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Rep:MgO - OHC15

2018-02-14T10:43:28Z

Ohc15: /* Quasi-Harmonic Approximation */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal expansivity of MgO was evaluated under Quasi-harmonic approximation (QHA) using a 2-atom primitive cell and molecular dynamics (MD) using a 64-atom cell in temperature range from 0K to 3000K. Optimal grid size for MgO QHA calculation was determined to be 32x32x32 where free energy is converged and gave a smooth density of states plot. Expansion coefficients of <math>3.0022\times10^{-5}\,\pm\,2.7146\times10^{-9}\,K^{-1}\,(QHA)</math> and <math>3.0740\times 10^{-5}\,\pm\,1.0717\times 10^{-7}\,K^{-1}\,(MD)</math> obtained agree closely with literature value of <math>3.12\times 10^{-5}\,K^{-1}</math> <ref name='love'/> from 300K to 1300K. MD was found to be a better calculation method over a wider temperature range, while QHA break down at temperature approximately greater than 2000K.

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail at high temperatures. Stronger vibrations at high temperatures result in erratic motion of atoms in the crystal, where bonds are lengthened to region beyond the cut-off sphere. Especially at temperatures near phase transition, simulations in this temperature rergion would not give meaningful rersults as aproximation 1) would no longer holds. Bond break during phase transition lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure 1. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure 2. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot.The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 of 1 longitudinal and 2 transverse acoustic modes and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure 3. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure 3. showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 15-25 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure 4. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure 5. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure 6. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure 7. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure 8. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure 9. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure 10. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure 11. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure 12. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, energy difference decreases greatly with increasing shrinking factor. It is noted that the shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking factor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure 13. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation 4).

[[File:LD Helmholtz.png|center|thumb|Figure 13. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods are comparable in conventional cell unit. Molecular dynamic calculation cannot be executed at zero kelvin as atoms in lattice has zero kinetic energy at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereature, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure 14.)

Under QHA, temperature rise leads to volume expansion of lattice to minimise Free Energy of system, which lowers vibration frequencies due to larger lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value due to bond break and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure 16. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure 17&18).

[[File:Cell Vol both methods.png|center|thumb|Figure. 14 Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\delta V}{\delta T})_{P}</math>

where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref name='love'>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure 16.) The almost zero standard deviations of best fit line gradients of lattice volumes from 300K to 1300K supports that the simplification is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure 15. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure 16. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}

===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step. Increasing Production step would also improve relability of averaged data by reducing the effect of non-equlibrated data on the final averrage. Both settings were changed to 700 and 900 steps for investigation. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less rounds of lattice volume and energy calculations would be performed on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure 17. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure 18. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure 19. <ref name='son'>A. Erba, M. Shahrokhi, R. Moradian and R. Dovesi, On how differently the quasi-harmonic approximation works for two isostructural crystals: Thermal properties of periclase and lime, J. Chem. Phys., 2015, 142, 44114.</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. The idea is similar to having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relative motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure 19. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref name='son'/>)|400px]]

==Conclusion==
Quasi-harmonic approximation is derived from harmonic approximation with anharmonic effect accounted for, while molecular dynamics follow atom trajectories in the equilibrated lattice. Cell volume data obtained from both methods were converted to same unit and gradirent of linear fit from 300K to 1300K was calculated to give volumetric thermal expansion coefficients of <math>3.0022e-05\,\pm\,2.7146e-09\,K^{-1}</math> and <math>3.0740\times 10^{-5}\,\pm\,1.0717\times 10^{-7}\,K^{-1}</math> respectively in QHA and MD. Experimental rersults matches with the literature value of <math>3.12\times 10^{-5}\,K^{-1}</math><ref name='love'/> with small standard dedvation values. Investigation at high temperature failed using QHA as cell volume was overestimated. On the other hand, molecular dynamic is a better model that can operate overwider temperature range, even going beyond the melting point of MgO and expand scope of investigation to phase transition region.

Extension on improving results obtained within molecular dynamics with higher equilibration and production time was shown to be successful in reducing random fluctuations in thermal expansion coefficients as cell volume data are closer to converged value. The effect of increasing equilibration time to 900 steps from 500 steps did not seem to be effective enough. If more time is available, individual experiment on finding the optimal equilibration and production step, as well as time step could beb carried out to maximise the accuracy nad reliability of data.

==Reference==

Rep:MgO - OHC15

2018-02-14T10:35:26Z

Ohc15: /* Conclusion */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal expansivity of MgO was evaluated under Quasi-harmonic approximation (QHA) using a 2-atom primitive cell and molecular dynamics (MD) using a 64-atom cell in temperature range from 0K to 3000K. Optimal grid size for MgO QHA calculation was determined to be 32x32x32 where free energy is converged and gave a smooth density of states plot. Expansion coefficients of <math>3.0022\times10^{-5}\,\pm\,2.7146\times10^{-9}\,K^{-1}\,(QHA)</math> and <math>3.0740\times 10^{-5}\,\pm\,1.0717\times 10^{-7}\,K^{-1}\,(MD)</math> obtained agree closely with literature value of <math>3.12\times 10^{-5}\,K^{-1}</math> <ref name='love'/> from 300K to 1300K. MD was found to be a better calculation method over a wider temperature range, while QHA break down at temperature approximately greater than 2000K.

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure 1. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure 2. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot.The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 of 1 longitudinal and 2 transverse acoustic modes and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure 3. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure 3. showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 15-25 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure 4. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure 5. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure 6. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure 7. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure 8. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure 9. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure 10. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure 11. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure 12. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, energy difference decreases greatly with increasing shrinking factor. It is noted that the shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking factor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure 13. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation 4).

[[File:LD Helmholtz.png|center|thumb|Figure 13. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods are comparable in conventional cell unit. Molecular dynamic calculation cannot be executed at zero kelvin as atoms in lattice has zero kinetic energy at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereature, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure 14.)

Under QHA, temperature rise leads to volume expansion of lattice to minimise Free Energy of system, which lowers vibration frequencies due to larger lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value due to bond break and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure 16. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure 17&18).

[[File:Cell Vol both methods.png|center|thumb|Figure. 14 Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\delta V}{\delta T})_{P}</math>

where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref name='love'>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure 16.) The almost zero standard deviations of best fit line gradients of lattice volumes from 300K to 1300K supports that the simplification is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure 15. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure 16. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}

===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step. Increasing Production step would also improve relability of averaged data by reducing the effect of non-equlibrated data on the final averrage. Both settings were changed to 700 and 900 steps for investigation. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less rounds of lattice volume and energy calculations would be performed on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure 17. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure 18. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure 19. <ref name='son'>A. Erba, M. Shahrokhi, R. Moradian and R. Dovesi, On how differently the quasi-harmonic approximation works for two isostructural crystals: Thermal properties of periclase and lime, J. Chem. Phys., 2015, 142, 44114.</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. The idea is similar to having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relative motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure 19. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref name='son'/>)|400px]]

==Conclusion==
Quasi-harmonic approximation is derived from harmonic approximation with anharmonic effect accounted for, while molecular dynamics follow atom trajectories in the equilibrated lattice. Cell volume data obtained from both methods were converted to same unit and gradirent of linear fit from 300K to 1300K was calculated to give volumetric thermal expansion coefficients of <math>3.0022e-05\,\pm\,2.7146e-09\,K^{-1}</math> and <math>3.0740\times 10^{-5}\,\pm\,1.0717\times 10^{-7}\,K^{-1}</math> respectively in QHA and MD. Experimental rersults matches with the literature value of <math>3.12\times 10^{-5}\,K^{-1}</math><ref name='love'/> with small standard dedvation values. Investigation at high temperature failed using QHA as cell volume was overestimated. On the other hand, molecular dynamic is a better model that can operate overwider temperature range, even going beyond the melting point of MgO and expand scope of investigation to phase transition region.

Extension on improving results obtained within molecular dynamics with higher equilibration and production time was shown to be successful in reducing random fluctuations in thermal expansion coefficients as cell volume data are closer to converged value. The effect of increasing equilibration time to 900 steps from 500 steps did not seem to be effective enough. If more time is available, individual experiment on finding the optimal equilibration and production step, as well as time step could beb carried out to maximise the accuracy nad reliability of data.

==Reference==

Rep:MgO - OHC15

2018-02-14T10:31:14Z

Ohc15: /* Conclusion */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal expansivity of MgO was evaluated under Quasi-harmonic approximation (QHA) using a 2-atom primitive cell and molecular dynamics (MD) using a 64-atom cell in temperature range from 0K to 3000K. Optimal grid size for MgO QHA calculation was determined to be 32x32x32 where free energy is converged and gave a smooth density of states plot. Expansion coefficients of <math>3.0022\times10^{-5}\,\pm\,2.7146\times10^{-9}\,K^{-1}\,(QHA)</math> and <math>3.0740\times 10^{-5}\,\pm\,1.0717\times 10^{-7}\,K^{-1}\,(MD)</math> obtained agree closely with literature value of <math>3.12\times 10^{-5}\,K^{-1}</math> <ref name='love'/> from 300K to 1300K. MD was found to be a better calculation method over a wider temperature range, while QHA break down at temperature approximately greater than 2000K.

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure 1. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure 2. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot.The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 of 1 longitudinal and 2 transverse acoustic modes and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure 3. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure 3. showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 15-25 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure 4. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure 5. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure 6. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure 7. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure 8. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure 9. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure 10. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure 11. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure 12. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, energy difference decreases greatly with increasing shrinking factor. It is noted that the shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking factor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure 13. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation 4).

[[File:LD Helmholtz.png|center|thumb|Figure 13. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods are comparable in conventional cell unit. Molecular dynamic calculation cannot be executed at zero kelvin as atoms in lattice has zero kinetic energy at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereature, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure 14.)

Under QHA, temperature rise leads to volume expansion of lattice to minimise Free Energy of system, which lowers vibration frequencies due to larger lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value due to bond break and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure 16. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure 17&18).

