. Therefore the QHA model can not provide an accurate description of MgO close to its melting point.

Molecular dynamics provided a better description of atomic motion. Temperatures from 2500 through 4000 K were analysed, with incremental steps of 500 K. It was found that, as temperature increased, not only did the vibrational amplitudes increase, but so did the type of vibrations; this was not seen in QHA. At 2500 K, 625 K below melting, there was significant vibrational motion in the super cell. However, the amplitudes of ions remained relatively tame, with no substantial displacements from equilibrium. Vibrations appeared to be random and no long range order in these vibrations were observed. As the temperature increase to and beyond melting, these two observations changed. A temperature of 4000 K, which is significantly over melting, resulting in substantial vibrations; where an ion could be seen to approach the equilibrium position of another ion. Furthermore, long range vibrations had appeared. At some points in time, for short periods, all ions would move in the same direction. As a liquid can not be described by a unit cell, owing to the lack of long range periodic order, the calculations can not be performed for temperatures above, or near melting. For a more accurate description, significantly larger numbers of atoms would be required, which causes the expense of the calculation to increase drastically.

'''APPROXIMATIONS'''

'''DIATOMIC'''

In a diatomic molecule, assuming a perfect harmonic potential, an increase in temperature corresponds to no change in equilibrium bond length, owing to the symmetry of the harmonic oscillator. The maximum and minimum bond length achieved by the atoms would increase and decrease with temperature, respectively. If the Morse potential was utilised instead, then the bond length of the diatomic molecule would increase with temperature as the potential is not symmetrical .

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion. If you compare the 0-400 K region QHA takes on the shape of a quadratic curve whereas the MD approach continues along the line described in 400-1000 K.

AO Pupto-5

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T12:06:28Z

Sa4213: /* Thermal Expansion of MgO */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.
 
 
=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of [FIGURE SORT IT OUT] as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>

<s>These values compare well to literature values: between 301.15 to 1273.15 K</s>

<s>28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.</s>

<s>Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.</s>

________________________________________________________________________________________________________________________________________________________________________

Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion.

'''MELTING'''

When high temperature calculations were performed with the QHA, the ions were significantly displaced. The amplitudes were so large that ions were almost encroaching into the next cell. Therefore the QHA model can not provide an accurate description of MgO close to its melting point.

Molecular dynamics provided a better description of atomic motion. Temperatures from 2500 through 4000 K were analysed, with incremental steps of 500 K. It was found that, as temperature increased, not only did the vibrational amplitudes increase, but so did the type of vibrations; this was not seen in QHA. At 2500 K, 625 K below melting, there was significant vibrational motion in the super cell. However, the amplitudes of ions remained relatively tame, with no substantial displacements from equilibrium. Vibrations appeared to be random and no long range order in these vibrations were observed. As the temperature increase to and beyond melting, these two observations changed. A temperature of 4000 K, which is significantly over melting, resulting in substantial vibrations; where an ion could be seen to approach the equilibrium position of another ion. Furthermore, long range vibrations had appeared. At some points in time, for short periods, all ions would move in the same direction. As a liquid can not be described by a unit cell, owing to the lack of long range periodic order, the calculations can not be performed for temperatures above, or near melting. For a more accurate description, significantly larger numbers of atoms would be required, which causes the expense of the calculation to increase drastically.

'''APPROXIMATIONS'''

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. In this model, the phonon frequencies are volume dependent which is a simplified way to compute anharmonicity. This in turn means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

'''DIATOMIC'''

In a diatomic molecule, assuming a perfect harmonic potential, an increase in temperature corresponds to no change in equilibrium bond length, owing to the symmetry of the harmonic oscillator. The maximum and minimum bond length achieved by the atoms would increase and decrease with temperature, respectively. If the Morse potential was utilised instead, then the bond length of the diatomic molecule would increase with temperature as the potential is not symmetrical .

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion. If you compare the 0-400 K region QHA takes on the shape of a quadratic curve whereas the MD approach continues along the line described in 400-1000 K.

AO Pupto-5

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T12:05:53Z

Sa4213: /* Thermal Expansion of MgO */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|400px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeVolumeQH_Formula_SA4213MgO.png|thumb|400px|Figure 11. Lattice Volume dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.

=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of [FIGURE SORT IT OUT] as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>

<s>These values compare well to literature values: between 301.15 to 1273.15 K</s>

<s>28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.</s>

<s>Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.</s>

________________________________________________________________________________________________________________________________________________________________________

Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion.

'''MELTING'''

When high temperature calculations were performed with the QHA, the ions were significantly displaced. The amplitudes were so large that ions were almost encroaching into the next cell. Therefore the QHA model can not provide an accurate description of MgO close to its melting point.

Molecular dynamics provided a better description of atomic motion. Temperatures from 2500 through 4000 K were analysed, with incremental steps of 500 K. It was found that, as temperature increased, not only did the vibrational amplitudes increase, but so did the type of vibrations; this was not seen in QHA. At 2500 K, 625 K below melting, there was significant vibrational motion in the super cell. However, the amplitudes of ions remained relatively tame, with no substantial displacements from equilibrium. Vibrations appeared to be random and no long range order in these vibrations were observed. As the temperature increase to and beyond melting, these two observations changed. A temperature of 4000 K, which is significantly over melting, resulting in substantial vibrations; where an ion could be seen to approach the equilibrium position of another ion. Furthermore, long range vibrations had appeared. At some points in time, for short periods, all ions would move in the same direction. As a liquid can not be described by a unit cell, owing to the lack of long range periodic order, the calculations can not be performed for temperatures above, or near melting. For a more accurate description, significantly larger numbers of atoms would be required, which causes the expense of the calculation to increase drastically.

'''APPROXIMATIONS'''

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. In this model, the phonon frequencies are volume dependent which is a simplified way to compute anharmonicity. This in turn means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

'''DIATOMIC'''

In a diatomic molecule, assuming a perfect harmonic potential, an increase in temperature corresponds to no change in equilibrium bond length, owing to the symmetry of the harmonic oscillator. The maximum and minimum bond length achieved by the atoms would increase and decrease with temperature, respectively. If the Morse potential was utilised instead, then the bond length of the diatomic molecule would increase with temperature as the potential is not symmetrical .

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion. If you compare the 0-400 K region QHA takes on the shape of a quadratic curve whereas the MD approach continues along the line described in 400-1000 K.

AO Pupto-5

== Conclusion ==

File:LatticeVolumeQH Formula SA4213MgO.png

2016-02-24T12:05:16Z

Sa4213:

File:LatticeConstantQH Formula SA4213MgO.png

2016-02-24T12:04:25Z

Sa4213:

Rep:Mod:SA4213MgO

2016-02-24T11:53:29Z

Sa4213: /* Calculating the Thermal Expansion Coefficients */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|600px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_SA4213MgO.png|thumb|600px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.

=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K and the gradient of [FIGURE SORT IT OUT] as dL/dT 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>

<s>These values compare well to literature values: between 301.15 to 1273.15 K</s>

<s>28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.</s>

<s>Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.</s>

________________________________________________________________________________________________________________________________________________________________________

Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion.

