Frequency Spectra in Disordered Alloys: An Interpolation Formula

The phonon-frequency spectrum of a mass-disordered alloy is calculated using the approximation of Elliott and Taylor for a reference lattice of mean-inverse atomic mass. The result is a relatively simple interpolation formula for the frequency spectrum which is symmetric in the two atomic types, is exact at both limits of concentration, and works qualitatively throughout the concentration range. A comparison is made with computer experiments in one and three dimensions. Good agreement is obtained in three-dimensional systems; the high-frequency structure coming from isolated clusters of atoms, which is important in one-dimensional systems, is not reproduced. The method is also used to calculate the weighted frequency spectrum from the displacement Green's function for a reference lattice of mean atomic mass. The results are contrasted with the true frequency spectrum. A comparison with inelastic neutron scattering data for 3 at.% W in Cr is made.