[[File:Cell Vol both methods.png|center|thumb|Figure. 14 Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\delta V}{\delta T})_{P}</math>

where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref name='love'>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure 16.) The almost zero standard deviations of best fit line gradients of lattice volumes from 300K to 1300K supports that the simplification is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure 15. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure 16. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}

===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step. Increasing Production step would also improve relability of averaged data by reducing the effect of non-equlibrated data on the final averrage. Both settings were changed to 700 and 900 steps for investigation. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less rounds of lattice volume and energy calculations would be performed on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure 17. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure 18. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure 19. <ref name='son'>A. Erba, M. Shahrokhi, R. Moradian and R. Dovesi, On how differently the quasi-harmonic approximation works for two isostructural crystals: Thermal properties of periclase and lime, J. Chem. Phys., 2015, 142, 44114.</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. The idea is similar to having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relative motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure 19. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref name='son'/>)|400px]]

==Conclusion==
Quasi-harmonic approximation is derived from harmonic approximation with anharmonic effect accounted for, while molecular dynamics follow atom trajectories in the equilibrated lattice. Cell volume data obtained from both methods were converted to same unit and gradirent of linear fit from 300K to 1300K was calculated to give volumetric thermal expansion coefficients of <math>3.0022e-05\,\pm\,2.7146e-09\,K^{-1}</math> and <math>3.0740\times 10^{-5}\,\pm\,1.0717\times 10^{-7}\,K^{-1}</math> respectively in QHA and MD. Experimental rersults matches with the literature value of <math>3.12\times 10^{-5}\,K^{-1}</math><ref name='love'/> with small standard dedvation values. Investigation at high temperature failed using QHA as cell volume was overestimated. On the other hand, molecular dynamic is a better model that can operate overwider temperature range, even going beyond the melting point of MgO and expand scope of investigation to phase transition region.

Extension on improving results obtained within molecular dynamics with higher equilibration and production time was shown to be successful in reducing random fluctuations in thermal expansion coefficients and therefore cell volume data being closer to converged value. If more time is available, individual experiment on finding the optimal equilibration and production step, as well as time step could beb carried out to maximise the accuracy nad reliability of data.

==Reference==

Rep:MgO - OHC15

2018-02-14T10:22:38Z

Ohc15: /* Abstract */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal expansivity of MgO was evaluated under Quasi-harmonic approximation (QHA) using a 2-atom primitive cell and molecular dynamics (MD) using a 64-atom cell in temperature range from 0K to 3000K. Optimal grid size for MgO QHA calculation was determined to be 32x32x32 where free energy is converged and gave a smooth density of states plot. Expansion coefficients of <math>3.0022\times10^{-5}\,\pm\,2.7146\times10^{-9}\,K^{-1}\,(QHA)</math> and <math>3.0740\times 10^{-5}\,\pm\,1.0717\times 10^{-7}\,K^{-1}\,(MD)</math> obtained agree closely with literature value of <math>3.12\times 10^{-5}\,K^{-1}</math> <ref name='love'/> from 300K to 1300K. MD was found to be a better calculation method over a wider temperature range, while QHA break down at temperature approximately greater than 2000K.

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure 1. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure 2. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot.The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 of 1 longitudinal and 2 transverse acoustic modes and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure 3. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure 3. showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 15-25 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure 4. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure 5. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure 6. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure 7. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure 8. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure 9. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure 10. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure 11. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure 12. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, energy difference decreases greatly with increasing shrinking factor. It is noted that the shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking factor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure 13. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation 4).

[[File:LD Helmholtz.png|center|thumb|Figure 13. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods are comparable in conventional cell unit. Molecular dynamic calculation cannot be executed at zero kelvin as atoms in lattice has zero kinetic energy at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereature, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure 14.)

Under QHA, temperature rise leads to volume expansion of lattice to minimise Free Energy of system, which lowers vibration frequencies due to larger lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value due to bond break and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure 16. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure 17&18).

[[File:Cell Vol both methods.png|center|thumb|Figure. 14 Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\delta V}{\delta T})_{P}</math>

where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref name='love'>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure 16.) The almost zero standard deviations of best fit line gradients of lattice volumes from 300K to 1300K supports that the simplification is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure 15. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure 16. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}

===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step. Increasing Production step would also improve relability of averaged data by reducing the effect of non-equlibrated data on the final averrage. Both settings were changed to 700 and 900 steps for investigation. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less rounds of lattice volume and energy calculations would be performed on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure 17. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure 18. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure 19. <ref name='son'>A. Erba, M. Shahrokhi, R. Moradian and R. Dovesi, On how differently the quasi-harmonic approximation works for two isostructural crystals: Thermal properties of periclase and lime, J. Chem. Phys., 2015, 142, 44114.</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. The idea is similar to having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relative motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure 19. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref name='son'/>)|400px]]

==Conclusion==

Quasi-harmonic approximation is derived from harmonic approximation with anharmonic effect accounted for, while molecular dynamics follow atom trajectories in the equilibrated lattice. Cell volume data obtained from both methods were converted to same unit and gradirent of linear fit from 300K to 1300K was calculated to give volumetric thermal expansion coefficients of <math>3.0022e-05\,\pm\,2.7146e-09\,K^{-1}</math> and <math>3.0740\times 10^{-5}\,\pm\,1.0717\times 10^{-7}\,K^{-1}</math> respectively in QHA and MD. Experimental rersults matches with the literature value of <math>3.12\times 10^{-5}\,K^{-1}</math><ref name='love'/> with small standard dedvation values. Investigation at high temperature failed using QHA as cell volume was overestimated. On the other hand, molecular dynamic is a better model that can operate overwider temperature range, even going beyond the melting point of MgO and show more information on phase transition.

==Reference==

Rep:MgO - OHC15

2018-02-14T10:22:04Z

Ohc15: /* Conclusion */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal expansivity of MgO was evaluated under Quasi-harmonic approximation (QHA) using a 2-atom primitive cell and molecular dynamics (MD) using a 64-atom cell in temperature range from 0K to 3000K. Optimal grid size for MgO QHA calculation was determined to be 32x32x32 where free energy is converged and gave a smooth density of states plot. Expansion coefficients of <math>3.0022\times10^{-5}\,\pm\,2.7146\times10^{-9}\,(QHA)</math> and <math>3.0740\times 10^{-5}\,\pm\,1.0717\times 10^{-7}\,(MD)</math> obtained agree closely with literature value of <math>3.12\times 10^{-5}</math> <ref name='love'/> from 300K to 1300K. MD was found to be a better calculation method over a wider temperature range, while QHA break down at temperature approximately greater than 2000K.

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure 1. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure 2. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot.The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 of 1 longitudinal and 2 transverse acoustic modes and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure 3. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure 3. showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 15-25 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure 4. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure 5. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure 6. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure 7. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure 8. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure 9. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure 10. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure 11. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure 12. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, energy difference decreases greatly with increasing shrinking factor. It is noted that the shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking factor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure 13. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation 4).

[[File:LD Helmholtz.png|center|thumb|Figure 13. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods are comparable in conventional cell unit. Molecular dynamic calculation cannot be executed at zero kelvin as atoms in lattice has zero kinetic energy at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereature, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure 14.)

Under QHA, temperature rise leads to volume expansion of lattice to minimise Free Energy of system, which lowers vibration frequencies due to larger lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value due to bond break and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure 16. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure 17&18).

[[File:Cell Vol both methods.png|center|thumb|Figure. 14 Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\delta V}{\delta T})_{P}</math>

where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref name='love'>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure 16.) The almost zero standard deviations of best fit line gradients of lattice volumes from 300K to 1300K supports that the simplification is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure 15. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure 16. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}

===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step. Increasing Production step would also improve relability of averaged data by reducing the effect of non-equlibrated data on the final averrage. Both settings were changed to 700 and 900 steps for investigation. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less rounds of lattice volume and energy calculations would be performed on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure 17. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure 18. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure 19. <ref name='son'>A. Erba, M. Shahrokhi, R. Moradian and R. Dovesi, On how differently the quasi-harmonic approximation works for two isostructural crystals: Thermal properties of periclase and lime, J. Chem. Phys., 2015, 142, 44114.</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. The idea is similar to having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relative motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure 19. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref name='son'/>)|400px]]

==Conclusion==

Quasi-harmonic approximation is derived from harmonic approximation with anharmonic effect accounted for, while molecular dynamics follow atom trajectories in the equilibrated lattice. Cell volume data obtained from both methods were converted to same unit and gradirent of linear fit from 300K to 1300K was calculated to give volumetric thermal expansion coefficients of <math>3.0022e-05\,\pm\,2.7146e-09\,K^{-1}</math> and <math>3.0740\times 10^{-5}\,\pm\,1.0717\times 10^{-7}\,K^{-1}</math> respectively in QHA and MD. Experimental rersults matches with the literature value of <math>3.12\times 10^{-5}\,K^{-1}</math><ref name='love'/> with small standard dedvation values. Investigation at high temperature failed using QHA as cell volume was overestimated. On the other hand, molecular dynamic is a better model that can operate overwider temperature range, even going beyond the melting point of MgO and show more information on phase transition.

==Reference==

Rep:MgO - OHC15

2018-02-14T10:10:53Z

Ohc15: /* Computational Calculation of MgO Free Energy and Thermal Expansion */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal expansivity of MgO was evaluated under Quasi-harmonic approximation (QHA) using a 2-atom primitive cell and molecular dynamics (MD) using a 64-atom cell in temperature range from 0K to 3000K. Optimal grid size for MgO QHA calculation was determined to be 32x32x32 where free energy is converged and gave a smooth density of states plot. Expansion coefficients of <math>3.0022\times10^{-5}\,\pm\,2.7146\times10^{-9}\,(QHA)</math> and <math>3.0740\times 10^{-5}\,\pm\,1.0717\times 10^{-7}\,(MD)</math> obtained agree closely with literature value of <math>3.12\times 10^{-5}</math> <ref name='love'/> from 300K to 1300K. MD was found to be a better calculation method over a wider temperature range, while QHA break down at temperature approximately greater than 2000K.

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure 1. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure 2. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot.The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 of 1 longitudinal and 2 transverse acoustic modes and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure 3. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure 3. showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 15-25 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure 4. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure 5. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure 6. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure 7. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure 8. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure 9. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure 10. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure 11. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure 12. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, energy difference decreases greatly with increasing shrinking factor. It is noted that the shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking factor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure 13. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation 4).

