ChemWiki - User contributions [en]

Rep:Mod:ZLY9512

2017-02-09T23:20:25Z

Lz2914: /* Thermal expansion of MgO */

= ''Thermal expansion of MgO'' =

== Introduction ==
Thermal Expansion is a very common phenomenon. It is common knowledge that within a certain range of temperatures, when the temperature goes up some materials expand. some materials expand Some materials expand when the temperature rises. This is due to an increase in atom vibrations resulting in a greater separation between the atoms on a microscopic scale, and on a macroscopic scale leads to an expansion of the materials.<ref name=":0">Paul A., Tipler; Gene Mosca (2008). ''Physics for Scientists and Engineers, Volume 1'' (6th ed.). New York, NY: Worth Publishers. pp. 666–670. ISBN 1-4292-0132-0.</ref> macroscopically makes a material to expand.[1] Although there are some materials which behave the opposite exactly the opposite, these specific cases will not be discussed here, certain cases would not be discussed here.

phenomenon. In this experiment, a simple MgO compound is studied to examine this expansive behaviour.
{| class="wikitable"
! colspan="2" |Table 1: Representations of MgO unit cells (FCC)
|-
|[[File:MgOpri.png|none|thumb|296x296px]]Primitive cell
|[[File:MgO conventional.jpg|none|thumb|294x294px]]Conventional cell
|}

The vibrational state is simulated using a computer in order to model the thermal expansion of MgO. The thermal expansion coefficient α is given by Equation (1)<ref name=":0" />:
[[File:Thermal exp coef.png|thumb|Equation (1) : Thermal Expansion Coefficient
|none]]

Two methods were used to calculate the thermal expansion coefficient, the Quasi-Harmonic approximation (LD) and molecular dynamics (MD). The former method is used to compute the vibrational energy levels of MgO. This method allows us to understand the phonon dispersion and the vibrational density of state (DOS). The latter method is used to simulate the vibrations as random motions of atoms inside a cell. It tells us how atoms respond over time under certain conditions. In this experiment, a General Utility Lattice Program (GULP) on the DLVisualize platform (Linux based) is used and by increasing the temperature, we can see how MgO atoms in a cell respond, what physical properties can be ascribed to this response and how it can be quantified.

== Results and Discussion ==

=== Phonon dispersion and Density of States ===
The phonon dispersion graph of MgO was produced by simulation to study the number of vibrational states which exist at k-points.
[[File:MgOPD.png|thumb|425x425px|Figure 1: Phonon Dispersion graph of MgO|centre]]Followed by the local phonon DOS with different grid sizes.
{| class="wikitable"
! colspan="4" |Table 2: MgO '''phonon DOS with variation of grid sizes'''
|-
|1x1x1
|[[File:DOS 111.JPG|thumb]]
|2x2x2
|[[File:DOS 222.JPG|thumb]]
|-
|3x3x3
|[[File:DOS 333.JPG|thumb]]
|16x16x16
|[[File:DOS 161616.JPG|thumb]]
|-
|20x20x20
|[[File:DOS 202020.JPG|thumb]]
|25x25x25
|[[File:DOS 252525.JPG|thumb]]
|}

From the 1x1x1 grid four frequencies match the symmetry point L which indicates that the k-point is equal to (1/2, 1/2, 1/2). In the 1x1x1 grid DOS graph, the two lower frequency numbers are twice as large as the two higher frequency numbers. This matches the fact that in the dispersion curve at the symmetry point L the two lower frequency numbers are doubly degenerated. The DOS resembles the dispersion curves. It contains the information at certain frequency numbers with respected to a symmetry point.

By comparing different grid sizes' dispersion curves, we can conclude that when the grid size increases, the DOS increases because more points in k-space are taken and used to calculate the DOS. The larger the grid size, the finer, smoother the DOS curve becomes as there are more atoms being examined more details are given. Since the 20x20x20 graph is extremely similar to the 25x25x25 graph, for a reasonable approximation to the DOS, the 20x20x20 grid size is a sufficiently accurate approximation.

The optimal grid size 20x20x20 can be used for other crystals such as CaO. CaO unit cells have a similar size as that of MgO hence the same grid can be used, whilist for Faujasite or lithium metal, a size change is required.
[[File:A*=2pi-a.png|thumb|Equation 2: Periodic direction in the reciprocal space, where '''a''' is the periodic direction in the direct space]]
Based on Equation 2, the periodic direction in the reciprocal space is inversely proportional to the periodic direction in the direct space. Hence for a larger cell such as Faujasite a smaller grid can be used while for a smaller cell such as lithium metal a larger grid is needed.

{| class="wikitable"
|+Table 3: Helmholtz free energy change with grid size
!Grid size
!Free Energy/ eV
!ΔA/ meV (with respected to 25x25x25)
|-
|1x1x1
|<nowiki>-40.930301</nowiki>
|<nowiki>-3.188</nowiki>
|-
|2x2x2
|<nowiki>-40.926609</nowiki>
|<nowiki>-0.126</nowiki>
|-
|3x3x3
|<nowiki>-40.926432</nowiki>
|0.051
|-
|4x4x4
|<nowiki>-40.926450</nowiki>
|0.033
|-
|8x8x8
|<nowiki>-40.926478</nowiki>
|0.005
|-
|10x10x10
|<nowiki>-40.926480</nowiki>
|0.003
|-
|15x15x15
|<nowiki>-40.926482</nowiki>
|0.001
|-
|20x20x20
|<nowiki>-40.926483</nowiki>
|0.000
|-
|25x25x25
|<nowiki>-40.926483</nowiki>
|0.000
|}

The Helmholtz free energy varies with the grid size. As the size increases, the free energy increases and it reaches a maximum at 3x3x3 grid and then decreases.

In this table the energy difference is calculated between each grid size and the most precise grid size (in this case which is the 25x25x25 grid). For a calculation accurate to 1 meV, 1x1x1 could be used. For an acuuracy of 0.5 meV, 2x2x2 could be used and for an acuuracy of 0.1 meV, 3x3x3 could be used.

=== Quasi-Harmonic Approximation (LD) ===
{| class="wikitable"
|+
Table 4: Effects of temperature on the Helmholtz free energy, the lattice constant and the cell volume.
!Temperature/ K
!Helmholtz free energy/ eV
!Lattice constant/ Å
!Cell Volume/ Å³

|-
|0
|<nowiki>-40.900978</nowiki>
|2.986563
|18.836496

|-
|100
|<nowiki>-40.902420</nowiki>
|2.986563
|18.836494

|-
|200
|<nowiki>-40.909377</nowiki>
|2.987605
|18.856202

|-
|300
|<nowiki>-40.928125</nowiki>
|2.989390
|18.890025

|-
|400
|<nowiki>-40.958594</nowiki>
|2.991630
|18.932508

|-
|500
|<nowiki>-40.999436</nowiki>
|2.994135
|18.980112

|-
|600
|<nowiki>-41.049315</nowiki>
|2.996821
|19.031223

|-
|700
|<nowiki>-41.1071190</nowiki>
|2.999644
|19.085059

|-
|800
|<nowiki>-41.171892</nowiki>
|3.002589
|19.141318

|-
|900
|<nowiki>-41.243018</nowiki>
|3.005636
|19.199640

|-
|1000
|<nowiki>-41.319848</nowiki>
|3.008785
|19.260044
|}
Quasi-Harmonic approximation suggests that the Morse potential can be used to represent the potential between atoms in a unit cell. This model is idealized by considering a perfect crystal. Each atom is an individual harmonic oscillator that does not interact with each other and that is being treated independently. Taken into account of the anharmonicity the vibrational energy can be found.<ref name=":1">Martin T. Dove, ''Introduction to Lattice Dynamics, ''University of Cambridge Press, ISBN-13: 9780521392938.
</ref>

The relationship between the Helmholtz free energy and temperature is given by Equation 3<ref name=":1" />:
[[File:A=U+TS.png|thumb|284x284px|Equation 3. Where ''U'' is the internal energy, ''EZP'' is the vibrational zero-point energy, ''T'' is the absolute temperature, ''V'' is the volume and ''S'' is the entropy due to the vibrational degrees of freedom.|none]]
[[File:(LD) A with T.png|thumb|449x449px|Figure 2: Helmholtz free energy versus temperature (LD)|none]]

The Helmholtz free energy becomes more negative, i.e. decreases, when the temperature increases. It is not hard to see that the result matches Equation 3. As T increases, the entropy term becomes more positive, hence the free energy decreases.

[[File:(LD) LC with T.png|thumb|487x487px|Figure 3: Lattice constant versus temperature (LD)|none]][[File:(LD) V with T.png|thumb|492x492px|Figure 4: Unit cell volume versus temperature (LD)|left]]

As expected, the lattice constant increases with temperature, as does the unit cell volume. At higher temperatures, atoms have a higher internal energy and hence the vibrational frequency increases. This causes a greater deviation from the static separation and therefore the material expands. However, in a diatomic molecule with an exactly harmonic potential the bond length does not increase with temperature as the time average interatomic separation remains constant.

Initially all three curves have a parabolic character, while at a higher temperatures they tend towards a more linear character. From Equation 1 and the gradient calculated from Figure 4 the thermal expansion coefficient is found to be 2.687x10-5 K-1. The literature value is 3.12x10-5 K-1 (at 300 K)<ref>O. L. Anderson and K. Zou, J. ''Thermodynamic Functions and Properties of MgO at High Compression and High Temperarture.'' Phys. Chem. Ref. Data, 1990, '''19''', 69.</ref>. The difference is 13% which is considerably large. Nevertheless it is good enough coming from an approximation.

As the temperature of MgO approaches the melting point, the phonon modes does not represent the actual motions of the ions well, simply because at the melting point the bond breaks and it changes the crystal structure which it is no longer perfect. This violates the approximation and hence the modes are not be able to represent the motions well.

.

.
=== Molecular Dynamics (MD) ===
[[File:(MD) V with T.png|thumb|493x493px|Figure 5: Unit cell volume versus temperature (MD)
]]
In this molecular dynamics method, a supercell containing 32 MgO units is used.

By doing a geometry optimization, the atomic position with a minimum energy is found and used to measure the lattice constant and the cell volume. Since the method is based on solving Newton's equations, the system is being treated classically<ref>Alder, B. J.; Wainwright, T. E. (1959). "Studies in Molecular Dynamics. I. General Method". ''J. Chem. Phys''. '''31''' (2): 459. Bibcode:1959JChPh..31..459A. doi:10.1063/1.1730376.</ref>. A larger system is used to compromise the loss of its periodicity due to the nature of random atom movements.

Again using Equation 1 the thermal expansion coefficient is found to be 3.086x10-5 K-1. The difference is 1.1% which is considerably smaller compared to the quasi-harmonic model.

Figure 6 shows a comparison between the results of using the two different methods. Both curves have a very similar shape while the MD method gives a smaller value at the same temperature (within 0-1000 K). It is worth mentioning that with increasing temperatures the two curves are getting closer and closer. [[File:LD and MD Comparison.png|thumb|493x493px|Figure 6: Comparison of using LD and MD]]The foundation of the Quasi-Harmonic approximation is that the system is being treated quantum mechanically<ref name=":1" />, whilst in the Molecular dynamic method the system is being treated classically. Due to this reason at very high temperatures (i.e. close to or beyond the melting point of the substance) the Quasi-Harmonic approximation no longer applies as the periodicity is being removed from the system due to the bond dissociation. On the contrary, Molecular dynamics applies at high temperatures as the method only considers the velocity of each individual atom. However, as can be seen in Figure 5, the molecular dynamic method was not able to give a value at 0 K simply because at 0 K atoms do not move.

.

.

.

.

.

.

.

.

.

.

== Conclusion ==
The thermal expansion coefficient of MgO was found by two using different methods. The Quasi-Harmonic approxiamtion and the molecular dynamics. By optimising the grid size of the MgO crystal a fairly accurate size 20x20x20 was used and the MgO density of states were found. This optimal grid size was then used in the Quasi-Harmonic approximation for finding the relationships between the lattice constant, the volume and the Helmholtz free energy with temperature. From the relationship between the volume and the temperature the thermal expansion coefficient of MgO was calculated. Unlike this method, the molecular dynamic approach does not rely on finding the optimal grid size of MgO. Based on the calculations the molecular dynamic method shows a greater agreement with the literature value. Also if the measurement is conducted at a very high temperature, the Molecular dynamics method should be used as the Quasi-Harmonic approximation method does not work.

== References ==

Rep:Mod:ZLY9512

2017-02-09T23:15:21Z

Lz2914: /* Molecular Dynamics (MD) */

= ''Thermal expansion of MgO'' =

== Introduction ==
Thermal Expansion is a very common phenomenon. It is common knowledge that within a certain range of temperatures, when the temperature goes up some materials expand. some materials expand Some materials expand when the temperature rises. This is due to an increase in atom vibrations resulting in a greater separation between the atoms on a microscopic scale, and on a macroscopic scale leads to an expansion of the materials.<ref name=":0">Paul A., Tipler; Gene Mosca (2008). ''Physics for Scientists and Engineers, Volume 1'' (6th ed.). New York, NY: Worth Publishers. pp. 666–670. ISBN 1-4292-0132-0.</ref> macroscopically makes a material to expand.[1] Although there are some materials which behave the opposite exactly the opposite, these specific cases will not be discussed here, certain cases would not be discussed here.

phenomenon. In this experiment, a simple MgO compound is studied to examine this expansive behaviour.
{| class="wikitable"
! colspan="2" |Table 1: Representations of MgO unit cells (FCC)
|-
|[[File:MgOpri.png|none|thumb|296x296px]]Primitive cell
|[[File:MgO conventional.jpg|none|thumb|294x294px]]Conventional cell
|}

The vibrational state is simulated using a computer in order to model the thermal expansion of MgO. The thermal expansion coefficient α is given by Equation (1)<ref name=":0" />:
[[File:Thermal exp coef.png|thumb|Equation (1) : Thermal Expansion Coefficient
|none]]

Two methods were used to calculate the thermal expansion coefficient, the Quasi-Harmonic approximation (LD) and molecular dynamics (MD). The former method is used to compute the vibrational energy levels of MgO. This method allows us to understand the phonon dispersion and the vibrational density of state (DOS). The latter method is used to simulate the vibrations as random motions of atoms inside a cell. It tells us how atoms respond over time under certain conditions. In this experiment, a General Utility Lattice Program (GULP) on the DLVisualize platform (Linux based) is used and by increasing the temperature, we can see how MgO atoms in a cell respond, what physical properties can be ascribed to this response and how it can be quantified.

== Results and Discussion ==

=== Phonon dispersion and Density of States ===
The phonon dispersion graph of MgO was produced by simulation to study the number of vibrational states which exist at k-points.
[[File:MgOPD.png|thumb|425x425px|Figure 1: Phonon Dispersion graph of MgO|centre]]Followed by the local phonon DOS with different grid sizes.
{| class="wikitable"
! colspan="4" |Table 2: MgO '''phonon DOS with variation of grid sizes'''
|-
|1x1x1
|[[File:DOS 111.JPG|thumb]]
|2x2x2
|[[File:DOS 222.JPG|thumb]]
|-
|3x3x3
|[[File:DOS 333.JPG|thumb]]
|16x16x16
|[[File:DOS 161616.JPG|thumb]]
|-
|20x20x20
|[[File:DOS 202020.JPG|thumb]]
|25x25x25
|[[File:DOS 252525.JPG|thumb]]
|}

From the 1x1x1 grid four frequencies match the symmetry point L which indicates that the k-point is equal to (1/2, 1/2, 1/2). In the 1x1x1 grid DOS graph, the two lower frequency numbers are twice as large as the two higher frequency numbers. This matches the fact that in the dispersion curve at the symmetry point L the two lower frequency numbers are doubly degenerated. The DOS resembles the dispersion curves. It contains the information at certain frequency numbers with respected to a symmetry point.

By comparing different grid sizes' dispersion curves, we can conclude that when the grid size increases, the DOS increases because more points in k-space are taken and used to calculate the DOS. The larger the grid size, the finer, smoother the DOS curve becomes as there are more atoms being examined more details are given. Since the 20x20x20 graph is extremely similar to the 25x25x25 graph, for a reasonable approximation to the DOS, the 20x20x20 grid size is a sufficiently accurate approximation.

