File:XWCELLVQHAMD.png

2016-01-25T20:24:16Z

Xw6613:

Rep:MOD:XWMGO

2016-01-25T20:06:55Z

Xw6613:

== Introduction ==

The face-centered cubic structure of MgO leads to four Mg2+ and four O2- contained in one concentional cell. For primitive cell of MgO, the structure becomes rhombohedron.

By considering the basic MgO molecule, the ionic interactions can be the basic atomic interations.

Phonon is a quantum representation of elementary vibration motion where the atoms or lattices oscillate uniformly at a single frequency.<ref name="Phonon"> WIKIPEDIA, ''Phonon'', Available from: https://en.wikipedia.org/wiki/Phonon [Accessed: 24th January 2016] </ref> As vibrational modes can be thermally excited, so phonons can be thermally excited.

In the computational MgO experiment, the crystal structure of MgO was investigated by using the simple models, '''DLVisualize''' and '''GULP''' for calculations.

'''DLVisulize''' is a sofware tool which allows the properties of MgO crystals to be calculated. In the output files, information like free energy, lattice constant and cell volume can be obtained.

The energy and vibrations of MgO were calculated from the atomic interations first,which was then used to obtain the free energy of the MgO crystals and therefore to investigate the thermal expansion behavior of MgO.

When investigating the thermal expansion behavior of MgO using the software, there were two ways for the prediction which are harmonic/quasi-harmonic approximation and molecular dynamics.

Harmonic approxiamation allows the independent vibrational modes to be used in describing the vibrational motions of the whole crystal and those independent vibrational modes can be simplely considered with 1D harmonic potential, which then allows the free energy to be considered the sum of vibrational modes of infinite crystals.

On the other hand, molecular dynamics is used to produce the actual vibrations of the atoms and a cell which contains 32 MgO molecules was used.

The comparison of the two methods can be discussed based on the Volume of MgO unit against Temperature graphs plotted using the data obtained from each method.

== The Initial Calculation on MgO ==

The '''Single Point''' of GULO was run and the output file contained information like the lattice vectors of primitive cell.

The properties of a single lattice cell of MgO were shown in the file. For example, the cell parameter was shown to be 2.9783 Å with internal angle of 60 degrees, which was a proof of rhombohedron structure of the MgO primitive cell as shown in the '''Table 1''' below.

<center>
{| class="wikitable" border="1"
|+ '''Table 1. The conventional and primitive cells of MgO'''
! Conventional Cell !! Primitive Cell !! Output File
|-
| [[File:ConventionalMGO.jpg|200px]]|| [[File:PrimitiveMGO.jpg|200px]] || [[File:MgO-model_1.out|calculated MgO-model]]
|}
</center>

The use of software basic tools such as structure display and cell size changing was practiced and familiarized in this part by following the script.

== The Calculation Of The Phonon Modes of MgO ==

=== Phonon Dispersion curve calculation ===

In this part, the calculation of phonon modes/vibrational modes were carried out using the '''Phonon Dispersion''' of '''GULP'''.

The points along the concentional path on '''k'''-space were shown to be W(1/2 1/4 3/4), L(1/2 1/2 1/2), G(0 0 0), X(1/2 0 1/2), W(1/2 1/4 3/4) and K(3/8 3/8 3/4). 50 points of phonons were computed through the W-L-G-W-X-K path.

The phonon dispersion graph was shown in '''Figure 1''' below, and the intersections between the curve and each '''k'''-point line can be explained as that the phonon modes can be found at that '''k'''-point and at the frequency value of the intersection.

For example, for '''k'''-point '''L(1/2 1/2 1/2)''', there were four intersections where the frequency values were around 290, 350 cm-1 which were degenerate and 680, 805 cm-1 which were singlet. An the specific frequency values can be found in the output file of the phonon dispersion calculation.

<gallery mode=packed widths="400px" heights="400px">
File:MgOdispersion.jpg‎|'''Figure 1. The Phonon Dispersion varies with the frequencies in k-space'''
</gallery>

Output File: [[File:MgOdisperC.out|MgO Phonon Dispersion]]

After obtaining the curve, the phonon modes listed in the by the panel can be visualized using '''Animate Model'''. The vibration mode 117 (GULP, phonon 4, 399.8 cm-1, '''0.000 0.000 0.000''') occurred inside the primitive cell due to its '''k'''-space point coordinate, and the vibration was shown to be the oxygen atom oscillating within the cell while the 8 magnesium atoms remaining still.