[[File:LD Helmholtz.png|center|thumb|Figure 13. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods are comparable in conventional cell unit. Molecular dynamic calculation cannot be executed at zero kelvin as atoms in lattice has zero kinetic energy at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereature, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure 14.)

Under QHA, temperature rise leads to volume expansion of lattice to minimise Free Energy of system, which lowers vibration frequencies due to larger lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value due to bond break and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure 16. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure 17&18).

[[File:Cell Vol both methods.png|center|thumb|Figure. 14 Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\delta V}{\delta T})_{P}</math>

where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref name='love'>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure 16.) The almost zero standard deviations of best fit line gradients of lattice volumes from 300K to 1300K supports that the simplification is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure 15. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure 16. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}

===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step. Increasing Production step would also improve relability of averaged data by reducing the effect of non-equlibrated data on the final averrage. Both settings were changed to 700 and 900 steps for investigation. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less rounds of lattice volume and energy calculations would be performed on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure 17. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure 18. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure 19. <ref name='son'>A. Erba, M. Shahrokhi, R. Moradian and R. Dovesi, On how differently the quasi-harmonic approximation works for two isostructural crystals: Thermal properties of periclase and lime, J. Chem. Phys., 2015, 142, 44114.</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. The idea is similar to having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relative motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure 19. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref name='son'/>)|400px]]

==Conclusion==

Quasi-harmonic approximation, derived from harmonic approximation with anharmonic effect accounted for, yielded a volumetric thermal expansion coefficient of <math>3.0022e-05\,\pm\,2.7146e-09</math> in temperature range of 300K -1300K at 0GPa, which agrees with the literature value of <math>3.12\times 10^{-5}</math><ref name='love'/>. However, data beyond approximately 2000K deviates greatly from reality.

On the other hand, molecular dynamics is a better model that can operate overwider temperature range, even going beyond the melting point of MgO. This method yielded a volumetric coefficient of <math>3.0740\times 10^{-5}\,\pm\,1.0717\times 10^{-7}</math>, again agrees with the literature value even closer htan that in QHA.

==Reference==

Rep:MgO - OHC15

2018-02-14T10:01:59Z

Ohc15:

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal expansivity of MgO was evaluated under Quasi-harmonic approximation (QHA) using a 2-atom primitive cell and molecular dynamics (MD) using a 64-atom cell in temperature range from 0K to 3000K. Optimal grid size for MgO QHA calculation was determined to be 32x32x32 where free energy is converged and gave a smooth density of states plot. Expansion coefficients of <math>3.0022\times10^{-5}\,\pm\,2.7146\times10^{-9}\,(QHA)</math> and <math>3.0740\times 10^{-5}\,\pm\,1.0717\times 10^{-7}\,(MD)</math> obtained agree closely with literature value of <math>3.12\times 10^{-5}</math> <nowiki><ref name='son'/></nowiki> from 300K to 1300K. MD was found to be a better calculation method over a wider temperature range, while QHA break down at temperature approximately greater than 2000K.

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure 1. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure 2. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot.The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 of 1 longitudinal and 2 transverse acoustic modes and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure 3. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure 3. showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 10-15 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure 4. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure 5. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure 6. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure 7. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure 8. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure 9. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure 10. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure 11. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure 12. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, energy difference decreases greatly with increasing shrinking factor. It is noted that the shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking factor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure 13. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation 4).

[[File:LD Helmholtz.png|center|thumb|Figure 13. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods are comparable in conventional cell unit. Molecular dynamic calculation cannot be executed at zero kelvin as atoms in lattice has zero kinetic energy at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereature, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure 14.)

Under QHA, temperature rise leads to volume expansion of lattice to minimise Free Energy of system, which lowers vibration frequencies due to larger lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value due to bond break and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure 16. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure 17&18).

[[File:Cell Vol both methods.png|center|thumb|Figure. 14 Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\delta V}{\delta T})_{P}</math>

where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure 16.) The almost zero standard deviations of best fit line gradients of lattice volumes from 300K to 1300K supports that the simplification is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure 15. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure 16. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}

===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step. Increasing Production step would also improve relability of averaged data by reducing the effect of non-equlibrated data on the final averrage. Both settings were changed to 700 and 900 steps for investigation. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less rounds of lattice volume and energy calculations would be performed on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure 17. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure 18. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure 19. <ref name='son'>A. Erba, M. Shahrokhi, R. Moradian and R. Dovesi, On how differently the quasi-harmonic approximation works for two isostructural crystals: Thermal properties of periclase and lime, J. Chem. Phys., 2015, 142, 44114.</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. The idea is similar to having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relative motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure 19. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref name='son'/>)|400px]]

==Conclusion==

Quasi-harmonic approximation, derived from harmonic approximation with anharrmonic effect accounted for, yielded volumetric thermal expansion coefficient of <math>3.0022e-05\,\pm\,2.7146e-09</math> in temperaturre range of 300K -1300K at 0GPa, which agrees with the literrature value. However, data beyond approximately 2000K deviates greatly from reality.

On the other hand, molecular dynamics is a better model that can operates overwider temperature range, even going beyond the melting point. The fact that molecular dynamics calculate lattice volume based on averaged mechanical properties extracted from atom trajectory is a more direct

==Reference==

Rep:MgO - OHC15

2018-02-14T10:00:37Z

Ohc15: /* Abstract */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal expansivity of MgO was evaluated under Quasi-harmonic approximation (QHA) using a 2-atom primitive cell and molecular dynamics (MD) using a 64-atom cell in temperature range from 0K to 3000K. Optimal grid size for MgO QHA calculation was determined to be 32x32x32 where free energy is converged and gave a smooth density of states plot. Expansion coefficients of <math>3.0022\times10^{-5}\,\pm\,2.7146\times10^{-9}\,(QHA)</math> and <math>3.0740\times 10^{-5}\,\pm\,1.0717\times 10^{-7}\,(MD)</math> obtained agree closely with literature value of <math>3.12\times 10^{-5}</math> from 300K to 1300K. MD was found to be a better calculation method over a wider temperature range, while QHA break down at temperature approximately greater than 2000K.

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure 1. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure 2. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot.The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 of 1 longitudinal and 2 transverse acoustic modes and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure 3. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure 3. showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 10-15 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure 4. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure 5. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure 6. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure 7. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure 8. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure 9. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure 10. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure 11. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure 12. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, energy difference decreases greatly with increasing shrinking factor. It is noted that the shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking factor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure 13. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation 4).

[[File:LD Helmholtz.png|center|thumb|Figure 13. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods are comparable in conventional cell unit. Molecular dynamic calculation cannot be executed at zero kelvin as atoms in lattice has zero kinetic energy at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereature, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure 14.)

Under QHA, temperature rise leads to volume expansion of lattice to minimise Free Energy of system, which lowers vibration frequencies due to larger lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value due to bond break and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure 16. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure 17&18).

[[File:Cell Vol both methods.png|center|thumb|Figure. 14 Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\delta V}{\delta T})_{P}</math>

where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure 16.) The almost zero standard deviations of best fit line gradients of lattice volumes from 300K to 1300K supports that the simplification is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure 15. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure 16. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}

===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step. Increasing Production step would also improve relability of averaged data by reducing the effect of non-equlibrated data on the final averrage. Both settings were changed to 700 and 900 steps for investigation. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less rounds of lattice volume and energy calculations would be performed on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure 17. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure 18. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure 19. <ref name='son'>A. Erba, M. Shahrokhi, R. Moradian and R. Dovesi, On how differently the quasi-harmonic approximation works for two isostructural crystals: Thermal properties of periclase and lime, J. Chem. Phys., 2015, 142, 44114.</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. The idea is similar to having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relative motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure 19. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref name='son'/>)|400px]]

==Conclusion==

Quasi-harmonic approximation, derived from harmonic approximation with anharrmonic effect accounted for, yielded volumetric thermal expansion coefficient of <math>3.0022e-05\,\pm\,2.7146e-09</math> in temperaturre range of 300K -1300K at 0GPa, which agrees with the literrature value. However, data beyond approximately 2000K deviates greatly from reality.

On the other hand, molecular dynamics is a better model that can operates overwider temperature range, even going beyond the melting point. The fact that molecular dynamics calculate lattice volume based on averaged mechanical properties extracted from atom trajectory is a more direct

==Reference==

Rep:MgO - OHC15

2018-02-14T09:58:59Z

Ohc15: /* Conclusion */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal Expansion of MgO calculated using Lattice Dynamic of a primitive MgO cell and Molecular Dynamic 32 MgO unit supercell model were calculated to be XXX and XXX respectively, which agrees with literature values of XXX.(ref)

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure 1. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure 2. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot.The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 of 1 longitudinal and 2 transverse acoustic modes and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure 3. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure 3. showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 10-15 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure 4. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure 5. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure 6. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure 7. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure 8. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure 9. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure 10. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure 11. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure 12. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, energy difference decreases greatly with increasing shrinking factor. It is noted that the shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking factor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure 13. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation 4).

[[File:LD Helmholtz.png|center|thumb|Figure 13. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods are comparable in conventional cell unit. Molecular dynamic calculation cannot be executed at zero kelvin as atoms in lattice has zero kinetic energy at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereature, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure 14.)

Under QHA, temperature rise leads to volume expansion of lattice to minimise Free Energy of system, which lowers vibration frequencies due to larger lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value due to bond break and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure 16. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure 17&18).

