  We study a fermion field coupled to a scalar via a Yukawa term. The scalar
field is the $\phi^4$ model with an impurity that preserves half of the BPS
property. We analyze the spectrum of the defects of the model and collisions
between them both close to the BPS regime and not. As the fermion binds to
these defects, it may be transferred from one to the other, which we quantify
via overlaps, known as Bogoliubov coefficients. BPS collisions are less likely
to transfer the fermion between defects and can be adiabatic for
non-relativistic velocities, especially for small coupling constants. Moreover,
closer to the BPS limit only a small fraction of the fermion number is radiated
away. In contrast, non-BPS collisions lead to more radiation in the fermion
field and excitation of the fermion to higher bound states, and the result is
more sensitive to the parameters.