=== The Phonon Density of States (DOS) ===

The DOS against Frequency grpahs were camputed using '''Phonon DOS''' calculation of '''GULP'''.

Different shrinking factors indicated different curve behaviors in the graphs

<center>
{| class="wikitable" border="1"
|+ '''Table 2. Phonon DOS against Frequency graphs for different shrinking factors'''
! Shrinking Factor !! 1x1x1 !! 2x2x2 !! 4x4x4 !! 6x6x6
|-
| Data Graph|| [[File:DOS1-MgO.jpg|250px]] || [[File:DOS2-MgO.jpg|250px]] || [[File:DOS4-MgO.jpg|250px]] || [[File:DOS6-MgO.jpg|250px]]
|-
| Output Files||[[File:XwDOS1.out|1x1x1 DOS]] || [[File:XWDOS2.out|2x2x2 DOS]] || [[File:XWDOS4.out|4x4x4 DOS]] || [[File:XWDOS6.out|6x6x6 DOS]]
|-
! Shrinking Factor !! 8x8x8 !! 12x12x12 !! 20x20x20 !! 30x30x30
|-
| Data Graph|| [[File:XWDOS8-MgO.jpg|250px]] || [[File:XWDOS12-MgO.jpg|250px]] || [[File:XWDOS20-MgO.jpg|250px]] || [[File:XWDOS30-MgO.jpg|250px]]
|-
| Output Files||[[File:XWDOS8.out|8x8x8 DOS]] || [[File:XWDOS12.out|12x12x12 DOS]] || [[File:XWDOS20.out|20x20x20 DOS]] || [[File:XWDOS30.out|30x30x30 DOS]]
|}
</center>

First by comparing the DOS vs Frequency graph of 1x1x1 shrinking factor in '''Table 2''' with the Phonon Dispersion curves in '''Figure 1''', it could be worked out that the DOS for 1x1x1 grid was computed from the '''k'''-point of '''L'''(1/2 1/2 1/2) which had four intersections where the frequency values were around 290, 350, 680 and 805 cm-1.

Also, the DOS intensity of the frequencies of 290 and 350 cm-1 are approximately twice the DOS intensity of 680 and 805 cm-1 in the DOS graphes. And this coule be explained by the double degeneracy of 290 and 350 cm-1.

The frequency values of the four peaks in DOS graph of 1x1x1 grid were the same as the four intersections of the point '''L'''.

As shown in the '''Table 2''', as the shrinking factor increases, the number of peaks in the DOS graphs increases and peaks starts to become spread out from 4x4x4.

Until the 8x8x8, the DOS shape shows the obvious increase of peak numbers with density spread out, which means more and more vibrational modes are available.

From 16x16x16, distinct peaks over the frequency range start to emerge and a curve appears instead.

The curve throught the frequency range indicates all frequencies in the range can lead to the corresponding vibrational modes.

The shrinking factors are used to define the size of grid, which indicates that as the size of grid increases, the DOS become spread out through the wavelength range as a curve rather than just peaks present.

The smooth shapes of DOS curve of 30x30x30 and 20x20x20 have little difference, and both of them resemble the shape of 16x16x16 which is a little bit noisy. This means after 16x16x16, there could be another shrinking factor which can give a good approximation of the system.

This optical shrinking factor can be a good point for the calculation of energies and other related properties with a reasonable accuracy.

== Computing the Free Energy with The Harmonic Approximation ==

<center>
{| class="wikitable" border="1"
|+ '''Table 3. The Free Energy of different shrinking factors'''
! Shrinking Factor !! Free Energy/ eV !! Shrinking Factor !! Free Energy/ eV
|-
| 1x1x1 || -40.930301 || 12x12x12 || -40.926481
|-
| 2x2x2 || -40.926609 || 13x13x13 || -40.926482
|-
| 3x3x3 || -40.926432 || 14x14x14 || -40.926482
|-
| 4x4x4 || -40.926450 || 15x15x15 || -40.926482
|-
| 5x5x5 || -40.926463 || 16x16x16 || -40.926482
|-
| 6x6x6 || -40.926471 || 17x17x17 || -40.926482
|-
| 7x7x7 || -40.926475 || 18x18x18 || -40.926483
|-
| 8x8x8 || -40.926478 || 19x19x19 || -40.926483
|-
| 9x9x9 || -40.926479 || 20x20x20 || -40.926483
|-
| 10x10x10 || -40.926480 || 30x30x30 || -40.926483
|-
| 11x11x11 || -40.926481 || 50x50x50 || -40.926483
|}
</center>

As shown in '''Table 3''', the free energy values increases as the shrinking factor increases, and the values are convergent to a value which is -40.926483 as shown above.