The optimal grid size 20x20x20 can be used for other crystals such as CaO. CaO unit cells have a similar size as that of MgO hence the same grid can be used, whilist for Faujasite or lithium metal, a size change is required.
[[File:A*=2pi-a.png|thumb|Equation 2: Periodic direction in the reciprocal space, where '''a''' is the periodic direction in the direct space]]
Based on Equation 2, the periodic direction in the reciprocal space is inversely proportional to the periodic direction in the direct space. Hence for a larger cell such as Faujasite a smaller grid can be used while for a smaller cell such as lithium metal a larger grid is needed.

{| class="wikitable"
|+Table 3: Helmholtz free energy change with grid size
!Grid size
!Free Energy/ eV
!ΔA/ meV (with respected to 25x25x25)
|-
|1x1x1
|<nowiki>-40.930301</nowiki>
|<nowiki>-3.188</nowiki>
|-
|2x2x2
|<nowiki>-40.926609</nowiki>
|<nowiki>-0.126</nowiki>
|-
|3x3x3
|<nowiki>-40.926432</nowiki>
|0.051
|-
|4x4x4
|<nowiki>-40.926450</nowiki>
|0.033
|-
|8x8x8
|<nowiki>-40.926478</nowiki>
|0.005
|-
|10x10x10
|<nowiki>-40.926480</nowiki>
|0.003
|-
|15x15x15
|<nowiki>-40.926482</nowiki>
|0.001
|-
|20x20x20
|<nowiki>-40.926483</nowiki>
|0.000
|-
|25x25x25
|<nowiki>-40.926483</nowiki>
|0.000
|}

The Helmholtz free energy varies with the grid size. As the size increases, the free energy increases and it reaches a maximum at 3x3x3 grid and then decreases.

In this table the energy difference is calculated between each grid size and the most precise grid size (in this case which is the 25x25x25 grid). For a calculation accurate to 1 meV, 1x1x1 could be used. For an acuuracy of 0.5 meV, 2x2x2 could be used and for an acuuracy of 0.1 meV, 3x3x3 could be used.

=== Quasi-Harmonic Approximation (LD) ===
{| class="wikitable"
|+
Table 4: Effects of temperature on the Helmholtz free energy, the lattice constant and the cell volume.
!Temperature/ K
!Helmholtz free energy/ eV
!Lattice constant/ Å
!Cell Volume/ Å³

|-
|0
|<nowiki>-40.900978</nowiki>
|2.986563
|18.836496

|-
|100
|<nowiki>-40.902420</nowiki>
|2.986563
|18.836494

|-
|200
|<nowiki>-40.909377</nowiki>
|2.987605
|18.856202

|-
|300
|<nowiki>-40.928125</nowiki>
|2.989390
|18.890025

|-
|400
|<nowiki>-40.958594</nowiki>
|2.991630
|18.932508

|-
|500
|<nowiki>-40.999436</nowiki>
|2.994135
|18.980112

|-
|600
|<nowiki>-41.049315</nowiki>
|2.996821
|19.031223

|-
|700
|<nowiki>-41.1071190</nowiki>
|2.999644
|19.085059

|-
|800
|<nowiki>-41.171892</nowiki>
|3.002589
|19.141318

|-
|900
|<nowiki>-41.243018</nowiki>
|3.005636
|19.199640

|-
|1000
|<nowiki>-41.319848</nowiki>
|3.008785
|19.260044
|}
Quasi-Harmonic approximation suggests that the Morse potential can be used to represent the potential between atoms in a unit cell. This model is idealized by considering a perfect crystal. Each atom is an individual harmonic oscillator that does not interact with each other and that is being treated independently. Taken into account of the anharmonicity the vibrational energy can be found.<ref name=":1">Martin T. Dove, ''Introduction to Lattice Dynamics, ''University of Cambridge Press, ISBN-13: 9780521392938.
</ref>

The relationship between the Helmholtz free energy and temperature is given by Equation 3<ref name=":1" />:
[[File:A=U+TS.png|thumb|284x284px|Equation 3. Where ''U'' is the internal energy, ''EZP'' is the vibrational zero-point energy, ''T'' is the absolute temperature, ''V'' is the volume and ''S'' is the entropy due to the vibrational degrees of freedom.|none]]
[[File:(LD) A with T.png|thumb|449x449px|Figure 2|none]]

The Helmholtz free energy becomes more negative, i.e. decreases, when the temperature increases. It is not hard to see that the result matches Equation 3. As T increases, the entropy term becomes more positive, hence the free energy decreases.

[[File:(LD) LC with T.png|thumb|487x487px|Figure 3|none]][[File:(LD) V with T.png|thumb|492x492px|Figure 4|left]]

As expected, the lattice constant increases with temperature, as does the unit cell volume. At higher temperatures, atoms have a higher internal energy and hence the vibrational frequency increases. This causes a greater deviation from the static separation and therefore the material expands. However, in a diatomic molecule with an exactly harmonic potential the bond length does not increase with temperature as the time average interatomic separation remains constant.

Initially all three curves have a parabolic character, while at a higher temperatures they tend towards a more linear character. From Equation 1 and the gradient calculated from Figure 4 the thermal expansion coefficient is found to be 2.687x10-5 K-1. The literature value is 3.12x10-5 K-1 (at 300 K)<ref>O. L. Anderson and K. Zou, J. ''Thermodynamic Functions and Properties of MgO at High Compression and High Temperarture.'' Phys. Chem. Ref. Data, 1990, '''19''', 69.</ref>. The difference is 13% which is considerably large. Nevertheless it is good enough coming from an approximation.

As the temperature of MgO approaches the melting point, the phonon modes does not represent the actual motions of the ions well, simply because at the melting point the bond breaks and it changes the crystal structure which it is no longer perfect. This violates the approximation and hence the modes are not be able to represent the motions well.

.

.
=== Molecular Dynamics (MD) ===
[[File:(MD) V with T.png|thumb|493x493px|Figure 5
]]
In this molecular dynamics method, a supercell containing 32 MgO units is used.

By doing a geometry optimization, the atomic position with a minimum energy is found and used to measure the lattice constant and the cell volume. Since the method is based on solving Newton's equations, the system is being treated classically<ref>Alder, B. J.; Wainwright, T. E. (1959). "Studies in Molecular Dynamics. I. General Method". ''J. Chem. Phys''. '''31''' (2): 459. Bibcode:1959JChPh..31..459A. doi:10.1063/1.1730376.</ref>. A larger system is used to compromise the loss of its periodicity due to the nature of random atom movements.

Again using Equation 1 the thermal expansion coefficient is found to be 3.086x10-5 K-1. The difference is 1.1% which is considerably smaller compared to the quasi-harmonic model.

Figure 6 shows a comparison between the results of using the two different methods. Both curves have a very similar shape while the MD method gives a smaller value at the same temperature (within 0-1000 K). It is worth mentioning that with increasing temperatures the two curves are getting closer and closer. [[File:LD and MD Comparison.png|thumb|493x493px|Figure 6]]The foundation of the Quasi-Harmonic approximation is that the system is being treated quantum mechanically<ref name=":1" />, whilst in the Molecular dynamic method the system is being treated classically. Due to this reason at very high temperatures (i.e. close to or beyond the melting point of the substance) the Quasi-Harmonic approximation no longer applies as the periodicity is being removed from the system due to the bond dissociation. On the contrary, Molecular dynamics applies at high temperatures as the method only considers the velocity of each individual atom. However, as can be seen in Figure 5, the molecular dynamic method was not able to give a value at 0 K simply because at 0 K atoms do not move.

.

.

.

.

.

.

.

.

.

== Conclusion ==
The thermal expansion coefficient of MgO was found by two using different methods. The Quasi-Harmonic approxiamtion and the molecular dynamics. By optimising the grid size of the MgO crystal a fairly accurate size 20x20x20 was used and the MgO density of states were found. This optimal grid size was then used in the Quasi-Harmonic approximation for finding the relationships between the lattice constant, the volume and the Helmholtz free energy with temperature. From the relationship between the volume and the temperature the thermal expansion coefficient of MgO was calculated. Unlike this method, the molecular dynamic approach does not rely on finding the optimal grid size of MgO. Based on the calculations the molecular dynamic method shows a greater agreement with the literature value. Also if the measurement is conducted at a very high temperature, the Molecular dynamics method should be used as the Quasi-Harmonic approximation method does not work.

== References ==

Rep:Mod:ZLY9512

2017-02-09T23:12:31Z

Lz2914: /* Molecular Dynamics (MD) */

= ''Thermal expansion of MgO'' =

== Introduction ==
Thermal Expansion is a very common phenomenon. It is common knowledge that within a certain range of temperatures, when the temperature goes up some materials expand. some materials expand Some materials expand when the temperature rises. This is due to an increase in atom vibrations resulting in a greater separation between the atoms on a microscopic scale, and on a macroscopic scale leads to an expansion of the materials.<ref name=":0">Paul A., Tipler; Gene Mosca (2008). ''Physics for Scientists and Engineers, Volume 1'' (6th ed.). New York, NY: Worth Publishers. pp. 666–670. ISBN 1-4292-0132-0.</ref> macroscopically makes a material to expand.[1] Although there are some materials which behave the opposite exactly the opposite, these specific cases will not be discussed here, certain cases would not be discussed here.

phenomenon. In this experiment, a simple MgO compound is studied to examine this expansive behaviour.
{| class="wikitable"
! colspan="2" |Table 1: Representations of MgO unit cells (FCC)
|-
|[[File:MgOpri.png|none|thumb|296x296px]]Primitive cell
|[[File:MgO conventional.jpg|none|thumb|294x294px]]Conventional cell
|}

The vibrational state is simulated using a computer in order to model the thermal expansion of MgO. The thermal expansion coefficient α is given by Equation (1)<ref name=":0" />:
[[File:Thermal exp coef.png|thumb|Equation (1) : Thermal Expansion Coefficient
|none]]

Two methods were used to calculate the thermal expansion coefficient, the Quasi-Harmonic approximation (LD) and molecular dynamics (MD). The former method is used to compute the vibrational energy levels of MgO. This method allows us to understand the phonon dispersion and the vibrational density of state (DOS). The latter method is used to simulate the vibrations as random motions of atoms inside a cell. It tells us how atoms respond over time under certain conditions. In this experiment, a General Utility Lattice Program (GULP) on the DLVisualize platform (Linux based) is used and by increasing the temperature, we can see how MgO atoms in a cell respond, what physical properties can be ascribed to this response and how it can be quantified.

== Results and Discussion ==

=== Phonon dispersion and Density of States ===
The phonon dispersion graph of MgO was produced by simulation to study the number of vibrational states which exist at k-points.
[[File:MgOPD.png|thumb|425x425px|Figure 1: Phonon Dispersion graph of MgO|centre]]Followed by the local phonon DOS with different grid sizes.
{| class="wikitable"
! colspan="4" |Table 2: MgO '''phonon DOS with variation of grid sizes'''
|-
|1x1x1
|[[File:DOS 111.JPG|thumb]]
|2x2x2
|[[File:DOS 222.JPG|thumb]]
|-
|3x3x3
|[[File:DOS 333.JPG|thumb]]
|16x16x16
|[[File:DOS 161616.JPG|thumb]]
|-
|20x20x20
|[[File:DOS 202020.JPG|thumb]]
|25x25x25
|[[File:DOS 252525.JPG|thumb]]
|}

From the 1x1x1 grid four frequencies match the symmetry point L which indicates that the k-point is equal to (1/2, 1/2, 1/2). In the 1x1x1 grid DOS graph, the two lower frequency numbers are twice as large as the two higher frequency numbers. This matches the fact that in the dispersion curve at the symmetry point L the two lower frequency numbers are doubly degenerated. The DOS resembles the dispersion curves. It contains the information at certain frequency numbers with respected to a symmetry point.

By comparing different grid sizes' dispersion curves, we can conclude that when the grid size increases, the DOS increases because more points in k-space are taken and used to calculate the DOS. The larger the grid size, the finer, smoother the DOS curve becomes as there are more atoms being examined more details are given. Since the 20x20x20 graph is extremely similar to the 25x25x25 graph, for a reasonable approximation to the DOS, the 20x20x20 grid size is a sufficiently accurate approximation.

The optimal grid size 20x20x20 can be used for other crystals such as CaO. CaO unit cells have a similar size as that of MgO hence the same grid can be used, whilist for Faujasite or lithium metal, a size change is required.
[[File:A*=2pi-a.png|thumb|Equation 2: Periodic direction in the reciprocal space, where '''a''' is the periodic direction in the direct space]]
Based on Equation 2, the periodic direction in the reciprocal space is inversely proportional to the periodic direction in the direct space. Hence for a larger cell such as Faujasite a smaller grid can be used while for a smaller cell such as lithium metal a larger grid is needed.

{| class="wikitable"
|+Table 3: Helmholtz free energy change with grid size
!Grid size
!Free Energy/ eV
!ΔA/ meV (with respected to 25x25x25)
|-
|1x1x1
|<nowiki>-40.930301</nowiki>
|<nowiki>-3.188</nowiki>
|-
|2x2x2
|<nowiki>-40.926609</nowiki>
|<nowiki>-0.126</nowiki>
|-
|3x3x3
|<nowiki>-40.926432</nowiki>
|0.051
|-
|4x4x4
|<nowiki>-40.926450</nowiki>
|0.033
|-
|8x8x8
|<nowiki>-40.926478</nowiki>
|0.005
|-
|10x10x10
|<nowiki>-40.926480</nowiki>
|0.003
|-
|15x15x15
|<nowiki>-40.926482</nowiki>
|0.001
|-
|20x20x20
|<nowiki>-40.926483</nowiki>
|0.000
|-
|25x25x25
|<nowiki>-40.926483</nowiki>
|0.000
|}

The Helmholtz free energy varies with the grid size. As the size increases, the free energy increases and it reaches a maximum at 3x3x3 grid and then decreases.

In this table the energy difference is calculated between each grid size and the most precise grid size (in this case which is the 25x25x25 grid). For a calculation accurate to 1 meV, 1x1x1 could be used. For an acuuracy of 0.5 meV, 2x2x2 could be used and for an acuuracy of 0.1 meV, 3x3x3 could be used.

=== Quasi-Harmonic Approximation (LD) ===
{| class="wikitable"
|+
Table 4: Effects of temperature on the Helmholtz free energy, the lattice constant and the cell volume.
!Temperature/ K
!Helmholtz free energy/ eV
!Lattice constant/ Å
!Cell Volume/ Å³

|-
|0
|<nowiki>-40.900978</nowiki>
|2.986563
|18.836496

|-
|100
|<nowiki>-40.902420</nowiki>
|2.986563
|18.836494

|-
|200
|<nowiki>-40.909377</nowiki>
|2.987605
|18.856202

|-
|300
|<nowiki>-40.928125</nowiki>
|2.989390
|18.890025

|-
|400
|<nowiki>-40.958594</nowiki>
|2.991630
|18.932508

|-
|500
|<nowiki>-40.999436</nowiki>
|2.994135
|18.980112

|-
|600
|<nowiki>-41.049315</nowiki>
|2.996821
|19.031223

|-
|700
|<nowiki>-41.1071190</nowiki>
|2.999644
|19.085059

|-
|800
|<nowiki>-41.171892</nowiki>
|3.002589
|19.141318

|-
|900
|<nowiki>-41.243018</nowiki>
|3.005636
|19.199640

|-
|1000
|<nowiki>-41.319848</nowiki>
|3.008785
|19.260044
|}
Quasi-Harmonic approximation suggests that the Morse potential can be used to represent the potential between atoms in a unit cell. This model is idealized by considering a perfect crystal. Each atom is an individual harmonic oscillator that does not interact with each other and that is being treated independently. Taken into account of the anharmonicity the vibrational energy can be found.<ref name=":1">Martin T. Dove, ''Introduction to Lattice Dynamics, ''University of Cambridge Press, ISBN-13: 9780521392938.
</ref>

The relationship between the Helmholtz free energy and temperature is given by Equation 3<ref name=":1" />:
[[File:A=U+TS.png|thumb|284x284px|Equation 3. Where ''U'' is the internal energy, ''EZP'' is the vibrational zero-point energy, ''T'' is the absolute temperature, ''V'' is the volume and ''S'' is the entropy due to the vibrational degrees of freedom.|none]]
[[File:(LD) A with T.png|thumb|449x449px|Figure 2|none]]

The Helmholtz free energy becomes more negative, i.e. decreases, when the temperature increases. It is not hard to see that the result matches Equation 3. As T increases, the entropy term becomes more positive, hence the free energy decreases.