[[File:Cell Vol both methods.png|center|thumb|Figure. 14 Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\delta V}{\delta T})_{P}</math>

where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure 16.) The almost zero standard deviations of best fit line gradients of lattice volumes from 300K to 1300K supports that the simplification is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure 15. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure 16. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}

===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step. Increasing Production step would also improve relability of averaged data by reducing the effect of non-equlibrated data on the final averrage. Both settings were changed to 700 and 900 steps for investigation. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less rounds of lattice volume and energy calculations would be performed on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure 17. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure 18. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure 19. <ref name='son'>A. Erba, M. Shahrokhi, R. Moradian and R. Dovesi, On how differently the quasi-harmonic approximation works for two isostructural crystals: Thermal properties of periclase and lime, J. Chem. Phys., 2015, 142, 44114.</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. The idea is similar to having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relative motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure 19. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref name='son'/>)|400px]]

==Conclusion==

Quasi-harmonic approximation, derived from harmonic approximation with anharrmonic effect accounted for, yielded volumetric thermal expansion coefficient of <math>3.0022e-05\,\pm\,2.7146e-09</math> in temperaturre range of 300K -1300K at 0GPa, which agrees with the literrature value. However, data beyond approximately 2000K deviates greatly from reality.

On the other hand, molecular dynamics is a better model that can operates overwider temperature range, even going beyond the melting point. The fact that molecular dynamics calculate lattice volume based on averaged mechanical properties extracted from atom trajectory is a more direct

==Reference==

Rep:MgO - OHC15

2018-02-14T09:25:18Z

Ohc15: /* Extension */

Rep:MgO - OHC15

2018-02-14T09:13:39Z

Ohc15: /* Extension */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal Expansion of MgO calculated using Lattice Dynamic of a primitive MgO cell and Molecular Dynamic 32 MgO unit supercell model were calculated to be XXX and XXX respectively, which agrees with literature values of XXX.(ref)

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure 1. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure 2. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot.The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 of 1 longitudinal and 2 transverse acoustic modes and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure 3. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure 3. showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 10-15 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure 4. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure 5. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure 6. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure 7. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure 8. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure 9. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure 10. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure 11. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure 12. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, energy difference decreases greatly with increasing shrinking factor. It is noted that the shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking factor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure 13. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation 4).

[[File:LD Helmholtz.png|center|thumb|Figure 13. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods are comparable in conventional cell unit. Molecular dynamic calculation cannot be executed at zero kelvin as atoms in lattice has zero kinetic energy at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereature, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure 14.)

Under QHA, temperature rise leads to volume expansion of lattice to minimise Free Energy of system, which lowers vibration frequencies due to larger lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value due to bond break and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure 16. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure 17&18).

[[File:Cell Vol both methods.png|center|thumb|Figure. 14 Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\delta V}{\delta T})_{P}</math>

where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure 16.) The almost zero standard deviations of best fit line gradients of lattice volumes from 300K to 1300K supports that the simplification is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure 15. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure 16. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}

===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step and Production step both to 700 and 900 steps. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less calculations of lattice volume and energy would be wasted on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure 17. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure 18. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure 19. <ref name='son'>A. Erba, M. Shahrokhi, R. Moradian and R. Dovesi, On how differently the quasi-harmonic approximation works for two isostructural crystals: Thermal properties of periclase and lime, J. Chem. Phys., 2015, 142, 44114.</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. Similar idea as having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relatuve motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref name='son'/>)|400px]]

==Conclusion==

==Reference==

Rep:MgO - OHC15

2018-02-14T08:47:09Z

Ohc15: /* Extension */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal Expansion of MgO calculated using Lattice Dynamic of a primitive MgO cell and Molecular Dynamic 32 MgO unit supercell model were calculated to be XXX and XXX respectively, which agrees with literature values of XXX.(ref)

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure 1. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure 2. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot.The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 of 1 longitudinal and 2 transverse acoustic modes and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure 3. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure 3. showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 10-15 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure 4. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure 5. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure 6. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure 7. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure 8. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure 9. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure 10. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure 11. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure 12. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, energy difference decreases greatly with increasing shrinking factor. It is noted that the shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking factor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure 13. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation 4).

[[File:LD Helmholtz.png|center|thumb|Figure 13. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods are comparable in conventional cell unit. Molecular dynamic calculation cannot be executed at zero kelvin as atoms in lattice has zero kinetic energy at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereature, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure 14.)

Under QHA, temperature rise leads to volume expansion of lattice to minimise Free Energy of system, which lowers vibration frequencies due to larger lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value due to bond break and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure 16. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure 17&18).

[[File:Cell Vol both methods.png|center|thumb|Figure. 14 Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\delta V}{\delta T})_{P}</math>

where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure 16.) The almost zero standard deviations of best fit line gradients of lattice volumes from 300K to 1300K supports that the simplification is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure 15. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure 16. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}

===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step and Production step both to 700 and 900 steps. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less calculations of lattice volume and energy would be wasted on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure 17. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure 18. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure 19. <ref>A. Erba, M. Shahrokhi, R. Moradian and R. Dovesi, On how differently the quasi-harmonic approximation works for two isostructural crystals: Thermal properties of periclase and lime, J. Chem. Phys., 2015, 142, 44114.</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. Similar idea as having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relatuve motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref> https://doi.org/10.1063/1.4906422</ref>)|400px]]

==Conclusion==

==Reference==

Rep:MgO - OHC15

2018-02-14T08:46:40Z

Ohc15: /* Extension */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal Expansion of MgO calculated using Lattice Dynamic of a primitive MgO cell and Molecular Dynamic 32 MgO unit supercell model were calculated to be XXX and XXX respectively, which agrees with literature values of XXX.(ref)

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure 1. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure 2. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot.The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 of 1 longitudinal and 2 transverse acoustic modes and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure 3. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure 3. showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 10-15 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure 4. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure 5. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure 6. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure 7. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure 8. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure 9. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure 10. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure 11. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure 12. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, energy difference decreases greatly with increasing shrinking factor. It is noted that the shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking factor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure 13. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation 4).

[[File:LD Helmholtz.png|center|thumb|Figure 13. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods are comparable in conventional cell unit. Molecular dynamic calculation cannot be executed at zero kelvin as atoms in lattice has zero kinetic energy at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereature, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure 14.)

Under QHA, temperature rise leads to volume expansion of lattice to minimise Free Energy of system, which lowers vibration frequencies due to larger lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value due to bond break and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure 16. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure 17&18).

[[File:Cell Vol both methods.png|center|thumb|Figure. 14 Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\delta V}{\delta T})_{P}</math>

where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure 16.) The almost zero standard deviations of best fit line gradients of lattice volumes from 300K to 1300K supports that the simplification is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure 15. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure 16. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}

===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step and Production step both to 700 and 900 steps. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less calculations of lattice volume and energy would be wasted on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure 19. <ref>A. Erba, M. Shahrokhi, R. Moradian and R. Dovesi, On how differently the quasi-harmonic approximation works for two isostructural crystals: Thermal properties of periclase and lime, J. Chem. Phys., 2015, 142, 44114.</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. Similar idea as having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relatuve motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref> https://doi.org/10.1063/1.4906422</ref>)|400px]]

==Conclusion==

==Reference==

Rep:MgO - OHC15

2018-02-14T08:40:07Z

Ohc15: /* Comparison of Thermal Expansion Results of Different Methods */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal Expansion of MgO calculated using Lattice Dynamic of a primitive MgO cell and Molecular Dynamic 32 MgO unit supercell model were calculated to be XXX and XXX respectively, which agrees with literature values of XXX.(ref)

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure 1. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure 2. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot.The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 of 1 longitudinal and 2 transverse acoustic modes and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure 3. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure 3. showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 10-15 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure 4. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure 5. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure 6. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure 7. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure 8. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure 9. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure 10. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure 11. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure 12. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, energy difference decreases greatly with increasing shrinking factor. It is noted that the shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking factor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure 13. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation 4).

[[File:LD Helmholtz.png|center|thumb|Figure 13. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods are comparable in conventional cell unit. Molecular dynamic calculation cannot be executed at zero kelvin as atoms in lattice has zero kinetic energy at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereature, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure 14.)

Under QHA, temperature rise leads to volume expansion of lattice to minimise Free Energy of system, which lowers vibration frequencies due to larger lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value due to bond break and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure 16. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure 17&18).

[[File:Cell Vol both methods.png|center|thumb|Figure. 14 Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\delta V}{\delta T})_{P}</math>

where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure 16.) The almost zero standard deviations of best fit line gradients of lattice volumes from 300K to 1300K supports that the simplification is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure 15. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure 16. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}

===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step and Production step both to 700 and 900 steps. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less calculations of lattice volume and energy would be wasted on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure. <ref> https://doi.org/10.1063/1.4906422</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. Similar idea as having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relatuve motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref> https://doi.org/10.1063/1.4906422</ref>)|400px]]

==Conclusion==

==Reference==

Rep:MgO - OHC15

2018-02-14T08:21:02Z

Ohc15: /* Helmholtz-Free Energy (Quasi-Harmonic Approximation) */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal Expansion of MgO calculated using Lattice Dynamic of a primitive MgO cell and Molecular Dynamic 32 MgO unit supercell model were calculated to be XXX and XXX respectively, which agrees with literature values of XXX.(ref)

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure 1. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure 2. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot.The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 of 1 longitudinal and 2 transverse acoustic modes and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure 3. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure 3. showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 10-15 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure 4. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure 5. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure 6. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure 7. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure 8. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure 9. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure 10. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure 11. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure 12. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, energy difference decreases greatly with increasing shrinking factor. It is noted that the shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking factor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure 13. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation 4).

[[File:LD Helmholtz.png|center|thumb|Figure 13. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods would be commparable in conventional cell unit. Molecular dynamic calculation cannot be performed at zero kelvin as atoms in lattice has zero vibration at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereaturer, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure.) This overestimation of cell volume is Under QHA, temperature rise leads to volume expansion of lattice, which lowers vibration frequencies due to smaller lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure.).

