Effects of isotopic disorder on the Raman spectra of crystals: Theory and ab initio calculations for diamond and germanium

We present a method to study the effects of isotopic composition on the Ram an spectra of crystals, in which disorder is treated exactly without resort ing to any ;ind of mean-field approximation. The Raman cross section is exp ressed in terms of a suitable diagonal element of the vibrational Green's f unction, which is accurately and efficiently calculated using the recursion technique. This method can be used in conjunction with both semiempirical lattice-dynamical models and with first-principles interatomic force consta nts. We have applied our technique to diamond and germanium using the most accurate interatomic force constants presently available, obtained from den sity-functional perturbation theory. Our method correctly reproduces the li ght scattering in diamond-where isotopic effects dominates over the anharmo nic ones-as well as in germanium, where anharmonic effects are larger.