General matrix equations for diffusion drift velocity in a driving force

From a complete-path approach in which defects are followed from the time of their creation at defect sources to the time of their destruction at defect sinks, general matrix equations are developed which relate the atom drift velocity ${\nu }_k$ for diffusion in a driving force to simple matrices whose elements can be written by inspection. Previous equations for ${\nu }_k$ have been restricted either to crystals and defects having high symmetry or to cases where the $x$ components of any atom jumps were either ± $b$ or 0. By contrast, the present equations involve no such restrictions. They apply to diffusion by interstitialcy, divacancy, and other more complex mechanisms, independent of the presence of symmetry planes, and allow calculation of diffusion drift velocities even when the individual atom jumps provide a variety of jump distances along the diffusion direction. These equations apply, for example, even when jumps to both nearest- and next-nearest-neighbor sites are allowed and to diffusion in arbitrary crystallographic directions even in noncubic crystals. The present equations can be expressed in terms of a general matrix $S$. The relation of $S$ to the less general matrices $S_{+}$ and $S_{-}$ defined previously for crystals having mirror symmetry planes normal to the diffusion direction is discussed.