[[File:(LD) LC with T.png|thumb|487x487px|Figure 3|none]][[File:(LD) V with T.png|thumb|492x492px|Figure 4|left]]

As expected, the lattice constant increases with temperature, as does the unit cell volume. At higher temperatures, atoms have a higher internal energy and hence the vibrational frequency increases. This causes a greater deviation from the static separation and therefore the material expands. However, in a diatomic molecule with an exactly harmonic potential the bond length does not increase with temperature as the time average interatomic separation remains constant.

Initially all three curves have a parabolic character, while at a higher temperatures they tend towards a more linear character. From Equation 1 and the gradient calculated from Figure 4 the thermal expansion coefficient is found to be 2.687x10-5 K-1. The literature value is 3.12x10-5 K-1 (at 300 K)<ref>O. L. Anderson and K. Zou, J. ''Thermodynamic Functions and Properties of MgO at High Compression and High Temperarture.'' Phys. Chem. Ref. Data, 1990, '''19''', 69.</ref>. The difference is 13% which is considerably large. Nevertheless it is good enough coming from an approximation.

As the temperature of MgO approaches the melting point, the phonon modes does not represent the actual motions of the ions well, simply because at the melting point the bond breaks and it changes the crystal structure which it is no longer perfect. This violates the approximation and hence the modes are not be able to represent the motions well.

.

.
=== Molecular Dynamics (MD) ===
[[File:(MD) V with T.png|thumb|493x493px|Figure 5
]]
In this molecular dynamics method, a supercell containing 32 MgO units is used.

By doing a geometry optimization, the atomic position with a minimum energy is found and used to measure the lattice constant and the cell volume. Since the method is based on solving Newton's equations, the system is being treated classically<ref>Alder, B. J.; Wainwright, T. E. (1959). "Studies in Molecular Dynamics. I. General Method". ''J. Chem. Phys''. '''31''' (2): 459. Bibcode:1959JChPh..31..459A. doi:10.1063/1.1730376.</ref>. A larger system is used to compromise the loss of its periodicity due to the nature of random atom movements.

Again using Equation 1 the thermal expansion coefficient is found to be 3.086x10-5 K-1. The difference is 1.1% which is considerably smaller compared to the quasi-harmonic model.

Figure 6 shows a comparison between the results of using the two different methods. Both curves have a very similar shape while the MD method gives a smaller value at the same temperature (within 0-1000 K). It is worth mentioning that with increasing temperatures the two curves are getting closer and closer. [[File:LD and MD Comparison.png|thumb|493x493px|Figure 6]]The foundation of the Quasi-Harmonic approximation is that the system is being treated quantum mechanically<ref name=":1" />, whilst in the Molecular dynamic method the system is being treated classically. Due to this reason at very high temperatures (i.e. close to or beyond the melting point of the substance) the Quasi-Harmonic approximation no longer applies as the periodicity is being removed from the system due to the bond dissociation. On the contrary, Molecular dynamics applies at high temperatures as the method only considers the velocity of each individual atom. However, as can be seen in Figure 5, the molecular dynamic method was not able to give a value at 0 K simply because at 0 K atoms do not move.

== Conclusion ==
The thermal expansion coefficient of MgO was found by two using different methods. The Quasi-Harmonic approxiamtion and the molecular dynamics. By optimising the grid size of the MgO crystal a fairly accurate size 20x20x20 was used and the MgO density of states were found. This optimal grid size was then used in the Quasi-Harmonic approximation for finding the relationships between the lattice constant, the volume and the Helmholtz free energy with temperature. From the relationship between the volume and the temperature the thermal expansion coefficient of MgO was calculated. Unlike this method, the molecular dynamic approach does not rely on finding the optimal grid size of MgO. Based on the calculations the molecular dynamic method shows a greater agreement with the literature value.

Either of

== References ==

Rep:Mod:ZLY9512

2017-02-09T23:08:04Z

Lz2914: /* Thermal expansion of MgO */

= ''Thermal expansion of MgO'' =

== Introduction ==
Thermal Expansion is a very common phenomenon. It is common knowledge that within a certain range of temperatures, when the temperature goes up some materials expand. some materials expand Some materials expand when the temperature rises. This is due to an increase in atom vibrations resulting in a greater separation between the atoms on a microscopic scale, and on a macroscopic scale leads to an expansion of the materials.<ref name=":0">Paul A., Tipler; Gene Mosca (2008). ''Physics for Scientists and Engineers, Volume 1'' (6th ed.). New York, NY: Worth Publishers. pp. 666–670. ISBN 1-4292-0132-0.</ref> macroscopically makes a material to expand.[1] Although there are some materials which behave the opposite exactly the opposite, these specific cases will not be discussed here, certain cases would not be discussed here.

phenomenon. In this experiment, a simple MgO compound is studied to examine this expansive behaviour.
{| class="wikitable"
! colspan="2" |Table 1: Representations of MgO unit cells (FCC)
|-
|[[File:MgOpri.png|none|thumb|296x296px]]Primitive cell
|[[File:MgO conventional.jpg|none|thumb|294x294px]]Conventional cell
|}

The vibrational state is simulated using a computer in order to model the thermal expansion of MgO. The thermal expansion coefficient α is given by Equation (1)<ref name=":0" />:
[[File:Thermal exp coef.png|thumb|Equation (1) : Thermal Expansion Coefficient
|none]]

Two methods were used to calculate the thermal expansion coefficient, the Quasi-Harmonic approximation (LD) and molecular dynamics (MD). The former method is used to compute the vibrational energy levels of MgO. This method allows us to understand the phonon dispersion and the vibrational density of state (DOS). The latter method is used to simulate the vibrations as random motions of atoms inside a cell. It tells us how atoms respond over time under certain conditions. In this experiment, a General Utility Lattice Program (GULP) on the DLVisualize platform (Linux based) is used and by increasing the temperature, we can see how MgO atoms in a cell respond, what physical properties can be ascribed to this response and how it can be quantified.

== Results and Discussion ==

=== Phonon dispersion and Density of States ===
The phonon dispersion graph of MgO was produced by simulation to study the number of vibrational states which exist at k-points.
[[File:MgOPD.png|thumb|425x425px|Figure 1: Phonon Dispersion graph of MgO|centre]]Followed by the local phonon DOS with different grid sizes.
{| class="wikitable"
! colspan="4" |Table 2: MgO '''phonon DOS with variation of grid sizes'''
|-
|1x1x1
|[[File:DOS 111.JPG|thumb]]
|2x2x2
|[[File:DOS 222.JPG|thumb]]
|-
|3x3x3
|[[File:DOS 333.JPG|thumb]]
|16x16x16
|[[File:DOS 161616.JPG|thumb]]
|-
|20x20x20
|[[File:DOS 202020.JPG|thumb]]
|25x25x25
|[[File:DOS 252525.JPG|thumb]]
|}

From the 1x1x1 grid four frequencies match the symmetry point L which indicates that the k-point is equal to (1/2, 1/2, 1/2). In the 1x1x1 grid DOS graph, the two lower frequency numbers are twice as large as the two higher frequency numbers. This matches the fact that in the dispersion curve at the symmetry point L the two lower frequency numbers are doubly degenerated. The DOS resembles the dispersion curves. It contains the information at certain frequency numbers with respected to a symmetry point.

By comparing different grid sizes' dispersion curves, we can conclude that when the grid size increases, the DOS increases because more points in k-space are taken and used to calculate the DOS. The larger the grid size, the finer, smoother the DOS curve becomes as there are more atoms being examined more details are given. Since the 20x20x20 graph is extremely similar to the 25x25x25 graph, for a reasonable approximation to the DOS, the 20x20x20 grid size is a sufficiently accurate approximation.

The optimal grid size 20x20x20 can be used for other crystals such as CaO. CaO unit cells have a similar size as that of MgO hence the same grid can be used, whilist for Faujasite or lithium metal, a size change is required.
[[File:A*=2pi-a.png|thumb|Equation 2: Periodic direction in the reciprocal space, where '''a''' is the periodic direction in the direct space]]
Based on Equation 2, the periodic direction in the reciprocal space is inversely proportional to the periodic direction in the direct space. Hence for a larger cell such as Faujasite a smaller grid can be used while for a smaller cell such as lithium metal a larger grid is needed.

{| class="wikitable"
|+Table 3: Helmholtz free energy change with grid size
!Grid size
!Free Energy/ eV
!ΔA/ meV (with respected to 25x25x25)
|-
|1x1x1
|<nowiki>-40.930301</nowiki>
|<nowiki>-3.188</nowiki>
|-
|2x2x2
|<nowiki>-40.926609</nowiki>
|<nowiki>-0.126</nowiki>
|-
|3x3x3
|<nowiki>-40.926432</nowiki>
|0.051
|-
|4x4x4
|<nowiki>-40.926450</nowiki>
|0.033
|-
|8x8x8
|<nowiki>-40.926478</nowiki>
|0.005
|-
|10x10x10
|<nowiki>-40.926480</nowiki>
|0.003
|-
|15x15x15
|<nowiki>-40.926482</nowiki>
|0.001
|-
|20x20x20
|<nowiki>-40.926483</nowiki>
|0.000
|-
|25x25x25
|<nowiki>-40.926483</nowiki>
|0.000
|}

The Helmholtz free energy varies with the grid size. As the size increases, the free energy increases and it reaches a maximum at 3x3x3 grid and then decreases.

In this table the energy difference is calculated between each grid size and the most precise grid size (in this case which is the 25x25x25 grid). For a calculation accurate to 1 meV, 1x1x1 could be used. For an acuuracy of 0.5 meV, 2x2x2 could be used and for an acuuracy of 0.1 meV, 3x3x3 could be used.

=== Quasi-Harmonic Approximation (LD) ===
{| class="wikitable"
|+
Table 4: Effects of temperature on the Helmholtz free energy, the lattice constant and the cell volume.
!Temperature/ K
!Helmholtz free energy/ eV
!Lattice constant/ Å
!Cell Volume/ Å³

|-
|0
|<nowiki>-40.900978</nowiki>
|2.986563
|18.836496

|-
|100
|<nowiki>-40.902420</nowiki>
|2.986563
|18.836494

|-
|200
|<nowiki>-40.909377</nowiki>
|2.987605
|18.856202

|-
|300
|<nowiki>-40.928125</nowiki>
|2.989390
|18.890025

|-
|400
|<nowiki>-40.958594</nowiki>
|2.991630
|18.932508

|-
|500
|<nowiki>-40.999436</nowiki>
|2.994135
|18.980112

|-
|600
|<nowiki>-41.049315</nowiki>
|2.996821
|19.031223

|-
|700
|<nowiki>-41.1071190</nowiki>
|2.999644
|19.085059

|-
|800
|<nowiki>-41.171892</nowiki>
|3.002589
|19.141318

|-
|900
|<nowiki>-41.243018</nowiki>
|3.005636
|19.199640

|-
|1000
|<nowiki>-41.319848</nowiki>
|3.008785
|19.260044
|}
Quasi-Harmonic approximation suggests that the Morse potential can be used to represent the potential between atoms in a unit cell. This model is idealized by considering a perfect crystal. Each atom is an individual harmonic oscillator that does not interact with each other and that is being treated independently. Taken into account of the anharmonicity the vibrational energy can be found.<ref name=":1">Martin T. Dove, ''Introduction to Lattice Dynamics, ''University of Cambridge Press, ISBN-13: 9780521392938.
</ref>

The relationship between the Helmholtz free energy and temperature is given by Equation 3<ref name=":1" />:
[[File:A=U+TS.png|thumb|284x284px|Equation 3. Where ''U'' is the internal energy, ''EZP'' is the vibrational zero-point energy, ''T'' is the absolute temperature, ''V'' is the volume and ''S'' is the entropy due to the vibrational degrees of freedom.|none]]
[[File:(LD) A with T.png|thumb|449x449px|Figure 2|none]]

The Helmholtz free energy becomes more negative, i.e. decreases, when the temperature increases. It is not hard to see that the result matches Equation 3. As T increases, the entropy term becomes more positive, hence the free energy decreases.

[[File:(LD) LC with T.png|thumb|487x487px|Figure 3|none]][[File:(LD) V with T.png|thumb|492x492px|Figure 4|left]]

As expected, the lattice constant increases with temperature, as does the unit cell volume. At higher temperatures, atoms have a higher internal energy and hence the vibrational frequency increases. This causes a greater deviation from the static separation and therefore the material expands. However, in a diatomic molecule with an exactly harmonic potential the bond length does not increase with temperature as the time average interatomic separation remains constant.

Initially all three curves have a parabolic character, while at a higher temperatures they tend towards a more linear character. From Equation 1 and the gradient calculated from Figure 4 the thermal expansion coefficient is found to be 2.687x10-5 K-1. The literature value is 3.12x10-5 K-1 (at 300 K)<ref>O. L. Anderson and K. Zou, J. ''Thermodynamic Functions and Properties of MgO at High Compression and High Temperarture.'' Phys. Chem. Ref. Data, 1990, '''19''', 69.</ref>. The difference is 13% which is considerably large. Nevertheless it is good enough coming from an approximation.

As the temperature of MgO approaches the melting point, the phonon modes does not represent the actual motions of the ions well, simply because at the melting point the bond breaks and it changes the crystal structure which it is no longer perfect. This violates the approximation and hence the modes are not be able to represent the motions well. [[File:(MD) V with T.png|thumb|493x493px|Figure 5
]]
=== Molecular Dynamics (MD) ===
In this molecular dynamics method, a supercell containing 32 MgO units is used.

By doing a geometry optimization, the atomic position with a minimum energy is found and used to measure the lattice constant and the cell volume. Since the method is based on solving Newton's equations, the system is being treated classically<ref>Alder, B. J.; Wainwright, T. E. (1959). "Studies in Molecular Dynamics. I. General Method". ''J. Chem. Phys''. '''31''' (2): 459. Bibcode:1959JChPh..31..459A. doi:10.1063/1.1730376.</ref>. A larger system is used to compromise the loss of its periodicity due to the nature of random atom movements.

Again using Equation 1 the thermal expansion coefficient is found to be 3.086x10-5 K-1. The difference is 1.1% which is considerably smaller compared to the quasi-harmonic model.

Figure 6 shows a comparison between the results of using the two different methods. Both curves have a very similar shape while the MD method gives a smaller value at the same temperature (within 0-1000 K). It is worth mentioning that with increasing temperatures the two curves are getting closer and closer. [[File:LD and MD Comparison.png|thumb|493x493px|Figure 6]]The foundation of the Quasi-Harmonic approximation is that the system is being treated quantum mechanically<ref name=":1" />, whilst in the Molecular dynamic method the system is being treated classically. Due to this reason at very high temperatures (i.e. close to or beyond the melting point of the substance) the Quasi-Harmonic approximation no longer applies as the periodicity is being removed from the system due to the bond dissociation. On the contrary, Molecular dynamics applies at high temperatures as the method only considers the velocity of each individual atom. However, as can be seen in Figure 5, the molecular dynamic method was not able to give a value at 0 K simply because at 0 K atoms do not move.
== Conclusion ==
The thermal expansion coefficient of MgO was found by two using different methods. The Quasi-Harmonic approxiamtion and the molecular dynamics. By optimising the grid size of the MgO crystal a fairly accurate size 20x20x20 was used and the MgO density of states were found. This optimal grid size was then used in the Quasi-Harmonic approximation for finding the relationships between the lattice constant, the volume and the Helmholtz free energy with temperature. From the relationship between the volume and the temperature the thermal expansion coefficient of MgO was calculated. Unlike this method, the molecular dynamic approach does not rely on finding the optimal grid size of MgO. Based on the calculations the molecular dynamic method shows a greater agreement with the literature value.

Either of

== References ==

Rep:Mod:ZLY9512

2017-02-09T20:34:15Z

Lz2914: /* Conclusion */

= ''Thermal expansion of MgO'' =

== Introduction ==
Thermal Expansion is a very common phenomenon in real life. It is a common knowledge that within a certain range of temperature, when the temperature goes up some materials expand. This is due to an increment of vibrations of atoms which results a greater separation between atoms microscopically and hence macroscopically makes a material to expand.<ref name=":0">Paul A., Tipler; Gene Mosca (2008). ''Physics for Scientists and Engineers, Volume 1'' (6th ed.). New York, NY: Worth Publishers. pp. 666–670. ISBN 1-4292-0132-0.</ref> Although there are some materials which behave the opposite, certain cases would not be discussed here.