[[File:Cell Vol both methods.png|center|thumb|Figure. Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

{| class='wikitable' boarder='0.5'
|-
|align='center'|<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\partial V}{\partial T})_{P}</math>
|-
|}
where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure.) The gradient of the linear fittings of lattice volumes from 300K to 1300K is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>, which is supported by the small gradient standard deviation values.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}
===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step and Production step both to 700 and 900 steps. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less calculations of lattice volume and energy would be wasted on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure. <ref> https://doi.org/10.1063/1.4906422</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. Similar idea as having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relatuve motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref> https://doi.org/10.1063/1.4906422</ref>)|400px]]

==Conclusion==

==Reference==

Rep:MgO - OHC15

2018-02-14T08:04:31Z

Ohc15: /* Phonon Dispersion and Density of States */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal Expansion of MgO calculated using Lattice Dynamic of a primitive MgO cell and Molecular Dynamic 32 MgO unit supercell model were calculated to be XXX and XXX respectively, which agrees with literature values of XXX.(ref)

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure 1. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure 2. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot.The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 of 1 longitudinal and 2 transverse acoustic modes and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure 3. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure 3. showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 10-15 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure 4. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure 5. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure 6. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure 7. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure 8. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure 9. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure 10. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure 11. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure 12. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, it is noted that shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking fafctor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure.. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation XX).

[[File:LD Helmholtz.png|center|thumb|Figure. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods would be commparable in conventional cell unit. Molecular dynamic calculation cannot be performed at zero kelvin as atoms in lattice has zero vibration at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereaturer, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure.) This overestimation of cell volume is Under QHA, temperature rise leads to volume expansion of lattice, which lowers vibration frequencies due to smaller lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure.).

[[File:Cell Vol both methods.png|center|thumb|Figure. Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

{| class='wikitable' boarder='0.5'
|-
|align='center'|<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\partial V}{\partial T})_{P}</math>
|-
|}
where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure.) The gradient of the linear fittings of lattice volumes from 300K to 1300K is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>, which is supported by the small gradient standard deviation values.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}
===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step and Production step both to 700 and 900 steps. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less calculations of lattice volume and energy would be wasted on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure. <ref> https://doi.org/10.1063/1.4906422</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. Similar idea as having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relatuve motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref> https://doi.org/10.1063/1.4906422</ref>)|400px]]

==Conclusion==

==Reference==

Rep:MgO - OHC15

2018-02-14T07:51:43Z

Ohc15: /* Experimental */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal Expansion of MgO calculated using Lattice Dynamic of a primitive MgO cell and Molecular Dynamic 32 MgO unit supercell model were calculated to be XXX and XXX respectively, which agrees with literature values of XXX.(ref)

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783\,Angstrom;\, \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone (FBZ) was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches, where <math>N</math> equals to the number of atoms in the system. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot. The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 10-15 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|-
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|-
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, it is noted that shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking fafctor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure.. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation XX).

[[File:LD Helmholtz.png|center|thumb|Figure. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods would be commparable in conventional cell unit. Molecular dynamic calculation cannot be performed at zero kelvin as atoms in lattice has zero vibration at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereaturer, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure.) This overestimation of cell volume is Under QHA, temperature rise leads to volume expansion of lattice, which lowers vibration frequencies due to smaller lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure.).

[[File:Cell Vol both methods.png|center|thumb|Figure. Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

{| class='wikitable' boarder='0.5'
|-
|align='center'|<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\partial V}{\partial T})_{P}</math>
|-
|}
where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure.) The gradient of the linear fittings of lattice volumes from 300K to 1300K is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>, which is supported by the small gradient standard deviation values.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}
===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step and Production step both to 700 and 900 steps. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less calculations of lattice volume and energy would be wasted on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure. <ref> https://doi.org/10.1063/1.4906422</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. Similar idea as having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relatuve motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref> https://doi.org/10.1063/1.4906422</ref>)|400px]]

==Conclusion==

==Reference==

Rep:MgO - OHC15

2018-02-14T07:42:35Z

Ohc15: /* Quasi-Harmonic Approximation */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal Expansion of MgO calculated using Lattice Dynamic of a primitive MgO cell and Molecular Dynamic 32 MgO unit supercell model were calculated to be XXX and XXX respectively, which agrees with literature values of XXX.(ref)

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions conditions. The major assumptions are 1) small atom displacements about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to values greater than zero, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783 Angstrom; \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches, where <math>N</math> equals to the number of atoms in the system. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot. The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 10-15 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|-
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|-
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, it is noted that shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking fafctor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure.. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation XX).

[[File:LD Helmholtz.png|center|thumb|Figure. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods would be commparable in conventional cell unit. Molecular dynamic calculation cannot be performed at zero kelvin as atoms in lattice has zero vibration at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereaturer, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure.) This overestimation of cell volume is Under QHA, temperature rise leads to volume expansion of lattice, which lowers vibration frequencies due to smaller lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure.).

[[File:Cell Vol both methods.png|center|thumb|Figure. Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

{| class='wikitable' boarder='0.5'
|-
|align='center'|<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\partial V}{\partial T})_{P}</math>
|-
|}
where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure.) The gradient of the linear fittings of lattice volumes from 300K to 1300K is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>, which is supported by the small gradient standard deviation values.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}
===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step and Production step both to 700 and 900 steps. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less calculations of lattice volume and energy would be wasted on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure. <ref> https://doi.org/10.1063/1.4906422</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. Similar idea as having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relatuve motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref> https://doi.org/10.1063/1.4906422</ref>)|400px]]

==Conclusion==

==Reference==

Rep:MgO - OHC15

2018-02-14T07:37:00Z

Ohc15: /* Lattice Dynamics - Vibration in Solid Crystals */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal Expansion of MgO calculated using Lattice Dynamic of a primitive MgO cell and Molecular Dynamic 32 MgO unit supercell model were calculated to be XXX and XXX respectively, which agrees with literature values of XXX.(ref)

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency <math>\omega_{k}</math>. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,1)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,2)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion per dimension of the system. Vibration modes with atomic displacement in the direction parallel to the wave vector <math>k</math> are known as longitudinal modes. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These branches are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of electromagnetic waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 3 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions. The major assumptions are 1) atom displacements are small about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to non-zero values, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783 Angstrom; \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches, where <math>N</math> equals to the number of atoms in the system. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot. The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 10-15 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|-
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|-
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, it is noted that shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking fafctor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure.. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation XX).

[[File:LD Helmholtz.png|center|thumb|Figure. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods would be commparable in conventional cell unit. Molecular dynamic calculation cannot be performed at zero kelvin as atoms in lattice has zero vibration at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereaturer, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure.) This overestimation of cell volume is Under QHA, temperature rise leads to volume expansion of lattice, which lowers vibration frequencies due to smaller lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure.).

[[File:Cell Vol both methods.png|center|thumb|Figure. Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

{| class='wikitable' boarder='0.5'
|-
|align='center'|<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\partial V}{\partial T})_{P}</math>
|-
|}
where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure.) The gradient of the linear fittings of lattice volumes from 300K to 1300K is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>, which is supported by the small gradient standard deviation values.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}
===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step and Production step both to 700 and 900 steps. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less calculations of lattice volume and energy would be wasted on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure. <ref> https://doi.org/10.1063/1.4906422</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. Similar idea as having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relatuve motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref> https://doi.org/10.1063/1.4906422</ref>)|400px]]

==Conclusion==

==Reference==

Rep:MgO - OHC15

2018-02-14T07:23:10Z

Ohc15: /* Introduction */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal Expansion of MgO calculated using Lattice Dynamic of a primitive MgO cell and Molecular Dynamic 32 MgO unit supercell model were calculated to be XXX and XXX respectively, which agrees with literature values of XXX.(ref)

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism are analysed in the 'Results and Discussion' section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is usually modelled as a spring between 2 masses under Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency<math>\omega_{k}</math> values. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,2)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,3)</math><ref name="name">Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion each in every dimension of the system. Vibration mode with atomic displacement in the direction parallel to the wave vector <math>k</math> is known as a longitudinal mode. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These modes are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of EM waves, typically in the infrared region of 1-30THz.<ref name="name" /> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 4 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribution.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions. The major assumptions are 1) atom displacements are small about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to non-zero values, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Lattice model is initially set to the provided conditions and allowed to evolve with time under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity trajectories of atoms every timestep from its old position and velocity in the previous step in the lattice for the number of set equilibration and production step. Once euilibriation steps are copmleted, lattice properties (cell parameters, volume, pressure, kinetic energy, potential energy, temperature and total energies) are reported in each production steps and averaged. Converged average energy and temperature of system acts as an indicator to whether the system is well-equilibriated to produce accurate results.

===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783 Angstrom; \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches, where <math>N</math> equals to the number of atoms in the system. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot. The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 10-15 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|-
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|-
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, it is noted that shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking fafctor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure.. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation XX).

[[File:LD Helmholtz.png|center|thumb|Figure. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods would be commparable in conventional cell unit. Molecular dynamic calculation cannot be performed at zero kelvin as atoms in lattice has zero vibration at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereaturer, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure.) This overestimation of cell volume is Under QHA, temperature rise leads to volume expansion of lattice, which lowers vibration frequencies due to smaller lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure.).

[[File:Cell Vol both methods.png|center|thumb|Figure. Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

{| class='wikitable' boarder='0.5'
|-
|align='center'|<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\partial V}{\partial T})_{P}</math>
|-
|}
where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure.) The gradient of the linear fittings of lattice volumes from 300K to 1300K is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>, which is supported by the small gradient standard deviation values.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}
===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step and Production step both to 700 and 900 steps. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less calculations of lattice volume and energy would be wasted on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure. <ref> https://doi.org/10.1063/1.4906422</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. Similar idea as having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relatuve motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref> https://doi.org/10.1063/1.4906422</ref>)|400px]]

==Conclusion==

==Reference==

Rep:MgO - OHC15

2018-02-14T06:58:12Z

Ohc15: /* Lattice Dynamics - Vibration in Solid Crystals */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal Expansion of MgO calculated using Lattice Dynamic of a primitive MgO cell and Molecular Dynamic 32 MgO unit supercell model were calculated to be XXX and XXX respectively, which agrees with literature values of XXX.(ref)

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be an extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism is analysed in the results and discussion section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is always modelled as a spring connecting 2 masses in Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode, illustrated by an equation of motion, has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency<math>\omega_{k}</math> values. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,2)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,3)</math><ref>Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion each in every dimension of the system. Vibration mode with atomic displacement in the direction parallel to the wave vector <math>k</math> is known as a longitudinal mode. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These modes are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of EM waves, typically in the infrared region of 1-30THz.<ref>Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

As energy is a function of frequency, the sum of all vibration branches, <math>j</math> at all different wave vector <math>k</math> points gives the free energy contribution of these vibration modes. The Helmholtz energy is given to be:

<math>F = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,3)</math>

Where <math>E_{0}</math> is the lattice energy. The second term in equation 4 is the sum of all ground state energy of vibrational states occupied, while the third term is the entropic term, depending on the probability of all accessible states being populated using Boltzmann distribu-tion.

