A monatomic chain with an impurity in mass and Hooke constant

Abstract

In this paper, a diatomic chain composed of classical harmonic oscillators and an impurity with different mass and Hooke constant is studied by means of the recurrence relations method. The Laplace transform of the momentum autocorrelation function of the impurity has three pairs of resonant poles and three separate branch cuts; however, one pair of poles was found nonphysical, so only two pairs of poles contribute to the momentum autocorrelation function. The three branch cuts give the acoustic and optical branches that are given by a convolution of a sum of sines and an expansion of even-order Bessel functions. The expansion coefficients are integrals of a Jacobian elliptic function along the real axis in a complex plane for the acoustic branch and those along a contour parallel to the imaginary axis for the optical branch, respectively. The ergodicity of momentum of the impurity is also discussed.