In this experiment, a simple compound MgO is used to study this phenomenon.
{| class="wikitable"
! colspan="2" |Table 1: Representations of MgO unit cells (FCC)
|-
|[[File:MgOpri.png|none|thumb|296x296px]]Primitive cell
|[[File:MgO conventional.jpg|none|thumb|294x294px]]Conventional cell
|}

The vibrational state is stimulated by computer in order to model the thermal expansion of MgO. The thermal expansion coefficient α is found by Equation (1)<ref name=":0" />:
[[File:Thermal exp coef.png|thumb|Equation (1) : Thermal Expansion Coefficient
|none]]

Two methods were used to calculate the thermal expansion coefficient, the quasi-harmonic approximation (LD) and molecular dynamics (MD). The former method is used to compute the vibrational energy levels of MgO. This method allows us to understand the phonon dispersion and the vibrational density of state (DOS). The latter method is used to simulate the vibrations as random motions of atoms inside a cell. It tells us how atoms respond over time under a certain condition. In this experiment, General Utility Lattice Program (GULP) on the DLVisualize platform (Linux based) is used and by increasing the temperature, we can see how MgO atoms in a cell respond, what physical property does this respond give and how we can quantify it.

== Results and Discussion ==

=== Phonon dispersion and Density of States ===
First of all, the phonon dispersion graph of MgO was produced by simulation to study the number of vibrational states which exist at k-points.
[[File:MgOPD.png|thumb|425x425px|Figure 1: Phonon Dispersion graph of MgO|centre]]Followed by the local phonon DOS with different grid sizes.
{| class="wikitable"
! colspan="4" |Table 2: MgO '''phonon DOS with variation of grid sizes'''
|-
|1x1x1
|[[File:DOS 111.JPG|thumb]]
|2x2x2
|[[File:DOS 222.JPG|thumb]]
|-
|3x3x3
|[[File:DOS 333.JPG|thumb]]
|16x16x16
|[[File:DOS 161616.JPG|thumb]]
|-
|20x20x20
|[[File:DOS 202020.JPG|thumb]]
|25x25x25
|[[File:DOS 252525.JPG|thumb]]
|}

From the 1x1x1 grid the four frequencies match the symmetry point L which indicates that the k-point is equal to (1/2, 1/2, 1/2). In the 1x1x1 grid DOS graph, the two lower frequency numbers are twice as large as the two higher frequency numbers. This matches the fact that in the dispersion curve at symmetry point L the two lower frequency numbers are doubly degenerated. The DOS resembles the dispersion curves. It contains the information at certain frequency numbers with respected to a symmetry point.

By comparing different grid sizes' dispersion curves, we can conclude that when the grid size increases, the DOS increases because there are more points in k-space are taken and used to calculate the DOS. The larger the grid size, the finer, smoother the DOS curve becomes as there are more atoms being examined more details are given. Since the 20x20x20 graph is extremely similar to the 25x25x25 graph, for a reasonable approximation to the DOS, 20x20x20 is accurate enough for a good approximation.

The optimal grid size 20x20x20 can be used for other crystal such as CaO. CaO unit cell has a similar size as that of MgO hence the same grid can be used. While for Faujasite or lithium metal, a size change is needed.
[[File:A*=2pi-a.png|thumb|Equation 2: Periodic direction in the reciprocal space, where '''a''' is the periodic direction in the direct space]]
Based on Equation 2, the periodic direction in the reciprocal space is inversely proportional to the periodic direction in the direct space. Hence for a lager cell such as Faujasite a smaller grid can be used while for a smaller cell such as lithium metal a larger grid is needed.

{| class="wikitable"
|+Table 3: Helmholtz free energy change with grid size
!Grid size
!Free Energy/ eV
!ΔA/ meV (with respected to 25x25x25)
|-
|1x1x1
|<nowiki>-40.930301</nowiki>
|<nowiki>-3.188</nowiki>
|-
|2x2x2
|<nowiki>-40.926609</nowiki>
|<nowiki>-0.126</nowiki>
|-
|3x3x3
|<nowiki>-40.926432</nowiki>
|0.051
|-
|4x4x4
|<nowiki>-40.926450</nowiki>
|0.033
|-
|8x8x8
|<nowiki>-40.926478</nowiki>
|0.005
|-
|10x10x10
|<nowiki>-40.926480</nowiki>
|0.003
|-
|15x15x15
|<nowiki>-40.926482</nowiki>
|0.001
|-
|20x20x20
|<nowiki>-40.926483</nowiki>
|0.000
|-
|25x25x25
|<nowiki>-40.926483</nowiki>
|0.000
|}

The Helmholtz free energy varies with the grid size. As the size increases, the free energy increases and it reaches a maximum at 3x3x3 grid and then decreases.

In this table the energy difference is also calculated between each grid size and the most precise grid size. In this case which is the 25x25x25 grid. For a calculation accurate to 1 meV, 1x1x1 could be used. To 0.5 meV, 2x2x2 could be used. To 0.1 meV, 3x3x3 could be used.

=== Quasi-Harmonic Approximation (LD) ===
{| class="wikitable"
|+
Table 4: Effects of temperature on the Helmholtz free energy, the lattice constant and the cell volume.
!Temperature/ K
!Helmholtz free energy/ eV
!Lattice constant/ Å
!Cell Volume/ Å³

|-
|0
|<nowiki>-40.900978</nowiki>
|2.986563
|18.836496

|-
|100
|<nowiki>-40.902420</nowiki>
|2.986563
|18.836494

|-
|200
|<nowiki>-40.909377</nowiki>
|2.987605
|18.856202

|-
|300
|<nowiki>-40.928125</nowiki>
|2.989390
|18.890025

|-
|400
|<nowiki>-40.958594</nowiki>
|2.991630
|18.932508

|-
|500
|<nowiki>-40.999436</nowiki>
|2.994135
|18.980112

|-
|600
|<nowiki>-41.049315</nowiki>
|2.996821
|19.031223

|-
|700
|<nowiki>-41.1071190</nowiki>
|2.999644
|19.085059

|-
|800
|<nowiki>-41.171892</nowiki>
|3.002589
|19.141318

|-
|900
|<nowiki>-41.243018</nowiki>
|3.005636
|19.199640

|-
|1000
|<nowiki>-41.319848</nowiki>
|3.008785
|19.260044
|}
Quasi-Harmonic approximation suggests that the Morse potential can be used to represent the potential between atoms in a unit cell. This model is idealized by considering a perfect crystal. Each atom is an individual harmonic oscillators and they do not interact with each other and they are being treated independently. Taken into account of the anharmonicity the vibrational energy can be found.<ref name=":1">Martin T. Dove, ''Introduction to Lattice Dynamics, ''University of Cambridge Press, ISBN-13: 9780521392938.
</ref>

The Relationship between the Helmholtz free energy with temperature is shown in Equation 3<ref name=":1" />:
[[File:A=U+TS.png|thumb|284x284px|Equation 3. Where ''U'' is the internal energy, ''EZP'' is the vibrational zero-point energy, ''T'' is the absolute temperature, ''V'' is the volume and ''S'' is the entropy due to the vibrational degrees of freedom.|none]]
[[File:(LD) A with T.png|thumb|449x449px|Figure 2|none]]

The Helmholtz free energy becomes more negative, in other words decreases when temperature increases. It is not hard to see that the result matches Equation 3. As T increases, the entropy term becomes more positive, hence the free energy decreases.

[[File:(LD) LC with T.png|thumb|487x487px|Figure 3|none]][[File:(LD) V with T.png|thumb|492x492px|Figure 4|left]]

As expected, the lattice constant increases with temperature. So does the unit cell volume. At higher temperature, atoms have a higher internal energy hence the vibrational frequency increases, which causes a greater deviation from the static separation and therefore the material expands. However, in a diatomic molecule with an exactly harmonic potential the bond length would not increase with temperature as the time average interatomic separation remains constant.

Initially all three curves have a parabolic character, while at a higher temperature they towards more to a linearity. From Equation 1 and the gradient calculated from Figure 4 the thermal expansion coefficient is found to be 2.687x10-5 K-1. The literature value is 3.12x10-5 K-1 (at 300 K)<ref>O. L. Anderson and K. Zou, J. ''Thermodynamic Functions and Properties of MgO at High Compression and High Temperarture.'' Phys. Chem. Ref. Data, 1990, '''19''', 69.</ref>. The difference is 13% which is considerably large, nevertheless it is good enough coming from an approximation.

As the temperature approaches the melting point of MgO, the phonon modes would not represent the actual motions of the ions well. Simply because at the melting point the bond breaks and it changes the crystal structure which it is no longer perfect. This violates the approximation and hence the modes would be able to represent the motions well. [[File:(MD) V with T.png|thumb|493x493px|Figure 5
]]
=== Molecular Dynamics (MD) ===
In this molecular dynamics method, a supercell containing 32 MgO units is used.

By doing a geometry optimization, the atomic position with a minimum energy is found and used to measure the the lattice constant and the cell volume. Since the method is based on solving the Newton's equations, the system is being treated classically<ref>Alder, B. J.; Wainwright, T. E. (1959). "Studies in Molecular Dynamics. I. General Method". ''J. Chem. Phys''. '''31''' (2): 459. Bibcode:1959JChPh..31..459A. doi:10.1063/1.1730376.</ref>. A larger system is used to compromise the lost of its periodicity due to the nature of random atom movements.

Again apply Equation 1 the thermal expansion coefficient is found to be 3.086x10-5 K-1. The difference is 1.1% which is considerably smaller compared to the quasi-harmonic model.

Figure 6 shows a comparison between the results of using two methods. Both curves have the very similar shape while the MD method gives a smaller values at the same temperature (within 0-1000 K at least can be seen). It is worth mentioning that with increasing temperatures two curves are getting closer and closer. [[File:LD and MD Comparison.png|thumb|493x493px|Figure 6]]The foundation of the Quasi-Harmonic approximation is that the system is being treated quantum mechanically<ref name=":1" />. Whilst in the Molecular dynamic method the system is being treated classically. Due to this reason at a very high temperature (i.e. close to or beyond the melting point of the substance) the Quasi-Harmonic approximation no longer applies as the periodicity is being removed from the system due to the bond dissociation. On the contrary, Molecular dynamics applies at high temperatures as the method only considers the velocity of each individual atom. However, as can be seen on Figure 5, the molecular dynamic method was not able to give a datum at 0 K simply because at 0 K atoms do not move.
== Conclusion ==
The thermal expansion coefficient of MgO was found by two using different methods. The Quasi-Harmonic approxiamtion and the molecular dynamics. By optimising the grid size of the MgO crystal a fairly accurate size 20x20x20 was used and the MgO density of states were found. This optimal grid size was then used in the Quasi-Harmonic approximation for finding the relationships between the lattice constant, the volume and the Helmholtz free energy with temperature. From the relationship between the volume and the temperature the thermal expansion coefficient of MgO was calculated. Unlike this method, the molecular dynamic approach does not rely on finding the optimal grid size of MgO. Based on the calculations the molecular dynamic method shows a greater agreement with the literature value.

Either of

== References ==

Rep:Mod:ZLY9512

2017-02-09T19:44:00Z

Lz2914: /* An opportunity to speculate */

= ''Thermal expansion of MgO'' =

== Introduction ==
Thermal Expansion is a very common phenomenon in real life. It is a common knowledge that within a certain range of temperature, when the temperature goes up some materials expand. This is due to an increment of vibrations of atoms which results a greater separation between atoms microscopically and hence macroscopically makes a material to expand. Although there are some materials which behave the opposite, certain cases would not be discussed here.

In this experiment, a simple compound MgO is used to study this phenomenon.
{| class="wikitable"
! colspan="2" |Table 1: Representations of MgO unit cells (FCC)
|-
|[[File:MgOpri.png|none|thumb|296x296px]]Primitive cell
|[[File:MgO conventional.jpg|none|thumb|294x294px]]Conventional cell
|}

The vibrational state is stimulated by computer in order to model the thermal expansion of MgO. The thermal expansion coefficient α is found by Equation (1):
[[File:Thermal exp coef.png|thumb|Equation (1) : Thermal Expansion Coefficient
|none]]

Two methods were used to calculate the thermal expansion coefficient, the quasi-harmonic approximation (LD) and molecular dynamics (MD). The former method is used to compute the vibrational energy levels of MgO. This method allows us to understand the phonon dispersion and the vibrational density of state (DOS). The latter method is used to simulate the vibrations as random motions of atoms inside a cell. It tells us how atoms respond over time under a certain condition. In this experiment, General Utility Lattice Program (GULP) on the DLVisualize platform (Linux based) is used and by increasing the temperature, we can see how MgO atoms in a cell respond, what physical property does this respond give and how we can quantify it.

== Results and Discussion ==

=== Phonon dispersion and Density of States ===
First of all, the phonon dispersion graph of MgO was produced by simulation to study the number of vibrational states which exist at k-points.
[[File:MgOPD.png|thumb|425x425px|Figure 1: Phonon Dispersion graph of MgO|centre]]Followed by the local phonon DOS with different grid sizes.
{| class="wikitable"
! colspan="4" |Table 2: MgO '''phonon DOS with variation of grid sizes'''
|-
|1x1x1
|[[File:DOS 111.JPG|thumb]]
|2x2x2
|[[File:DOS 222.JPG|thumb]]
|-
|3x3x3
|[[File:DOS 333.JPG|thumb]]
|16x16x16
|[[File:DOS 161616.JPG|thumb]]
|-
|20x20x20
|[[File:DOS 202020.JPG|thumb]]
|25x25x25
|[[File:DOS 252525.JPG|thumb]]
|}

From the 1x1x1 grid the four frequencies match the symmetry point L which indicates that the k-point is equal to (1/2, 1/2, 1/2). In the 1x1x1 grid DOS graph, the two lower frequency numbers are twice as large as the two higher frequency numbers. This matches the fact that in the dispersion curve at symmetry point L the two lower frequency numbers are doubly degenerated. The DOS resembles the dispersion curves. It contains the information at certain frequency numbers with respected to a symmetry point.

By comparing different grid sizes' dispersion curves, we can conclude that when the grid size increases, the DOS increases because there are more points in k-space are taken and used to calculate the DOS. The larger the grid size, the finer, smoother the DOS curve becomes as there are more atoms being examined more details are given. Since the 20x20x20 graph is extremely similar to the 25x25x25 graph, for a reasonable approximation to the DOS, 20x20x20 is accurate enough for a good approximation.

The optimal grid size 20x20x20 can be used for other crystal such as CaO. CaO unit cell has a similar size as that of MgO hence the same grid can be used. While for Faujasite or lithium metal, a size change is needed.
[[File:A*=2pi-a.png|thumb|Equation 2: Periodic direction in the reciprocal space, where '''a''' is the periodic direction in the direct space]]
Based on Equation 2, the periodic direction in the reciprocal space is inversely proportional to the periodic direction in the direct space. Hence for a lager cell such as Faujasite a smaller grid can be used while for a smaller cell such as lithium metal a larger grid is needed.

{| class="wikitable"
|+Table 3: Helmholtz free energy change with grid size
!Grid size
!Free Energy/ eV
!ΔA/ meV (with respected to 25x25x25)
|-
|1x1x1
|<nowiki>-40.930301</nowiki>
|<nowiki>-3.188</nowiki>
|-
|2x2x2
|<nowiki>-40.926609</nowiki>
|<nowiki>-0.126</nowiki>
|-
|3x3x3
|<nowiki>-40.926432</nowiki>
|0.051
|-
|4x4x4
|<nowiki>-40.926450</nowiki>
|0.033
|-
|8x8x8
|<nowiki>-40.926478</nowiki>
|0.005
|-
|10x10x10
|<nowiki>-40.926480</nowiki>
|0.003
|-
|15x15x15
|<nowiki>-40.926482</nowiki>
|0.001
|-
|20x20x20
|<nowiki>-40.926483</nowiki>
|0.000
|-
|25x25x25
|<nowiki>-40.926483</nowiki>
|0.000
|}

The Helmholtz free energy varies with the grid size. As the size increases, the free energy increases and it reaches a maximum at 3x3x3 grid and then decreases.

In this table the energy difference is also calculated between each grid size and the most precise grid size. In this case which is the 25x25x25 grid. For a calculation accurate to 1 meV, 1x1x1 could be used. To 0.5 meV, 2x2x2 could be used. To 0.1 meV, 3x3x3 could be used.