The energy values of shrinking factors that are greater or equal to 3x3x3 are accurate to 1meV which is 0.001 eV.

The energy values of shrinking factors that are greater or equal to 4x4x4 are accurate to 0.1meV which is 0.0001 eV.

As the value -40.926483 was first obtained in 18x18x18 shrinking factor, so 18x18x18 is the good starting value for the lastter thermal properties' calculations.

== Thermal Expansion of MgO ==

By setting the shrinking factor as 18x18x18, the free energies, lattice constants and the cell volumes were calculated from 0 K to 2800 K in steps of 100 K for 0-1000 K and 200 K for 1000-2800 K.

The calculations were simply carried out using '''Phonon DOS''' but with different temperature values.

<center>
{| class="wikitable" border="1"
|+ '''Table 4. The Free Energy of different shrinking factors'''
! Temperature/ K !! Free Energy/ eV !! Lattice Constant/ Å !! Cell Volume/ Å3
|-
| 0 || 0.172485 || 2.986563 || 18.36496
|-
| 100 || -40.902420 || 2.986563 || 18.836494
|-
| 200 || -40.909377 || 2.987605 || 18.856202
|-
| 300 || -40.928124 || 2.989391 || 18.890025
|-
| 400 || -40.958594 || 2.991630 || 18.932508
|-
| 500 || -40.999435 || 2.994136 || 18.980113
|-
| 600 || -41.049315 || 2.996821 || 19.031224
|-
| 700 || -41.107119 || 2.999645 || 19.085060
|-
| 800 || -41.171891 || 3.002590 || 19.141319
|-
| 900 || -41.243017 || 3.005637 || 19.199641
|-
| 1000 || -41.319848 || 3.008786 || 19.260045
|-
| 1200 || -41.488739 || 3.015392 || 19.387171
|-
| 1400 || -41.675513 || 3.022436 || 19.523334
|-
| 1600 || -41.877959 || 3.029977 || 19.669831
|-
| 1800 || -42.094427 || 3.038113 || 19.828684
|-
| 2000 || -42.323671 || 3.046989 || 20.002960
|-
| 2200 || -42.564750 || 3.056836 || 20.197505
|-
| 2400 || -42.816992 || 3.068052 || 20.420640
|-
| 2600 || -43.079960 || 3.081440 || 20.689113
|-
| 2800 || -43.353556 || 3.099261 || 21.050132
|}
</center>

<gallery mode=packed widths="250px" heights="250px">
File:XWEvsT.png‎|'''Figure 2. The plot of Free Energy against Temperature'''
</gallery>

<gallery mode=packed widths="250px" heights="250px">
File:XWLCvsT.png|'''Figure 3. The plot of Lattice Constant against Temperature'''
</gallery>

<gallery mode=packed widths="250px" heights="250px">
File:XWVvsT.png‎|'''Figure 4. The plot of Cell Volume against Temperature'''
</gallery>

All of the plots showed smooth curves as '''Figure 1''', '''Figure 2''' and '''Figure 3''' shown above.

The T=0 K data points were not plotted inside the graphs, this is to the zero-point energy values appeared. To obtain more reliable free energy against T graph, the calculations for 0-100 K should be carried out.

The description of the curve lines in the plot can be expressed by each equation if the trend line could be used. In the three plots, the relationships are not completely linear as the observable different increase with each same T interval change.

The free energy decreases as the temperature increases, while the lattice constant and the cell volume increases as the temperature increases.

For the Cell Volume against T plot, a trend line can be used to find out the coefficient of thermal expansion as shown below.

<gallery mode=packed widths="300px" heights="300px">
File:XWTRENDLINE.png‎|'''Figure 5. The plot of Cell Volume against Temperature with a trend line'''
</gallery>

The R2 value 0.9978 which is very close to 1 indicates that this trend line is a good expression of the relationship between V and T.