'''MELTING'''

When high temperature calculations were performed with the QHA, the ions were significantly displaced. The amplitudes were so large that ions were almost encroaching into the next cell. Therefore the QHA model can not provide an accurate description of MgO close to its melting point.

Molecular dynamics provided a better description of atomic motion. Temperatures from 2500 through 4000 K were analysed, with incremental steps of 500 K. It was found that, as temperature increased, not only did the vibrational amplitudes increase, but so did the type of vibrations; this was not seen in QHA. At 2500 K, 625 K below melting, there was significant vibrational motion in the super cell. However, the amplitudes of ions remained relatively tame, with no substantial displacements from equilibrium. Vibrations appeared to be random and no long range order in these vibrations were observed. As the temperature increase to and beyond melting, these two observations changed. A temperature of 4000 K, which is significantly over melting, resulting in substantial vibrations; where an ion could be seen to approach the equilibrium position of another ion. Furthermore, long range vibrations had appeared. At some points in time, for short periods, all ions would move in the same direction. As a liquid can not be described by a unit cell, owing to the lack of long range periodic order, the calculations can not be performed for temperatures above, or near melting. For a more accurate description, significantly larger numbers of atoms would be required, which causes the expense of the calculation to increase drastically.

'''APPROXIMATIONS'''

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. In this model, the phonon frequencies are volume dependent which is a simplified way to compute anharmonicity. This in turn means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

'''DIATOMIC'''

In a diatomic molecule, assuming a perfect harmonic potential, an increase in temperature corresponds to no change in equilibrium bond length, owing to the symmetry of the harmonic oscillator. The maximum and minimum bond length achieved by the atoms would increase and decrease with temperature, respectively. If the Morse potential was utilised instead, then the bond length of the diatomic molecule would increase with temperature as the potential is not symmetrical .

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion. If you compare the 0-400 K region QHA takes on the shape of a quadratic curve whereas the MD approach continues along the line described in 400-1000 K.

AO Pupto-5

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T11:34:31Z

Sa4213: /* Molecular Dynamics */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|600px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_SA4213MgO.png|thumb|600px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.

=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>

<s>These values compare well to literature values: between 301.15 to 1273.15 K</s>

<s>28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.</s>

<s>Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.</s>

________________________________________________________________________________________________________________________________________________________________________

Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion.

'''MELTING'''

When high temperature calculations were performed with the QHA, the ions were significantly displaced. The amplitudes were so large that ions were almost encroaching into the next cell. Therefore the QHA model can not provide an accurate description of MgO close to its melting point.

Molecular dynamics provided a better description of atomic motion. Temperatures from 2500 through 4000 K were analysed, with incremental steps of 500 K. It was found that, as temperature increased, not only did the vibrational amplitudes increase, but so did the type of vibrations; this was not seen in QHA. At 2500 K, 625 K below melting, there was significant vibrational motion in the super cell. However, the amplitudes of ions remained relatively tame, with no substantial displacements from equilibrium. Vibrations appeared to be random and no long range order in these vibrations were observed. As the temperature increase to and beyond melting, these two observations changed. A temperature of 4000 K, which is significantly over melting, resulting in substantial vibrations; where an ion could be seen to approach the equilibrium position of another ion. Furthermore, long range vibrations had appeared. At some points in time, for short periods, all ions would move in the same direction. As a liquid can not be described by a unit cell, owing to the lack of long range periodic order, the calculations can not be performed for temperatures above, or near melting. For a more accurate description, significantly larger numbers of atoms would be required, which causes the expense of the calculation to increase drastically.

'''APPROXIMATIONS'''

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. In this model, the phonon frequencies are volume dependent which is a simplified way to compute anharmonicity. This in turn means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

'''DIATOMIC'''

In a diatomic molecule, assuming a perfect harmonic potential, an increase in temperature corresponds to no change in equilibrium bond length, owing to the symmetry of the harmonic oscillator. The maximum and minimum bond length achieved by the atoms would increase and decrease with temperature, respectively. If the Morse potential was utilised instead, then the bond length of the diatomic molecule would increase with temperature as the potential is not symmetrical .

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [13] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we should have a linear decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion. If you compare the 0-400 K region QHA takes on the shape of a quadratic curve whereas the MD approach continues along the line described in 400-1000 K.

AO Pupto-5

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-24T11:29:39Z

Sa4213: /* Molecular Dynamics */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|600px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_SA4213MgO.png|thumb|600px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.

=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>

<s>These values compare well to literature values: between 301.15 to 1273.15 K</s>

<s>28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.</s>

<s>Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.</s>

________________________________________________________________________________________________________________________________________________________________________

Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion.

'''MELTING'''

When high temperature calculations were performed with the QHA, the ions were significantly displaced. The amplitudes were so large that ions were almost encroaching into the next cell. Therefore the QHA model can not provide an accurate description of MgO close to its melting point.

Molecular dynamics provided a better description of atomic motion. Temperatures from 2500 through 4000 K were analysed, with incremental steps of 500 K. It was found that, as temperature increased, not only did the vibrational amplitudes increase, but so did the type of vibrations; this was not seen in QHA. At 2500 K, 625 K below melting, there was significant vibrational motion in the super cell. However, the amplitudes of ions remained relatively tame, with no substantial displacements from equilibrium. Vibrations appeared to be random and no long range order in these vibrations were observed. As the temperature increase to and beyond melting, these two observations changed. A temperature of 4000 K, which is significantly over melting, resulting in substantial vibrations; where an ion could be seen to approach the equilibrium position of another ion. Furthermore, long range vibrations had appeared. At some points in time, for short periods, all ions would move in the same direction. As a liquid can not be described by a unit cell, owing to the lack of long range periodic order, the calculations can not be performed for temperatures above, or near melting. For a more accurate description, significantly larger numbers of atoms would be required, which causes the expense of the calculation to increase drastically.

'''APPROXIMATIONS'''

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. In this model, the phonon frequencies are volume dependent which is a simplified way to compute anharmonicity. This in turn means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

'''DIATOMIC'''

In a diatomic molecule, assuming a perfect harmonic potential, an increase in temperature corresponds to no change in equilibrium bond length, owing to the symmetry of the harmonic oscillator. The maximum and minimum bond length achieved by the atoms would increase and decrease with temperature, respectively. If the Morse potential was utilised instead, then the bond length of the diatomic molecule would increase with temperature as the potential is not symmetrical .

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

{|align="center"
|[[File:Energy_QHvsMD_SA4213MgO.png|thumb|400px|Figure 12. Free Energy vs Temperature for MD compared to QHA]]
|[[File:Volume_QHvsMD_SA4213MgO.png|thumb|400px|Figure 13. Lattice Constant vs Temperature for MD compared to QHA]]
|}

Figure [INSERT number] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we have a decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion. If you compare the 0-400 K region QHA takes on the shape of a quadratic curve whereas the MD approach continues along the line described in 400-1000 K.

AO Pupto-5

== Conclusion ==

File:Volume QHvsMD SA4213MgO.png

2016-02-24T11:28:03Z

Sa4213:

File:Energy QHvsMD SA4213MgO.png

2016-02-24T11:26:19Z

Sa4213:

Rep:Mod:SA4213MgO

2016-02-24T11:09:24Z

Sa4213: /* Calculating the Thermal Expansion Coefficients */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|600px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_SA4213MgO.png|thumb|600px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.