=== Quasi-Harmonic Approximation (LD) ===
{| class="wikitable"
|+
Table 4: Effects of temperature on the Helmholtz free energy, the lattice constant and the cell volume.
!Temperature/ K
!Helmholtz free energy/ eV
!Lattice constant/ Å
!Cell Volume/ Å³

|-
|0
|<nowiki>-40.900978</nowiki>
|2.986563
|18.836496

|-
|100
|<nowiki>-40.902420</nowiki>
|2.986563
|18.836494

|-
|200
|<nowiki>-40.909377</nowiki>
|2.987605
|18.856202

|-
|300
|<nowiki>-40.928125</nowiki>
|2.989390
|18.890025

|-
|400
|<nowiki>-40.958594</nowiki>
|2.991630
|18.932508

|-
|500
|<nowiki>-40.999436</nowiki>
|2.994135
|18.980112

|-
|600
|<nowiki>-41.049315</nowiki>
|2.996821
|19.031223

|-
|700
|<nowiki>-41.1071190</nowiki>
|2.999644
|19.085059

|-
|800
|<nowiki>-41.171892</nowiki>
|3.002589
|19.141318

|-
|900
|<nowiki>-41.243018</nowiki>
|3.005636
|19.199640

|-
|1000
|<nowiki>-41.319848</nowiki>
|3.008785
|19.260044

|-
|1500
|<nowiki>-41.774892</nowiki>
|3.026138
|19.595184

|-
|2000
|<nowiki>-42.323671</nowiki>
|3.046987
|20.002966

|}
Quasi-Harmonic approximation suggests that the Morse potential can be used to represent the potential between atoms in a unit cell. This model is idealized by considering a perfect crystal. Each atom is an individual harmonic oscillators and they do not interact with each other and they are being treated independently. Taken into account of the anharmonicity the vibrational energy can be found.

The Relationship between the Helmholtz free energy with temperature is shown in Equation 3:
[[File:A=U+TS.png|thumb|284x284px|Equation 3. Where ''U'' is the internal energy, ''EZP'' is the vibrational zero-point energy, ''T'' is the absolute temperature, ''V'' is the volume and ''S'' is the entropy due to the vibrational degrees of freedom.|none]]
[[File:(LD) A with T.png|thumb|449x449px|Figure 2|none]]

The Helmholtz free energy becomes more negative, in other words decreases when temperature increases. It is not hard to see that the result matches Equation 3. As T increases, the entropy term becomes more positive, hence the free energy decreases.

[[File:(LD) LC with T.png|thumb|487x487px|Figure 3|none]][[File:(LD) V with T.png|thumb|492x492px|Figure 4|left]]

As expected, the lattice constant increases with temperature. So does the unit cell volume. At higher temperature, atoms have a higher internal energy hence the vibrational frequency increases, which causes a greater deviation from the static separation and therefore the material expands. However, in a diatomic molecule with an exactly harmonic potential the bond length would not increase with temperature as the time average interatomic separation remains constant.

Initially all three curves have a parabolic character, while at a higher temperature they towards more to a linearity. From Equation 1 and the gradient calculated from Figure 4 the thermal expansion coefficient is found to be 2.687x10-5 K-1. The literature value is 3.12x10-5 K-1 (at 300 K)<ref>O. L. Anderson and K. Zou, J. ''Thermodynamic Functions and Properties of MgO at High Compression and High Temperarture.'' Phys. Chem. Ref. Data, 1990, '''19''', 69.</ref>. The difference is 13% which is considerably large, nevertheless it is good enough coming from an approximation.

As the temperature approaches the melting point of MgO, the phonon modes would not represent the actual motions of the ions well. Simply because at the melting point the bond breaks and it changes the crystal structure which it is no longer perfect. This violates the approximation and hence the modes would be able to represent the motions well.

=== Molecular Dynamics (MD) ===
[[File:(MD) V with T.png|thumb|493x493px|Figure 5
]]
In this molecular dynamics method, a supercell containing 32 MgO units is used.

By doing a geometry optimization, the atomic position with a minimum energy is found and used to measure the the lattice constant and the cell volume. Since the method is based on solving the Newton's equations, the system is being treated classically. A larger system is used to compromise the lost of its periodicity due to the nature of random atom movements.

Again apply Equation 1 the thermal expansion coefficient is found to be 3.086x10-5 K-1. The difference is 1.1% which is considerably smaller compared to the quasi-harmonic model.

Figure 6 shows a comparison between the results of using two methods. Both curves have the very similar shape while the MD method gives a smaller values at the same temperature (within 0-1000 K at least can be seen). It is worth mentioning that with increasing temperatures two curves are getting closer and closer. [[File:LD and MD Comparison.png|thumb|493x493px|Figure 6]]The foundation of the Quasi-Harmonic approximation is that the system is being treated quantum mechanically. Whilst in the Molecular dynamic method the system is being treated classically. Due to this reason at a very high temperature (i.e. close to or beyond the melting point of the substance) the Quasi-Harmonic approximation no longer applies as the periodicity is being removed from the system due to the bond dissociation. On the contrary, Molecular dynamics applies at high temperatures as the method only considers the velocity of each individual atom.
== Conclusion ==
AAAAAAAA

== References ==

Rep:Mod:ZLY9512

2017-02-09T03:53:14Z

Lz2914: /* Molecular Dynamics (MD) */

= ''Thermal expansion of MgO'' =

== Introduction ==
Thermal Expansion is a very common phenomenon in real life. It is a common knowledge that within a certain range of temperature, when the temperature goes up some materials expand. This is due to an increment of vibrations of atoms which results a greater separation between atoms microscopically and hence macroscopically makes a material to expand. Although there are some materials which behave the opposite, certain cases would not be discussed here.

In this experiment, a simple compound MgO is used to study this phenomenon.
{| class="wikitable"
! colspan="2" |Table 1: Representations of MgO unit cells (FCC)
|-
|[[File:MgOpri.png|none|thumb|296x296px]]Primitive cell
|[[File:MgO conventional.jpg|none|thumb|294x294px]]Conventional cell
|}

The vibrational state is stimulated by computer in order to model the thermal expansion of MgO. The thermal expansion coefficient α is found by Equation (1):
[[File:Thermal exp coef.png|thumb|Equation (1) : Thermal Expansion Coefficient
|none]]

Two methods were used to calculate the thermal expansion coefficient, the quasi-harmonic approximation (LD) and molecular dynamics (MD). The former method is used to compute the vibrational energy levels of MgO. This method allows us to understand the phonon dispersion and the vibrational density of state (DOS). The latter method is used to simulate the vibrations as random motions of atoms inside a cell. It tells us how atoms respond over time under a certain condition. In this experiment, General Utility Lattice Program (GULP) on the DLVisualize platform (Linux based) is used and by increasing the temperature, we can see how MgO atoms in a cell respond, what physical property does this respond give and how we can quantify it.

== Results and Discussion ==

=== Phonon dispersion and Density of States ===
First of all, the phonon dispersion graph of MgO was produced by simulation to study the number of vibrational states which exist at k-points.
[[File:MgOPD.png|thumb|425x425px|Figure 1: Phonon Dispersion graph of MgO|centre]]Followed by the local phonon DOS with different grid sizes.
{| class="wikitable"
! colspan="4" |Table 2: MgO '''phonon DOS with variation of grid sizes'''
|-
|1x1x1
|[[File:DOS 111.JPG|thumb]]
|2x2x2
|[[File:DOS 222.JPG|thumb]]
|-
|3x3x3
|[[File:DOS 333.JPG|thumb]]
|16x16x16
|[[File:DOS 161616.JPG|thumb]]
|-
|20x20x20
|[[File:DOS 202020.JPG|thumb]]
|25x25x25
|[[File:DOS 252525.JPG|thumb]]
|}

From the 1x1x1 grid the four frequencies match the symmetry point L which indicates that the k-point is equal to (1/2, 1/2, 1/2). In the 1x1x1 grid DOS graph, the two lower frequency numbers are twice as large as the two higher frequency numbers. This matches the fact that in the dispersion curve at symmetry point L the two lower frequency numbers are doubly degenerated. The DOS resembles the dispersion curves. It contains the information at certain frequency numbers with respected to a symmetry point.

By comparing different grid sizes' dispersion curves, we can conclude that when the grid size increases, the DOS increases because there are more points in k-space are taken and used to calculate the DOS. The larger the grid size, the finer, smoother the DOS curve becomes as there are more atoms being examined more details are given. Since the 20x20x20 graph is extremely similar to the 25x25x25 graph, for a reasonable approximation to the DOS, 20x20x20 is accurate enough for a good approximation.

The optimal grid size 20x20x20 can be used for other crystal such as CaO. CaO unit cell has a similar size as that of MgO hence the same grid can be used. While for Faujasite or lithium metal, a size change is needed.
[[File:A*=2pi-a.png|thumb|Equation 2: Periodic direction in the reciprocal space, where '''a''' is the periodic direction in the direct space]]
Based on Equation 2, the periodic direction in the reciprocal space is inversely proportional to the periodic direction in the direct space. Hence for a lager cell such as Faujasite a smaller grid can be used while for a smaller cell such as lithium metal a larger grid is needed.

{| class="wikitable"
|+Table 3: Helmholtz free energy change with grid size
!Grid size
!Free Energy/ eV
!ΔA/ meV (with respected to 25x25x25)
|-
|1x1x1
|<nowiki>-40.930301</nowiki>
|<nowiki>-3.188</nowiki>
|-
|2x2x2
|<nowiki>-40.926609</nowiki>
|<nowiki>-0.126</nowiki>
|-
|3x3x3
|<nowiki>-40.926432</nowiki>
|0.051
|-
|4x4x4
|<nowiki>-40.926450</nowiki>
|0.033
|-
|8x8x8
|<nowiki>-40.926478</nowiki>
|0.005
|-
|10x10x10
|<nowiki>-40.926480</nowiki>
|0.003
|-
|15x15x15
|<nowiki>-40.926482</nowiki>
|0.001
|-
|20x20x20
|<nowiki>-40.926483</nowiki>
|0.000
|-
|25x25x25
|<nowiki>-40.926483</nowiki>
|0.000
|}

The Helmholtz free energy varies with the grid size. As the size increases, the free energy increases and it reaches a maximum at 3x3x3 grid and then decreases.

In this table the energy difference is also calculated between each grid size and the most precise grid size. In this case which is the 25x25x25 grid. For a calculation accurate to 1 meV, 1x1x1 could be used. To 0.5 meV, 2x2x2 could be used. To 0.1 meV, 3x3x3 could be used.

=== Quasi-Harmonic Approximation (LD) ===
{| class="wikitable"
|+
Table 4: Effects of temperature on the Helmholtz free energy, the lattice constant and the cell volume.
!Temperature/ K
!Helmholtz free energy/ eV
!Lattice constant/ Å
!Cell Volume/ Å³

|-
|0
|<nowiki>-40.900978</nowiki>
|2.986563
|18.836496

|-
|100
|<nowiki>-40.902420</nowiki>
|2.986563
|18.836494

|-
|200
|<nowiki>-40.909377</nowiki>
|2.987605
|18.856202

|-
|300
|<nowiki>-40.928125</nowiki>
|2.989390
|18.890025

|-
|400
|<nowiki>-40.958594</nowiki>
|2.991630
|18.932508

|-
|500
|<nowiki>-40.999436</nowiki>
|2.994135
|18.980112

|-
|600
|<nowiki>-41.049315</nowiki>
|2.996821
|19.031223

|-
|700
|<nowiki>-41.1071190</nowiki>
|2.999644
|19.085059

|-
|800
|<nowiki>-41.171892</nowiki>
|3.002589
|19.141318

|-
|900
|<nowiki>-41.243018</nowiki>
|3.005636
|19.199640

|-
|1000
|<nowiki>-41.319848</nowiki>
|3.008785
|19.260044

|-
|1500
|<nowiki>-41.774892</nowiki>
|3.026138
|19.595184

|-
|2000
|<nowiki>-42.323671</nowiki>
|3.046987
|20.002966

|}
Quasi-Harmonic approximation suggests that the Morse potential can be used to represent the potential between atoms in a unit cell. This model is idealized by considering a perfect crystal. Each atom is an individual harmonic oscillators and they do not interact with each other and they are being treated independently. Taken into account of the anharmonicity the vibrational energy can be found.

The Relationship between the Helmholtz free energy with temperature is shown in Equation 3:
[[File:A=U+TS.png|thumb|284x284px|Equation 3. Where ''U'' is the internal energy, ''EZP'' is the vibrational zero-point energy, ''T'' is the absolute temperature, ''V'' is the volume and ''S'' is the entropy due to the vibrational degrees of freedom.|none]]
[[File:(LD) A with T.png|thumb|449x449px|Figure 2|none]]

The Helmholtz free energy becomes more negative, in other words decreases when temperature increases. It is not hard to see that the result matches Equation 3. As T increases, the entropy term becomes more positive, hence the free energy decreases.

[[File:(LD) LC with T.png|thumb|487x487px|Figure 3|none]][[File:(LD) V with T.png|thumb|492x492px|Figure 4|left]]

As expected, the lattice constant increases with temperature. So does the unit cell volume. At higher temperature, atoms have a higher internal energy hence the vibrational frequency increases, which causes a greater deviation from the static separation and therefore the material expands. However, in a diatomic molecule with an exactly harmonic potential the bond length would not increase with temperature as the time average interatomic separation remains constant.

Initially all three curves have a parabolic character, while at a higher temperature they towards more to a linearity. From Equation 1 and the gradient calculated from Figure 4 the thermal expansion coefficient is found to be 2.687x10-5 K-1. The literature value is 3.12x10-5 K-1 (at 300 K)<ref>O. L. Anderson and K. Zou, J. ''Thermodynamic Functions and Properties of MgO at High Compression and High Temperarture.'' Phys. Chem. Ref. Data, 1990, '''19''', 69.</ref>. The difference is 13% which is considerably large, nevertheless it is good enough coming from an approximation.

As the temperature approaches the melting point of MgO, the phonon modes would not represent the actual motions of the ions well. Simply because at the melting point the bond breaks and it changes the crystal structure which it is no longer perfect. This violates the approximation and hence the modes would be able to represent the motions well.

.

=== Molecular Dynamics (MD) ===
[[File:(MD) V with T.png|thumb|493x493px|Figure 5
]]
In this molecular dynamics method, a supercell containing 32 MgO units is used.

By doing a geometry optimization, the atomic position with a minimum energy is found and used to measure the the lattice constant and the cell volume. Since the method is based on solving the Newton's equations, the system is being treated classically. A larger system is used to compromise the lost of its periodicity due to the nature of random atom movements.

Again apply Equation 1 the thermal expansion coefficient is found to be 3.086x10-5 K-1. The difference is 1.1% which is considerably smaller compared to the quasi-harmonic model.

Figure 6 shows a comparison between the results of using two methods. Both curves have the very similar shape while the MD method gives a smaller values at the same temperature (within 0-1000 K at least can be seen). It is worth mentioning that with increasing temperatures two curves are getting closer and closer. [[File:LD and MD Comparison.png|thumb|493x493px|Figure 6]]The reason
* ''Why do the two methods produce different answers ? - how does the difference depend on temperature ?''

=== ''An opportunity to speculate'' ===
* ''From your experience of computing an accurate Free Energy how large a cell should be used to perform reliably MD for MgO ?''
* ''What would happen to the cell volume at high temperature in the quasi-harmonic approximation and in the MD ?''

== Conclusion ==

== References ==

File:A*=2pi-a.png

2017-02-09T03:48:33Z

Lz2914:

File:(MD) V with T.png

2017-02-09T03:35:04Z

Lz2914: Lz2914 uploaded a new version of File:(MD) V with T.png

Rep:Mod:ZLY9512

2017-02-09T03:33:30Z

Lz2914: /* Molecular Dynamics */

= ''Thermal expansion of MgO'' =

== Introduction ==
Thermal Expansion is a very common phenomenon in real life. It is a common knowledge that within a certain range of temperature, when the temperature goes up some materials expand. This is due to an increment of vibrations of atoms which results a greater separation between atoms microscopically and hence macroscopically makes a material to expand. Although there are some materials which behave the opposite, certain cases would not be discussed here.

In this experiment, a simple compound MgO is used to study this phenomenon.
{| class="wikitable"
! colspan="2" |Table 1: Representations of MgO unit cells (FCC)
|-
|[[File:MgOpri.png|none|thumb|296x296px]]Primitive cell
|[[File:MgO conventional.jpg|none|thumb|294x294px]]Conventional cell
|}

The vibrational state is stimulated by computer in order to model the thermal expansion of MgO. The thermal expansion coefficient α is found by Equation (1):
[[File:Thermal exp coef.png|thumb|Equation (1) : Thermal Expansion Coefficient
|none]]

Two methods were used to calculate the thermal expansion coefficient, the quasi-harmonic approximation (LD) and molecular dynamics (MD). The former method is used to compute the vibrational energy levels of MgO. This method allows us to understand the phonon dispersion and the vibrational density of state (DOS). The latter method is used to simulate the vibrations as random motions of atoms inside a cell. It tells us how atoms respond over time under a certain condition. In this experiment, General Utility Lattice Program (GULP) on the DLVisualize platform (Linux based) is used and by increasing the temperature, we can see how MgO atoms in a cell respond, what physical property does this respond give and how we can quantify it.