To find the thermal expansion coefficient, the equation of this plot is required.

According to the general form of the coefficient of thermal expansion αV =(∂V/∂T)/V with unit of K-1,<ref name="Thernal Expansion"> WIKIPEDIA, ''Thermal Expansion'', Available from: https://en.wikipedia.org/wiki/Thermal_expansion#General_volumetric_thermal_expansion_coefficient [Accessed: 25th January 2016] </ref> and the equation obtained in '''Figure 5''' which is y = 2*10-7*x2 + 0.0002*x + 18.826 , therfore, ∂V/∂T = 4.0*10-7*T + 0.0002 which was then substituted back to the general form of the coefficient of thermal expansion to obtain the value for at T.

Therefore, '''Excel''' was used to calculate each coefficient value of each V against T data point, and the plot of coefficient of thermal expansion against T was obtained below.

<gallery mode=packed widths="300px" heights="300px">
File:XWLINEAR.png|'''Figure 6. Linear trend line of αV against T'''
File:XWnonLinearTEC.png|'''Figure 7. Non-linear trend line of αV against T'''
</gallery>

The closer the R2 value to 1, the better the expression is.

As shown in the '''Figure 6''' and '''Figure 7''', the non-linear equation was better one to describe the relationship between the coefficient of thermal expansion and T.

The theoretical value of the Linear Coefficient of Thermal Expansion is (12.6 +/- 0.5)*10-6 K-1 between the Temperature range of 1000-2000 celsius degrees.<ref name="Orbital">CHARLES J. ENGBERG, ERNEST H. ZEHMS, ''Thermal Expansion of AI,O,, BeO, MgO, B,C, Sic, and
Tic Above 1000°C '', '''Journal of the American Ceramic Society.''', 1959, '''42(6)''', pp 300-305. '''DOI:''' 10.1111/j.1151-2916.1959.tb12958.x </ref> And 1000-2000 celsius degrees is the same as 1275-2275 K.

The coefficient values in the '''Figure 6''' and '''Figure 7''' was in the range of 12.7412*10-6-62.707*10-6 K-1 with the T range of 100-2800 K.

The 1275-2275 K is within the range of 100-2800 K and the coefficient range (12.6 +/- 0.5)*10-6 K-1 is within 12.7412*10-6-62.707*10-6 K-1, which indicates some extent of the accuracy from the calculations.

In the quasi-harmonic approximation which was used in the calculations above, the harmonicity

== Molecular Dynamics Calculations ==

In this calculation method, the parameter of ''dT'' and ''size'' were considered.<ref name="MD"> 3rd Year MgO Computational Script, ''Molecular Dynamics'', Available from: http://www.ch.ic.ac.uk/harrison/Teaching/Thermal_Expansion/md.html [Accessed: 25th January 2016] </ref> Therefore, a model with 32 units of MgO was used instead, which allowed the flexibility to be performed.

<center>
{| class="wikitable" border="1"
|+ '''Table 4. The The Average Volume of 32 units of MgO at different T'''
! Temperature/ K !!Average Volume/ Å3
|-
| 100 || 599.552364
|-
| 200 || 600.513626
|-
| 300 || 602.899441
|-
| 400 || 603.241540
|-
| 500 || 605.731599
|-
| 600 || 607.831884
|-
| 700 || 609.326722
|-
| 800 || 612.059646
|-
| 900 || 613.477026
|-
| 1000 || 615.053673
|-
| 1200 || 620.019685
|-
| 1400 || 622.667240
|-
| 1600 || 626.171861
|-
| 1800 || 630.981406
|-
| 2000 || 632.416616
|-
| 2200 || 637.036302
|-
| 2400 || 642.621784
|-
| 2600 || 648.409448
|-
| 2800 || 655.021355
|}
</center>

The cell volume per formula unit for the data in '''Table 4''' can be calculated using Cell Volume(per formula) = Average Volume/32.

The calculations for the Coefficient of Thermal Expansion wsa the same as the one used in the quasi-harmonic approximation method which used αV =(∂V/∂T)/V.

