Rate Processes in Solids

The jump rate $k$ for atomic diffusion via the vacancy mechanism in crystals is evaluated by treating a jump as the result of a particular type of thermal fluctuation in a crystal. The occurrence rate of this fluctuation is found (a) using a dynamical method similar in content to that of Rice and Manley, and (b) using statistical mechanics directly as was done by Vineyard. If the jumps are effected only when the surrounding atoms take up one single distorted configuration (the first approach being limited to describing one configuration only) then both methods are exactly equivalent and give the same $k$ (Sec. II)

The total $k$ is then obtained by summing the contributions to $k$ from the most significant configurations using a saddle point integration (Sec. III). This is done without introducing new (incalculable) "saddle-point" frequencies which in the Vineyard theory, leaves the pre-exponential factor imprecisely defined. In $k$, all parameters are well defined in terms of the force constants and its mass dependence is always exactly $m^{-\frac{1}{2}}$ [Eq. (27)].

In the discussion, contact is made with earlier forms of the frequency factor, and, by way of example, the expression for $k$ is used to evaluate the self-diffusion coefficient in solid argon giving a value in agreement with experiment (Sec. IV)