Phonon density of states is the number of states per unit frequency. It is a normalised plot show-ing the fraction of all sampled phonon states that at each frequency value at specific <math>k</math> space points determined by size of shrinking factor.

===Quasi-Harmonic Approximation===
Quasi-Harmonic approximation is an approximation built upon harmonic approximation and nearest-neighbour interactions. The major assumptions are 1) atom displacements are small about its equilibrium position and 2) only interactions with neighbour atoms within the cut-off sphere are considered.
In harmonic approximation, Taylor expansion about the equilibrium position is applied to approximate the harmonic energy of system up to the second derivative term. The first derivative of potential energy is zero at equilibrium unit cell length, leaving the harmonic energy equation with only the dominating second derivative term of the converging series. This approximation is representative enough to explain most physical properties, but not thermal expansion of crystal solid. This is because properties like thermal expansion involves the anharmonic shifting of equilibrium position to non-zero values, resulting in volume expansion. In Quasi-Harmonic approximation, anharmonic corrections modified the free energy equation under harmonic approximation by including the volume dependence of phonon frequency, giving the QHA Helmholtz Free energy equation:

<math>F_{QHA} (V,T) = E_{0} + \frac{1}{2}\sum_{k,j} \frac{h\omega_{j,k}(V)}{2\pi} + k_{B}T \sum_{j,k} ln[1-exp(\frac{-h\omega_{j,k}(V)}{2\pi k_{B}T})]\,\,\, \,\,\,(Eq. \,4)</math>

Although anharmonicity of higher-energy phonon states are covered in QHA, the approximation would fail once bond breaking occur near phase transition temperature. Bond break lengthens internuclear distance between atoms beyond the cut-off sphere and this kind of interactions would be neglected automatically during calculation.

===Molecular Dynamics===
Noting that QHA does not provide a full explanation to thermal expansion at temperatures near to and beyond the melting point. Molecular dynamics is a better alternative that gives reasonable thermal expansion predictions even at high temperatures. Lattice model is initially set to provided conditions and allowed to evolve within the timestep under Newtonian mechanics. The Verlet algorithm is used to calculate position and velocity information of atoms from its old position and velocity in the previous step in the lattice after each time step. In this model, the regular packing structure of lattice no longer exists. Atoms in the lattice can translate through each other and collide elastically. Averaged total energy of the system is calculated as a sum of kinetic and potential energy after each step, as well as temperature of system to ensure properties of lattice is equilibrated and converged.
===Thermal Expansion===
Thermal expansion (or contraction) is the tendency of a material to undergo volumetric change in response to temperature change. Normally materials would expand as temperature increases, however, material contraction with temperature rise is possible. Generally, temperature increase would result in the increase in the average internuclear separation between atoms in the material. This is caused by phonon excitation to higher energy phonon levels, where anharmonic effect comes into play, shifting the equilibrium position of atoms to greater than zero. This temperature dependence of a material is quantified by the thermal expansion coefficient, <math>\alpha</math>, which is defined as:

<math>\alpha _{V}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P}\,\,\,\,\,\,\,\,(Eq.\,5)</math>

This is an extensively useful indicator to evaluate the suitability of a material to be used for certain purpose where inevitable temperature variation is involved. In reality, long experimental time and extreme environment conditions, like high pressure and high temperatures, made the execution of experiment and the data collection process difficult. With the aid of the mathematical models described above, an alternative pathway is available to gain insights to thermal expansivity of newly developed materials in research from computational predictions, fast-tracking the development process.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783 Angstrom; \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches, where <math>N</math> equals to the number of atoms in the system. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot. The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 10-15 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|-
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|-
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, it is noted that shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking fafctor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure.. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation XX).

[[File:LD Helmholtz.png|center|thumb|Figure. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods would be commparable in conventional cell unit. Molecular dynamic calculation cannot be performed at zero kelvin as atoms in lattice has zero vibration at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereaturer, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure.) This overestimation of cell volume is Under QHA, temperature rise leads to volume expansion of lattice, which lowers vibration frequencies due to smaller lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure.).

[[File:Cell Vol both methods.png|center|thumb|Figure. Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

{| class='wikitable' boarder='0.5'
|-
|align='center'|<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\partial V}{\partial T})_{P}</math>
|-
|}
where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure.) The gradient of the linear fittings of lattice volumes from 300K to 1300K is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>, which is supported by the small gradient standard deviation values.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}
===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step and Production step both to 700 and 900 steps. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less calculations of lattice volume and energy would be wasted on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure. <ref> https://doi.org/10.1063/1.4906422</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. Similar idea as having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relatuve motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref> https://doi.org/10.1063/1.4906422</ref>)|400px]]

==Conclusion==

==Reference==

Rep:MgO - OHC15

2018-02-14T03:30:49Z

Ohc15: /* Introduction */

=Computational Calculation of MgO Free Energy and Thermal Expansion=

==Abstract==
Thermal Expansion of MgO calculated using Lattice Dynamic of a primitive MgO cell and Molecular Dynamic 32 MgO unit supercell model were calculated to be XXX and XXX respectively, which agrees with literature values of XXX.(ref)

==Introduction==
In the study of lattice dynamics, atoms are noted to be vibrating about their equilibrium position, even at absolute zero temperature. These vibrations in solid crystals are correlated to many important physical principles characteristic of the lattice, for instance, bond length, heat capacity, thermal conductivity, thermal expansivity and free energy. Therefore, being able to understand and obtain a mathematical description of atom vibrations would be an extremely helpful in research to provide quick insight to lattice behaviour. In this experiment, calculations were performed on MgO (fcc) crystal using Quasi-Harmonic Approximation and Molecular Dynamic methods. Results of both calculation mechanism is analysed in the results and discussion section.

===Lattice Dynamics - Vibration in Solid Crystals===
Vibrations are sometimes called phonons, due to its wave-particle duality property in the language of quantum mechanics. Phonon interacts with particles by propagating in the form of travelling waves through lattice via the presence of a repulsive interaction between atoms, which is always modelled as a spring connecting 2 masses in Harmonic approximation and nearest-neighbour interaction condition. The force of such interaction is defined by the harmonic force constant, <math>J</math>, analogous to ‘spring constant’. A vibration mode, illustrated by an equation of motion, has its characteristic wavevector, <math>k</math>, wavelength <math>\lambda</math>, where <math>\lambda=\frac{2\pi}{k}</math>, and frequency<math>\omega_{k}</math> values. The number of vibration modes increases with the number of atoms in the system. Therefore, in an infinite ionic crystal, continuum of vibrational modes, aka branches, are observed. The characteristic wave vector and frequency allow each vibrational mode to be presented in terms of its normal modes under Fourier transform in the reciprocal <math>k</math> space. An example would be the graph of <math>\omega_{k}</math>, and thus the energy, in reciprocal space, which is also known as a dispersion curve. The equation of <math>\omega_{k}=\omega_{k}(k)</math> as a function of <math>k</math> is:

<math>\omega_{k}=\sqrt {\frac{4J}{M}} |\sin(\frac{ka}{2})|\,\,\,\,\,\,\, (1D\,Monoatomic) \,\,\,\,(Eq. \,2)</math><ref>Dove, M. (1993). The harmonic approximation and lattice dynamics of very simple systems. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 18-35). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.004</ref>

<math>\omega_{k}^{2}=\frac{J(M+m)}{Mm} \pm\frac{J}{Mm}\sqrt{(M+m)^{2}-4Mm\sin(ka/2)} \,\,\,\,\,\,\,\,(3D\,Diatomic) \,\,\,\,(Eq. \,3)</math><ref>Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref>

On the dispersion curve, each branch represents the band of vibrational modes in the crystal. The number of branch on the dispersion curve is given by the product of dimensionality and number of different atoms in the lattice. In the case of MgO, <math>3\times2=6</math> branches is expected. Usually only the first Brillouin zone, in the range of <math>-\frac{\pi}{a}<k\leq\frac{\pi}{a}</math>, is displayed as the equation of motion of a second wave in the repeated second Brillouin zone is identical to that of the original wave. The repeated Brillouin zone would become useful for investigation in anharmonic interactions and neutron scattering.

Among the <math>3N</math> branches of vibrational modes in a system, <math>N</math> atoms give <math>N</math> number of equations of motion each in every dimension of the system. Vibration mode with atomic displacement in the direction parallel to the wave vector <math>k</math> is known as a longitudinal mode. The other 2 sets of vibrational modes with atomic displacement in the 2 orthogonal directions are known as transverse modes. These modes are further categorised as acoustic and optic vibrations. Acoustic vibration mode is characterised by a dispersion curve that converges to zero in the limit of small <math>k</math>. Meanwhile, optic mode is characterised by the fact that it has a frequency lying in the optical region of EM waves, typically in the infrared region of 1-30THz.<ref>Dove, M. (1993). Dynamics of diatomic crystals: General principles. In Introduction to Lattice Dynamics (Cambridge Topics in Mineral Physics and Chemistry, pp. 36-54). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511619885.005</ref> Analysis of vibration mode degeneracy for a certain wave vector on the dispersion curve could provide geometry symmetry information of the crystal.

==Experimental==
In this experiment, vibration modes of MgO crystal and its energies were calculated on a graphical user interface, DLVisualize, using the molecular mdelling code General Utility Lattice Program (GULP) in RedHat Linux environment. These information was then used to obtain Free Energy of the crystal for prediction of thermal expansitivity of the lattice.

All simulations carried out under constant zero pressure condition using ionic.lib force field model with Include Shells option de-selected on the GULP Potential parameters panel. Under this setting, calculations were performed on crystal using Buckingham repulsive potential between a simplified +2e charge Mg ion and a -2e O ion. Include Shells setting is useful in allowing the study of polaristion of the system, but this is not considered in the experiment.

