  We generalize the analysis of the normal modes for a rotating ionic Coulomb
crystal in a Penning trap to allow for inhomogeneities in the system. