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 ${\omega }_1$ leading to migration and another mode $\omega$ which is representative of the remaining defect lattice modes. The frequency $\omega$ is allowed to vary along the migration path. The resonances obtained in the equilibrium configuration (EC) are given by $2{\omega }_0=n{\omega }_1$, where ${\omega }_0$ is $\omega$ 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 $\approx 2\times {10}^{-13\mathrm{}}$ 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 ${\phi }_{0r}\left(\mathrm{Cu}\right)\approx 6.5\times {10}^3{n}^2$ dyn/cm, where $n$ is the order of the resonance.