Theory of the Vibrations of Dilute Alloys with Short-Range Order

Using thermally averaged double-time Green's functions, we develop a theory to calculate the effect of a small concentration of nonrandom mass defects on the vibrational properties of a monatomic crystal. The low-concentration approximation used is shown to be equivalent to that used by Elliott and Taylor for the random-impurity case, correct to first order but only approximately correct to higher orders in the concentration. This simple theory is used to find a shift of the resonant or local-mode peak due to short-range order among defects, which might be seen by infrared absorption in imperfect insulators. The integrated absorption is shown to be independent of the ordering for charged defects, but not for uncharged defects which induce optical absorption through atomic deformations. The general expression derived for the inelastic coherent neutron scattering cross section includes a branch-mixing term which disappears for scattering vectors of high symmetry. Using the Debye approximation (for which the cross section can be written in a self-energy form) and the appropriate short-range order parameters from the linear theory of Clapp and Moss, we calculate the shifts and widths of the neutron scattering peaks for ${\mathrm{Cu}}_{0.907}$ ${\mathrm{Au}}_{0.093}$. The agreement with the experimental results of Svensson, Brockhouse, and Rowe is not good. A small clustering of light mass defects, represented approximately by nearest-neighbor correlations, is shown to broaden a low-frequency impurity band but to have relatively little effect on a high-frequency local mode.