PHONON THEORY OF SOLID-STATE DIFFUSION INCLUDING ANHARMONIC EFFECTS

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 D0. 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 D0 were estimated numerically for Cu, using a nearest-neighbor Debye approximation. The values of D0 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 D0, 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 0.865ΘD for Cu, there is a 7% decrease of D0 due to quantum terms. The anharmonic terms introduced no direct effects on the mass dependence of D0. 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 D0, 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. © 1969 The American Physical Society.