Diffusion via Transition State Theory

Unlike the analysis of the structure and equilibrium concentrations of point defects, questions of defect motion take us directly to the heart of some of the most important unsolved questions in nonequilibrium physics. In particular, while there are a host of useful empirical constructs built around the idea of Arrhenius activated processes with rates determined generically by expressions of the form 

In anticipation of the work we will do quantitatively below, let s first work up a qualitative understanding of the process. We begin by imagining the diffusing atom to occupy one of the wells as depicted in fig. 7.21. By virtue of thermal excitation, it is known that the atom is vibrating about within that well. The question we pose is how likely is it that the diffusing atom will reach the saddle point between the two wells The answer to that question allows us to determine the flux across the saddle point, and thereby, the rate of traversal of the saddle configuration which can then be tied to the diffusion constant. 

Our one-dimensional example imitates the presentation given by Eshelby (1975a) of the Vineyard (1957) rate theory. The transition rate may be computed as the ratio of two quantities that may themselves be evaluated on the basis of notions familiar from equilibrium statistical mechanics. The numerator of the expression of interest is given by the total number of particles crossing the saddle per unit time, while the denominator reflects the number of particles available to make this transition in the well from which the particles depart. In particular, the transition rate is 

In the context of fig. 7.21 if we consider the process of atomic hopping from left to right, the numerator requires us to evaluate properties at the saddle point, while the denominator demands an analysis of the properties of the well to the left of the saddle point. 

Our intention is to invoke the tools of statistical mechanics developed earlier to quantify these ideas. From the standpoint of our one-dimensional example, the transition rate of eqn (7.54) may be rewritten mathematically as 

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