Analytical Mechanics of Atomic Migration Including Nonadiabatic Effects

A theoretical analysis of the analytical mechanics of atomic migration in many-body condensed systems is given together with numerical calculations for Cu. In order to simplify the many-body problem, a realistic Morse-type potential is designed for metals for nuclear motions orthogonal to the migration direction. Then the well-developed classical theory of three-body chemical reactions is utilized. The vibrational energy for motions during a migration event is derived including nonadiabatic terms. The resonances are isolated, in part, between the localized mode frequency ω₁ leading to migration and another mode ω which is representative of the remaining defect lattice modes. The frequency ω is allowed to vary along the migration path. The resonances obtained in the equilibrium configuration (EC) are given by 2ω₀=nω₁, where ω₀ is ω in the EC and n is an integer. More complicated resonance relations are obtained for the saddle-point configuration (SPC) and for an arbitrary point between the EC and SPC. The jump time is derived, and a numerical evaluation for Cu yields a value ≈2×10^-13 sec for the time between the EC and SPC at 888 ^∘{}C. The jump time is found not to be strongly dependent on the ratio of force constants in the EC and SPC. For impurity diffusion a set of values of the impurity force constants at which resonance occurs is derived. For Cu, this is given by φ₀ᵣ(Cu)≈6.5×10³n² dyn/cm, where n is the order of the resonance.