== References ==

File:XWLINEAR.png

2016-01-25T20:06:44Z

Xw6613:

File:XWTRENDLINE.png

2016-01-25T20:06:10Z

Xw6613: Xw6613 uploaded a new version of File:XWTRENDLINE.png

File:XWTRENDLINE.png

2016-01-25T20:05:32Z

Xw6613: Xw6613 uploaded a new version of File:XWTRENDLINE.png

Rep:MOD:XWMGO

2016-01-25T20:03:27Z

Xw6613: /* Thermal Expansion of MgO */

== Introduction ==

The face-centered cubic structure of MgO leads to four Mg2+ and four O2- contained in one concentional cell. For primitive cell of MgO, the structure becomes rhombohedron.

By considering the basic MgO molecule, the ionic interactions can be the basic atomic interations.

Phonon is a quantum representation of elementary vibration motion where the atoms or lattices oscillate uniformly at a single frequency.<ref name="Phonon"> WIKIPEDIA, ''Phonon'', Available from: https://en.wikipedia.org/wiki/Phonon [Accessed: 24th January 2016] </ref> As vibrational modes can be thermally excited, so phonons can be thermally excited.

In the computational MgO experiment, the crystal structure of MgO was investigated by using the simple models, '''DLVisualize''' and '''GULP''' for calculations.

'''DLVisulize''' is a sofware tool which allows the properties of MgO crystals to be calculated. In the output files, information like free energy, lattice constant and cell volume can be obtained.

The energy and vibrations of MgO were calculated from the atomic interations first,which was then used to obtain the free energy of the MgO crystals and therefore to investigate the thermal expansion behavior of MgO.

When investigating the thermal expansion behavior of MgO using the software, there were two ways for the prediction which are harmonic/quasi-harmonic approximation and molecular dynamics.

Harmonic approxiamation allows the independent vibrational modes to be used in describing the vibrational motions of the whole crystal and those independent vibrational modes can be simplely considered with 1D harmonic potential, which then allows the free energy to be considered the sum of vibrational modes of infinite crystals.

On the other hand, molecular dynamics is used to produce the actual vibrations of the atoms and a cell which contains 32 MgO molecules was used.

The comparison of the two methods can be discussed based on the Volume of MgO unit against Temperature graphs plotted using the data obtained from each method.

== The Initial Calculation on MgO ==

The '''Single Point''' of GULO was run and the output file contained information like the lattice vectors of primitive cell.

The properties of a single lattice cell of MgO were shown in the file. For example, the cell parameter was shown to be 2.9783 Å with internal angle of 60 degrees, which was a proof of rhombohedron structure of the MgO primitive cell as shown in the '''Table 1''' below.

<center>
{| class="wikitable" border="1"
|+ '''Table 1. The conventional and primitive cells of MgO'''
! Conventional Cell !! Primitive Cell !! Output File
|-
| [[File:ConventionalMGO.jpg|200px]]|| [[File:PrimitiveMGO.jpg|200px]] || [[File:MgO-model_1.out|calculated MgO-model]]
|}
</center>

The use of software basic tools such as structure display and cell size changing was practiced and familiarized in this part by following the script.

== The Calculation Of The Phonon Modes of MgO ==

=== Phonon Dispersion curve calculation ===

In this part, the calculation of phonon modes/vibrational modes were carried out using the '''Phonon Dispersion''' of '''GULP'''.

The points along the concentional path on '''k'''-space were shown to be W(1/2 1/4 3/4), L(1/2 1/2 1/2), G(0 0 0), X(1/2 0 1/2), W(1/2 1/4 3/4) and K(3/8 3/8 3/4). 50 points of phonons were computed through the W-L-G-W-X-K path.

The phonon dispersion graph was shown in '''Figure 1''' below, and the intersections between the curve and each '''k'''-point line can be explained as that the phonon modes can be found at that '''k'''-point and at the frequency value of the intersection.

For example, for '''k'''-point '''L(1/2 1/2 1/2)''', there were four intersections where the frequency values were around 290, 350 cm-1 which were degenerate and 680, 805 cm-1 which were singlet. An the specific frequency values can be found in the output file of the phonon dispersion calculation.

<gallery mode=packed widths="400px" heights="400px">
File:MgOdispersion.jpg‎|'''Figure 1. The Phonon Dispersion varies with the frequencies in k-space'''
</gallery>

Output File: [[File:MgOdisperC.out|MgO Phonon Dispersion]]

After obtaining the curve, the phonon modes listed in the by the panel can be visualized using '''Animate Model'''. The vibration mode 117 (GULP, phonon 4, 399.8 cm-1, '''0.000 0.000 0.000''') occurred inside the primitive cell due to its '''k'''-space point coordinate, and the vibration was shown to be the oxygen atom oscillating within the cell while the 8 magnesium atoms remaining still.

