= MgO Thermal Expansion =

==Introduction==

===Aim===
This investigation aims at studying the thermal expansion of magnesium oxide crystal using quasi-harmonic approximation and molecular dynamics. The phonon modes of MgO crystal at different temperature were studied and relevant information, e.g. internal energy, density of states (DOS) and lattice parameters were extracted.

===System===

{| style="text-align: center; margin-left: auto; margin-right: auto; border: none;"
|| [[File:rhl primitive.jpg|thumb|300x300px|Figure 1. Primitive Cell of MgO]]|| [[File:rhl conventional.jpg|thumb|300x300px|Figure 2. Conventional Cell of MgO]]
|}

The crystal lattice of MgO has a FCC structure similar to that of NaCl and many simple metal oxides. The primitive cell of MgO has one atom of oxygen sitting in the middle of a rhombohedron and eight atoms of magnesium on all eight corners which contribute to 1/8 * 8 =1 atom. The conventional cell has four times the size of a primitive cell and a supercell contains 32 times the size of a primitive cell. The difference in sizes will determine which cell type is the most appropriate for a certain computational method.

==Methodology==
===Quasi-Harmonic Approximation===
Assuming the MgO lattice to be a perfect crystal with no defect whatsoever, the entire lattice can be approximated into infinite continuum of unit cells along x, y and z axis and hence the vibrations of the entire lattice can be broken down into vibrations along 1-D chains on x, y and z axis. Each type of vibration is governed by one individual wavevector k=2pi/lambda, which in turn defines the vibrational frequency and hence energy as functions of k.
The lattice structure in real space is converted into reciprocal space (k-space).
By summing up all k values for each vibrational band, the total vibrational energy of a crystal can be computed.
The plot of frequency over k values is called a dispersion curve and k values of special interests: the symmetry points are labelled.

The quasi-harmonic approximation is based on the assumption that each atom on the lattice oscillate around its equilibrium position in simple harmonic motion when the surrounding temperature does not exceed a certain value (otherwise the bonding in the lattice will dissociate). However, the quasi-harmonic motion differs from simple harmonic motion that it allows the change in atomic distance and hence the change in volume (thermal expansion) is made possible.

===Molecular Dynamics===
The mechanism of molecular dynamics involves assigning each particle in the lattice an initial configuration and a random velocity to make up the given temperature. The initial properties will be used in computing the force and hence acceleration experienced by each atom. The acceleration value is then used to compute a new velocity and hence a new location of each atom. As the system tends to equilibrium, other properties such as temperature and energy will be extracted.

==Software==
Linux platform was chosen over windows due to its efficiency in performing calculations. The lattice structure was displayed using DLV, which also helps with illustrating lattice properties. The calculations were performed using General Utility Lattice Program (GULP), which________.

==Results and Discussion==
===Phonon Modes===
{|
|[[File:RHL Dispersion curve.png|400 px|left|Figure 2. Phonon dispersion curve of MgO lattice.]]
|}
In solid state physics/chemistry, a phonon refers to a collective periodic and elastic excitation/vibration of atoms or molecules. The phonon mode of MgO lattice in k-space along the conventional path is simulated by GULP to support the calculation of free energy by quasi-harmonic model.
The simulation produces various phonon dispersion curves and they collectively display the vibrational band structure of MgO crystal.

===Density of States (DOS)===
{|
|[[File:rhl1.png|thumb|Figure 3. Density of states of MgO phonon, shrinking factors: 1x1x1, k-point considered is L.]]
|[[File:2.png |thumb|Figure 4. Density of states of MgO phonon, shrinking factors: 2x2x2.]]
|[[File:4.png |thumb|Figure 5. Density of states of MgO phonon, shrinking factors: 4x4x4.]]
|-
|[[File:rhl8.png |thumb|Figure 6. Density of states of MgO phonon, shrinking factors: 8x8x8.]]
|[[File:rhl16.png|thumb|Figure 7. Density of States of MgO phonon, shrinking factors: 16x16x16.]]
|[[File:rhl32.png |thumb|Figure 8. Density of states of MgO phonon, shrinking factors: 32x32x32.]]
|}

The density of states is defined by ____, i.e. number of levels between two energies. It can be roughly described as a 90 degree rotation of a dispersion diagram, because each point on a dispersion curve is a state defined by its k value and frequency, i.e. energy. This is to say, the flatter the dispersion curve, the higher the density of states, i.e. more states on the same energy level.

The shrinking factors are multiplied by 2 each time a new DOS is obtained in order to ________.
The DOS maintains a good level of details after shrinking factor=16.

For the 1*1*1 DOS, the peaks are located near 280, 350, 670 and 810 cm-1 and these correspond to point L in the dispersion curve. To obtain a reliable display of DOS, input shrinking factors are varied until the resulted density of state diagram shows all necessary details because the shrinking factor is the number of k values computed within a brillouin zone. Larger shrinking factor will naturally give more data points within the brillouin zone and hence more details about the density of states.

SPECULATION: what grid sizes are suitable?

===Free Energy Calculation by Harmonic Approximation===

{|class="wikitable" style="text-align: center;"
|+ Table 2: Helmholtz Free Energy of MgO at various grid sizes
(6 d.p. for comparison)
|-
!Shrinking Factors
!Phonon Helmholtz Free Energy (eV)
!Accuracy(compared with grid size 32^3 (meV)
|-
|1x1x1||-40.930301||3.818
|-
|2x2x2||-40.926609||0.126
|-
|4x4x4||-40.926452||0.033
|-
|8x8x8||-40.926478||0.005
|-
|16x16x16||-40.926482||0.001
|-
|32x32x32||-40.926483||0
|}

As shown in the table above, the difference between two consecutive Helmholtz Free Energy steadily decreases as the shrinking factor grows. An 1*1*1 grid is accurate enough for accuracy down to 1 meV and a 2*2*2 grid is sufficient for accuracy to 0.5 meV.

The change in free energy when different shrinking factors are used is due to the addition of more details with increasing number of shrinking factor as the energy is computed by summing up the energy related to each k value and the shrinking factor refers to how many k values are sampled during calculation.

The MgO model simulated above would be suitable for computing properties for crystals of similar structures such as most simple oxides as they mostly have fcc structure and comparable lattice parameters and hence similar brillouin zone and naturally k values. However, simulating other crystal structures that drastically differ from MgO while still using MgO model will be largely inaccurate as they will take different spatial arrangement in reciprocal space and hence different k values.

===Thermal Expansion===

{|
|[[File:Rhl free energy.PNG|500 px|left|Figure 2. Helmholtz Free Energy vs Temperature]]
|}
{|
|[[File:Rhl lattice parameter.PNG|500 px|left|Figure 2. Helmholtz Free Energy vs Temperature]]
|}

The calculation done for thermal expansion is based on quasi-harmonic approximation rather than a complete harmonic approximation due to the fact that harmonic approximation does not allow any shift in equilibrium position of atoms on a lattice as any energy input will only lead to the occupation of higher vibrational energy level rather than any shift in position. Using quasi-harmonic model, which is a combination of harmonic oscillator, Coulombic repulsion, etc., the slight shift in equilibrium position of phonons will be simulated and only then can the thermal expansion, which is essentially the change in bond distance, be fully illustrated.

====Free Energy====
The Helmholtz Free Energy increases substantially with an increasing temperature as predicted by its definition: <math>A=U-TS</math>. The actual value is computed by<math>
F = E_0 + \frac{1}{2}\sum_{k,j} \bar{h}\omega + k_B T\sum_{k,j} ln[1-exp(-\bar{h}\omega /k_B T).
</math>
The definition of Helmholtz Free Energy indicates that at low temperature, Helmholtz Free Energy is dominated by internal energy term and any change in temperature, which contributes to the entropy term, is insignificant. This explains why the curve of free energy vs temperature shows a flat curve at low temperature before becoming steeper. The entire curve illustrates how temperature, i.e. entropic term gradually becomes dominating in Helmholtz Free Energy.

====Lattice Parameter====
As temperature increases, the unit cells receive more energy and can therefore populate higher vibrational states and shift from their original equilibrium position. This shift in equilibrium position constitutes in the change in bond distance and hence the expansion of lattice.

As temperature increases near the melting point of MgO, it is obvious that the distance between two neighbouring atoms will reach the dissociation limit and the harmonic approximation will break down as the vibrating atom will no longer return to its equilibrium position but drift away. This is demonstrated by the fact that the calculation could not be achieved in 3000 K because the vibration is no longer possible.

====Expansion Coefficient====
The expansion coefficient is defined as: <math>
\alpha_V = \frac{1}{V}\,\left(\frac{\partial V}{\partial T}\right)_p
</math> and in this study the expansion of MgO is obtained by applying this equation to the linear region of volume vs temperature diagram. <math>\alpha_v=(1/18.89)*(19.26-18.89)/(1000-300)=2.80*10^-5 K^-1</math> The result differs slightly from experimental value as expected since the assumption does not include any consideration to the actual lattice structure of a crystal, which must contain a certain level of defects and impurities. OTHER ASSUMPTIONS?

===Molecular Dynamics===
[[File:Rhl QH vs MD.PNG|500 px|left|QH and MD prediction of volume vs temperature]]

The thermal expansion predicted by molecular dynamics is generally in good agreement with that by quasi-harmonic approximation in higher temperatures, but the results do differ significantly in lower tempeartures. The difference can be rationalised by the fact that QH has taken into consideration of zero energy of a harmonic oscillator and the effect of zero energy is more pronounced in lower temperatures. As molecular dynamics approximation is totally Newtonian, it does not take into consideration of zero point energy when T=0 and hence has no zero energy contribution to the volume of the lattice. In higher temperatures, the contribution of zero point energy becomes insignificant as ____ dominates.

It must be noticed that since MD is totally Newtonian and does not consider the dissociation of bonding as QH does, the cell volume simulated by MD will keep increasing with temperature even when the calculation by QH is no longer possible due to bond dissociation.

