A Diatomic Chain with A Mass Impurity

Abstract

A classic diatomic chain with one mass impurity is studied using the recurrence relations method. The momentum autocorrelation function of the impurity results from contributions of two pairs of resonant poles and three branch cuts. The pole contribution is given by cosine function(s) and the cut contribution is the acoustic and optical branches. The acoustic and optical branches are given by expansions of even-order Bessel function. The expansion coefficients are integrals of elliptic functions in the real axis in a complex plane for the acoustic branch and along a contour parallel to the imaginary axis for the optical branch, respectively. An integral 22 22 1 2 0 d / ( r sin )( r sin ) 1 1 ϕ θ − θ− θ ∫ (r2 2 >r1 2 >1) is evaluated.