=== The Phonon Density of States (DOS) ===

The DOS against Frequency grpahs were camputed using '''Phonon DOS''' calculation of '''GULP'''.

Different shrinking factors indicated different curve behaviors in the graphs

<center>
{| class="wikitable" border="1"
|+ '''Table 2. Phonon DOS against Frequency graphs for different shrinking factors'''
! Shrinking Factor !! 1x1x1 !! 2x2x2 !! 4x4x4 !! 6x6x6
|-
| Data Graph|| [[File:DOS1-MgO.jpg|250px]] || [[File:DOS2-MgO.jpg|250px]] || [[File:DOS4-MgO.jpg|250px]] || [[File:DOS6-MgO.jpg|250px]]
|-
| Output Files||[[File:XwDOS1.out|1x1x1 DOS]] || [[File:XWDOS2.out|2x2x2 DOS]] || [[File:XWDOS4.out|4x4x4 DOS]] || [[File:XWDOS6.out|6x6x6 DOS]]
|-
! Shrinking Factor !! 8x8x8 !! 12x12x12 !! 20x20x20 !! 30x30x30
|-
| Data Graph|| [[File:XWDOS8-MgO.jpg|250px]] || [[File:XWDOS12-MgO.jpg|250px]] || [[File:XWDOS20-MgO.jpg|250px]] || [[File:XWDOS30-MgO.jpg|250px]]
|-
| Output Files||[[File:XWDOS8.out|8x8x8 DOS]] || [[File:XWDOS12.out|12x12x12 DOS]] || [[File:XWDOS20.out|20x20x20 DOS]] || [[File:XWDOS30.out|30x30x30 DOS]]
|}
</center>

First by comparing the DOS vs Frequency graph of 1x1x1 shrinking factor in '''Table 2''' with the Phonon Dispersion curves in '''Figure 1''', it could be worked out that the DOS for 1x1x1 grid was computed from the '''k'''-point of '''L'''(1/2 1/2 1/2) which had four intersections where the frequency values were around 290, 350, 680 and 805 cm-1.

Also, the DOS intensity of the frequencies of 290 and 350 cm-1 are approximately twice the DOS intensity of 680 and 805 cm-1 in the DOS graphes. And this coule be explained by the double degeneracy of 290 and 350 cm-1.

The frequency values of the four peaks in DOS graph of 1x1x1 grid were the same as the four intersections of the point '''L'''.

As shown in the '''Table 2''', as the shrinking factor increases, the number of peaks in the DOS graphs increases and peaks starts to become spread out from 4x4x4.

Until the 8x8x8, the DOS shape shows the obvious increase of peak numbers with density spread out, which means more and more vibrational modes are available.

From 16x16x16, distinct peaks over the frequency range start to emerge and a curve appears instead.

The curve throught the frequency range indicates all frequencies in the range can lead to the corresponding vibrational modes.

The shrinking factors are used to define the size of grid, which indicates that as the size of grid increases, the DOS become spread out through the wavelength range as a curve rather than just peaks present.

The smooth shapes of DOS curve of 30x30x30 and 20x20x20 have little difference, and both of them resemble the shape of 16x16x16 which is a little bit noisy. This means after 16x16x16, there could be another shrinking factor which can give a good approximation of the system.

This optical shrinking factor can be a good point for the calculation of energies and other related properties with a reasonable accuracy.

== Computing the Free Energy with The Harmonic Approximation ==

<center>
{| class="wikitable" border="1"
|+ '''Table 3. The Free Energy of different shrinking factors'''
! Shrinking Factor !! Free Energy/ eV !! Shrinking Factor !! Free Energy/ eV
|-
| 1x1x1 || -40.930301 || 12x12x12 || -40.926481
|-
| 2x2x2 || -40.926609 || 13x13x13 || -40.926482
|-
| 3x3x3 || -40.926432 || 14x14x14 || -40.926482
|-
| 4x4x4 || -40.926450 || 15x15x15 || -40.926482
|-
| 5x5x5 || -40.926463 || 16x16x16 || -40.926482
|-
| 6x6x6 || -40.926471 || 17x17x17 || -40.926482
|-
| 7x7x7 || -40.926475 || 18x18x18 || -40.926483
|-
| 8x8x8 || -40.926478 || 19x19x19 || -40.926483
|-
| 9x9x9 || -40.926479 || 20x20x20 || -40.926483
|-
| 10x10x10 || -40.926480 || 30x30x30 || -40.926483
|-
| 11x11x11 || -40.926481 || 50x50x50 || -40.926483
|}
</center>

As shown in '''Table 3''', the free energy values increases as the shrinking factor increases, and the values are convergent to a value which is -40.926483 as shown above.