=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[Thermal_Expansion_CoeffQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>

<s>These values compare well to literature values: between 301.15 to 1273.15 K</s>

<s>28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.</s>

<s>Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.</s>

________________________________________________________________________________________________________________________________________________________________________

Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion.

'''MELTING'''

When high temperature calculations were performed with the QHA, the ions were significantly displaced. The amplitudes were so large that ions were almost encroaching into the next cell. Therefore the QHA model can not provide an accurate description of MgO close to its melting point.

Molecular dynamics provided a better description of atomic motion. Temperatures from 2500 through 4000 K were analysed, with incremental steps of 500 K. It was found that, as temperature increased, not only did the vibrational amplitudes increase, but so did the type of vibrations; this was not seen in QHA. At 2500 K, 625 K below melting, there was significant vibrational motion in the super cell. However, the amplitudes of ions remained relatively tame, with no substantial displacements from equilibrium. Vibrations appeared to be random and no long range order in these vibrations were observed. As the temperature increase to and beyond melting, these two observations changed. A temperature of 4000 K, which is significantly over melting, resulting in substantial vibrations; where an ion could be seen to approach the equilibrium position of another ion. Furthermore, long range vibrations had appeared. At some points in time, for short periods, all ions would move in the same direction. As a liquid can not be described by a unit cell, owing to the lack of long range periodic order, the calculations can not be performed for temperatures above, or near melting. For a more accurate description, significantly larger numbers of atoms would be required, which causes the expense of the calculation to increase drastically.

'''APPROXIMATIONS'''

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. In this model, the phonon frequencies are volume dependent which is a simplified way to compute anharmonicity. This in turn means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

'''DIATOMIC'''

In a diatomic molecule, assuming a perfect harmonic potential, an increase in temperature corresponds to no change in equilibrium bond length, owing to the symmetry of the harmonic oscillator. The maximum and minimum bond length achieved by the atoms would increase and decrease with temperature, respectively. If the Morse potential was utilised instead, then the bond length of the diatomic molecule would increase with temperature as the potential is not symmetrical .

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

[Table]

Figure [INSERT number] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we have a decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion. If you compare the 0-400 K region QHA takes on the shape of a quadratic curve whereas the MD approach continues along the line described in 400-1000 K.

AO Pupto-5

== Conclusion ==

File:Thermal Expansion CoeffQH SA4213MgO.png

2016-02-24T11:09:07Z

Sa4213:

Rep:Mod:SA4213MgO

2016-02-23T17:23:23Z

Sa4213: /* Molecular Dynamics */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|600px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_SA4213MgO.png|thumb|600px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.

=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[ThermalExpansionCoefficientQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>

<s>These values compare well to literature values: between 301.15 to 1273.15 K</s>

<s>28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.</s>

<s>Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.</s>

________________________________________________________________________________________________________________________________________________________________________

Temperature is a measure of the kinetic energy of the molecules in the cell, therefore as temperature increases kinetic energy and velocity will increase. An increase in velocity will cause a larger maximum amplitude in the quantum harmonic approximation. This effect across all the atoms leads to an expansion.

'''MELTING'''

When high temperature calculations were performed with the QHA, the ions were significantly displaced. The amplitudes were so large that ions were almost encroaching into the next cell. Therefore the QHA model can not provide an accurate description of MgO close to its melting point.

Molecular dynamics provided a better description of atomic motion. Temperatures from 2500 through 4000 K were analysed, with incremental steps of 500 K. It was found that, as temperature increased, not only did the vibrational amplitudes increase, but so did the type of vibrations; this was not seen in QHA. At 2500 K, 625 K below melting, there was significant vibrational motion in the super cell. However, the amplitudes of ions remained relatively tame, with no substantial displacements from equilibrium. Vibrations appeared to be random and no long range order in these vibrations were observed. As the temperature increase to and beyond melting, these two observations changed. A temperature of 4000 K, which is significantly over melting, resulting in substantial vibrations; where an ion could be seen to approach the equilibrium position of another ion. Furthermore, long range vibrations had appeared. At some points in time, for short periods, all ions would move in the same direction. As a liquid can not be described by a unit cell, owing to the lack of long range periodic order, the calculations can not be performed for temperatures above, or near melting. For a more accurate description, significantly larger numbers of atoms would be required, which causes the expense of the calculation to increase drastically.

'''APPROXIMATIONS'''

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. In this model, the phonon frequencies are volume dependent which is a simplified way to compute anharmonicity. This in turn means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

'''DIATOMIC'''

In a diatomic molecule, assuming a perfect harmonic potential, an increase in temperature corresponds to no change in equilibrium bond length, owing to the symmetry of the harmonic oscillator. The maximum and minimum bond length achieved by the atoms would increase and decrease with temperature, respectively. If the Morse potential was utilised instead, then the bond length of the diatomic molecule would increase with temperature as the potential is not symmetrical .

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

[Table]

Figure [INSERT number] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we have a decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion. If you compare the 0-400 K region QHA takes on the shape of a quadratic curve whereas the MD approach continues along the line described in 400-1000 K.

AO Pupto-5

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-23T16:53:57Z

Sa4213: /* Calculating the Thermal Expansion Coefficients */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|600px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_SA4213MgO.png|thumb|600px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.

=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[ThermalExpansionCoefficientQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>\alpha_V</math> = <math>3 \alpha_L</math>

These values compare well to literature values: between 301.15 to 1273.15 K

28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.

Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.

<nowiki>===Speculations===</nowiki>
* What is the physical origin of thermal expansion ?
* As the temperature approaches the melting point of MgO how well do the phonon modes represent the actual motions of the ions ?
* In a diatomic molecule with an exactly harmonic potential would you expect the bond length to increase with temperature ? Why does it increase in the solid when we are using an quasi-harmonic approximation ?

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. In this model, the phonon frequencies are volume dependent which is a simplified way to compute anharmonicity. This in turn means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

The physical origin of thermal expansion arises from the fact that as the temperature increases the amplitude of atomic vibrations increases. A higher vibrational amplitude means that the average atomic distance in a cell increases, hence resulting in expansion. Additionally, as the temperature approaches the melting point of MgO the quasi-harmonic approximation breaks down since this approximation only takes into account linear motion. Therefore, the quasi-harmonic model prevents movement of free ions, which is present in molten MgO and hence this model is not appropriate for MgO near the melting point. Furthermore, as the temperature approaches the melting point the phonon modes become less representative of the actual motion of the ions since at higher temperatures the quasi-harmonic approximation tends to neglect phonon interactions.

For a diatomic molecule with an exactly harmonic potential, the bond length would not be expected in increase with temperature because the potential is symmetric and hence the average distance between atoms is constant. However, in the quasi-harmonic model for a solid the bond length increases with temperature (thermal expansion) due to the additional anharmonic factor included where bond dissociation is possible.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

[Table]

Figure [INSERT number] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we have a decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion. If you compare the 0-400 K region QHA takes on the shape of a quadratic curve whereas the MD approach continues along the line described in 400-1000 K.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-23T16:52:12Z

Sa4213: /* Calculating the Thermal Expansion Coefficients */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|600px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_SA4213MgO.png|thumb|600px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.

