Phonon Theory of Solid-State Diffusion Including Anharmonic Effects

Abstract

An anharmonic-lattice-vibration theory of solid-state diffusion, based on classical theory developed previously, is presented in second-quantization form and is evaluated for Cu. Equilibrium statistical mechanics was used, and the Goldstone diagrams were isolated for the interacting-phonon events which contribute to $D_0$. A displacement transformation suitable for the inhomogeneous strain inherent in the migration mechanism was used. The results of a comprehensive theoretical analysis of the migration portion of $D_0$ were estimated numerically for Cu, using a nearest-neighbor Debye approximation. The values of $D_0$ obtained, including an experimental value of the entropy of vacancy formation, were 0.106, 0.091, and 0.078 for temperatures of 293, 793, and 1293°K, respectively. The temperature dependence shown by $D_0$, in the classical limit, was due to the $T$ dependence of the atomic force constants, thermal expansion, and $T$-dependent anharmonic terms. At lower temperatures, quantum effects introduced terms with inverse powers of $T$ and mass. At 293°K, which is $0.865{\Theta }_D$ for Cu, there is a 7% decrease of $D_0$ due to quantum terms. The anharmonic terms introduced no direct effects on the mass dependence of $D_0$. If the first few anharmonic terms are included in an expansion of the activation energy, the first and most important term is linear in $T$ in the classical limit. Hence, it appears in the experimentally measured $D_0$, rather than in the activation energy. Therefore, harmonic-lattice-vibration theories or elastic theories of the activation energy should be realistic. A suggestion is made regarding the possibility of controlling the diffusion process by the artificial stimulation of phonons using laser radiation in a selected frequency range.