Diffusion-reaction model for positron trapping and annihilation at spherical extended defects and in precipitate--matrix composites

The exact solution of a diffusion$-$reaction model for the trapping and annihilation of positrons in small extended spherical defects (clusters, voids, small precipitates) with competitive rate-limited trapping in vacancy-type point defects is presented. Closed-form expressions are obtained for the mean positron lifetime and for the intensities of the two positron lifetime components associated with trapping at defects. The exact solutions can be conveniently applied for the analysis of experimental data and allow an assessment in how far the usual approach, which takes diffusion limitation into account by means of effective diffusion-trapping rates, is appropriate. The model is further extended for application to larger precipitates where diffusion- and reaction limited trapping is not only considered for the trapping from the matrix into the precipitate$-$matrix interface but also for the trapping from inside the precipitates into the interfaces. This makes the model applicable to all type of composite structures where spherical objects are embedded in a matrix irrespective of their size and their number density.