=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[ThermalExpansionCoefficientQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math> α_V = 3α_L </math>

These values compare well to literature values: between 301.15 to 1273.15 K

28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.

Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.

<nowiki>===Speculations===</nowiki>
* What is the physical origin of thermal expansion ?
* As the temperature approaches the melting point of MgO how well do the phonon modes represent the actual motions of the ions ?
* In a diatomic molecule with an exactly harmonic potential would you expect the bond length to increase with temperature ? Why does it increase in the solid when we are using an quasi-harmonic approximation ?

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. In this model, the phonon frequencies are volume dependent which is a simplified way to compute anharmonicity. This in turn means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

The physical origin of thermal expansion arises from the fact that as the temperature increases the amplitude of atomic vibrations increases. A higher vibrational amplitude means that the average atomic distance in a cell increases, hence resulting in expansion. Additionally, as the temperature approaches the melting point of MgO the quasi-harmonic approximation breaks down since this approximation only takes into account linear motion. Therefore, the quasi-harmonic model prevents movement of free ions, which is present in molten MgO and hence this model is not appropriate for MgO near the melting point. Furthermore, as the temperature approaches the melting point the phonon modes become less representative of the actual motion of the ions since at higher temperatures the quasi-harmonic approximation tends to neglect phonon interactions.

For a diatomic molecule with an exactly harmonic potential, the bond length would not be expected in increase with temperature because the potential is symmetric and hence the average distance between atoms is constant. However, in the quasi-harmonic model for a solid the bond length increases with temperature (thermal expansion) due to the additional anharmonic factor included where bond dissociation is possible.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

[Table]

Figure [INSERT number] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we have a decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion. If you compare the 0-400 K region QHA takes on the shape of a quadratic curve whereas the MD approach continues along the line described in 400-1000 K.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-23T16:51:52Z

Sa4213: /* Calculating the Thermal Expansion Coefficients */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|600px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_SA4213MgO.png|thumb|600px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.

=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[ThermalExpansionCoefficientQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: <math>α_V = 3α_L </math>

These values compare well to literature values: between 301.15 to 1273.15 K

28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.

Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.

<nowiki>===Speculations===</nowiki>
* What is the physical origin of thermal expansion ?
* As the temperature approaches the melting point of MgO how well do the phonon modes represent the actual motions of the ions ?
* In a diatomic molecule with an exactly harmonic potential would you expect the bond length to increase with temperature ? Why does it increase in the solid when we are using an quasi-harmonic approximation ?

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. In this model, the phonon frequencies are volume dependent which is a simplified way to compute anharmonicity. This in turn means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

The physical origin of thermal expansion arises from the fact that as the temperature increases the amplitude of atomic vibrations increases. A higher vibrational amplitude means that the average atomic distance in a cell increases, hence resulting in expansion. Additionally, as the temperature approaches the melting point of MgO the quasi-harmonic approximation breaks down since this approximation only takes into account linear motion. Therefore, the quasi-harmonic model prevents movement of free ions, which is present in molten MgO and hence this model is not appropriate for MgO near the melting point. Furthermore, as the temperature approaches the melting point the phonon modes become less representative of the actual motion of the ions since at higher temperatures the quasi-harmonic approximation tends to neglect phonon interactions.

For a diatomic molecule with an exactly harmonic potential, the bond length would not be expected in increase with temperature because the potential is symmetric and hence the average distance between atoms is constant. However, in the quasi-harmonic model for a solid the bond length increases with temperature (thermal expansion) due to the additional anharmonic factor included where bond dissociation is possible.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

[Table]

Figure [INSERT number] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we have a decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion. If you compare the 0-400 K region QHA takes on the shape of a quadratic curve whereas the MD approach continues along the line described in 400-1000 K.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-23T16:42:34Z

Sa4213: /* Thermal Expansion of MgO */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As before this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the vibration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy as: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the utility of the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches combine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}
Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. This replicates when we tried to find the optimum grid size that best compromised accuracy and computation time, that beyond a certain limit the increase in accuracy reduces to an acceptable level such that it is unnecessary to study larger grid sizes. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

== Thermal Expansion of MgO ==
The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|600px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_SA4213MgO.png|thumb|600px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

When Free energy against Temperature was plotted (Figure 10) it showed that the free energy becomes more negative with increasing temperature. In the Quasi-Harmonic approximation free energy is calculated via: <math> A = U - TS </math>

Therefore we have a linear decrease in free energy as temperature increases. This general shape of the graph is a curve, which suggests more variables are affecting the free energy. If we're to look at Gibb's Free energy:

<math> G = H - TS </math>

inserting <math>H = U + PV</math> into the above:

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

As <math> U = q + w </math>, and <math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

We can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature change. The initial slow decrease in free energy suggests that the change in pressure isn't constant, as we expect: <math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.

=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[ThermalExpansionCoefficientQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>K-1

Using L0 as the lattice constant at 0K 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>K-1

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as: αV = 3αL

These values compare well to literature values: between 301.15 to 1273.15 K

28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.

Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.

<nowiki>===Speculations===</nowiki>
* What is the physical origin of thermal expansion ?
* As the temperature approaches the melting point of MgO how well do the phonon modes represent the actual motions of the ions ?
* In a diatomic molecule with an exactly harmonic potential would you expect the bond length to increase with temperature ? Why does it increase in the solid when we are using an quasi-harmonic approximation ?

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. In this model, the phonon frequencies are volume dependent which is a simplified way to compute anharmonicity. This in turn means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

The physical origin of thermal expansion arises from the fact that as the temperature increases the amplitude of atomic vibrations increases. A higher vibrational amplitude means that the average atomic distance in a cell increases, hence resulting in expansion. Additionally, as the temperature approaches the melting point of MgO the quasi-harmonic approximation breaks down since this approximation only takes into account linear motion. Therefore, the quasi-harmonic model prevents movement of free ions, which is present in molten MgO and hence this model is not appropriate for MgO near the melting point. Furthermore, as the temperature approaches the melting point the phonon modes become less representative of the actual motion of the ions since at higher temperatures the quasi-harmonic approximation tends to neglect phonon interactions.

For a diatomic molecule with an exactly harmonic potential, the bond length would not be expected in increase with temperature because the potential is symmetric and hence the average distance between atoms is constant. However, in the quasi-harmonic model for a solid the bond length increases with temperature (thermal expansion) due to the additional anharmonic factor included where bond dissociation is possible.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

[Table]

Figure [INSERT number] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we have a decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion. If you compare the 0-400 K region QHA takes on the shape of a quadratic curve whereas the MD approach continues along the line described in 400-1000 K.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-23T16:05:43Z

Sa4213: /* Molecular Dynamics */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As befere this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

It is important to note that every possible vibration is associated with a k value.

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the viration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy via: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches comine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}

== Thermal Expansion of MgO ==

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|600px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_SA4213MgO.png|thumb|600px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

When Gibbs Free energy against Temperature was plotted it showed that the free energy becomes more negative with increasing temperature. This relationship can be derived as shown below:

<math> G = H - TS </math>

<math>H = U + PV</math>

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

<math> U = q + w </math>

<math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

Thus we can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature values. Initially the graph shows a slow decrease in free energy which has a more negative gradient as temperature increases. This suggests that the change in pressure isn't constant, as we expect:

<math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>

which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.

