Momentum autocorrelation function of an impurity in a classical oscillator chain with alternating masses— I. General theory

A chain composed of classic harmonic oscillators with alternating masses and one mass impurity is studied using the recurrence relations method. The dynamics of the momentum autocorrelation function of the impurity mass is contributed by three branch cuts corresponding to acoustic and optic modes and by two pairs of poles for resonant modes. One pair of the poles are located beyond the optic branch and the other in the gaps between the branches. Criteria for the resonances are established. General expressions for frequencies and amplitudes of the resonant modes are derived. Ergodic behavior of the momentum of the impurity is analyzed.