File:Rhl conventional.jpg

2017-03-09T18:12:34Z

Rl2014:

2017-03-09T17:15:29Z

Rl2014: /* Phonon Modes */

Rep:MOD:order66

2017-03-09T17:15:10Z

Rl2014: /* Thermal Expansion */

== MgO Thermal Expansion ==

==Introduction==

===Aim===
This investigation aims at studying the thermal expansion properties of magnesium oxide crystal using quasi-harmonic approximation and molecular dynamics. The simulation results will be used in calculating the thermal expansion coefficient of MgO at different conditions and predicting the outcome of thermal expansion.

===System===
The crystal lattice of MgO has a FCC structure similar to that of NaCl. Its primitive cell has one atom of oxygen sitting in the middle of the rhombohedron and eight atoms of magnesium on all eight corners which contribute to 1/8 * 8 =1 atom. The conventional cell has four times the size of a primitive cell and a supercell would contain 32 times the size of a primitive cell. The difference in size will significantly influence the outcome of calculation as it will be shown later.

===Origin of Thermal Expansion===
The expansion of crystal lattice can be illustrated by several means. The most straightforward reasoning is that due to the increase in internal energy, the average bonding distance of atoms in a lattice/solid increases, which is in turn due to the increased amplitude of vibration. The higher amplitude of vibration causes an increase in energy among atoms in the original lattice and hence the atoms tend to stay away from each other to accommodate the extra vibrational energy.

==Methodology==
===Quasi-Harmonic Approximation===
Assuming the MgO lattice to be a perfect crystal with no defect whatsoever, the entire lattice can be approximated into infinite continuum of unit cells along x, y and z axis and hence the vibrations of the entire lattice can be broken down into vibrations along 1-D chains on x, y and z axis. Each type of vibration is governed by one individual wavevector k=2pi/lambda, which in turn defines the vibrational frequency and hence energy as functions of k.
The lattice structure in real space is converted into reciprocal space (k-space).
By summing up all k values for each vibrational band, the total vibrational energy of a crystal can be computed.
The plot of frequency over k values is called a dispersion curve and k values of special interests: the symmetry points are labelled.

The quasi-harmonic approximation is based on the assumption that each atom on the lattice oscillate around its equilibrium position in simple harmonic motion when the surrounding temperature does not exceed a certain value (otherwise the bonding in the lattice will dissociate). However, the quasi-harmonic motion differs from simple harmonic motion that it allows the change in atomic distance and hence the change in volume (thermal expansion) is made possible.

===Molecular Dynamics===
The mechanism of molecular dynamics involves assigning each particle in the lattice an initial configuration and a random velocity to make up the given temperature. The initial properties will be used in computing the force and hence acceleration experienced by each atom. The acceleration value is then used to compute a new velocity and hence a new location of each atom. As the system tends to equilibrium, other properties such as temperature and energy will be extracted.

==Software==
Linux platform was chosen over windows due to its efficiency in performing calculations. The lattice structure was displayed using DLV, which also helps with illustrating lattice properties. The calculations were performed using General Utility Lattice Program (GULP), which________.

==Results and Discussion==
===Phonon Modes===
{|
|[[File:RHL Dispersion curve.png|thumb|left|Figure 2. Phonon dispersion curve of MgO lattice.]]
|}
In solid state physics/chemistry, a phonon refers to a collective periodic and elastic excitation/vibration of atoms or molecules. The phonon mode of MgO lattice in k-space along the conventional path is simulated by GULP to support the calculation of free energy by quasi-harmonic model.
The simulation produces various phonon dispersion curves and they collectively display the vibrational band structure of MgO crystal.

===Density of States (DOS)===
{|
|[[File:rhl1.png|thumb|Figure 3. Density of states of MgO phonon, shrinking factors: 1x1x1, k-point considered is L.]]
|[[File:2.png |thumb|Figure 4. Density of states of MgO phonon, shrinking factors: 2x2x2.]]
|[[File:4.png |thumb|Figure 5. Density of states of MgO phonon, shrinking factors: 4x4x4.]]
|-
|[[File:rhl8.png |thumb|Figure 6. Density of states of MgO phonon, shrinking factors: 8x8x8.]]
|[[File:rhl16.png|thumb|Figure 7. Density of States of MgO phonon, shrinking factors: 16x16x16.]]
|[[File:rhl32.png |thumb|Figure 8. Density of states of MgO phonon, shrinking factors: 32x32x32.]]
|}

The density of states is defined by ____, i.e. number of levels between two energies. It can be roughly described as a 90 degree rotation of a dispersion diagram, because each point on a dispersion curve is a state defined by its k value and frequency, i.e. energy. This is to say, the flatter the dispersion curve, the higher the density of states, i.e. more states on the same energy level.

The shrinking factors are multiplied by 2 each time a new DOS is obtained in order to ________.
The DOS maintains a good level of details after shrinking factor=16.

For the 1*1*1 DOS, the peaks are located near 280, 350, 670 and 810 cm-1 and these correspond to point L in the dispersion curve. To obtain a reliable display of DOS, input shrinking factors are varied until the resulted density of state diagram shows all necessary details because the shrinking factor is the number of k values computed within a brillouin zone. Larger shrinking factor will naturally give more data points within the brillouin zone and hence more details about the density of states.

SPECULATION: what grid sizes are suitable?