=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[ThermalExpansionCoefficientQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>

Using L0 as the lattice constant at 0K
 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as αV = 3αL

These values compare well to literature values: between 301.15 to 1273.15 K

28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.

Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.

<nowiki>===Speculations===</nowiki>
* What is the physical origin of thermal expansion ?
* As the temperature approaches the melting point of MgO how well do the phonon modes represent the actual motions of the ions ?
* In a diatomic molecule with an exactly harmonic potential would you expect the bond length to increase with temperature ? Why does it increase in the solid when we are using an quasi-harmonic approximation ?

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. In this model, the phonon frequencies are volume dependent which is a simplified way to compute anharmonicity. This in turn means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

The physical origin of thermal expansion arises from the fact that as the temperature increases the amplitude of atomic vibrations increases. A higher vibrational amplitude means that the average atomic distance in a cell increases, hence resulting in expansion. Additionally, as the temperature approaches the melting point of MgO the quasi-harmonic approximation breaks down since this approximation only takes into account linear motion. Therefore, the quasi-harmonic model prevents movement of free ions, which is present in molten MgO and hence this model is not appropriate for MgO near the melting point. Furthermore, as the temperature approaches the melting point the phonon modes become less representative of the actual motion of the ions since at higher temperatures the quasi-harmonic approximation tends to neglect phonon interactions.

For a diatomic molecule with an exactly harmonic potential, the bond length would not be expected in increase with temperature because the potential is symmetric and hence the average distance between atoms is constant. However, in the quasi-harmonic model for a solid the bond length increases with temperature (thermal expansion) due to the additional anharmonic factor included where bond dissociation is possible.

== Molecular Dynamics ==

Next the crystal was studied via Molecular Dynamics (MD), this required a different cell to that in the QHA. In QHA we were able to use a primitive unit cell with 1 MgO unit, this wouldn't produce meaningful data as every cell of the crystal would be moving in phase. Therefore we are using a 2x2x2 supercell of conventional unit cells, therefore containing 32 MgO units. We could have used a larger cell for more accurate results, but as before we face a trade off between information gained and computational time spent.

[Table]

Figure [INSERT number] shows that as the temperature increases in MD calculations the energy and cell volume (per formula unit) increased linearly. This is because the MD calculations treats the system classically under <math> F = ma </math> and as such: <math> E = \frac{3}{2} k_b T </math>. This is different to the QHA approach which as discussed above, computes energy via: <math> A = U - TS </math>
Therefore we have a decrease in free energy as temperature increases.

When comparing the cell volume per formula unit, we see that in the range 400-1000 K both methods produce a very similar change in volume per unit increase in temperature - therefore similar coefficient of thermal expansion. If you compare the 0-400 K region QHA takes on the shape of a quadratic curve whereas the MD approach continues along the line described in 400-1000 K.

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-23T15:34:37Z

Sa4213: /* Computing the Phonon Dispersion Curves */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As befere this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

It is important to note that every possible vibration is associated with a k value.

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the viration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity. The equation relating phonon wavelength and wavenumber: <math>k= \frac{2\pi}{\lambda}</math> tells us that as lambda approaches infinity, k approaches 0 at Γ. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy via: <math>E = \hbar\omega \qquad</math>
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches comine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As <math> a*= \frac{2\pi}{a}</math> large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}

== Thermal Expansion of MgO ==

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|600px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_SA4213MgO.png|thumb|600px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

When Gibbs Free energy against Temperature was plotted it showed that the free energy becomes more negative with increasing temperature. This relationship can be derived as shown below:

<math> G = H - TS </math>

<math>H = U + PV</math>

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

<math> U = q + w </math>

<math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

Thus we can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature values. Initially the graph shows a slow decrease in free energy which has a more negative gradient as temperature increases. This suggests that the change in pressure isn't constant, as we expect:

<math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>

which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.

=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[ThermalExpansionCoefficientQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>

Using L0 as the lattice constant at 0K
 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as αV = 3αL

These values compare well to literature values: between 301.15 to 1273.15 K

28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.

Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.

<nowiki>===Speculations===</nowiki>
* What is the physical origin of thermal expansion ?
* As the temperature approaches the melting point of MgO how well do the phonon modes represent the actual motions of the ions ?
* In a diatomic molecule with an exactly harmonic potential would you expect the bond length to increase with temperature ? Why does it increase in the solid when we are using an quasi-harmonic approximation ?

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. In this model, the phonon frequencies are volume dependent which is a simplified way to compute anharmonicity. This in turn means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

The physical origin of thermal expansion arises from the fact that as the temperature increases the amplitude of atomic vibrations increases. A higher vibrational amplitude means that the average atomic distance in a cell increases, hence resulting in expansion. Additionally, as the temperature approaches the melting point of MgO the quasi-harmonic approximation breaks down since this approximation only takes into account linear motion. Therefore, the quasi-harmonic model prevents movement of free ions, which is present in molten MgO and hence this model is not appropriate for MgO near the melting point. Furthermore, as the temperature approaches the melting point the phonon modes become less representative of the actual motion of the ions since at higher temperatures the quasi-harmonic approximation tends to neglect phonon interactions.

For a diatomic molecule with an exactly harmonic potential, the bond length would not be expected in increase with temperature because the potential is symmetric and hence the average distance between atoms is constant. However, in the quasi-harmonic model for a solid the bond length increases with temperature (thermal expansion) due to the additional anharmonic factor included where bond dissociation is possible.

== Molecular Dynamics ==

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-22T15:51:53Z

Sa4213:

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As befere this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

It is important to note that every possible vibration is associated with a k value.

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the viration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity and thus k approaches 0 at Γ. This can be rationalised via: k= \frac{2\pi}{\lambda}. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy via: E = \hbar\omega \qquad
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches comine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As a*= \frac{2\pi}{a} large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}

== Thermal Expansion of MgO ==

{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|600px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_SA4213MgO.png|thumb|600px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the thermal expansion coefficient, x is the (initial) dimension under study'', ∂x is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
From this we can see that the greater the expansion per unit increase in temperature the larger the thermal expansion coefficient. We expect MgO to have a low volumetric thermal expansion coefficient, due to the strong ionic bonding present.

When Gibbs Free energy against Temperature was plotted it showed that the free energy becomes more negative with increasing temperature. This relationship can be derived as shown below:

<math> G = H - TS </math>

<math>H = U + PV</math>

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

<math> U = q + w </math>

<math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

Thus we can explain the decrease in the Gibbs free energy, as despite having an increase in volume, the entropic contribution wins due to the large temperature values. Initially the graph shows a slow decrease in free energy which has a more negative gradient as temperature increases. This suggests that the change in pressure isn't constant, as we expect:

<math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>

which would give a linear negative gradient. These disparities from the above equations could be due to limitations of the approximation.

=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[ThermalExpansionCoefficientQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>

Using L0 as the lattice constant at 0K
 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as αV = 3αL

These values compare well to literature values: between 301.15 to 1273.15 K

28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.

Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.

