Theory of alloys. I. Embedded-cluster calculations of phonon spectra for a one-dimensional binary alloy

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 $A_c{B}_{1-c}$ with mass disorder. The theory is especially applicable to concentrations $0.05\lesssim c\lesssim 0.95$ 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 $c\to 0$ and $c\to 1$. Its greatest virtue is that it is computationally efficient, because it does not require large clusters.