Study about Diatomic Chain with a Mass Impurity

In order to explore how an impurity affects a harmonic oscillator chain, a classic diatomic chain with one mass impurity is studied by means of the recurrence relations method. The momentum Autocor- relation function of the impurity results from two pairs of resonant poles and three branch cuts. Cosine(s) provides the pole contribution, while the acoustic and optic branches provide the cut contribution. Both branches are expressed as an expansion of even-order Bessel functions. The expansion coefficients are integrals of Jacobian elliptic functions along the real axis in a complex plane for the acoustic branch and integrals along a contour parallel to the imaginary axis for the optical branch, respectively. The computed pole and cut contributions indicate how the mass impurity impacts the momentum autocorrelation function of the mass impurity in a typical diatomic chain, and so provide more insight into the momentum autocorrelation function of the mass impurity.