<nowiki>===Speculations===</nowiki>
* What is the physical origin of thermal expansion ?
* As the temperature approaches the melting point of MgO how well do the phonon modes represent the actual motions of the ions ?
* In a diatomic molecule with an exactly harmonic potential would you expect the bond length to increase with temperature ? Why does it increase in the solid when we are using an quasi-harmonic approximation ?

In this calculation, the main approximation is to do with the anharmonic contributions to the harmonic approximation. In this model, the phonon frequencies are volume dependent which is a simplified way to compute anharmonicity. This in turn means that at higher temperatures the anharmonic factor increases. Other approximations include the Born-Oppenheimer Approximation which assumes that the motion of atomic nuclei and electron in a molecule can be separated. These approximations thus limit the precision and validity of the model used at higher temperatures.

The physical origin of thermal expansion arises from the fact that as the temperature increases the amplitude of atomic vibrations increases. A higher vibrational amplitude means that the average atomic distance in a cell increases, hence resulting in expansion. Additionally, as the temperature approaches the melting point of MgO the quasi-harmonic approximation breaks down since this approximation only takes into account linear motion. Therefore, the quasi-harmonic model prevents movement of free ions, which is present in molten MgO and hence this model is not appropriate for MgO near the melting point. Furthermore, as the temperature approaches the melting point the phonon modes become less representative of the actual motion of the ions since at higher temperatures the quasi-harmonic approximation tends to neglect phonon interactions.

For a diatomic molecule with an exactly harmonic potential, the bond length would not be expected in increase with temperature because the potential is symmetric and hence the average distance between atoms is constant. However, in the quasi-harmonic model for a solid the bond length increases with temperature (thermal expansion) due to the additional anharmonic factor included where bond dissociation is possible.

== Molecular Dynamics ==

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-22T15:49:11Z

Sa4213: /* Thermal Expansion of MgO */

Rep:Mod:SA4213MgO

2016-02-22T14:59:32Z

Sa4213: /* Thermal Expansion of MgO */

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As befere this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

It is important to note that every possible vibration is associated with a k value.

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the viration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity and thus k approaches 0 at Γ. This can be rationalised via: k= \frac{2\pi}{\lambda}. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy via: E = \hbar\omega \qquad
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches comine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As a*= \frac{2\pi}{a} large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}

== Thermal Expansion of MgO ==


+Plot the lattice constant against temperature 
+Comment on the shape of these curves. 
Compute the coefficient of thermal expansion for MgO 
How does this compare to that measured ? Find a measurement in the literature or on the web - at what temperature was the measurement made ? 
What are the main approximations in your calculation ?


{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|600px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_SA4213MgO.png|thumb|600px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_x = \frac{1}{x}\,\left(\frac{\partial x}{\partial T}\right)_p
</math> ''where αx is the volumetric thermal expansion coefficient, x is the (initial) dimension under study'', ∂V is the partial derivative of that dimension, ∂T is the partial derivative of temperature (at constant pressure) 
:
: 
Therefore, from this it can be said that the higher the expansion coefficient the greater the thermal expansion. Additionally, since MgO is an isotropic solid one would expect that the volumetric thermal expansion coefficient is low, resulting from the strong ionic bonding present.

Plotting the Gibbs Free energy against the Temperature also yields the expected result that being the Free energy of the system decreases with increasing Temperature, this relationship can be derived mathematically as seen below:

<math> G = H - TS </math>

<math>H = U + PV</math>

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

<math> U = q + w </math>

<math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

Thus despite the Gibbs free energy increasing slightly with increasing volume, it is heavily outweighed by the entropic contribution due to the large temperature values, checking the log files confirms the expected positive entropy for the system and thus increasing the temperature further lowers the Gibbs Free energy. This can be seen in the graph by the initial gradual change in Free energy followed by a sharp near linear decrease with temperature suggesting a strong negative correlation. This is further enforced by an R 2 value of 0.9204 for all data points and a value of 0.9261 when isolating the mostly linear part of the curve. The fact that the curve isn't perfectly linear does suggest that the Pressure isn't constant as if it was the gradient would be exactly linear with a gradient of -S given by

<math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>

Of course this could also be due to a limitation of the approximation however as it impossible to see from the log files if the pressure had changed it is hard to determine this.
 
 
=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[ThermalExpansionCoefficientQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>
Using L0 as the lattice constant at 0K
 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as αV = 3αL

Comparing these values to literature values one paper[7] given a range of 28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.

Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.

== Molecular Dynamics ==

zz/as/sg/jb

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-22T14:58:29Z

Sa4213:

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As befere this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

It is important to note that every possible vibration is associated with a k value.

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the viration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity and thus k approaches 0 at Γ. This can be rationalised via: k= \frac{2\pi}{\lambda}. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy via: E = \hbar\omega \qquad
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches comine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As a*= \frac{2\pi}{a} large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}

== Thermal Expansion of MgO ==


+Plot the lattice constant against temperature 
+Comment on the shape of these curves. 
Compute the coefficient of thermal expansion for MgO 
How does this compare to that measured ? Find a measurement in the literature or on the web - at what temperature was the measurement made ? 
What are the main approximations in your calculation ?


{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|600px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_SA4213MgO.png|thumb|600px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation.

The coefficient of thermal expansion measures the dependence of size on temperature, standardised by dividing by the dimensionality under study, for example by volume if αV. The general equation is:

:<math>
\alpha_V = \frac{1}{V}\,\left(\frac{\partial V}{\partial T}\right)_p
</math> ''where αV is the volumetric thermal expansion coefficient, V is the (initial) volume'', ∂V is the partial derivative of volume, ∂T is the partial derivative of temperature (at constant pressure) 
:
: 
Therefore, from this it can be said that the higher the expansion coefficient the greater the thermal expansion. Additionally, since MgO is an isotropic solid one would expect that the volumetric thermal expansion coefficient is low, resulting from the strong ionic bonding present.

Plotting the Gibbs Free energy against the Temperature also yields the expected result that being the Free energy of the system decreases with increasing Temperature, this relationship can be derived mathematically as seen below:

<math> G = H - TS </math>

<math>H = U + PV</math>

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

<math> U = q + w </math>

<math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

Thus despite the Gibbs free energy increasing slightly with increasing volume, it is heavily outweighed by the entropic contribution due to the large temperature values, checking the log files confirms the expected positive entropy for the system and thus increasing the temperature further lowers the Gibbs Free energy. This can be seen in the graph by the initial gradual change in Free energy followed by a sharp near linear decrease with temperature suggesting a strong negative correlation. This is further enforced by an R 2 value of 0.9204 for all data points and a value of 0.9261 when isolating the mostly linear part of the curve. The fact that the curve isn't perfectly linear does suggest that the Pressure isn't constant as if it was the gradient would be exactly linear with a gradient of -S given by

<math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>

Of course this could also be due to a limitation of the approximation however as it impossible to see from the log files if the pressure had changed it is hard to determine this.
 
 
=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[ThermalExpansionCoefficientQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>
Using L0 as the lattice constant at 0K
 

Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as αV = 3αL

Comparing these values to literature values one paper[7] given a range of 28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.

Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.