A rhombohedron primitive unit cell of MgO crystal <math>(a=b=c=2.9783 Angstrom; \alpha=\beta=\gamma=60^{o})</math> was used in most calculations, while a 32-primitive cell unit supercell was used for molecular dynamic simulations.

===Phonon Dispersion and Density of States Calculation of MgO===
Phonon properties, including the dispersion curve and density of states, were investigated at 0K. Phonon dispersion shows the frequency <math>\omega</math> of each vibrational mode at wavevector k. The phonon dispersion plot of 50 k points in the First Brillouin Zone was obtained and compared with literature results. Density of states is the counts of number of vibrational modes at each frequency <math>\omega</math>. Both calculations were performed using a 1x1x1 grid size.

===K-Space Grid Sizes and Free Energy===
In addition to the 1x1x1 grid size, other k-space grid sizes (i.e. 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 8x8x8, 16x16x16, 32x32x32 and 64x64x64) at 300K were performed to see its effect on density of states within the Quasi-Harmonic approximation. Helmholtz-free energy values of MgO at each grid size was analysed. 32x32x32 grid size was determined to be an appropriate grid size, where the balance between accuracy and computational time is best suited for the rest of the experiment.

===Thermal Expansion Calculation of MgO===
Gibbs-free energy optimisation was calculated using 32x32x32 grid size at temperatures ranging from 0K to 2900K every 100K under Quasi-harmonic approximation. Lattice constant and free energy values of primitive unit cell were plotted against temperature. Lattice constant was further used to obtain linear thermal expansion coefficient of MgO crystal.

===Molecular Dynamics Calculation of MgO===
Molecular Dynamic method was applied to calculate averaged crystal properties of a 32-MgO unit superecell in the NPT ensemble (fixed particle number, external pressure and temperature) at temperatures ranging from 100K to 3000K in step of 100K. Simulations were set to perform at time step of 1.0 femtoseconds, Equilibration steps and Production steps both to be 500. Sampling steps and Trajectory write steps were set to 5. Averaged values of energies and lattice volume at varying temperratures were analysed. Volumetric thermal expansion coefficient of MgO was obtained from the lattice volume against temperature plot and compared with results of the previous section using Quasi-harmonic approximation.

Futher work was performed to reduce fluctuations in thermal expansion coefficients at high temperatures by increasing Equilibration steps and Production steps both to 700 and 900.

==Results and Discussion==
===Phonon Dispersion and Density of States===
{| class='wikitable' boarder='0.5'
|-
|align='left'|[[File:GULP phonon dispersion.PNG|center|thumb|Figure. Phonon Dispersion Curve of MgO with labeled Optical and Acoustic Branches|300px]]
|align='left'|[[File:Phonon DOS.PNG|center|thumb|Figure. Phonon Density of States at 1x1x1 Shrinking Factor|300px]]
|-
|}
6 vibrational bands were observed in the dispersion curve above for our 2-atom primitive cell agreeing with our expectation of <math>3N=3\times2=6</math> branches, where <math>N</math> equals to the number of atoms in the system. These 6 vibrbational bands consist of 3 acoustic and 3 optical branches, as identified in the phonon dispersion plot. The phonon dispersion curve follows a k-space path
going from <math>W-\,L-\,\Gamma-\,X-\,W-\,K</math> in the FBZ. The actual frequency values and coordinates in the reciprocal k realm of the points marked on the curve could be found in the log file by matching the number of degenerate and non-degenerate states and their approximate values base on the dispersion curves as shown below.

{| class='wikitable' boarder='0.5'
|+ Frequencies of vibrational modes at sepcific k-points
|-
!align='left'|k-point
!align='left'|Log file
|-
|align='center'| <math>W[\frac{\pi}{a}\frac{\pi}{2a}\frac{3\pi}{2a}]</math>

*Density of states at this k-point would give only 4 observable peaks instead of 6
*As there are 2 degenerate states at 2 frequencies, 454.91 cm-1 and 608.81cm-1
|align='left'|[[File:Vib w.PNG|center|400px]][[File:Vib W 1.PNG|center|400px]]
|-
|align='center'| <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math>

*There are 4 states that are degenerate 2 each at 288.49cm-1 and 351.76cm-1
*Only 4 observable peaks in DOS at this point
|align='left'|[[File:Vib L.PNG|center|400px]]
|-
|align='center'| <math>\Gamma[000]</math>

*3 degenerate modes at 0cm-1 and 2 degenerate states at 399.79cm-1, so DOS would only show 3 observable peaks
|align='left'|[[File:Vib T.PNG|center|400px]]
|-
|align='center'| <math>X[\frac{\pi}{a}0\frac{\pi}{a}]</math>

*2 degenerate phonons at 344.86cm-1 and another 2 degenerate phonons at 457.04cm-1
|align='left'|[[File:Vib X.PNG|center|400px]]
|-
|align='center'| <math>K[\frac{3\pi}{4a}\frac{3\pi}{4a}\frac{3\pi}{2a}]</math>

*All 6 phonons have distinct frequencies
|align='left'|[[File:Vib K.PNG|center|400px]]
|-
|}

Figure. combined the phonon dispersion plot with the density of states using a 1x1x1 grid size. This indicated that the DOS obtained corresponds to <math>L[\frac{\pi}{a}\frac{\pi}{a}\frac{\pi}{a}]</math> on the dispersion curve. Peak intensity at 288.49cm-1 and 351.76cm-1 are twice of that at 676.23cm-1 and 818.82cm-1 showing the presence of degenerate phonons.

[[File:L matching.png|center|thumb|Figure showing that the DOS obtained correspond to point L|400px]]

DOS copmuted at larger grid size were observed to have better resolution. Spikey peaks start to join together and become broader as grid size increases. A continuous spectrum start to emerge as grid size increase from 8x8x8 to 16x16x16. Larger shrinking factor/ grid size essentially increasees the number of k-space points visited within the FBZ and thus giving more representative DOS, where phonon distribution at intermediate frequencies become observable. Further increasing the shrinking factor to 32x32x32 gives a much smoother curve with similar shapes as that at grid size of 16x16x16. DOS of 64x64x64 grid size has barely any difference from the one at 32x32x32, showing that DOS is representative enough at 32x32x32 level. Regarding the time of calculation, higher grid size does take longer time for calculation to reach completion. Comprimising between accuracy and calculation time, 32x32x32 is the most appropriate grid size to give high resolution DOS while the computaional time lies within a mintue. 64x64x64 is not a practical grid size to use as one calculation could take up to 10-15 minutes and the improvement in accuracy is not significant enough.

{| class='wikitable' boarder='0.5'
|+ Density of States at Different Shrinking Factor
|-
|align='left'|[[File:Phonon DOS 2x2x2.PNG|center|thumb|Figure. Phonon Density of States at 2x2x2 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 3x3x3.PNG|center|thumb|Figure. Phonon Density of States at 3x3x3 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 4x4x4.PNG|center|thumb|Figure. Phonon Density of States at 4x4x4 Shrinking Factor|300px]]
|align='left'|[[ File:Phonon DOS 5x5x5.PNG|center|thumb|Figure. Phonon Density of States at 5x5x5 Shrinking Factor|300px]]
|-
|align='left'|[[ File:Phonon DOS 6x6x6.PNG|center|thumb|Figure. Phonon Density of States at 6x6x6 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 8x.PNG|center|thumb|Figure. Phonon Density of States at 8x8x8 Shrinking Factor|300px]]
|-
|align='left'|[[File:Phonon DOS 16x.PNG|center|thumb|Figure. Phonon Density of States at 16x16x16 Shrinking Factor|300px]]
|align='left'|[[File:Phonon DOS 32x.PNG|center|thumb|Figure. Phonon Density of States at 32x32x32 Shrinking Factor|300px]]
|-
|align='left'|[[File:Phonon DOS 64x.PNG|center|thumb|Figure. Phonon Density of States at 64x64x64 Shrinking Factor|300px]]
|-
|}

===Helmholtz-Free Energy (Quasi-Harmonic Approximation)===

Effect of Grid Size on Helmholtz-Free Energy

From the DOS analysis above, it is noted that shape of DOS plot converges with increasing shrinking factor and does not change much as shrinking fafctor increases from 32x32x32 to 64x64x64, thus, concluding that 32x32x32 is the optimal grid size for later calculations. The Helmholtz-Free energy values resulted from calculations at each different grid sizes tested in the previous section were listed in the table below. Energy values were found to converge to a value of -40.926483 eV (to 6 decimal places)with increasing grid size. The converged free energy value was first obtained at 32x32x32 level, supporting the conclusion that 32x32x32 is the optimal grid size for MgO, where calculations are accurate to 1 micro-eV. Depending on the level of precision required for the purpose of the experiment, lower grid size level might be better fit as they are less computationally demanding. Grid size 3x3x3 is suitable for calculation accurate up to 1 meV as well as 0.5 meV, while for accuracy up to 0.1 meV a grid size of 4x4x4 would be needed.

{| class='wikitable' boarder='0.5'
|+Helmholtz-Free Energy at Different Grid Sizes
|-
|align='left'|[[File:Shrinking E Plot.png|center|600px]]
|align='left'|

{| class='wikitable' boarder='0.5'
|-
!align='left'|Shrinking Factor
!align='left'|Helmholtz-Free Energy / eV
!align='left'|Energy Difference / meV
|-
|align='left'|1
|align='left'|-40.930301
|align='left'|0
|-
|align='left'|2
|align='left'|-40.926609
|align='left'|3.692
|-
|align='left'|3
|align='left'|-40.926432
|align='left'|0.177
|-
|align='left'|4
|align='left'|-40.926450
|align='left'|-0.018
|-
|align='left'|5
|align='left'|-40.926463
|align='left'|-0.013
|-
|align='left'|6
|align='left'|-40.926471
|align='left'|-0.008
|-
|align='left'|8
|align='left'|-40.926478
|align='left'|-0.007
|-
|align='left'|16
|align='left'|-40.926482
|align='left'|-0.004
|-
|align='left'|32
|align='left'|-40.926483
|align='left'|-0.001
|-
|align='left'|64
|align='left'|-40.926483
|align='left'|0
|-
|}
|-
|}

If we consider other species, the optimal grid size of 32x32x32 might no longer be applicable. For an ionic compound similar to MgO, like CaO, it is expected that the optimal grid size would not be too far off from 32x32x32 although CaO has a slightly larger internuclear distance and thus larger lattice parameter than MgO. It is reasonable for a lattice species with larger lattice parameter to have a smaller reciprocal k-space containing less k points in its FBZ (i.e. <math>-\frac{\pi}{a}\leq k\leq+\frac{\pi}{a}</math>). Thus, for compounds of metals with much smaller ionic radii, like lithium, would require a much bigger grid size to accommodate a representative number of k points, in order to reach the same level of accuracy in lattice property calculations. Vice versa, for ionic compounds with larger lattice parameter (e.g. a Zeolite), a smaller grid size is expected to yield same level of accuracy as the population of k points is smaller in a smaller FBZ.