===Free Energy Calculation by Harmonic Approximation===

{|class="wikitable" style="text-align: center;"
|+ Table 2: Helmholtz Free Energy of MgO at various grid sizes
(6 d.p. for comparison)
|-
!Shrinking Factors
!Phonon Helmholtz Free Energy (eV)
!Accuracy(compared with grid size 32^3 (meV)
|-
|1x1x1||-40.930301||3.818
|-
|2x2x2||-40.926609||0.126
|-
|4x4x4||-40.926452||0.033
|-
|8x8x8||-40.926478||0.005
|-
|16x16x16||-40.926482||0.001
|-
|32x32x32||-40.926483||0
|}

As shown in the table above, the difference between two consecutive Helmholtz Free Energy steadily decreases as the shrinking factor grows. An 1*1*1 grid is accurate enough for accuracy down to 1 meV and a 2*2*2 grid is sufficient for accuracy to 0.5 meV.

The change in free energy when different shrinking factors are used is due to the addition of more details with increasing number of shrinking factor as the energy is computed by summing up the energy related to each k value and the shrinking factor refers to how many k values are sampled during calculation.

The MgO model simulated above would be suitable for computing properties for crystals of similar structures such as most simple oxides as they mostly have fcc structure and comparable lattice parameters and hence similar brillouin zone and naturally k values. However, simulating other crystal structures that drastically differ from MgO while still using MgO model will be largely inaccurate as they will take different spatial arrangement in reciprocal space and hence different k values.

===Thermal Expansion===

{|
|[[File:Rhl free energy.PNG|500 px|left|Figure 2. Helmholtz Free Energy vs Temperature]]
|}
{|
|[[File:Rhl lattice parameter.PNG|500 px|left|Figure 2. Helmholtz Free Energy vs Temperature]]
|}

The calculation done for thermal expansion is based on quasi-harmonic approximation rather than a complete harmonic approximation due to the fact that harmonic approximation does not allow any shift in equilibrium position of atoms on a lattice as any energy input will only lead to the occupation of higher vibrational energy level rather than any shift in position. Using quasi-harmonic model, which is a combination of harmonic oscillator, Coulombic repulsion, etc., the slight shift in equilibrium position of phonons will be simulated and only then can the thermal expansion, which is essentially the change in bond distance, be fully illustrated.

====Free Energy====
The Helmholtz Free Energy increases substantially with an increasing temperature as predicted by its definition: <math>A=U-TS</math>. The actual value is computed by<math>
F = E_0 + \frac{1}{2}\sum_{k,j} \bar{h}\omega + k_B T\sum_{k,j} ln[1-exp(-\bar{h}\omega /k_B T).
</math>
The definition of Helmholtz Free Energy indicates that at low temperature, Helmholtz Free Energy is dominated by internal energy term and any change in temperature, which contributes to the entropy term, is insignificant. This explains why the curve of free energy vs temperature shows a flat curve at low temperature before becoming steeper. The entire curve illustrates how temperature, i.e. entropic term gradually becomes dominating in Helmholtz Free Energy.

====Lattice Parameter====
As temperature increases, the unit cells receive more energy and can therefore populate higher vibrational states and shift from their original equilibrium position. This shift in equilibrium position constitutes in the change in bond distance and hence the expansion of lattice.

As temperature increases near the melting point of MgO, it is obvious that the distance between two neighbouring atoms will reach the dissociation limit and the harmonic approximation will break down as the vibrating atom will no longer return to its equilibrium position but drift away. This is demonstrated by the fact that the calculation could not be achieved in 3000 K because the vibration is no longer possible.

====Expansion Coefficient====
The expansion coefficient is defined as: <math>
\alpha_V = \frac{1}{V}\,\left(\frac{\partial V}{\partial T}\right)_p
</math> and in this study the expansion of MgO is obtained by applying this equation to the linear region of volume vs temperature diagram. <math>\alpha_v=(1/18.89)*(19.26-18.89)/(1000-300)=2.80*10^-5 K^-1</math> The result differs slightly from experimental value as expected since the assumption does not include any consideration to the actual lattice structure of a crystal, which must contain a certain level of defects and impurities. OTHER ASSUMPTIONS?

===Molecular Dynamics===
[[File:Rhl QH vs MD.PNG|500 px|left|QH and MD prediction of volume vs temperature]]

The thermal expansion predicted by molecular dynamics is generally in good agreement with that by quasi-harmonic approximation in higher temperatures, but the results do differ significantly in lower tempeartures. The difference can be rationalised by the fact that QH has taken into consideration of zero energy of a harmonic oscillator and the effect of zero energy is more pronounced in lower temperatures. As molecular dynamics approximation is totally Newtonian, it does not take into consideration of zero point energy when T=0 and hence has no zero energy contribution to the volume of the lattice. In higher temperatures, the contribution of zero point energy becomes insignificant as ____ dominates.

It must be noticed that since MD is totally Newtonian and does not consider the dissociation of bonding as QH does, the cell volume simulated by MD will keep increasing with temperature even when the calculation by QH is no longer possible due to bond dissociation.

