  The Tamura model is a particular type of 1$^{\text{st}}$ Born approximation
of the phonon-isotope scattering problem. The expression for the mode-resolved
phonon-isotope scattering rates in this model, derived in 1983, is still widely
used in ab initio transport calculations. While the original work emphasized
its applicability to low-energy acoustic phonons only, it has, nevertheless,
been also applied to optical phonons in the field of phonon transport. The
model has the salient feature of being a perturbation theory on top of a
virtual crystal background. As such, this approach does not correspond to the
proper methodology for solving the phonon-substitution defect problem in the
respective limit. Here, we explore three avenues to go beyond the Tamura model
and carry out calculations on a set of common materials to compare the
different approaches. This work allows a systematic improvement of the
treatment of phonon-isotope scattering in ab initio phonon transport, while
unifying it with the general phonon-substitution scattering problem.
