Figure 6. Kinetic Energy Plot for Vacancy and k-ratio 1/5

Figure 7. Kinetic Energy Plot of Beat Phenomenon

These figures illustrate that there is likely no attenuation occurring in the solid. The time pattern that does occur is called the beat phenomenon [5]. This occurs when the time signal is a sum of two sinusoids that are very near in frequency. Using a least squares best fit of this data, the frequencies of the two signals are 3.99 Thz and 4.06 Thz. The amplitudes of the two signals are not equal, the ratio of the former to the latter is approximately 0.6. The beat pattern may occur in simulation using a crystal repeat of 12, however the second waveform may have such a close frequency and small amplitude such that its effects are unnoticeable. This may suggest that as the vacancy concentration increases, the frequency difference of split waves increases and the amplitude ratio may move closer to unity.

From this simulation we do not expect to see any attenuation due to vacancies. One reason attenuation might not occur in the simulation is that the simulated time may be too small.

Nonlinear Vibrations

Phonon vibrations are assumed to be very low amplitude vibrations of crystalline solids which allows the use of linear force-displacement relations which give rise to sinudoidal waves. If the atoms are displaced far enough the forces and displacements are not linearly related and the vibrations become nonlinear.

An interesting point on the phonon dispersion curve is at the end of the Brillouin zone for the [100] and [111] directions. Here, the longitudinal and transverse waves of both directions converge to the same frequency. MD simulations of the nonlinearity show that the vibrations do not have a strong nonlinear behavior.

MD simulations were carried out using the simulation box used to create the phonon dispersion curves. The wave vector was set to the maximum, and a time step of 0.125 fs was used. All the simulations were run for 1.25 ps. The results show that the waves at even very high amplitudes (0.5 angstrom) only exhibit single frequency oscillation. Figure 10 shows the time response of the kinetic energy for a longitudinal wave in the [100] direction with a 0.5 angstrom amplitude. All simulations at the Brillouin zone wave number in the [100] and [111] directions showed this type of response, though the frequencies did vary some. To contrast this, figure 8 shows the kinetic energy response of a 0.5 angstrom wave in the [111] direction for a wave number of 1, the minimum for the box size. Table 1 below shows the frequencies of the waves, both longitudinal and transverse, as a function of the wave amplitude for the [100] and [111] directions.

Figure 8. ½ Angstrom Vibration in the [111] direction k-ratio 1/12

Figure 9. ½ Angstrom Vibration in the [111] direction k-ratio 1

Figure 10. ½ Angstrom Vibration in the [100] direction k-ratio 1

[100] Longitudinal		[100] Transverse		[111] Longitudinal		[111] Transverse
Amplitude	frequency	Amplitude	frequency	Amplitude	frequency	Amplitude	frequency
Angstroms	Thz	Angstroms	Thz	Angstroms	Thz	Angstroms	Thz
0.01	6.394	0.01	6.394	0.01	6.399	0.01	6.398
0.05	6.395	0.05	6.398	0.05	6.529	0.05	6.496
0.1	6.409	0.1	6.409	0.1	6.914	0.1	6.788
0.5	6.759	0.5	6.759	0.5	NA	0.5	NA