Rep:MOD:order66

2017-03-09T17:14:36Z

Rl2014: /* Thermal Expansion */

== MgO Thermal Expansion ==

==Introduction==

===Aim===
This investigation aims at studying the thermal expansion properties of magnesium oxide crystal using quasi-harmonic approximation and molecular dynamics. The simulation results will be used in calculating the thermal expansion coefficient of MgO at different conditions and predicting the outcome of thermal expansion.

===System===
The crystal lattice of MgO has a FCC structure similar to that of NaCl. Its primitive cell has one atom of oxygen sitting in the middle of the rhombohedron and eight atoms of magnesium on all eight corners which contribute to 1/8 * 8 =1 atom. The conventional cell has four times the size of a primitive cell and a supercell would contain 32 times the size of a primitive cell. The difference in size will significantly influence the outcome of calculation as it will be shown later.

===Origin of Thermal Expansion===
The expansion of crystal lattice can be illustrated by several means. The most straightforward reasoning is that due to the increase in internal energy, the average bonding distance of atoms in a lattice/solid increases, which is in turn due to the increased amplitude of vibration. The higher amplitude of vibration causes an increase in energy among atoms in the original lattice and hence the atoms tend to stay away from each other to accommodate the extra vibrational energy.

==Methodology==
===Quasi-Harmonic Approximation===
Assuming the MgO lattice to be a perfect crystal with no defect whatsoever, the entire lattice can be approximated into infinite continuum of unit cells along x, y and z axis and hence the vibrations of the entire lattice can be broken down into vibrations along 1-D chains on x, y and z axis. Each type of vibration is governed by one individual wavevector k=2pi/lambda, which in turn defines the vibrational frequency and hence energy as functions of k.
The lattice structure in real space is converted into reciprocal space (k-space).
By summing up all k values for each vibrational band, the total vibrational energy of a crystal can be computed.
The plot of frequency over k values is called a dispersion curve and k values of special interests: the symmetry points are labelled.

The quasi-harmonic approximation is based on the assumption that each atom on the lattice oscillate around its equilibrium position in simple harmonic motion when the surrounding temperature does not exceed a certain value (otherwise the bonding in the lattice will dissociate). However, the quasi-harmonic motion differs from simple harmonic motion that it allows the change in atomic distance and hence the change in volume (thermal expansion) is made possible.

===Molecular Dynamics===
The mechanism of molecular dynamics involves assigning each particle in the lattice an initial configuration and a random velocity to make up the given temperature. The initial properties will be used in computing the force and hence acceleration experienced by each atom. The acceleration value is then used to compute a new velocity and hence a new location of each atom. As the system tends to equilibrium, other properties such as temperature and energy will be extracted.

==Software==
Linux platform was chosen over windows due to its efficiency in performing calculations. The lattice structure was displayed using DLV, which also helps with illustrating lattice properties. The calculations were performed using General Utility Lattice Program (GULP), which________.

==Results and Discussion==
===Phonon Modes===
{|
|[[File:RHL Dispersion curve.png|thumb|left|Figure 2. Phonon dispersion curve of MgO lattice.]]
|}
In solid state physics/chemistry, a phonon refers to a collective periodic and elastic excitation/vibration of atoms or molecules. The phonon mode of MgO lattice in k-space along the conventional path is simulated by GULP to support the calculation of free energy by quasi-harmonic model.
The simulation produces various phonon dispersion curves and they collectively display the vibrational band structure of MgO crystal.

===Density of States (DOS)===
{|
|[[File:rhl1.png|thumb|Figure 3. Density of states of MgO phonon, shrinking factors: 1x1x1, k-point considered is L.]]
|[[File:2.png |thumb|Figure 4. Density of states of MgO phonon, shrinking factors: 2x2x2.]]
|[[File:4.png |thumb|Figure 5. Density of states of MgO phonon, shrinking factors: 4x4x4.]]
|-
|[[File:rhl8.png |thumb|Figure 6. Density of states of MgO phonon, shrinking factors: 8x8x8.]]
|[[File:rhl16.png|thumb|Figure 7. Density of States of MgO phonon, shrinking factors: 16x16x16.]]
|[[File:rhl32.png |thumb|Figure 8. Density of states of MgO phonon, shrinking factors: 32x32x32.]]
|}

The density of states is defined by ____, i.e. number of levels between two energies. It can be roughly described as a 90 degree rotation of a dispersion diagram, because each point on a dispersion curve is a state defined by its k value and frequency, i.e. energy. This is to say, the flatter the dispersion curve, the higher the density of states, i.e. more states on the same energy level.

The shrinking factors are multiplied by 2 each time a new DOS is obtained in order to ________.
The DOS maintains a good level of details after shrinking factor=16.

For the 1*1*1 DOS, the peaks are located near 280, 350, 670 and 810 cm-1 and these correspond to point L in the dispersion curve. To obtain a reliable display of DOS, input shrinking factors are varied until the resulted density of state diagram shows all necessary details because the shrinking factor is the number of k values computed within a brillouin zone. Larger shrinking factor will naturally give more data points within the brillouin zone and hence more details about the density of states.

SPECULATION: what grid sizes are suitable?

