Momentum autocorrelation function of an impurity in a classical oscillator chain with alternating masses II. Illustrations

• Contributions from 3 branch cuts and 2 pairs of resonant poles.
• Dependence of the frequency and amplitude upon two mass-ratios.
• Frequency theorem governing the upper and lower frequencies of the branches.
• Envelope of the integrand family of cut contribution.
• Convergence of numerical results to the analytical value of parameter WW.

A classic diatomic chain with a mass impurity is studied using the recurrence relations method. The dependence of frequency and amplitude upon two mass ratios is illustrated in detail. By examining the figures we can learn a lot about the model, such as the widths of the acoustic and optical branches as well as the gap between them, the frequency and amplitude of the resonant modes, etc. The momentum autocorrelation functions of the impurity in different regions of the configuration space are numerically calculated and illustrated. A theorem is obtained governing the upper and lower frequencies of the acoustic and optical branches. The envelope of the integrand of the cut contribution is determined. The convergence of numerical calculations of time integration for parameter WW to its theoretical value is confirmed.