== Results and Discussion ==

=== Phonon dispersion and Density of States ===
First of all, the phonon dispersion graph of MgO was produced by simulation to study the number of vibrational states which exist at k-points.
[[File:MgOPD.png|thumb|425x425px|Figure 1: Phonon Dispersion graph of MgO|centre]]Followed by the local phonon DOS with different grid sizes.
{| class="wikitable"
! colspan="4" |Table 2: MgO '''phonon DOS with variation of grid sizes'''
|-
|1x1x1
|[[File:DOS 111.JPG|thumb]]
|2x2x2
|[[File:DOS 222.JPG|thumb]]
|-
|3x3x3
|[[File:DOS 333.JPG|thumb]]
|16x16x16
|[[File:DOS 161616.JPG|thumb]]
|-
|20x20x20
|[[File:DOS 202020.JPG|thumb]]
|25x25x25
|[[File:DOS 252525.JPG|thumb]]
|}

From the 1x1x1 grid the four frequencies match the symmetry point L which indicates that the k-point is equal to (1/2, 1/2, 1/2). In the 1x1x1 grid DOS graph, the two lower frequency numbers are twice as large as the two higher frequency numbers. This matches the fact that in the dispersion curve at symmetry point L the two lower frequency numbers are doubly degenerated. The DOS resembles the dispersion curves. It contains the information at certain frequency numbers with respected to a symmetry point.

By comparing different grid sizes' dispersion curves, we can conclude that when the grid size increases, the DOS increases because there are more points in k-space are taken and used to calculate the DOS. The larger the grid size, the finer, smoother the DOS curve becomes as there are more atoms being examined more details are given. Since the 20x20x20 graph is extremely similar to the 25x25x25 graph, for a reasonable approximation to the DOS, 20x20x20 is accurate enough for a good approximation.

{| class="wikitable"
|+Table 3: Helmholtz free energy change with grid size
!Grid size
!Free Energy/ eV
!ΔA/ meV (with respected to 25x25x25)
|-
|1x1x1
|<nowiki>-40.930301</nowiki>
|<nowiki>-3.188</nowiki>
|-
|2x2x2
|<nowiki>-40.926609</nowiki>
|<nowiki>-0.126</nowiki>
|-
|3x3x3
|<nowiki>-40.926432</nowiki>
|0.051
|-
|4x4x4
|<nowiki>-40.926450</nowiki>
|0.033
|-
|8x8x8
|<nowiki>-40.926478</nowiki>
|0.005
|-
|10x10x10
|<nowiki>-40.926480</nowiki>
|0.003
|-
|15x15x15
|<nowiki>-40.926482</nowiki>
|0.001
|-
|20x20x20
|<nowiki>-40.926483</nowiki>
|0.000
|-
|25x25x25
|<nowiki>-40.926483</nowiki>
|0.000
|}

The Helmholtz free energy varies with the grid size. As the size increases, the free energy increases and it reaches a maximum at 3x3x3 grid and then decreases.

In this table the energy difference is also calculated between each grid size and the most precise grid size. In this case which is the 25x25x25 grid. For a calculation accurate to 1 meV, 1x1x1 could be used. To 0.5 meV, 2x2x2 could be used. To 0.1 meV, 3x3x3 could be used.

=== Quasi-Harmonic Approximation (LD) ===
{| class="wikitable"
|+
Table 4: Effects of temperature on the Helmholtz free energy, the lattice constant and the cell volume.
!Temperature/ K
!Helmholtz free energy/ eV
!Lattice constant/ Å
!Cell Volume/ Å³

|-
|0
|<nowiki>-40.900978</nowiki>
|2.986563
|18.836496

|-
|100
|<nowiki>-40.902420</nowiki>
|2.986563
|18.836494

|-
|200
|<nowiki>-40.909377</nowiki>
|2.987605
|18.856202

|-
|300
|<nowiki>-40.928125</nowiki>
|2.989390
|18.890025

|-
|400
|<nowiki>-40.958594</nowiki>
|2.991630
|18.932508

|-
|500
|<nowiki>-40.999436</nowiki>
|2.994135
|18.980112

|-
|600
|<nowiki>-41.049315</nowiki>
|2.996821
|19.031223

|-
|700
|<nowiki>-41.1071190</nowiki>
|2.999644
|19.085059

|-
|800
|<nowiki>-41.171892</nowiki>
|3.002589
|19.141318

|-
|900
|<nowiki>-41.243018</nowiki>
|3.005636
|19.199640

|-
|1000
|<nowiki>-41.319848</nowiki>
|3.008785
|19.260044

|-
|1500
|<nowiki>-41.774892</nowiki>
|3.026138
|19.595184

|-
|2000
|<nowiki>-42.323671</nowiki>
|3.046987
|20.002966

|}
Quasi-Harmonic approximation suggests that the Morse potential can be used to represent the potential between atoms in a unit cell. This model is idealized by considering a perfect crystal. Each atom is an individual harmonic oscillators and they do not interact with each other and they are being treated independently. Taken into account of the anharmonicity the vibrational energy can be found.

The Relationship between the Helmholtz free energy with temperature is shown in Equation 2:
[[File:A=U+TS.png|thumb|284x284px|Equation 2. Where ''U'' is the internal energy, ''EZP'' is the vibrational zero-point energy, ''T'' is the absolute temperature, ''V'' is the volume and ''S'' is the entropy due to the vibrational degrees of freedom.|none]]
[[File:(LD) A with T.png|thumb|449x449px|Figure 2|none]]

The Helmholtz free energy becomes more negative, in other words decreases when temperature increases. It is not hard to see that the result matches Equation 2. As T increases, the entropy term becomes more positive, hence the free energy decreases.

[[File:(LD) LC with T.png|thumb|487x487px|Figure 3|none]][[File:(LD) V with T.png|thumb|492x492px|Figure 4|left]]

As expected, the lattice constant increases with temperature. So does the unit cell volume. At higher temperature, atoms have a higher internal energy hence the vibrational frequency increases, which causes a greater deviation from the static separation and therefore the material expands. However, in a diatomic molecule with an exactly harmonic potential the bond length would not increase with temperature as the time average interatomic separation remains constant.

Initially all three curves have a parabolic character, while at a higher temperature they towards more to a linearity. From Equation 1 and the gradient calculated from Figure 4 the thermal expansion coefficient is found to be 2.687x10-5 K-1. The literature value is 3.12x10-5 K-1 (at 300 K)<ref>O. L. Anderson and K. Zou, J. ''Thermodynamic Functions and Properties of MgO at High Compression and High Temperarture.'' Phys. Chem. Ref. Data, 1990, '''19''', 69.</ref>. The difference is 13% which is considerably large, nevertheless it is good enough coming from an approximation.

As the temperature approaches the melting point of MgO, the phonon modes would not represent the actual motions of the ions well. Simply because at the melting point the bond breaks and it changes the crystal structure which it is no longer perfect. This violates the approximation and hence the modes would be able to represent the motions well.

.

=== Molecular Dynamics (MD) ===
[[File:(MD) V with T.png|thumb|493x493px|Figure 5
]]
In this molecular dynamics method, a supercell containing 32 MgO units is used.

By doing a geometry optimization, the atomic position with a minimum energy is found and used to measure the the lattice constant and the cell volume. Since the method is based on solving the Newton's equations, the system is being treated classically. A larger system is used to compromise the lost of its periodicity due to the nature of random atom movements.

Again apply Equation 1 the thermal expansion coefficient is found to be 3.010x10-5 K-1. The difference is 3.5% which is considerably smaller compared to the quasi-harmonic model.

Figure 6 shows a comparison between the results of using two methods. Both curves have the very similar shape while the MD method gives a smaller values at the same temperature (within 0-1000 K at least can be seen). It is worth mentioning that with increasing temperatures two curves are getting closer and closer. [[File:LD and MD Comparison.png|thumb|493x493px|Figure 6]]The reason
* ''Why do the two methods produce different answers ? - how does the difference depend on temperature ?''

=== ''An opportunity to speculate'' ===
* ''From your experience of computing an accurate Free Energy how large a cell should be used to perform reliably MD for MgO ?''
* ''What would happen to the cell volume at high temperature in the quasi-harmonic approximation and in the MD ?''

== Conclusion ==

== References ==

File:(MD) V with T.png

2017-02-09T03:33:16Z

Lz2914: Lz2914 uploaded a new version of File:(MD) V with T.png

File:LD and MD Comparison.png

2017-02-09T03:33:04Z

Lz2914: Lz2914 uploaded a new version of File:LD and MD Comparison.png

Rep:Mod:ZLY9512

2017-02-09T02:00:33Z

Lz2914: /* Molecular Dynamics */

= ''Thermal expansion of MgO'' =

== Introduction ==
Thermal Expansion is a very common phenomenon in real life. It is a common knowledge that within a certain range of temperature, when the temperature goes up some materials expand. This is due to an increment of vibrations of atoms which results a greater separation between atoms microscopically and hence macroscopically makes a material to expand. Although there are some materials which behave the opposite, certain cases would not be discussed here.

In this experiment, a simple compound MgO is used to study this phenomenon.
{| class="wikitable"
! colspan="2" |Table 1: Representations of MgO unit cells (FCC)
|-
|[[File:MgOpri.png|none|thumb|296x296px]]Primitive cell
|[[File:MgO conventional.jpg|none|thumb|294x294px]]Conventional cell
|}

The vibrational state is stimulated by computer in order to model the thermal expansion of MgO. The thermal expansion coefficient α is found by Equation (1):
[[File:Thermal exp coef.png|thumb|Equation (1) : Thermal Expansion Coefficient
|none]]

Two methods were used to calculate the thermal expansion coefficient, the quasi-harmonic approximation (LD) and molecular dynamics (MD). The former method is used to compute the vibrational energy levels of MgO. This method allows us to understand the phonon dispersion and the vibrational density of state (DOS). The latter method is used to simulate the vibrations as random motions of atoms inside a cell. It tells us how atoms respond over time under a certain condition. In this experiment, General Utility Lattice Program (GULP) on the DLVisualize platform (Linux based) is used and by increasing the temperature, we can see how MgO atoms in a cell respond, what physical property does this respond give and how we can quantify it.

== Results and Discussion ==

=== Phonon dispersion and Density of States ===
First of all, the phonon dispersion graph of MgO was produced by simulation to study the number of vibrational states which exist at k-points.
[[File:MgOPD.png|thumb|425x425px|Figure 1: Phonon Dispersion graph of MgO|centre]]Followed by the local phonon DOS with different grid sizes.
{| class="wikitable"
! colspan="4" |Table 2: MgO '''phonon DOS with variation of grid sizes'''
|-
|1x1x1
|[[File:DOS 111.JPG|thumb]]
|2x2x2
|[[File:DOS 222.JPG|thumb]]
|-
|3x3x3
|[[File:DOS 333.JPG|thumb]]
|16x16x16
|[[File:DOS 161616.JPG|thumb]]
|-
|20x20x20
|[[File:DOS 202020.JPG|thumb]]
|25x25x25
|[[File:DOS 252525.JPG|thumb]]
|}

From the 1x1x1 grid the four frequencies match the symmetry point L which indicates that the k-point is equal to (1/2, 1/2, 1/2). In the 1x1x1 grid DOS graph, the two lower frequency numbers are twice as large as the two higher frequency numbers. This matches the fact that in the dispersion curve at symmetry point L the two lower frequency numbers are doubly degenerated. The DOS resembles the dispersion curves. It contains the information at certain frequency numbers with respected to a symmetry point.

By comparing different grid sizes' dispersion curves, we can conclude that when the grid size increases, the DOS increases because there are more points in k-space are taken and used to calculate the DOS. The larger the grid size, the finer, smoother the DOS curve becomes as there are more atoms being examined more details are given. Since the 20x20x20 graph is extremely similar to the 25x25x25 graph, for a reasonable approximation to the DOS, 20x20x20 is accurate enough for a good approximation.

{| class="wikitable"
|+Table 3: Helmholtz free energy change with grid size
!Grid size
!Free Energy/ eV
!ΔA/ meV (with respected to 25x25x25)
|-
|1x1x1
|<nowiki>-40.930301</nowiki>
|<nowiki>-3.188</nowiki>
|-
|2x2x2
|<nowiki>-40.926609</nowiki>
|<nowiki>-0.126</nowiki>
|-
|3x3x3
|<nowiki>-40.926432</nowiki>
|0.051
|-
|4x4x4
|<nowiki>-40.926450</nowiki>
|0.033
|-
|8x8x8
|<nowiki>-40.926478</nowiki>
|0.005
|-
|10x10x10
|<nowiki>-40.926480</nowiki>
|0.003
|-
|15x15x15
|<nowiki>-40.926482</nowiki>
|0.001
|-
|20x20x20
|<nowiki>-40.926483</nowiki>
|0.000
|-
|25x25x25
|<nowiki>-40.926483</nowiki>
|0.000
|}

The Helmholtz free energy varies with the grid size. As the size increases, the free energy increases and it reaches a maximum at 3x3x3 grid and then decreases.

In this table the energy difference is also calculated between each grid size and the most precise grid size. In this case which is the 25x25x25 grid. For a calculation accurate to 1 meV, 1x1x1 could be used. To 0.5 meV, 2x2x2 could be used. To 0.1 meV, 3x3x3 could be used.

=== Quasi-Harmonic Approximation ===
{| class="wikitable"
|+
Table 4: Effects of temperature on the Helmholtz free energy, the lattice constant and the cell volume.
!Temperature/ K
!Helmholtz free energy/ eV
!Lattice constant/ Å
!Cell Volume/ Å³

|-
|0
|<nowiki>-40.900978</nowiki>
|2.986563
|18.836496

|-
|100
|<nowiki>-40.902420</nowiki>
|2.986563
|18.836494

|-
|200
|<nowiki>-40.909377</nowiki>
|2.987605
|18.856202

|-
|300
|<nowiki>-40.928125</nowiki>
|2.989390
|18.890025

|-
|400
|<nowiki>-40.958594</nowiki>
|2.991630
|18.932508

|-
|500
|<nowiki>-40.999436</nowiki>
|2.994135
|18.980112

|-
|600
|<nowiki>-41.049315</nowiki>
|2.996821
|19.031223

|-
|700
|<nowiki>-41.1071190</nowiki>
|2.999644
|19.085059

|-
|800
|<nowiki>-41.171892</nowiki>
|3.002589
|19.141318

|-
|900
|<nowiki>-41.243018</nowiki>
|3.005636
|19.199640

|-
|1000
|<nowiki>-41.319848</nowiki>
|3.008785
|19.260044

|-
|1500
|<nowiki>-41.774892</nowiki>
|3.026138
|19.595184

|-
|2000
|<nowiki>-42.323671</nowiki>
|3.046987
|20.002966

|}
Quasi-Harmonic approximation suggests that the Morse potential can be used to represent the potential between atoms in a unit cell. This model is idealized by considering a perfect crystal. Each atom is an individual harmonic oscillators and they do not interact with each other and they are being treated independently. Taken into account of the anharmonicity the vibrational energy can be found.

The Relationship between the Helmholtz free energy with temperature is shown in Equation 2:
[[File:A=U+TS.png|thumb|284x284px|Equation 2. Where ''U'' is the internal energy, ''EZP'' is the vibrational zero-point energy, ''T'' is the absolute temperature, ''V'' is the volume and ''S'' is the entropy due to the vibrational degrees of freedom.|none]]
[[File:(LD) A with T.png|thumb|449x449px|Figure 2|none]]

The Helmholtz free energy becomes more negative, in other words decreases when temperature increases. It is not hard to see that the result matches Equation 2. As T increases, the entropy term becomes more positive, hence the free energy decreases.

[[File:(LD) LC with T.png|thumb|487x487px|Figure 3|none]][[File:(LD) V with T.png|thumb|492x492px|Figure 4|left]]

As expected, the lattice constant increases with temperature. So does the unit cell volume. At higher temperature, atoms have a higher internal energy hence the vibrational frequency increases, which causes a greater deviation from the static separation and therefore the material expands. However, in a diatomic molecule with an exactly harmonic potential the bond length would not increase with temperature as the time average interatomic separation remains constant.