The energy values of shrinking factors that are greater or equal to 3x3x3 are accurate to 1meV which is 0.001 eV.

The energy values of shrinking factors that are greater or equal to 4x4x4 are accurate to 0.1meV which is 0.0001 eV.

As the value -40.926483 was first obtained in 18x18x18 shrinking factor, so 18x18x18 is the good starting value for the lastter thermal properties' calculations.

== Thermal Expansion of MgO ==

By setting the shrinking factor as 18x18x18, the free energies, lattice constants and the cell volumes were calculated from 0 K to 2800 K in steps of 100 K for 0-1000 K and 200 K for 1000-2800 K.

The calculations were simply carried out using '''Phonon DOS''' but with different temperature values.

<center>
{| class="wikitable" border="1"
|+ '''Table 4. The Free Energy of different shrinking factors'''
! Temperature/ K !! Free Energy/ eV !! Lattice Constant/ Å !! Cell Volume/ Å3
|-
| 0 || 0.172485 || 2.986563 || 18.36496
|-
| 100 || -40.902420 || 2.986563 || 18.836494
|-
| 200 || -40.909377 || 2.987605 || 18.856202
|-
| 300 || -40.928124 || 2.989391 || 18.890025
|-
| 400 || -40.958594 || 2.991630 || 18.932508
|-
| 500 || -40.999435 || 2.994136 || 18.980113
|-
| 600 || -41.049315 || 2.996821 || 19.031224
|-
| 700 || -41.107119 || 2.999645 || 19.085060
|-
| 800 || -41.171891 || 3.002590 || 19.141319
|-
| 900 || -41.243017 || 3.005637 || 19.199641
|-
| 1000 || -41.319848 || 3.008786 || 19.260045
|-
| 1200 || -41.488739 || 3.015392 || 19.387171
|-
| 1400 || -41.675513 || 3.022436 || 19.523334
|-
| 1600 || -41.877959 || 3.029977 || 19.669831
|-
| 1800 || -42.094427 || 3.038113 || 19.828684
|-
| 2000 || -42.323671 || 3.046989 || 20.002960
|-
| 2200 || -42.564750 || 3.056836 || 20.197505
|-
| 2400 || -42.816992 || 3.068052 || 20.420640
|-
| 2600 || -43.079960 || 3.081440 || 20.689113
|-
| 2800 || -43.353556 || 3.099261 || 21.050132
|}
</center>

<gallery mode=packed widths="250px" heights="250px">
File:XWEvsT.png‎|'''Figure 2. The plot of Free Energy against Temperature'''
</gallery>

<gallery mode=packed widths="250px" heights="250px">
File:XWLCvsT.png|'''Figure 3. The plot of Lattice Constant against Temperature'''
</gallery>

<gallery mode=packed widths="250px" heights="250px">
File:XWVvsT.png‎|'''Figure 4. The plot of Cell Volume against Temperature'''
</gallery>

All of the plots showed smooth curves as '''Figure 1''', '''Figure 2''' and '''Figure 3''' shown above.

The T=0 K data points were not plotted inside the graphs, this is to the zero-point energy values appeared. To obtain more reliable free energy against T graph, the calculations for 0-100 K should be carried out.

The description of the curve lines in the plot can be expressed by each equation if the trend line could be used. In the three plots, the relationships are not completely linear as the observable different increase with each same T interval change.

The free energy decreases as the temperature increases, while the lattice constant and the cell volume increases as the temperature increases.

For the Cell Volume against T plot, a trend line can be used to find out the coefficient of thermal expansion as shown below.

<gallery mode=packed widths="300px" heights="300px">
File:XWTRENDLINE.png‎|'''Figure 5. The plot of Cell Volume against Temperature with a trend line'''
</gallery>

The R2 value 0.9978 which is very close to 1 indicates that this trend line is a good expression of the relationship between V and T.

