Correlated Walk and Diffusion Equations in a Driving Force

The correlation factor for diffusion by a vacancy mechanism arises because an atom can exchange with a given vacancy more than once. This results in a series of correlated exchanges between the atom and vacancy. To include correlation effects in the diffusion equations, one must allow the atom sufficient time to complete any correlated series of jumps. With this in mind, the basic diffusion equations are derived in terms of the number $N$ of correlated series. A complete description of planar diffusion then can be presented in terms of effective jump frequencies, where each jump is independent of previous jumps. The resulting equations are correct to first order in small quantities. This is shown to be true even when there are driving forces and a diffusion-coefficient gradient. Final expressions for the diffusion coefficient, correlation factor, drift velocity, and diffusion flux are valid for diffusion in any direction in cubic crystals and along particular directions in noncubic crystals. As an example, effects from a chemical-concentration gradient are discussed.