== Molecular Dynamics ==

zz/as/sg/jb

== Conclusion ==

Rep:Mod:SA4213MgO

2016-02-22T14:51:31Z

Sa4213:

== Introduction ==
Introduction

== Internal Energy of an MgO Crystal ==

{|align="center"
|[[File:PrimitiveSA4213MgO.png|left|thumb|300px|Figure 1. Primitive Cell of MgO]]
|[[File:ConventionalSA4213MgO.png|left|thumb|300px|Figure 2. Conventional Cell of MgO]]
|}

It is necessary to define our unit cell for our MgO calculations, as such we have the primitive unit cell (Figure 1) and the conventional unit cell (figure 2). The primitive cell has a total of 2 atoms – Mg and O; thus is the simplest cell to describe the crystal. It's cell vector dimensions are shown in table 1. The cell takes the shape of a rhombohedron with a lattice constant of a = 2.978(3) Å and internal angle α = 60 [INSERT DEGREE SIGN]. The GULP calculation correlates with LCAO HF calculations found in literature (2.573 Å [INSERT REFERENCE doi: 10.1007/10681719_206 ]).

{|class="wikitable"
|+ Table 1ː Cell Vector Dimensions/Å
|----
| 0.00000
| 2.10597
| 2.10597
|---
| 2.10597
| 0.00000
| 2.10597
|---
| 2.10597
| 2.10597
| 0.00000
|---
|}

A simple calculation to find the total lattice energy was undertaken. In this, the Mg ion is given a charge of +2e, the O ion -2e and electrostatic potentials are considered, then the energy required to separate the ions of the lattice to infinite separation is calculated at absolute zero. This gave a value of -41.07 eV per primitive unit cell.

The conventional cell, Figure 2, is face centred cubic with a lattice constant of 4.212 ‎Å and internal angle of 90 degrees. As befere this also allies with literature (4.211 Å – [INSERT REFERENCE U. Rössler and R. Blachnik, Magnesium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-6]. The conventional cell is larger than the primitive, and contains 8 atoms – 4 of both Mg and O; as such it has quadruple the volume of the primitive cell.

== Computing the Phonon Dispersion Curves ==

{|align="center"
|[[File:MgO_Phonon_Dispersion_SA4213MgO.png|thumb|300px|Figure 3. Phonon Dispersion Graph of MgO]]
|}

It is important to note that every possible vibration is associated with a k value.

Next the phonon dispersion curve of MgO was computed using GULP – figure 3. Here we measured the frequency of a photon needed to excite the viration at 50 values of k along the path W, L, Γ, X, W, K. We can see 6 different phonon modes, or branches in the dispersion curve. This is due to each axis having acoustic (in-phase vibration) and optical (out-of-phase vibration) phonons arising from having 2 atoms in our primitive cell. [INSERT REFERENCE - G. E. Peckham. Phonon Dispersion Relations in Crystals. 1964: 1-5.] For the three optical modes (3N-3) as k approaches 0 their frequencies are non zero as they cause the atoms to move in opposite directions upon excitation. For the three acoustic modes, due to their in phase vibrations their wavelength approaches infinity and thus k approaches 0 at Γ. This can be rationalised via: k= \frac{2\pi}{\lambda}. Tracing the branches from Γ to L, and then W, the acoustic branches split into the three acoustic vibrational modes. These are seen as two transverse modes which are degenerate at L and a longitudinal mode.
 
 
=== Phonon Density of States (DOS) ===

From the phonon dispersion curves, we can find the number of available states at each K value we consider. The number of states can then be plotted as a function of frequency to obtain a density of states (DOS) relation. As such we receive DOS curves plotting the distribution of phonons in terms of vibration and by extension energy via: E = \hbar\omega \qquad
 The more k values we consider, the more detailed and accurate our Phonon DOS will be. To achieve the most accurate answer we would need to having an infinitely large nxnxn grid such that we sample over all k points in our cell, essentially meaning the spacing between k points is dk. This would lead to an infinitely long computation and be very expensive with very large values of n. Instead we must find a grid size large enough to provide enough k points to resemble the true value, and when increased doesn't provide as large an increase in the information we're getting out as the increase in computation time we're using to perform the calculation.

{|align="center"
|[[File:MgO_Phonon_DOS_1x1x1_SA4213MgO.png|thumb|300px|Figure 4. Phonon DOS of MgO using 1x1x1 grid size]]
|[[File:MgO_Phonon_DOS_2x2x2_SA4213MgO.png|thumb|300px|Figure 5. Phonon DOS of MgO using 2x2x2 grid size]]
|[[File:MgO_Phonon_DOS_16x16x16_SA4213MgO.png|thumb|300px|Figure 6. Phonon DOS of MgO using 16x16x16 grid size]]
|[[File:MgO_Phonon_DOS_32x32x32_SA4213MgO.png|thumb|300px|Figure 7. Phonon DOS of MgO using 32x32x32 grid size]]
|}

Figure 4 shows the Phonon DOS using a 1x1x1 grid, which samples one k value. By comparing the peak intensities and frequencies (288 cm-1 and 352 cm-1; 676 cm-1 and 819 cm-1) We see the pair of peaks at 300 cm-1 is roughly double the intensity of those around 700 cm-1. Implying branches have come together to be degenerate at that K value. Those frequencies match with the K point L, and it can be seen from W to L that 4 branches comine to 2. It was found that 16x16x16 (Figure 6) was both computationally cheap and able to accurately replicate the Phonon DOS of larger n value grids (figure 7 - 32x32x32). The larger the n value beyond 16 the smoother the graph of the DOS as more k points are being sampled it is more representative of the true Phonon DOS.

{|align="center"
|[[File:MgO_Phonon_Dispersion%2BDOS_SA4213MgO.png|thumb|600px|Figure 8. Phonon Dispersion next to Phonon DOS of MgO using 32x32x32 grid and Frequency as their common axis]]
|}

The Phonon DOS(frequency) is proportional to the inverse of the slope of Frequency(k) vs. k, which corresponds to flatter branches equal larger DOS values at that frequency. Figure 8 shows this visually. We can see that if we sample enough k points we will produce a DOS that samples from enough of the k values to accurately translate the dispersion graph. A 1x1x1 grid size only samples one k value and so doesn't accurately represent the DOS of MgO. Whereas the 32x32x32 grid accurately translated the inverse of the slope into its DOS plot.

The size of the grid is dependent on the size of the cell in real space. As a*= \frac{2\pi}{a} large values of a (large cells in real space) will give small values of a* (small cells in k space). If instead we were looking at the DOS of a metal such as lithium, which has a small cell in real space (a = 3.51‎ Å [INSERT REFERENCE: M. Nadler and C. Kempfer, Anal. Chem., 1959, 31, 2109]) - therefore large cell in k space, we will need large values of n so that the k values we sample will accurately represent the k values across all of the cell. Conversely a large repeat unit for example in a zeolite (a = 24.5 ‎Å. [INSERT REFERENCE: J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer Berlin Heidelberg, Berlin, 1999, 311], will have a small cell in k space, thus we can produce accurate results with small values of n as the k points we sample will be close together so the points we're missing won't add enough information to our DOS to outweigh the negative of doing a more computationally intensive calculation. We could perform this grid size DOS calculation on a crystal cell of similar size such as CaO as its cell will have many similarities with MgO. Most importantly the value of a = 4.800 ‎Å [INSERT REFERENCE: U. Rössler and R. Blachnik, Calcium Oxide Crystal Structure, Lattice Parameters, Thermal Expansion, In: II-VI and I-VII compounds; semimagnetic compounds, Springer, Berlin, 1999, 1-3] and the similarity of MgO to CaO will produce similar cells in k space, so we can expect a 16x16x16 grid size to produce a reasonable approximation of DOS.

