Transmission Properties of an Isotopically Disordered One-Dimensional Harmonic Crystal

Abstract

The amplitude ${\mathfrak{S}}_N\mathrm{(\omega )}$ of a wave of frequency ω which is transmitted by a disordered array of N isotopic defects in a one‐dimensional crystal has been investigated in the limit in which N → ∞ while the over‐all concentration of the defects in the array remains fixed. The transmitted amplitude ${\mathfrak{S}}_N\mathrm{(\omega )}$ is proportional to the reciprocal of the magnitude of an Nth‐order determinant whose elements depend explicitly upon the spacings between defects, the incident frequency ω, and the relative mass difference Q = (M − m)/m between the defect particles and the particles of the host crystal. ${\mathfrak{S}}_N\mathrm{(\omega )}$ is represented as exp [−Nα_N(ω, Q, C)], where C is the over‐all fractional concentration of defects; two types of estimates of α_N(ω, Q, C) are obtained. First, assuming that the spacings between nearest‐neighbor pairs of defects are independent random variables, upper and lower bounds are obtained on α_N(ω, Q, C) which are independent of N. Provided that C is sufficiently small, the lower bound is positive. Second, Monte Carlo estimates of α_N(ω, Q, C) are obtained in the cases Q = 1, C = 0.1 and Q = 1, C = 0.5, for arrays of 3 × 10⁴ defects. These Monte Carlo estimates are compared with the previously obtained bounds. It is also shown that at the special frequencies of Matsuda and for Q ≥ Q_crit, the limiting value of α_N(ω, Q, C) is positive in the entire concentration range 0 < C < 1. Explicit upper and lower bounds are obtained on α(sin (π/4), 1, C).