===Free Energy Calculation by Harmonic Approximation===

{|class="wikitable" style="text-align: center;"
|+ Table 2: Helmholtz Free Energy of MgO at various grid sizes
(6 d.p. for comparison)
|-
!Shrinking Factors
!Phonon Helmholtz Free Energy (eV)
!Accuracy(compared with grid size 32^3 (meV)
|-
|1x1x1||-40.930301||3.818
|-
|2x2x2||-40.926609||0.126
|-
|4x4x4||-40.926452||0.033
|-
|8x8x8||-40.926478||0.005
|-
|16x16x16||-40.926482||0.001
|-
|32x32x32||-40.926483||0
|}

As shown in the table above, the difference between two consecutive Helmholtz Free Energy steadily decreases as the shrinking factor grows. An 1*1*1 grid is accurate enough for accuracy down to 1 meV and a 2*2*2 grid is sufficient for accuracy to 0.5 meV.

The change in free energy when different shrinking factors are used is due to the addition of more details with increasing number of shrinking factor as the energy is computed by summing up the energy related to each k value and the shrinking factor refers to how many k values are sampled during calculation.

The MgO model simulated above would be suitable for computing properties for crystals of similar structures such as most simple oxides as they mostly have fcc structure and comparable lattice parameters and hence similar brillouin zone and naturally k values. However, simulating other crystal structures that drastically differ from MgO while still using MgO model will be largely inaccurate as they will take different spatial arrangement in reciprocal space and hence different k values.

===Thermal Expansion===

{|
|[[File:Rhl free energy.PNG|500 px|left|Figure 2. Helmholtz Free Energy vs Temperature]]
|[[File:Rhl lattice parameter.PNG|500 px|left|Figure 2. Helmholtz Free Energy vs Temperature]]
|}

The calculation done for thermal expansion is based on quasi-harmonic approximation rather than a complete harmonic approximation due to the fact that harmonic approximation does not allow any shift in equilibrium position of atoms on a lattice as any energy input will only lead to the occupation of higher vibrational energy level rather than any shift in position. Using quasi-harmonic model, which is a combination of harmonic oscillator, Coulombic repulsion, etc., the slight shift in equilibrium position of phonons will be simulated and only then can the thermal expansion, which is essentially the change in bond distance, be fully illustrated.

====Free Energy====
The Helmholtz Free Energy increases substantially with an increasing temperature as predicted by its definition: <math>A=U-TS</math>. The actual value is computed by<math>
F = E_0 + \frac{1}{2}\sum_{k,j} \bar{h}\omega + k_B T\sum_{k,j} ln[1-exp(-\bar{h}\omega /k_B T).
</math>
The definition of Helmholtz Free Energy indicates that at low temperature, Helmholtz Free Energy is dominated by internal energy term and any change in temperature, which contributes to the entropy term, is insignificant. This explains why the curve of free energy vs temperature shows a flat curve at low temperature before becoming steeper. The entire curve illustrates how temperature, i.e. entropic term gradually becomes dominating in Helmholtz Free Energy.

====Lattice Parameter====
As temperature increases, the unit cells receive more energy and can therefore populate higher vibrational states and shift from their original equilibrium position. This shift in equilibrium position constitutes in the change in bond distance and hence the expansion of lattice.

As temperature increases near the melting point of MgO, it is obvious that the distance between two neighbouring atoms will reach the dissociation limit and the harmonic approximation will break down as the vibrating atom will no longer return to its equilibrium position but drift away. This is demonstrated by the fact that the calculation could not be achieved in 3000 K because the vibration is no longer possible.

====Expansion Coefficient====
The expansion coefficient is defined as: <math>
\alpha_V = \frac{1}{V}\,\left(\frac{\partial V}{\partial T}\right)_p
</math> and in this study the expansion of MgO is obtained by applying this equation to the linear region of volume vs temperature diagram. <math>\alpha_v=(1/18.89)*(19.26-18.89)/(1000-300)=2.80*10^-5 K^-1</math> The result differs slightly from experimental value as expected since the assumption does not include any consideration to the actual lattice structure of a crystal, which must contain a certain level of defects and impurities. OTHER ASSUMPTIONS?

===Molecular Dynamics===
[[File:Rhl QH vs MD.PNG|500 px|left|QH and MD prediction of volume vs temperature]]

The thermal expansion predicted by molecular dynamics is generally in good agreement with that by quasi-harmonic approximation in higher temperatures, but the results do differ significantly in lower tempeartures. The difference can be rationalised by the fact that QH has taken into consideration of zero energy of a harmonic oscillator and the effect of zero energy is more pronounced in lower temperatures. As molecular dynamics approximation is totally Newtonian, it does not take into consideration of zero point energy when T=0 and hence has no zero energy contribution to the volume of the lattice. In higher temperatures, the contribution of zero point energy becomes insignificant as ____ dominates.

It must be noticed that since MD is totally Newtonian and does not consider the dissociation of bonding as QH does, the cell volume simulated by MD will keep increasing with temperature even when the calculation by QH is no longer possible due to bond dissociation.

