=The Free Energy and Thermal Expansion of MgO=

==Introduction==
The aims of this computational experiment are to compute the phonon dispersion curve, density of states, free energy and thermal expansion of magnesium oxide, MgO. Both the quasi-harmonic approximation and a molecular dynamics simulation are used, and the results from both compared.

The system being analysed is the MgO crystal. The conventional cell is made up of 8 atoms, has a face centred cubic (fcc) structure and parameters ac=bc=cc. The primitive cell is made up of 2 atoms, has the parameters ap=bp=cp and is the smallest possible way of representing the crystal lattice. Below is a diagram showing how the two cells are related.

{| class="wikitable" border="1"
|+ Types of MgO Lattice Cells
! Conventional Cell !! Primitive Cell !! Primitive Cell shown inside the Conventional
|-
| [[File:Conventional cell cet114.png|300px]] || [[File:Primitive cell cet114.png|300px]] || [[File:Primitive cell inside the conventional cet114.png|300px]]
|}
 
===Methodology===

==Results and Discussions==

===Computing the Phonons of the MgO Lattice===
The phonon dispersion curve of the MgO lattice was computed to inspect its normal modes of vibration, or phonon modes. A phonon dispersion curve shows the possible frequencies of the phonon modes at a particular wave-vector, '''k'''.

[[File:MgO Phonon dispersion cet114.png|thumb|centre|500px|Phonon Dispersion Curve of MgO]]
 

A Density of States (DOS) graph can be plotted when all the k-points are summed. A DOS was calculated for a grid with shrinking factors 1x1x1, which has a single '''k'''-point. The DOS graph has 4 peaks and relates to the '''L''' '''k'''-point in the dispersion curve, with 2 degenerate points at 288 and 352 cm-1, and 2 non-degenerate points at 676 and 819 cm-1. This degeneracy explains why the peaks at 288 and 352 cm-1 are twice the size as the other two peaks.

[[File:MgO Phonon DOS 1x1x1 cet114.png|thumb|centre|500px|DOS of MgO with shrinking factors 1x1x1]]
 

In order to plot an accurate DOS, all the '''k'''-points need to be included. This can be achieved by increasing the shrinking factors and enlarging the grid of '''k'''-points. The most accurate result would be obtained by using a grid of infinite size, however as this not possible the grid sizes 2x2x2, 3x3x3, 4x4x4, 8x8x8, 16x6x16, 32x32x32 and 64x64x64 were calculated.

{| class="wikitable" border="1"
|+ DOS of Different Grid Sizes
! 2x2x2 !! 3x3x3 !! 4x4x4 !! 8x8x8
|-
| [[File:MgO Phonon DOS 2x2x2 cet114.png|centre|250px]] || [[File:MgO Phonon DOS 3x3x3 cet114.png|centre|250px]] || [[File:MgO Phonon DOS 4x4x4 cet114.png|centre|250px]] || [[File:MgO Phonon DOS 8x8x8 cet114.png|centre|250px]]
|}

{| class="wikitable" border="1"
|+ DOS of Different Grid Sizes
!16x16x16 !! 32x32x32 !! 64x64x64
|-
| [[File:MgO Phonon DOS 16x16x16 cet114.png|centre|250px]] || [[File:MgO Phonon DOS 32x32x32 cet114.png|centre|250px]] || [[File:MgO Phonon DOS 64x64x64 cet114.png|centre|250px]]
|}
 
As the shrinking factors increase the DOS smooths out as more '''k'''-points are accessed. From comparing the graphs, it can be seen that the DOS with a 32x32x32 grid size gives a result similar to the 64x64x64 grid size DOS, and so is the minimum grid size that should be used in these calculations.

Calcium Oxide (CaO) has a very similar lattice to MgO. It is made up of Ca2+ and O2- ions held in an ionic lattice, with a lattice constant of 4.803 Å.<ref name="CaO" /> The lattice constant of MgO is 4.209 Å<ref name="MgO" />, meaning that the CaO is slightly larger. As it is larger in real space, it is smaller in reciprocal space and therefore requires less '''k'''-points meaning a marginally smaller grid size could be used, however 32x32x32 would still work well. The Zeolite Faujasite is far larger than MgO, with lattice constant of 24.66 Å.<ref name="Faujasite" /> It is so much lager in real space that a much smaller '''k'''-point grid would be sufficient.

