Quantum theory of impurity migration in crystals

A quantum theory of impurity migration in crystals is proposed. The impurity state is taken in the form of a wave packet constructed out of its Bloch states in the host lattice. Its time evolution is studied including its interaction with the host lattice phonons. A correspondence is established between the classical diffusion equation and the time evolution of the probability density arising out of the impurity wave packet. The diffusion coefficient D_T and trapping rate γ_T are related to the imaginary part of the energy shift of the impurity caused by its interaction with phonons. The detailed calculations are carried out using second order perturbation theory for the energy shift. The Debye model for the host lattice and effective mass approximation for the impurity band are used. At low temperature D_T is found to be proportional to_T3/2, and at high temperature the Arrhenius formula of Vineyard is obtained. The estimated migration energy for μ⁺ migration in bcc metals agrees reasonably with the experimental values.