Rep:MOD:order66

2017-03-09T17:13:30Z

Rl2014:

== MgO Thermal Expansion ==

==Introduction==

===Aim===
This investigation aims at studying the thermal expansion properties of magnesium oxide crystal using quasi-harmonic approximation and molecular dynamics. The simulation results will be used in calculating the thermal expansion coefficient of MgO at different conditions and predicting the outcome of thermal expansion.

===System===
The crystal lattice of MgO has a FCC structure similar to that of NaCl. Its primitive cell has one atom of oxygen sitting in the middle of the rhombohedron and eight atoms of magnesium on all eight corners which contribute to 1/8 * 8 =1 atom. The conventional cell has four times the size of a primitive cell and a supercell would contain 32 times the size of a primitive cell. The difference in size will significantly influence the outcome of calculation as it will be shown later.

===Origin of Thermal Expansion===
The expansion of crystal lattice can be illustrated by several means. The most straightforward reasoning is that due to the increase in internal energy, the average bonding distance of atoms in a lattice/solid increases, which is in turn due to the increased amplitude of vibration. The higher amplitude of vibration causes an increase in energy among atoms in the original lattice and hence the atoms tend to stay away from each other to accommodate the extra vibrational energy.

==Methodology==
===Quasi-Harmonic Approximation===
Assuming the MgO lattice to be a perfect crystal with no defect whatsoever, the entire lattice can be approximated into infinite continuum of unit cells along x, y and z axis and hence the vibrations of the entire lattice can be broken down into vibrations along 1-D chains on x, y and z axis. Each type of vibration is governed by one individual wavevector k=2pi/lambda, which in turn defines the vibrational frequency and hence energy as functions of k.
The lattice structure in real space is converted into reciprocal space (k-space).
By summing up all k values for each vibrational band, the total vibrational energy of a crystal can be computed.
The plot of frequency over k values is called a dispersion curve and k values of special interests: the symmetry points are labelled.

The quasi-harmonic approximation is based on the assumption that each atom on the lattice oscillate around its equilibrium position in simple harmonic motion when the surrounding temperature does not exceed a certain value (otherwise the bonding in the lattice will dissociate). However, the quasi-harmonic motion differs from simple harmonic motion that it allows the change in atomic distance and hence the change in volume (thermal expansion) is made possible.

===Molecular Dynamics===
The mechanism of molecular dynamics involves assigning each particle in the lattice an initial configuration and a random velocity to make up the given temperature. The initial properties will be used in computing the force and hence acceleration experienced by each atom. The acceleration value is then used to compute a new velocity and hence a new location of each atom. As the system tends to equilibrium, other properties such as temperature and energy will be extracted.

==Software==
Linux platform was chosen over windows due to its efficiency in performing calculations. The lattice structure was displayed using DLV, which also helps with illustrating lattice properties. The calculations were performed using General Utility Lattice Program (GULP), which________.

==Results and Discussion==
===Phonon Modes===
{|
|[[File:RHL Dispersion curve.png|thumb|left|Figure 2. Phonon dispersion curve of MgO lattice.]]
|}
In solid state physics/chemistry, a phonon refers to a collective periodic and elastic excitation/vibration of atoms or molecules. The phonon mode of MgO lattice in k-space along the conventional path is simulated by GULP to support the calculation of free energy by quasi-harmonic model.
The simulation produces various phonon dispersion curves and they collectively display the vibrational band structure of MgO crystal.

===Density of States (DOS)===
{|
|[[File:rhl1.png|thumb|Figure 3. Density of states of MgO phonon, shrinking factors: 1x1x1, k-point considered is L.]]
|[[File:2.png |thumb|Figure 4. Density of states of MgO phonon, shrinking factors: 2x2x2.]]
|[[File:4.png |thumb|Figure 5. Density of states of MgO phonon, shrinking factors: 4x4x4.]]
|-
|[[File:rhl8.png |thumb|Figure 6. Density of states of MgO phonon, shrinking factors: 8x8x8.]]
|[[File:rhl16.png|thumb|Figure 7. Density of States of MgO phonon, shrinking factors: 16x16x16.]]
|[[File:rhl32.png |thumb|Figure 8. Density of states of MgO phonon, shrinking factors: 32x32x32.]]
|}

The density of states is defined by ____, i.e. number of levels between two energies. It can be roughly described as a 90 degree rotation of a dispersion diagram, because each point on a dispersion curve is a state defined by its k value and frequency, i.e. energy. This is to say, the flatter the dispersion curve, the higher the density of states, i.e. more states on the same energy level.

The shrinking factors are multiplied by 2 each time a new DOS is obtained in order to ________.
The DOS maintains a good level of details after shrinking factor=16.

For the 1*1*1 DOS, the peaks are located near 280, 350, 670 and 810 cm-1 and these correspond to point L in the dispersion curve. To obtain a reliable display of DOS, input shrinking factors are varied until the resulted density of state diagram shows all necessary details because the shrinking factor is the number of k values computed within a brillouin zone. Larger shrinking factor will naturally give more data points within the brillouin zone and hence more details about the density of states.

SPECULATION: what grid sizes are suitable?

