Quantum theory of interstitial diffusion in crystals

Abstract

A quantum theory of diffusion is proposed based on the following premises. The impurity state is taken in the form of a wave packet constructed out of its Bloch states in the host lattice, and its time evolution is studied including its interaction with the host-lattice phonons. It is shown that a correspondence can be established between the classical diffusion equation and the time evolution of the probability density arising out of the impurity wave packet. The diffusion and trapping coefficients DT and gamma T are related to the imaginary part of the energy shift of the impurity caused by its interaction with phonons. The detailed calculations are based on second-order perturbation theory for the energy shift, the Debye model for the host lattice and the effective-mass approximation for the impurity band. At low temperatures DT is found to be proportional to T32/, and at high temperatures the Arrhenius formula of Vineyard (1957) is obtained. The estimated migration energies for mu + diffusion in BCC metals agree reasonably with the experimental values. The conditions for non-existence of gamma T are also discussed.