Vibrations of random dilute alloys

Abstract

The vibrational properties of a crystal with a randomly arranged small concentration c of impurities are considered by a thermal Green function technique. Detailed calculations are made for the case when the impurities have changed mass. The Green functions lead directly to average and weighted displacement and momentum correlations which may in turn be related to observable properties such as density of states, thermal energies, optical absorption, neutron scattering, etc. Detailed calculations are presented in a number of cases based on realistic models and compared with experiment. The method treats correctly multiple interactions of a vibrational quantum with a single defect but treats interference between defects only approximately. It is therefore correct to order c but only approximately correct to higher orders. Resonances of the impurities which occur within the phonon spectra are satisfactorily described. The impurity bands which occur around the frequencies of the localized modes of the isolated defects are also described but the approximations are less satisfactory in this case.