===Free Energy Calculation by Harmonic Approximation===

{|class="wikitable" style="text-align: center;"
|+ Table 2: Helmholtz Free Energy of MgO at various grid sizes
(6 d.p. for comparison)
|-
!Shrinking Factors
!Phonon Helmholtz Free Energy (eV)
!Accuracy(compared with grid size 32^3 (meV)
|-
|1x1x1||-40.930301||3.818
|-
|2x2x2||-40.926609||0.126
|-
|4x4x4||-40.926452||0.033
|-
|8x8x8||-40.926478||0.005
|-
|16x16x16||-40.926482||0.001
|-
|32x32x32||-40.926483||0
|}

As shown in the table above, the difference between two consecutive Helmholtz Free Energy steadily decreases as the shrinking factor grows. An 1*1*1 grid is accurate enough for accuracy down to 1 meV and a 2*2*2 grid is sufficient for accuracy to 0.5 meV.

The change in free energy when different shrinking factors are used is due to the addition of more details with increasing number of shrinking factor as the energy is computed by summing up the energy related to each k value and the shrinking factor refers to how many k values are sampled during calculation.

The MgO model simulated above would be suitable for computing properties for crystals of similar structures such as most simple oxides as they mostly have fcc structure and comparable lattice parameters and hence similar brillouin zone and naturally k values. However, simulating other crystal structures that drastically differ from MgO while still using MgO model will be largely inaccurate as they will take different spatial arrangement in reciprocal space and hence different k values.

===Thermal Expansion===

{|
|[[File:Rhl free energy.PNG|thumb|left|Figure 2. Helmholtz Free Energy vs Temperature]]
|[[File:Rhl lattice parameter.PNG|thumb|left|Figure 2. Helmholtz Free Energy vs Temperature]]
|}

The calculation done for thermal expansion is based on quasi-harmonic approximation rather than a complete harmonic approximation due to the fact that harmonic approximation does not allow any shift in equilibrium position of atoms on a lattice as any energy input will only lead to the occupation of higher vibrational energy level rather than any shift in position. Using quasi-harmonic model, which is a combination of harmonic oscillator, Coulombic repulsion, etc., the slight shift in equilibrium position of phonons will be simulated and only then can the thermal expansion, which is essentially the change in bond distance, be fully illustrated.

====Free Energy====
The Helmholtz Free Energy increases substantially with an increasing temperature as predicted by its definition: <math>A=U-TS</math>. The actual value is computed by<math>
F = E_0 + \frac{1}{2}\sum_{k,j} \bar{h}\omega + k_B T\sum_{k,j} ln[1-exp(-\bar{h}\omega /k_B T).
</math>
The definition of Helmholtz Free Energy indicates that at low temperature, Helmholtz Free Energy is dominated by internal energy term and any change in temperature, which contributes to the entropy term, is insignificant. This explains why the curve of free energy vs temperature shows a flat curve at low temperature before becoming steeper. The entire curve illustrates how temperature, i.e. entropic term gradually becomes dominating in Helmholtz Free Energy.

====Lattice Parameter====
As temperature increases, the unit cells receive more energy and can therefore populate higher vibrational states and shift from their original equilibrium position. This shift in equilibrium position constitutes in the change in bond distance and hence the expansion of lattice.

As temperature increases near the melting point of MgO, it is obvious that the distance between two neighbouring atoms will reach the dissociation limit and the harmonic approximation will break down as the vibrating atom will no longer return to its equilibrium position but drift away. This is demonstrated by the fact that the calculation could not be achieved in 3000 K because the vibration is no longer possible.

====Expansion Coefficient====
The expansion coefficient is defined as: <math>
\alpha_V = \frac{1}{V}\,\left(\frac{\partial V}{\partial T}\right)_p
</math> and in this study the expansion of MgO is obtained by applying this equation to the linear region of volume vs temperature diagram. <math>\alpha_v=(1/18.89)*(19.26-18.89)/(1000-300)=2.80*10^-5 K^-1</math> The result differs slightly from experimental value as expected since the assumption does not include any consideration to the actual lattice structure of a crystal, which must contain a certain level of defects and impurities. OTHER ASSUMPTIONS?

===Molecular Dynamics===
[[File:Rhl QH vs MD.PNG|500 px|left|QH and MD prediction of volume vs temperature]]

The thermal expansion predicted by molecular dynamics is generally in good agreement with that by quasi-harmonic approximation in higher temperatures, but the results do differ significantly in lower tempeartures. The difference can be rationalised by the fact that QH has taken into consideration of zero energy of a harmonic oscillator and the effect of zero energy is more pronounced in lower temperatures. As molecular dynamics approximation is totally Newtonian, it does not take into consideration of zero point energy when T=0 and hence has no zero energy contribution to the volume of the lattice. In higher temperatures, the contribution of zero point energy becomes insignificant as ____ dominates.

It must be noticed that since MD is totally Newtonian and does not consider the dissociation of bonding as QH does, the cell volume simulated by MD will keep increasing with temperature even when the calculation by QH is no longer possible due to bond dissociation.

File:Rhl QH vs MD.PNG

2017-03-09T17:03:16Z

Rl2014:

Rep:MOD:order66

2017-03-09T16:36:16Z

File:8.png

2017-03-08T21:31:00Z

Rl2014: Rl2014 uploaded a new version of File:8.png