Diffusion-reaction model for positron trapping and annihilation at grain boundaries of rectangular-shaped columnar grains

The present work aims at an extension of the diffusion-reaction model to two spatial variables for treating positron trapping and annihilation at grain boundaries of polycrystals with grains of rectangular column shape. Analytical solutions for the positron lifetime spectrum and the mean positron lifetime are derived taking into account concomitant transition-limited positron trapping at intragranular point defects. Despite the two-dimensional nature of the diffusion-reaction problem, single-sum expressions could be deduced for the characteristic positron annihilation parameters. The relevant limiting cases are considered, i.e., entirely reaction-limitation, yielding the solution of standard rate theory, entirely diffusion-limitation as well as entirely reaction-limitation in one direction when the extension of the rectangular column in this direction is much shorter than in the other. The present model contains the known solution for the platelet structure as the limiting case of infinite extension of the rectangular cross-section in one direction. In the regime of reaction-controlled trapping, the positron trapping rate at grain boundaries is determined by the ratio between perimeter and cross-section of the columnar-shaped grains. Besides this geometric ratio, the shape of the columns (square-, rectangular-, cylindrical-shaped) becomes relevant, when positron trapping rate at grain boundaries is in addition diffusion-limited. .