Initially all three curves have a parabolic character, while at a higher temperature they towards more to a linearity. From Equation 1 and the gradient calculated from Figure 4 the thermal expansion coefficient is found to be 2.687x10-5 K-1. The literature value is 3.12x10-5 K-1 (at 300 K)<ref>O. L. Anderson and K. Zou, J. ''Thermodynamic Functions and Properties of MgO at High Compression and High Temperarture.'' Phys. Chem. Ref. Data, 1990, '''19''', 69.</ref>. The difference is 13% which is considerably large, nevertheless it is good enough coming from an approximation.

As the temperature approaches the melting point of MgO, the phonon modes would not represent the actual motions of the ions well. Simply because at the melting point the bond breaks and it changes the crystal structure which it is no longer perfect. This violates the approximation and hence the modes would be able to represent the motions well.

.

=== Molecular Dynamics ===
[[File:(MD) V with T.png|thumb|493x493px|Figure 5
]]
In this molecular dynamics method, a supercell containing 32 MgO units is used.

By doing a geometry optimization, the atomic position with a minimum energy is found and used to measure the the lattice constant and the cell volume. As so, the system is being treated classically. A larger system is used to compromise the lost of its periodicity due to the nature of random atom movements.

Again apply Equation 1 the thermal expansion coefficient is found to be 3.010x10-5 K-1. The difference is 3.5% which is considerably smaller compared to the quasi-harmonic model.

[[File:LD and MD Comparison.png|thumb|498x498px|Figure 6]]In Figure 6,
*
* ''Why do the two methods produce different answers ? - how does the difference depend on temperature ?''

=== ''An opportunity to speculate'' ===
* ''From your experience of computing an accurate Free Energy how large a cell should be used to perform reliably MD for MgO ?''
* ''What would happen to the cell volume at high temperature in the quasi-harmonic approximation and in the MD ?''

== References ==

Rep:Mod:ZLY9512

2017-02-09T01:56:41Z

Lz2914:

= ''Thermal expansion of MgO'' =

== Introduction ==
Thermal Expansion is a very common phenomenon in real life. It is a common knowledge that within a certain range of temperature, when the temperature goes up some materials expand. This is due to an increment of vibrations of atoms which results a greater separation between atoms microscopically and hence macroscopically makes a material to expand. Although there are some materials which behave the opposite, certain cases would not be discussed here.

In this experiment, a simple compound MgO is used to study this phenomenon.
{| class="wikitable"
! colspan="2" |Table 1: Representations of MgO unit cells (FCC)
|-
|[[File:MgOpri.png|none|thumb|296x296px]]Primitive cell
|[[File:MgO conventional.jpg|none|thumb|294x294px]]Conventional cell
|}

The vibrational state is stimulated by computer in order to model the thermal expansion of MgO. The thermal expansion coefficient α is found by Equation (1):
[[File:Thermal exp coef.png|thumb|Equation (1) : Thermal Expansion Coefficient
|centre]]

Two methods were used to calculate the thermal expansion coefficient, the quasi-harmonic approximation (LD) and molecular dynamics (MD). The former method is used to compute the vibrational energy levels of MgO. This method allows us to understand the phonon dispersion and the vibrational density of state (DOS). The latter method is used to simulate the vibrations as random motions of atoms inside a cell. It tells us how atoms respond over time under a certain condition. In this experiment, General Utility Lattice Program (GULP) on the DLVisualize platform (Linux based) is used and by increasing the temperature, we can see how MgO atoms in a cell respond, what physical property does this respond give and how we can quantify it.

== Results and Discussion ==

=== Phonon dispersion and Density of States ===
First of all, the phonon dispersion graph of MgO was produced by simulation to study the number of vibrational states which exist at k-points.
[[File:MgOPD.png|thumb|425x425px|Figure 1: Phonon Dispersion graph of MgO|centre]]Followed by the local phonon DOS with different grid sizes.
{| class="wikitable"
! colspan="4" |Table 2: MgO '''phonon DOS with variation of grid sizes'''
|-
|1x1x1
|[[File:DOS 111.JPG|thumb]]
|2x2x2
|[[File:DOS 222.JPG|thumb]]
|-
|3x3x3
|[[File:DOS 333.JPG|thumb]]
|16x16x16
|[[File:DOS 161616.JPG|thumb]]
|-
|20x20x20
|[[File:DOS 202020.JPG|thumb]]
|25x25x25
|[[File:DOS 252525.JPG|thumb]]
|}

From the 1x1x1 grid the four frequencies match the symmetry point L which indicates that the k-point is equal to (1/2, 1/2, 1/2). In the 1x1x1 grid DOS graph, the two lower frequency numbers are twice as large as the two higher frequency numbers. This matches the fact that in the dispersion curve at symmetry point L the two lower frequency numbers are doubly degenerated. The DOS resembles the dispersion curves. It contains the information at certain frequency numbers with respected to a symmetry point.

By comparing different grid sizes' dispersion curves, we can conclude that when the grid size increases, the DOS increases because there are more points in k-space are taken and used to calculate the DOS. The larger the grid size, the finer, smoother the DOS curve becomes as there are more atoms being examined more details are given. Since the 20x20x20 graph is extremely similar to the 25x25x25 graph, for a reasonable approximation to the DOS, 20x20x20 is accurate enough for a good approximation.

{| class="wikitable"
|+Table 3: Helmholtz free energy change with grid size
!Grid size
!Free Energy/ eV
!ΔA/ meV (with respected to 25x25x25)
|-
|1x1x1
|<nowiki>-40.930301</nowiki>
|<nowiki>-3.188</nowiki>
|-
|2x2x2
|<nowiki>-40.926609</nowiki>
|<nowiki>-0.126</nowiki>
|-
|3x3x3
|<nowiki>-40.926432</nowiki>
|0.051
|-
|4x4x4
|<nowiki>-40.926450</nowiki>
|0.033
|-
|8x8x8
|<nowiki>-40.926478</nowiki>
|0.005
|-
|10x10x10
|<nowiki>-40.926480</nowiki>
|0.003
|-
|15x15x15
|<nowiki>-40.926482</nowiki>
|0.001
|-
|20x20x20
|<nowiki>-40.926483</nowiki>
|0.000
|-
|25x25x25
|<nowiki>-40.926483</nowiki>
|0.000
|}

The Helmholtz free energy varies with the grid size. As the size increases, the free energy increases and it reaches a maximum at 3x3x3 grid and then decreases.

In this table the energy difference is also calculated between each grid size and the most precise grid size. In this case which is the 25x25x25 grid. For a calculation accurate to 1 meV, 1x1x1 could be used. To 0.5 meV, 2x2x2 could be used. To 0.1 meV, 3x3x3 could be used.

=== Quasi-Harmonic Approximation ===
{| class="wikitable"
|+
Table 4: Effects of temperature on the Helmholtz free energy, the lattice constant and the cell volume.
!Temperature/ K
!Helmholtz free energy/ eV
!Lattice constant/ Å
!Cell Volume/ Å³

|-
|0
|<nowiki>-40.900978</nowiki>
|2.986563
|18.836496

|-
|100
|<nowiki>-40.902420</nowiki>
|2.986563
|18.836494

|-
|200
|<nowiki>-40.909377</nowiki>
|2.987605
|18.856202

|-
|300
|<nowiki>-40.928125</nowiki>
|2.989390
|18.890025

|-
|400
|<nowiki>-40.958594</nowiki>
|2.991630
|18.932508

|-
|500
|<nowiki>-40.999436</nowiki>
|2.994135
|18.980112

|-
|600
|<nowiki>-41.049315</nowiki>
|2.996821
|19.031223

|-
|700
|<nowiki>-41.1071190</nowiki>
|2.999644
|19.085059

|-
|800
|<nowiki>-41.171892</nowiki>
|3.002589
|19.141318

|-
|900
|<nowiki>-41.243018</nowiki>
|3.005636
|19.199640

|-
|1000
|<nowiki>-41.319848</nowiki>
|3.008785
|19.260044

|-
|1500
|<nowiki>-41.774892</nowiki>
|3.026138
|19.595184

|-
|2000
|<nowiki>-42.323671</nowiki>
|3.046987
|20.002966

|}
Quasi-Harmonic approximation suggests that the Morse potential can be used to represent the potential between atoms in a unit cell. This model is idealized by considering a perfect crystal. Each atom is an individual harmonic oscillators and they do not interact with each other and they are being treated independently. Taken into account of the anharmonicity the vibrational energy can be found.

The Relationship between the Helmholtz free energy with temperature is shown in Equation 2:
[[File:A=U+TS.png|thumb|284x284px|Equation 2. Where ''U'' is the internal energy, ''EZP'' is the vibrational zero-point energy, ''T'' is the absolute temperature, ''V'' is the volume and ''S'' is the entropy due to the vibrational degrees of freedom.]]
[[File:(LD) A with T.png|thumb|449x449px|Figure 2|none]]

The Helmholtz free energy becomes more negative, in other words decreases when temperature increases. It is not hard to see that the result matches Equation 2. As T increases, the entropy term becomes more positive, hence the free energy decreases.

[[File:(LD) LC with T.png|thumb|487x487px|Figure 3|none]][[File:(LD) V with T.png|thumb|492x492px|Figure 4|left]]

As expected, the lattice constant increases with temperature. So does the unit cell volume. At higher temperature, atoms have a higher internal energy hence the vibrational frequency increases, which causes a greater deviation from the static separation and therefore the material expands. However, in a diatomic molecule with an exactly harmonic potential the bond length would not increase with temperature as the time average interatomic separation remains constant.

Initially all three curves have a parabolic character, while at a higher temperature they towards more to a linearity. From Equation 1 and the gradient calculated from Figure 4 the thermal expansion coefficient is found to be 2.687x10-5 K-1. The literature value is 3.12x10-5 K-1 (at 300 K)<ref>O. L. Anderson and K. Zou, J. ''Thermodynamic Functions and Properties of MgO at High Compression and High Temperarture.'' Phys. Chem. Ref. Data, 1990, '''19''', 69.</ref>. The difference is 13% which is considerably large, nevertheless it is good enough coming from an approximation.

As the temperature approaches the melting point of MgO, the phonon modes would not represent the actual motions of the ions well. Simply because at the melting point the bond breaks and it changes the crystal structure which it is no longer perfect. This violates the approximation and hence the modes would be able to represent the motions well.

=== Molecular Dynamics ===
In this molecular dynamics method, a supercell containing 32 MgO units is used.

By doing a geometry optimization, the atomic position with a minimum energy is found and used to measure the the lattice constant and the cell volume. As so, the system is being treated classically. A larger system is used to compromise the lost of its periodicity due to the nature of random atom movements.
[[File:(MD) V with T.png|thumb|493x493px]]
Again apply Equation 1 the thermal expansion coefficient is found to be 3.010x10-5 K-1. The difference is 3.5% which is considerably smaller compared to the quasi-harmonic model.

[[File:LD and MD Comparison.png|thumb|498x498px]]
*
* ''Why do the two methods produce different answers ? - how does the difference depend on temperature ?''

=== ''An opportunity to speculate'' ===
* ''From your experience of computing an accurate Free Energy how large a cell should be used to perform reliably MD for MgO ?''
* ''What would happen to the cell volume at high temperature in the quasi-harmonic approximation and in the MD ?''

== References ==

File:LD and MD Comparison.png

2017-02-09T01:37:53Z

Lz2914:

File:Comparison.png

2017-02-09T01:36:47Z

Lz2914: Lz2914 uploaded a new version of File:Comparison.png

File:Comparison.png

2017-02-09T01:36:01Z

Lz2914: Lz2914 uploaded a new version of File:Comparison.png

File:(MD) V with T.png

2017-02-09T01:35:15Z

Lz2914:

Rep:Mod:ZLY9512

2017-02-09T01:09:49Z

Lz2914:

= ''Thermal expansion of MgO'' =

== Introduction ==
Thermal Expansion is a very common phenomenon in real life. It is a common knowledge that within a certain range of temperature, when the temperature goes up some materials expand. This is due to an increment of vibrations of atoms which results a greater separation between atoms microscopically and hence macroscopically makes a material to expand. Although there are some materials which behave the opposite, certain cases would not be discussed here.

In this experiment, a simple compound MgO is used to study this phenomenon.
{| class="wikitable"
! colspan="2" |Table 1: Representations of MgO unit cells (FCC)
|-
|[[File:MgOpri.png|none|thumb|296x296px]]Primitive cell
|[[File:MgO conventional.jpg|none|thumb|294x294px]]Conventional cell
|}

The vibrational state is stimulated by computer in order to model the thermal expansion of MgO. The thermal expansion coefficient α is found by Equation (1):
[[File:Thermal exp coef.png|none|thumb|Equation (1) : Thermal Expansion Coefficient
]]

Two methods were used to calculate the thermal expansion coefficient, the quasi-harmonic approximation (LD) and molecular dynamics (MD). The former method is used to compute the vibrational energy levels of MgO. This method allows us to understand the phonon dispersion and the vibrational density of state (DOS). The latter method is used to simulate the vibrations as random motions of atoms inside a cell. It tells us how atoms respond over time under a certain condition. In this experiment, General Utility Lattice Program (GULP) on the DLVisualize platform (Linux based) is used and by increasing the temperature, we can see how MgO atoms in a cell respond, what physical property does this respond give and how we can quantify it.

== Results and Discussion ==

=== Phonon dispersion and Density of States ===
First of all, the phonon dispersion graph of MgO was produced by simulation to study the number of vibrational states which exist at k-points.
[[File:MgOPD.png|none|thumb|425x425px|Figure 1: Phonon Dispersion graph of MgO]]Followed by the local phonon DOS with different grid sizes.
{| class="wikitable"
! colspan="4" |Table 2: MgO '''phonon DOS with variation of grid sizes'''
|-
|1x1x1
|[[File:DOS 111.JPG|thumb]]
|2x2x2
|[[File:DOS 222.JPG|thumb]]
|-
|3x3x3
|[[File:DOS 333.JPG|thumb]]
|16x16x16
|[[File:DOS 161616.JPG|thumb]]
|-
|20x20x20
|[[File:DOS 202020.JPG|thumb]]
|25x25x25
|[[File:DOS 252525.JPG|thumb]]
|}

From the 1x1x1 grid the four frequencies match the symmetry point L which indicates that the k-point is equal to (1/2, 1/2, 1/2). In the 1x1x1 grid DOS graph, the two lower frequency numbers are twice as large as the two higher frequency numbers. This matches the fact that in the dispersion curve at symmetry point L the two lower frequency numbers are doubly degenerated. The DOS resembles the dispersion curves. It contains the information at certain frequency numbers with respected to a symmetry point.

By comparing different grid sizes' dispersion curves, we can conclude that when the grid size increases, the DOS increases because there are more points in k-space are taken and used to calculate the DOS. The larger the grid size, the finer, smoother the DOS curve becomes as there are more atoms being examined more details are given. Since the 20x20x20 graph is extremely similar to the 25x25x25 graph, for a reasonable approximation to the DOS, 20x20x20 is accurate enough for a good approximation.

{| class="wikitable"
|+Table 3: Helmholtz free energy change with grid size
!Grid size
!Free Energy/ eV
!ΔA/ meV (with respected to 25x25x25)
|-
|1x1x1
|<nowiki>-40.930301</nowiki>
|<nowiki>-3.188</nowiki>
|-
|2x2x2
|<nowiki>-40.926609</nowiki>
|<nowiki>-0.126</nowiki>
|-
|3x3x3
|<nowiki>-40.926432</nowiki>
|0.051
|-
|4x4x4
|<nowiki>-40.926450</nowiki>
|0.033
|-
|8x8x8
|<nowiki>-40.926478</nowiki>
|0.005
|-
|10x10x10
|<nowiki>-40.926480</nowiki>
|0.003
|-
|15x15x15
|<nowiki>-40.926482</nowiki>
|0.001
|-
|20x20x20
|<nowiki>-40.926483</nowiki>
|0.000
|-
|25x25x25
|<nowiki>-40.926483</nowiki>
|0.000
|}

The Helmholtz free energy varies with the grid size. As the size increases, the free energy increases and it reaches a maximum at 3x3x3 grid and then decreases.

In this table the energy difference is also calculated between each grid size and the most precise grid size. In this case which is the 25x25x25 grid. For a calculation accurate to 1 meV, 1x1x1 could be used. To 0.5 meV, 2x2x2 could be used. To 0.1 meV, 3x3x3 could be used.

