Transmission Properties of an Isotopically Disordered One-Dimensional Harmonic Crystal. II. Solution of a Functional Equation

Abstract

The amplitude of a wave of frequency ω which is transmitted by a disordered array of N isotopic defects in a 1‐dimensional harmonic crystal is investigated in the limit N → ∞. In particular, the ratio T̂_N(ω) of the amplitude of the Nth defect to the amplitude of the first defect is represented as $\exp {\mathrm{\hspace{0.17em}[-N\alpha ̂}}_N{\mathrm{(\omega ,Q,\{a}}_n\mathrm{\})]}$ , where {a_n}, n = 2, ⋯, N, is the sequence of nearest‐neighbor spacings and Q = (M − m)/m. It is known from earlier work that ${\mathrm{\alpha ̂}}_N{\mathrm{(\omega ,Q,\{a}}_n\mathrm{\})}$ is the logarithm of the Nth root of the magnitude of a continuant determinant of order N. The value of the continuant is expressed formally as a product of N factors ĝ_n which are recursively related. In the present case, the ĝ_n happen to lie on a circle K₀ in the complex ĝ plane. Assuming that the spacings between defects are independent identically distributed random variables with the mean value c⁻¹ and going to the limit N → ∞, a functional equation for the limiting distribution function of the ĝ_n on K₀ is derived. The limiting value $\mathrm{\alpha (\omega ,Q,c)=}\lim {\mathrm{\alpha ̂}}_N{\mathrm{(\omega ,Q,\{a}}_n\mathrm{\})}$ , as N → ∞, can be determined from the limiting distribution function of the ĝ_n. We determine the solution of the functional equation in three different ways for three different cases: (a) In the case of the special frequency of Matsuda, ω = 2^−½ and Q = 1, we obtain exact values of the integral of the ĝ distribution function which are in excellent agreement with Monte Carlo estimates; (b) in the physically interesting case where the mean spacing between defects is small compared to the incident wavelength, i.e., c⁻¹ω ≪ 1, we obtain the solution of the functional equation correct to first order in c⁻¹ω and we calculate the lowest‐order nonzero value of α(ω, Q, c); (c) for the general case of moderate values of ω, Q, and c, we develop a numerical method for solving the functional equation and present the results of the numerical calculations in several representative cases. These numerical results are in good agreement with Monte Carlo estimates. One of the principal results, obtained by solving the functional equation, is that α(ω, Q, c) > 0 for ω ≪ c < 1 and ω(|Q| + c⁻¹) ≪ 1 with Q ≠ 0.