===Using the Harmonic Approximation to Calculate the Free Energy===
Using the quasi-harmonic approximation, the free energy is calculated in relation to an increasing k-point grid by summing all the normal modes of vibration. By comparing the results of increased grid sizes, the best compromise between experimental time and answer accuracy can be found.

[[File:Free Energy against K-point Grid Size cet114.png|thumb|right|450px|Free Energy against K-point Grid Size]]

{| class="wikitable" border="1"
|+ Free Energies of Different Grid Sizes
! k-space grid !! Free Energy
|-
| 1x1x1 || -40.930301
|-
| 2x2x2 || -40.926609
|-
| 3x3x3 || -40.926432
|-
| 4x4x4 || -40.926450
|-
| 8x8x8 || -40.926478
|-
| 16x16x16 || -40.926482
|-
| 32x32x32 || -40.926483
|-
| 64x64x64 || -40.926483
|}

 
 
It is seen from the table and graph above that initially there is a relatively dramatic change to the free energy with an increase in grid size. After a grid size of 8x8x8 the free energy is reasonably stable, however when you reach a size of 64x64x64 it remains the same as the 32x32x32 at 8 significant figures. For this reason, a grid size of 32x32x32 provides an accurate enough result and takes less time to run than the calculation with a 64x64x64 grid.
For free energy calculations that are accurate to 1meV and 0.5 meV, a grid size of 3x3x3 can be used, and for an energy accurate to 0.1meV a grid size of 8x8x8 could be used.

===The Thermal Expansion of MgO===
The equation for the Helmholtz free energy is <math>A=U-TS</math>. This equation can be simplified to <math>dA=-PdV-SdT</math>, showing that the free energy is dependent on both temperature and volume. Using the quasi-harmonic approximation, the MgO free energy was computed with respect to the temperature of the system. The free energy, lattice constants and primitive cell volume were plotted against temperature.

{| class="wikitable" border="1"
|+
! Free Energy against Temperature !! Lattice Constants against Temperature
|-
| [[File:Free Energy against Temperature cet114.png|centre|500px]] || [[File:Lattice Constants against Temperature cet114.png|centre|500px]]
|}

 

The Helmholtz equation shows that as the temperature increases, the magnitude of the negative term increases, which explains the increased negativity of the free energy. In the cases of the lattice parameter and volume, they increase as the temperature does. This is because these computations are run using the quasi-harmonic approximation, where an increase in temperature moves the system up an anharmonic, Morse-like potential and results in the bond lengths increasing. The plots are not completely linear due to the zero-point energy assumed in the system. At high temperatures the approximation breaks down, as it assumes that the bonds will never break and simply vibrate more and more, meaning even at the melting point of the crystal the bonds would still be unbroken. With an exactly harmonic diatomic molecule, the bond lengths do not change as the equilibrium position stays the same as the energy increases.

 
[[File:Primitive Cell Volume against Temperature cet114.png|thumb|centre|500px|Primitive Cell Volume against Temperature]]
 

By looking at the linear region, between 300 and 1000 K, of the graph where cell volume is plotted against temperature, the thermal expansion coefficient, <math> \alpha</math>, can be calculated using the equation <math> \alpha={1 \over V_0} \Bigg( {\partial V\over \partial T}\Bigg)_P </math>.

The calculated coefficient is 2.671x10-5 K-1. The <math>\alpha</math> coefficient found in literature is 4.47x10-5 K-1, which was taken at 1000 K. While the two are of the same order, there is a clear difference between them.
===Molecular Dynamics Calculations===
[[File:Quasi-Harmonic and Molecular Dynamics Primitive Cell Volumes against Temperature cet114.png|thumb|centre|500px|Quasi-Harmonic and Molecular Dynamics Primitive Cell Volumes against Temperature]]
 
==Conclusions==

== References ==
<references>
<ref name="CaO">K. Doll, M. Dolg and H. Stoll, ''Physical Review B'', 1996, '''54''', pp. 13529-13535.</ref>
<ref name="MgO">A. Cimino, P. Porta and M. Valigi, ''Journal of The American Ceramic Society'', '''49''', pp. 152-156.</ref>
<ref name="Faujasite">D. N. Stamires, ''Clays and Clay Minerals'', 1973, '''21''', pp. 379-389.</ref>

</references>

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File:Lattice Constants against Temperature cet114.png

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File:MgO Phonon DOS 1x1x1 cet114.png

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