Cut contribution to momentum autocorrelation function of an impurity in a classical diatomic chain

Abstract

A classic diatomic chain with a mass impurity is studied using the recurrence relations method. The momentum autocorrelation function of the impurity is a sum of contributions from two pairs of resonant poles and three branch cuts. The former results in cosine function and the latter in acoustic and optical branches. By use of convolution theorem, analytical expressions for the acoustic and optical branches are derived as even-order Bessel function expansions. The expansion coefficients are integrals of elliptic functions in the real axis for the acoustic branch and along a contour parallel to the imaginary axis for the optical branch, respectively. An integral is carried out for the calculation of optical branch: ∫0ϕdθ/√((1 − r12 sin2 θ)(1 − r22 sin2 θ)) = igsn−1 (sin ϕ) (r22 > r12 > 1, g is a constant).