=== Computing the Free Energy using the Quasi-Harmonic Approximation ===

Table 2 shows the variation of free energy as a function of grid size. As n increases the free energy value converges to -40.926483 eV. Beyond n=4 the variation in free energy quickly approaches 0. The variation in free energy between k=1 and k=48 is not large and accounts for 0.009% of the total free energy. The main contributions to the free energy, monopole interactions and inter-atomic potentials, are covered by the Buckingham potential and so the deviation from the converged value isn't large.

{|class="wikitable"
|+ Table 2ː Energy vs Grid size
! Grid Size nxnxn/n
! Free Energy/eV
! Accuracy/meV
! rowspan="9"|[[File:Internal_Energy_vs_Grid_Size_SA4213MgO.png|thumb|300px|Figure 9. Internal Energy vs Grid Size]]
|-
| 1
| -40.930301
| 4
|-
| 2
| -40.926609
| 0.2
|-
| 3
| -40.926432
| 0.1
|-
| 4
| -40.926450
| 0.5
|-
| 8
| -40.926478
| 0.5
|-
| 16
| -40.926482
| 0.1
|-
| 32
| -40.926483
| 0.1
|-
| 48
| -40.926483
| N/A
|}

== Thermal Expansion of MgO ==


+Plot the free energy against temperature 
+Plot the lattice constant against temperature 
+Comment on the shape of these curves. 
Compute the coefficient of thermal expansion for MgO 
How does this compare to that measured ? Find a measurement in the literature or on the web - at what temperature was the measurement made ? 
What are the main approximations in your calculation ?


{|align="center"
|[[File:EnergyQH_SA4213MgO.png|thumb|600px|Figure 10. Free Energy dependence on Temperature for the Quasi-Harmonic Approximation]]
|[[File:LatticeConstantQH_SA4213MgO.png|thumb|600px|Figure 11. Lattice Constant dependence on Temperature for the Quasi-Harmonic Approximation]]
|}

The structure of MgO was then optimised with respect to the free energy, whilst varying temperature between 0 to 1000 K. The free energy was then computed within the quasi-harmonic approximation. Additionally, the thermal expansion of MgO was computed using molecular dynamics and the results compared with that from the quasi-harmonic approximation. In both cases, the coefficient of thermal expansion was calculated.

The coefficient of thermal expansion describes how the size an object changes with temperature. More specifically, it measures the fractional change in size per degree change in temperature at constant pressure. There are many types of defined coefficients, for example linear, area and volumetric. The type to use depends on the particular application and the dimensions required. In this case, the volumetric thermal expansion coefficient was used, the equation of which is shown below.

:<math>
\alpha_V = \frac{1}{V}\,\left(\frac{\partial V}{\partial T}\right)_p
</math> ''where αV is the volumetric thermal expansion coefficient, V is the (initial) volume'', ∂V is the partial derivative of volume, ∂T is the partial derivative of temperature (at constant pressure) 
:
: 
Therefore, from this it can be said that the higher the expansion coefficient the greater the thermal expansion. Additionally, since MgO is an isotropic solid one would expect that the volumetric thermal expansion coefficient is low, resulting from the strong ionic bonding present.

Plotting the Gibbs Free energy against the Temperature also yields the expected result that being the Free energy of the system decreases with increasing Temperature, this relationship can be derived mathematically as seen below:

<math> G = H - TS </math>

<math>H = U + PV</math>

<math>G = U + PV - TS</math>

<math> dG = dU + PdV + VdP - TdS - SdT </math>

<math> U = q + w </math>

<math> dq = TdS </math> <math> dw = -PdV </math>

<math> \therefore dG = VdP - SdT </math>

Thus despite the Gibbs free energy increasing slightly with increasing volume, it is heavily outweighed by the entropic contribution due to the large temperature values, checking the log files confirms the expected positive entropy for the system and thus increasing the temperature further lowers the Gibbs Free energy. This can be seen in the graph by the initial gradual change in Free energy followed by a sharp near linear decrease with temperature suggesting a strong negative correlation. This is further enforced by an R 2 value of 0.9204 for all data points and a value of 0.9261 when isolating the mostly linear part of the curve. The fact that the curve isn't perfectly linear does suggest that the Pressure isn't constant as if it was the gradient would be exactly linear with a gradient of -S given by

<math> \left( \frac{\partial G}{\partial T} \right)_P = -S </math>

Of course this could also be due to a limitation of the approximation however as it impossible to see from the log files if the pressure had changed it is hard to determine this.
 
 
=== Calculating the Thermal Expansion Coefficients ===

{|align="center"
|[[ThermalExpansionCoefficientQH_SA4213MgO.png|thumb|600px|Figure 12. Linear Dependence of Lattice Constant with Temperature]]
|}

Linear Thermal expansion coefficient:

<math>\alpha_L = \frac{1}{L} \left(\frac{\partial L}{\partial T}\right)_P</math>

<math>\alpha_L = \frac{0.00002346}{2.986563} = 7.855\times 10^{-6} </math>
Using L0 as the lattice constant at 0K
 
Volumetric Thermal expansion coefficient:

<math>\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\alpha_V = \frac{0.00044678}{18.836496} = 2.372\times 10^{-5}</math>

It is interesting to note that αV is 3.020 times αL. This implies MgO is an isotropic material [INSERT REFERENCE: Vinson JR. Plate and Panel structures of Isotropic, Composite and Piezoelectric Materials, including Sandwich Construction. Delaware: Springer; 2005], as the value is essentially 3 - within in the error caused by limitations in the theory, which would manifest itself as equal expansion along each lattice constant a, b and c. Therefore we can express αV as αV = 3αL

Comparing these values to literature values one paper[7] given a range of 28-1000 °C or 301.15-1273.15 K which would result in a very similar <math> \Delta T </math> to that in the experiment gave an averaged value of <math>14.3 \times 10^{-6} K^{-1}</math> with the Linear and Volumetric coefficient at 293 <math>K = 7.88 \times 10^{-6} K^{-1}</math> and <math>2.36 \times 10^{-5} K^{-1} </math> which are very close to the values obtained from the experiment, the deviation in average value is most likely due to the fact that the model used assumes that at every lattice constant the harmonic oscillator is applicable to the system whereas in reality this may not be the case.

Interestingly the experimental values at near room temperature for both Linear and Volumetric are within 0.0000005 and 0.000001 respectively. This suggests that over the range in this experiment the highest contribution is that of approximately room temperature as it correlates to that value in real experiments.

== Molecular Dynamics ==

zz/as/sg/jb

== Conclusion ==

File:ThermalExpansionCoefficientQH SA4213MgO.png

2016-02-22T14:10:39Z

Sa4213:

File:LatticeConstantQH SA4213MgO.png

2016-02-22T13:41:54Z

Sa4213: