  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.
