Comparison of single-site approximations for the lattice dynamics of mass-disordered alloys

Calculations of phonon frequency distributions, spectral functions, and neutron-scattering cross sections have been carried out for the vibrations in monatomic fcc crystals with high concentrations of mass defects. The coherent-potential approximation (CPA), the average-$t$-matrix approximation (ATA), and a perturbation theory were used in the calculations. ATA predicts frequency distributions which can be markedly different from those for CPA; the spectral functions Im $〈G_j(\overrightarrow{\mathrm{q}},\omega )〉$ show an even greater difference. At all alloy concentrations and host-defect mass ratios, there are some ATA spectral functions which have $\delta$-function components, and thus an unphysical zero width. In some cases, ATA also predicts spectral functions near the top of the host band which have pronounced double peaks. These double peaks do not occur in CPA, and are inconsistent with the results of a neutron scattering experiment on ${\mathrm{Ni}}_{55}$ ${\mathrm{Pd}}_{45}$. However, for moderate concentrations ($c\simeq 0.2$) of light defects, ATA may have some advantages over CPA.