To find the thermal expansion coefficient, the equation of this plot is required.

According to the general form of the coefficient of thermal expansion αV =(∂V/∂T)/V with unit of K-1,<ref name="Thernal Expansion"> WIKIPEDIA, ''Thermal Expansion'', Available from: https://en.wikipedia.org/wiki/Thermal_expansion#General_volumetric_thermal_expansion_coefficient [Accessed: 25th January 2016] </ref> and the equation obtained in '''Figure 5''' which is y = 2*10-7*x2 + 0.0002*x + 18.826 , therfore, ∂V/∂T = 4.0*10-7*T + 0.0002 which was then substituted back to the general form of the coefficient of thermal expansion to obtain the value for at T.

Therefore, '''Excel''' was used to calculate each coefficient value of each V against T data point, and the plot of coefficient of thermal expansion against T was obtained below.

<gallery mode=packed widths="300px" heights="300px">
File:XWLinearTEC.png|'''Figure 6. Linear trend line of αV against T'''
File:XWnonLinearTEC.png|'''Figure 7. Non-linear trend line of αV against T'''
</gallery>

The closer the R2 value to 1, the better the expression is.

As shown in the '''Figure 6''' and '''Figure 7''', the non-linear equation was better one to describe the relationship between the coefficient of thermal expansion and T.

The theoretical value of the Linear Coefficient of Thermal Expansion is (12.6 +/- 0.5)*10-6 K-1 between the Temperature range of 1000-2000 celsius degrees.<ref name="Orbital">CHARLES J. ENGBERG, ERNEST H. ZEHMS, ''Thermal Expansion of AI,O,, BeO, MgO, B,C, Sic, and
Tic Above 1000°C '', '''Journal of the American Ceramic Society.''', 1959, '''42(6)''', pp 300-305. '''DOI:''' 10.1111/j.1151-2916.1959.tb12958.x </ref> And 1000-2000 celsius degrees is the same as 1275-2275 K.

The coefficient values in the '''Figure 6''' and '''Figure 7''' was in the range of 12.7412*10-6-62.707*10-6 K-1 with the T range of 100-2800 K.

The 1275-2275 K is within the range of 100-2800 K and the coefficient range (12.6 +/- 0.5)*10-6 K-1 is within 12.7412*10-6-62.707*10-6 K-1, which indicates some extent of the accuracy from the calculations.

In the quasi-harmonic approximation which was used in the calculations above, the harmonicity

== Molecular Dynamics Calculations ==

In this calculation method, the parameter of ''dT'' and ''size'' were considered.<ref name="MD"> 3rd Year MgO Computational Script, ''Molecular Dynamics'', Available from: http://www.ch.ic.ac.uk/harrison/Teaching/Thermal_Expansion/md.html [Accessed: 25th January 2016] </ref> Therefore, a model with 32 units of MgO was used instead, which allowed the flexibility to be performed.

<center>
{| class="wikitable" border="1"
|+ '''Table 4. The The Average Volume of 32 units of MgO at different T'''
! Temperature/ K !!Average Volume/ Å3
|-
| 100 || 599.552364
|-
| 200 || 600.513626
|-
| 300 || 602.899441
|-
| 400 || 603.241540
|-
| 500 || 605.731599
|-
| 600 || 607.831884
|-
| 700 || 609.326722
|-
| 800 || 612.059646
|-
| 900 || 613.477026
|-
| 1000 || 615.053673
|-
| 1200 || 620.019685
|-
| 1400 || 622.667240
|-
| 1600 || 626.171861
|-
| 1800 || 630.981406
|-
| 2000 || 632.416616
|-
| 2200 || 637.036302
|-
| 2400 || 642.621784
|-
| 2600 || 648.409448
|-
| 2800 || 655.021355
|}
</center>

The cell volume per formula unit for the data in '''Table 4''' can be calculated using Cell Volume(per formula) = Average Volume/32.

The calculations for the Coefficient of Thermal Expansion wsa the same as the one used in the quasi-harmonic approximation method which used αV =(∂V/∂T)/V.

== References ==

File:XWnonLinearTEC.png

2016-01-25T19:57:25Z

Xw6613: Xw6613 uploaded a new version of File:XWnonLinearTEC.png