Rep:Mod:HB2515MgORND2
For a 1D crystal, it would be expected to see 2 vibrational modes. But as we are looking in 3D we should be expecting 6. By summing over all the vibrational energy levels we can find the DOS. And summing over all the vibrational modes in the DOS will give the Helmholtz free energy.
A phonon dispersion curve is a was of plotting the variation in frequencies at all the different k values in reciprocal space. As the crystal has 3 dimensions, we will have k values in all of those three dimensions, therefore kx, ky and kz. When plotting this we encounter the problem that the plot will have to be 4D, which is not possible. Maybe a plot in 3D with a colour code could replace this and be a temporary solution to the problem, but this is not the best general solution. To solve this problem, kx, ky and kz will all vary to different values varying from -π/2 to π/2 (first brouillon zone) whilst describing the frequency for every normal mode at each set of k values. The number of normal mode here is 3N with N the number of atoms (no -6 as there is no translation and no rotation).
As in the equation above, a and a* are inversely proportional. Therefore if the unit cell in real space is smaller (smaller a parameter), the reciprocal space will be larger, and it may require more calculations and data points to get accurate results (calculating more wavevectors). The phonon dispersion plot give a relationship between the frequency and the wavevector k. While computing the phonon dispersion curve, 50 different wavevectors were calculated giving a plot as follows:
GRAPH OF DISPERSION PLOT. SCREENSHOT OF A PRIMITIVE AND CONVENTIONAL UNIT CELL OF MGO
Figure 1: Primitive unit cell of MgO
It is then possible to generate a density of states diagram, by summer over all the different states for a particular frequency, but only one wavevector. This is represented by the following plot with is for a 1x1x1 cubic cell.
DOS STATES PLOT
Figure 1: Primitive unit cell of MgO
This gives sharp peaks of the vibrations are at precise frequencies. We should be expecting 6 different normal modes, as mentioned previously. This is what we observe in the previous plot as the peaks come in a 2:2:1:1 ratio. Therefore we can identify that this corresponds to the set of wavevectors at the L point in the dispersion plot by comparing with the number of branches at that point in the dispersion plot. This can be checked by examining the log file of the output file, it confirms the ratio that was suspected, labeling the acoustic modes give peaks at 286 and 351 cm-1 and 0.333 and the optical modes give peaks at 676 and 806 cm-1 as 0.167. The two acoustic peaks are both doubly degenerate. The log file also reinforces this by examining the frequencies at point L matching the density of states plot at (0.5, 0.5, 0.5).
This is not however a good representation of reality because a crystal if formed of a infinite number of cells next to each other arranged periodically. To increase the grid size, large shrinking factors are selected, which will spread the frequency bands over a larger number of states in the DOS plot whilst increasing the number of wavevectors at which the frequency is calculated and displayed in the plot. Many different grid sizes were tested, and 40x40x40 was selected giving a smooth curve, whilst maintaining a realistic time per calculation.