Nonequilibrium Processes in Isotopically Disordered Crystals

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Abstract
The equations of motion of the atoms in an isotopically disordered crystal, which contains a fraction p of atoms of mass M1, and a fraction 1−p of atoms of mass M2, are expanded in terms of the normal coordinates of a monatomic lattice whose atoms all have mass M=pM1+(1−p)M2. The equations of motion of these normal coordinates are derived and are then solved by Laplace transform methods. The perturbed normal coordinates are found to decay exponentially into the future and into the past until an inverse power dependence on time becomes dominant. Calculations of the mean lifetime and frequency shift of each normal coordinate are carried out for the one‐dimensional case. A theory of the optical absorption spectrum of an isotopically disordered ionic crystal is obtained, and the distribution function for the energies of the normal modes and the mean energy in a normal mode are found. The generalization of the methods of this paper to three‐dimensional lattices is discussed.