Temperature dependence

Helmhotlz-Free Energy values collected from simulations performed using 32x32x32 grid size on a MgO primitive unit cell every 100K from 0K to 2900K were plotted in Figure.. In general, Helmholtz-free energy values get increasingly negative with temperature . It is noted that the trend started off exponentially. However, as temperature further increases, the non-linear temperature dependence of Helmholtz free energy becomes linear. This trend satisfies the equation of Helmholtz-Free Energy (Equation XX).

[[File:LD Helmholtz.png|center|thumb|Figure. Helmholtz-Free Energy of MgO in Tempereature Range from 0K to 2900K (Quasi-Harmonic)|400px]]

===Comparison of Thermal Expansion Results of Different Methods===

Lattice volumes of 2-atom MgO primitive cell calculated under Quasi-Harmonic Approximation (QHA) were multiplied by 4 to convert to 8-atom conventional cell unit, whereas, that of the 32-primitive cell unit supercell in molecular dynamic method is reduced by a factor of 8, so that results from both methods would be commparable in conventional cell unit. Molecular dynamic calculation cannot be performed at zero kelvin as atoms in lattice has zero vibration at 0K. In general, both methods show a trend of increasing cell volume with temerature. However, the rate of increment under QHA initially increases at low tempereaturer, then slows down in the region of 300K to 1300K. As temperature further increases, cell expansion grows more rapily.(Figure.) This overestimation of cell volume is Under QHA, temperature rise leads to volume expansion of lattice, which lowers vibration frequencies due to smaller lattice parameter. This narrows the energy gap between vibrational states as energy of phonon modes, <math>E=\frac{h\omega}{2}</math>, is lowered. This results in high energy phonon modes being more easily populated, leading to even greater atom displacements from equilibrium positions and bigger cell volumes. Near the melting temperature of approximately 3100K, QHA no longer give a true representation of the reality as some atom displacements would exceed the cutoff value and these interactions between those atoms would not be taken into consideration during calculation. Therefore, temperature investigation of lattice volume was stopped at 3000K.

On the other hand, results of molecular dynamic method show a relatively linear correlation between cell volume and temperature. Figure. shows random fluctuations in expansion coefficients with temperature. Flulctuations are more vigorous at higher temperatures. This is due to insufficient time for lattice properties of the 64-atom supercell model to reach equilibrium and converge. Improvement on such issue is further discussed below (Figure.).

[[File:Cell Vol both methods.png|center|thumb|Figure. Conventional cell lattice volume variation with temperature|400px]]

Using the lattice volume data, the volumetric thermal expansion coefficients of QHA and molecular dynamic method from 300K to 1300K were found to be <math>\alpha_{V(QHA)}=3.00\times10^{-5}K^{-1}</math> and <math>\alpha_{V(MD)}=3.07\times10^{-5}K^{-1}</math> respectively, using the equation:

{| class='wikitable' boarder='0.5'
|-
|align='center'|<math>\alpha_{V}=\frac{1}{V_{0}}(\frac{\partial V}{\partial T})_{P}</math>
|-
|}
where <math>V_{0}</math> is the initial volume and were taken as the lattice volumes at 300K for both methods. Both experimental expansion coefficient values agree with the literature <math>(\alpha_{V}=3.12\times10^{-5}K^{-1})</math><ref>Y. S. Touloukian, R. K. Kirdby, R. E. Taylor, and T. Y. R. Lee,
Thermophysical Properties of Matter (Plenum, New York,
1977), Vol. 13.</ref> at 300K. This temperature range was selected as the increment in thermal expansion coefficient with temperature is the least significant in this region. (Figure.) The gradient of the linear fittings of lattice volumes from 300K to 1300K is a valid approximate of the partial derivative, <math>\frac{\partial V}{\partial T}</math>, which is supported by the small gradient standard deviation values.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:QHA VS MD LV.png|center|thumb|Figure. Conventional cell lattice volume against temperature plot with volumetric thermal expansion coefficient reported for both calculation methods|400px]]
|align='left'|[[File:Volumetric alpha.png|center|thumb|Figure. Volumetric expansion coefficient variation with tempereature for both methods|400px]]
|-
|}
===Extension===

As discussed above, random fluctuations in expansion coefficients at high temperature under molecular dynamic method were significantly large, showing that lattice properties were not converged under the settings described in the experimental section. The most direct measure to resolve this issue would be to increase time allowed for equilibration. This could be done by raising the number of Equilibriation step and Production step both to 700 and 900 steps. Experimental data showed that the volume -temperature plot is less bumpy for larger step length. The difference between the highest and lowest expansion coefficent decreases as length of Equilibriation step and Production step increases from 500, 700 to 900. This proves that the great fluctuation in expansion coefficient at high temperatures correlated length of allowed time for equilibration and obtaining more averaging steps. However, longer Equilibriation step and Production step increases the calculation time for each simulation to end. A better solution would be just lengthening the number of Equilibriation step, so that less calculations of lattice volume and energy would be wasted on non-equilibrated state of the lattice.

{| class='wikitable' boarder='0.5'
|-
| align='left'|[[File:Step sizes volume.png|center|thumb|Figure. Conventional cell lattice volume against temperature at different equilibriation and production step lengths|400px]]
|align='left'|[[File:Step sizes.png|center|thumb|Figure. Volumetric expansion coefficient against temperature at different equilibriation and production step lengths|400px]]
|-
|}

The convergence of lattice properties were found to be correlated to the size of supercell used in the simulation. Although Figure. <ref> https://doi.org/10.1063/1.4906422</ref> shows results obtained under Quasi-Harmonic approximation, it illustrates that increase in supercell size converges lattice properties, like free energy and thermal expansion coefficient. Similar idea as having larger grid size, greater number of atoms in a larger supercell provide larger sample size and yield better averaged values, especially in the context of molecular dynamics. Simulation calculations are based on the relatuve motion of atoms in the model. Take a 2 atom and a 64 atom system as an example, where both systems are initiated by the movement of one atom due to an external force. The effect of such movement on the whole system would be much profound in a smaller cell, thus, giving less representative prediction of lattice properties.

[[File:Lit graph.PNG|center|thumb|Figure. Thermal expansion coefficient as a function of temperature at different supercell size of MgO (extracted from literature<ref> https://doi.org/10.1063/1.4906422</ref>)|400px]]

==Conclusion==

==Reference==

Rep:MgO - OHC15

2018-02-13T13:31:51Z

Ohc15: /* Extension */

Rep:MgO - OHC15

2018-02-13T04:06:35Z

Ohc15: /* Comparison of Thermal Expansion Results of Different Methods */

File:Lit graph.PNG

2018-02-13T03:55:54Z

Ohc15:

Rep:MgO - OHC15

2018-02-13T03:20:23Z

Ohc15: /* Comparison of Thermal Expansion Results of Different Methods */

Rep:MgO - OHC15

2018-02-13T02:47:38Z

Ohc15: /* References */

Rep:MgO - OHC15

2018-02-13T02:46:26Z

Ohc15: /* Results and Discussion */

File:Cell Vol both methods.png

2018-02-13T02:08:17Z

Ohc15:

File:QHA VS MD LV.png

2018-02-13T01:14:45Z

Ohc15: Ohc15 uploaded a new version of File:QHA VS MD LV.png

Rep:MgO - OHC15

2018-02-12T23:46:00Z

Ohc15: /* Comparison of Thermal Expansion Results of Different Methods */

Rep:MgO - OHC15

2018-02-12T23:28:07Z

Ohc15: /* Helmholtz-Free Energy (Quasi-Harmonic Approximation) */

Rep:MgO - OHC15

2018-02-12T23:27:52Z

Ohc15: /* Helmholtz-Free Energy (Quasi-Harmonic Approximation) */

Rep:MgO - OHC15

2018-02-12T23:26:49Z

Ohc15: /* Results and Discussion */

Rep:MgO - OHC15

2018-02-12T23:25:14Z

Ohc15: /* Comparison of Thermal Expansion Results of Different Methods */

File:Step sizes.png

2018-02-12T23:20:19Z

Ohc15:

File:Step sizes volume.png

2018-02-12T23:19:47Z

Ohc15:

File:Volumetric alpha.png

2018-02-12T23:17:42Z

Ohc15:

File:QHA VS MD LV.png

2018-02-12T23:16:19Z

Ohc15:

Rep:MgO - OHC15

2018-02-12T23:12:08Z

Ohc15: /* Comparison of Thermal Expansion Results of Different Methods */

Rep:MgO - OHC15

2018-02-12T21:59:50Z

Ohc15: /* Effect of Grid Size on Helmholtz-Free Energy */

Rep:MgO - OHC15

2018-02-12T21:53:18Z

Ohc15: /* Comparison of Thermal Expansion Results of Different Methods */

File:LD Helmholtz.png

2018-02-12T20:58:19Z

Ohc15: Ohc15 uploaded a new version of File:LD Helmholtz.png

File:LD Helmholtz.png

2018-02-12T20:55:12Z

Ohc15:

Rep:MgO - OHC15

2018-02-12T20:37:19Z

Ohc15: /* Effect of Grid Size on Helmholtz-Free Energy */

Rep:MgO - OHC15

2018-02-12T20:34:06Z

Ohc15: /* Comparison of Thermal Expansion Results of Different Methods */

Rep:MgO - OHC15

2018-02-12T20:30:35Z

Ohc15: /* Effect of Grid Size on Free Energy */

File:Shrinking E Plot.png

2018-02-12T19:56:40Z

Ohc15:

Rep:MgO - OHC15

2018-02-12T18:52:40Z

Ohc15: /* Phonon Dispersion and Density of States */

Rep:MgO - OHC15

2018-02-12T17:46:22Z

Ohc15: /* Results and Discussion */