=== Quasi-Harmonic Approximation ===
{| class="wikitable"
|+
Table 4: Effects of temperature on the Helmholtz free energy, the lattice constant and the cell volume.
!Temperature/ K
!Helmholtz free energy/ eV
!Lattice constant/ Å
!Cell Volume/ Å³

|-
|0
|<nowiki>-40.900978</nowiki>
|2.986563
|18.836496

|-
|100
|<nowiki>-40.902420</nowiki>
|2.986563
|18.836494

|-
|200
|<nowiki>-40.909377</nowiki>
|2.987605
|18.856202

|-
|300
|<nowiki>-40.928125</nowiki>
|2.989390
|18.890025

|-
|400
|<nowiki>-40.958594</nowiki>
|2.991630
|18.932508

|-
|500
|<nowiki>-40.999436</nowiki>
|2.994135
|18.980112

|-
|600
|<nowiki>-41.049315</nowiki>
|2.996821
|19.031223

|-
|700
|<nowiki>-41.1071190</nowiki>
|2.999644
|19.085059

|-
|800
|<nowiki>-41.171892</nowiki>
|3.002589
|19.141318

|-
|900
|<nowiki>-41.243018</nowiki>
|3.005636
|19.199640

|-
|1000
|<nowiki>-41.319848</nowiki>
|3.008785
|19.260044

|-
|1500
|<nowiki>-41.774892</nowiki>
|3.026138
|19.595184

|-
|2000
|<nowiki>-42.323671</nowiki>
|3.046987
|20.002966

|}
Quasi-Harmonic approximation suggests that the Morse potential can be used to represent the potential between atoms in a unit cell. This model is idealized by considering a perfect crystal. Each atom is an individual harmonic oscillators and they do not interact with each other and they are being treated independently. Taken into account of the anharmonicity the vibrational energy can be found.

The Relationship between the Helmholtz free energy with temperature is shown in Equation 2:
[[File:A=U+TS.png|none|thumb|284x284px|Equation 2. Where ''U'' is the internal energy, ''EZP'' is the vibrational zero-point energy, ''T'' is the absolute temperature, ''V'' is the volume and ''S'' is the entropy due to the vibrational degrees of freedom.]]
[[File:(LD) A with T.png|thumb|513x513px|Figure 2|none]]

<nowiki> </nowiki>The Helmholtz free energy becomes more negative, in other words decreases when temperature increases. It is not hard to see that the result matches Equation 2. As T increases, the entropy term becomes more positive, hence the free energy decreases.

[[File:(LD) LC with T.png|thumb|512x512px|left|Figure 3]][[File:(LD) V with T.png|thumb|516x516px|Figure 4|left]]

As expected, the lattice constant increases with temperature. So does the unit cell volume. At higher temperature, atoms have a higher internal energy hence the vibrational frequency increases, which causes a greater deviation from the static separation and therefore the material expands. However, in a diatomic molecule with an exactly harmonic potential the bond length would not increase with temperature as the time average interatomic separation remains constant.

Initially all three curves have a parabolic character, while at a higher temperature they towards more to a linearity. From Equation 1 and the gradient calculated from Figure 4 the thermal expansion coefficient is found to be 2.687x10-5 K-1. The literature value is 3.12x10-5 K-1 (at 300 K)<ref>O. L. Anderson and K. Zou, J. ''Thermodynamic Functions and Properties of MgO at High Compression and High Temperarture.'' Phys. Chem. Ref. Data, 1990, '''19''', 69.</ref>. The difference is 13% which is considerably large, nevertheless it is good enough coming from an approximation.

As the temperature approaches the melting point of MgO, the phonon modes would not represent the actual motions of the ions well. Simply because at the melting point the bond breaks and it changes the crystal structure which it is no longer perfect. This violates the approximation and hence the modes would be able to represent the motions well.

== Molecular Dynamics ==

=== ''Questions'' ===
* ''Replot your data from the quasi-harmonic approximation calculations as, cell volume per formula unit vs temperature, and add some points from the MD runs at a few suitable temperatures.''
* ''How does the thermal expansion predicted by MD compare to that predicted by the quasi-harmonic approximation ?''
* ''Why do the two methods produce different answers ? - how does the difference depend on temperature ?''

=== ''An opportunity to speculate'' ===
* ''From your experience of computing an accurate Free Energy how large a cell should be used to perform reliably MD for MgO ?''
* ''What would happen to the cell volume at high temperature in the quasi-harmonic approximation and in the MD ?''

Rep:Mod:ZLY9512

2017-02-09T01:07:44Z

Lz2914:

= ''Thermal expansion of MgO'' =

== Introduction ==
Thermal Expansion is a very common phenomenon in real life. It is a common knowledge that within a certain range of temperature, when the temperature goes up some materials expand. This is due to an increment of vibrations of atoms which results a greater separation between atoms microscopically and hence macroscopically makes a material to expand. Although there are some materials which behave the opposite, certain cases would not be discussed here.

In this experiment, a simple compound MgO is used to study this phenomenon.
{| class="wikitable"
! colspan="2" |Table 1: Representations of MgO unit cells (FCC)
|-
|[[File:MgOpri.png|none|thumb|296x296px]]Primitive cell
|[[File:MgO conventional.jpg|none|thumb|294x294px]]Conventional cell
|}

The vibrational state is stimulated by computer in order to model the thermal expansion of MgO. The thermal expansion coefficient α is found by Equation (1):
[[File:Thermal exp coef.png|none|thumb|Equation (1) : Thermal Expansion Coefficient
]]

Two methods were used to calculate the thermal expansion coefficient, the quasi-harmonic approximation (LD) and molecular dynamics (MD). The former method is used to compute the vibrational energy levels of MgO. This method allows us to understand the phonon dispersion and the vibrational density of state (DOS). The latter method is used to simulate the vibrations as random motions of atoms inside a cell. It tells us how atoms respond over time under a certain condition. In this experiment, General Utility Lattice Program (GULP) on the DLVisualize platform (Linux based) is used and by increasing the temperature, we can see how MgO atoms in a cell respond, what physical property does this respond give and how we can quantify it.

== Results and Discussion ==

=== Phonon dispersion and Density of States ===
First of all, the phonon dispersion graph of MgO was produced by simulation to study the number of vibrational states which exist at k-points.
[[File:MgOPD.png|none|thumb|425x425px|Figure 1: Phonon Dispersion graph of MgO]]Followed by the local phonon DOS with different grid sizes.
{| class="wikitable"
! colspan="4" |Table 2: MgO '''phonon DOS with variation of grid sizes'''
|-
|1x1x1
|[[File:DOS 111.JPG|thumb]]
|2x2x2
|[[File:DOS 222.JPG|thumb]]
|-
|3x3x3
|[[File:DOS 333.JPG|thumb]]
|16x16x16
|[[File:DOS 161616.JPG|thumb]]
|-
|20x20x20
|[[File:DOS 202020.JPG|thumb]]
|25x25x25
|[[File:DOS 252525.JPG|thumb]]
|}

From the 1x1x1 grid the four frequencies match the symmetry point L which indicates that the k-point is equal to (1/2, 1/2, 1/2). In the 1x1x1 grid DOS graph, the two lower frequency numbers are twice as large as the two higher frequency numbers. This matches the fact that in the dispersion curve at symmetry point L the two lower frequency numbers are doubly degenerated. The DOS resembles the dispersion curves. It contains the information at certain frequency numbers with respected to a symmetry point.

By comparing different grid sizes' dispersion curves, we can conclude that when the grid size increases, the DOS increases because there are more points in k-space are taken and used to calculate the DOS. The larger the grid size, the finer, smoother the DOS curve becomes as there are more atoms being examined more details are given. Since the 20x20x20 graph is extremely similar to the 25x25x25 graph, for a reasonable approximation to the DOS, 20x20x20 is accurate enough for a good approximation.

{| class="wikitable"
|+Table 3: Helmholtz free energy change with grid size
!Grid size
!Free Energy/ eV
!ΔA/ meV (with respected to 25x25x25)
|-
|1x1x1
|<nowiki>-40.930301</nowiki>
|<nowiki>-3.188</nowiki>
|-
|2x2x2
|<nowiki>-40.926609</nowiki>
|<nowiki>-0.126</nowiki>
|-
|3x3x3
|<nowiki>-40.926432</nowiki>
|0.051
|-
|4x4x4
|<nowiki>-40.926450</nowiki>
|0.033
|-
|8x8x8
|<nowiki>-40.926478</nowiki>
|0.005
|-
|10x10x10
|<nowiki>-40.926480</nowiki>
|0.003
|-
|15x15x15
|<nowiki>-40.926482</nowiki>
|0.001
|-
|20x20x20
|<nowiki>-40.926483</nowiki>
|0.000
|-
|25x25x25
|<nowiki>-40.926483</nowiki>
|0.000
|}

The Helmholtz free energy varies with the grid size. As the size increases, the free energy increases and it reaches a maximum at 3x3x3 grid and then decreases.

In this table the energy difference is also calculated between each grid size and the most precise grid size. In this case which is the 25x25x25 grid. For a calculation accurate to 1 meV, 1x1x1 could be used. To 0.5 meV, 2x2x2 could be used. To 0.1 meV, 3x3x3 could be used.

=== Quasi-Harmonic Approximation ===
{| class="wikitable"
|+
Table 4: Effects of temperature on the Helmholtz free energy, the lattice constant and the cell volume.
!Temperature/ K
!Helmholtz free energy/ eV
!Lattice constant/ Å
!Cell Volume/ Å³

|-
|0
|<nowiki>-40.900978</nowiki>
|2.986563
|18.836496

|-
|100
|<nowiki>-40.902420</nowiki>
|2.986563
|18.836494

|-
|200
|<nowiki>-40.909377</nowiki>
|2.987605
|18.856202

|-
|300
|<nowiki>-40.928125</nowiki>
|2.989390
|18.890025

|-
|400
|<nowiki>-40.958594</nowiki>
|2.991630
|18.932508

|-
|500
|<nowiki>-40.999436</nowiki>
|2.994135
|18.980112

|-
|600
|<nowiki>-41.049315</nowiki>
|2.996821
|19.031223

|-
|700
|<nowiki>-41.1071190</nowiki>
|2.999644
|19.085059

|-
|800
|<nowiki>-41.171892</nowiki>
|3.002589
|19.141318

|-
|900
|<nowiki>-41.243018</nowiki>
|3.005636
|19.199640

|-
|1000
|<nowiki>-41.319848</nowiki>
|3.008785
|19.260044

|-
|1500
|<nowiki>-41.774892</nowiki>
|3.026138
|19.595184

|-
|2000
|<nowiki>-42.323671</nowiki>
|3.046987
|20.002966

|}
Quasi-Harmonic approximation suggests that the Morse potential can be used to represent the potential between atoms in a unit cell. This model is idealized by considering a perfect crystal. Each atom is an individual harmonic oscillators and they do not interact with each other and they are being treated independently. Taken into account of the anharmonicity the vibrational energy can be found.

The Relationship between the Helmholtz free energy with temperature is shown in Equation 2:
[[File:A=U+TS.png|none|thumb|284x284px|Equation 2. Where ''U'' is the internal energy, ''EZP'' is the vibrational zero-point energy, ''T'' is the absolute temperature, ''V'' is the volume and ''S'' is the entropy due to the vibrational degrees of freedom.]]
[[File:(LD) A with T.png|left|thumb|513x513px|Figure 2]]

<nowiki> </nowiki>The Helmholtz free energy becomes more negative, in other words decreases when temperature increases. It is not hard to see that the result matches Equation 2. As T increases, the entropy term becomes more positive, hence the free energy decreases.

[[File:(LD) LC with T.png|thumb|512x512px|left|Figure 3]][[File:(LD) V with T.png|thumb|516x516px|centre|Figure 4]]

<nowiki> </nowiki>As expected, the lattice constant increases with temperature. So does the unit cell volume. At higher temperature, atoms have a higher internal energy hence the vibrational frequency increases, which causes a greater deviation from the static separation and therefore the material expands. However, in a diatomic molecule with an exactly harmonic potential the bond length would not increase with temperature as the time average interatomic separation remains constant.

Initially all three curves have a parabolic character, while at a higher temperature they towards more to a linearity. From Equation 1 and the gradient calculated from Figure 4 the thermal expansion coefficient is found to be 2.687x10-5 K-1. The literature value is 3.12x10-5 K-1 (at 300 K)<ref>O. L. Anderson and K. Zou, J. ''Thermodynamic Functions and Properties of MgO at High Compression and High Temperarture.'' Phys. Chem. Ref. Data, 1990, '''19''', 69.</ref>. The difference is 13% which is considerably large, nevertheless it is good enough coming from an approximation.

As the temperature approaches the melting point of MgO, the phonon modes would not represent the actual motions of the ions well. Simply because at the melting point the bond breaks and it changes the crystal structure which it is no longer perfect. This violates the approximation and hence the modes would be able to represent the motions well.

== Molecular Dynamics ==

=== ''Questions'' ===
* ''Replot your data from the quasi-harmonic approximation calculations as, cell volume per formula unit vs temperature, and add some points from the MD runs at a few suitable temperatures.''
* ''How does the thermal expansion predicted by MD compare to that predicted by the quasi-harmonic approximation ?''
* ''Why do the two methods produce different answers ? - how does the difference depend on temperature ?''

=== ''An opportunity to speculate'' ===
* ''From your experience of computing an accurate Free Energy how large a cell should be used to perform reliably MD for MgO ?''
* ''What would happen to the cell volume at high temperature in the quasi-harmonic approximation and in the MD ?''

File:A=U+TS.png

2017-02-08T23:00:35Z

Lz2914:

Rep:Mod:ZLY9512

2017-02-08T07:03:39Z

Lz2914:

File:(LD) LC with T.png

2017-02-08T06:58:43Z

Lz2914:

File:(LD) V with T.png

2017-02-08T06:54:46Z

Lz2914:

File:(LD) LP with T.png

2017-02-08T06:54:16Z

Lz2914:

File:(LD) A with T.png

2017-02-08T06:53:13Z

Lz2914:

Rep:Mod:ZLY9512

2017-02-08T06:45:57Z

Lz2914:

Rep:Mod:ZLY9512

2017-02-08T06:42:37Z

Lz2914: /* Thermal expansion of MgO */

Rep:Mod:ZLY9512

2017-02-08T06:35:39Z

Lz2914: /* Results and Discussion */

Rep:Mod:ZLY9512

2017-02-08T06:33:52Z

Lz2914: /* Quasi-Harmonic Approximation */

Rep:Mod:ZLY9512

2017-02-08T05:41:36Z

Lz2914:

Rep:Mod:ZLY9512

2017-02-08T02:45:36Z

Lz2914: /* Results and Discussion */

File:DOS 252525.JPG

2017-02-08T02:44:57Z

Lz2914:

File:DOS 202020.JPG

2017-02-08T02:43:53Z

Lz2914:

File:DOS 161616.JPG

2017-02-08T02:43:38Z

Lz2914:

File:DOS 333.JPG

2017-02-08T02:43:02Z

Lz2914:

File:DOS 222.JPG

2017-02-08T02:42:31Z

Lz2914:

File:DOS 111.JPG

2017-02-08T02:42:07Z

Lz2914:

Rep:Mod:ZLY9512

2017-02-07T02:50:40Z

Lz2914:

Rep:Mod:ZLY9512

2017-02-07T02:42:23Z

Lz2914:

File:MgOPD.png

2017-02-07T02:39:45Z

Lz2914:

File:PDMgOlyz.jpg

2017-02-07T02:38:40Z

Lz2914:

Rep:Mod:ZLY9512

2017-02-07T02:33:20Z

Lz2914:

Rep:Mod:ZLY9512

2017-02-07T02:29:26Z

Lz2914:

File:MgOpri.png

2017-02-07T02:27:56Z

Lz2914:

File:MgO primitive.jpg

2017-02-07T02:26:40Z

Lz2914: Lz2914 uploaded a new version of File:MgO primitive.jpg

File:MgO primitive.jpg

2017-02-07T02:25:57Z

Lz2914: Lz2914 uploaded a new version of File:MgO primitive.jpg

File:MgO primitive.jpg

2017-02-07T02:23:48Z

Lz2914: Lz2914 uploaded a new version of File:MgO primitive.jpg

Rep:Mod:ZLY9512

2017-02-07T02:23:37Z

Lz2914:

File:MgO conventional.jpg

2017-02-07T02:23:18Z

Lz2914: