Abstract
By treating a small cluster embedded in an effective medium (described here by the coherent-potential approximation) we can reproduce the "exact" numerical frequency-distribution spectra of the vibrating-linear-chain alloy ${A}_{c}{B}_{1\ensuremath{-}c}$ with mass disorder. The theory is especially applicable to concentrations $0.05\ensuremath{\lesssim}c\ensuremath{\lesssim}0.95$ throughout the alloy regime, where other theories are quantitatively unreliable. Unlike purely numerical calculations, the present method can be practically applied to real three-dimensional alloys. The theory is valid for all concentrations $c$ and all mass ratios $\frac{{m}_{B}}{{m}_{A}}$; it satisfies the oscillator strength sum rule, and it reduces to the exactly soluble single-defect theory in the limits $c\ensuremath{\rightarrow}0$ and $c\ensuremath{\rightarrow}1$. Its greatest virtue is that it is computationally efficient, because it does not require large clusters.
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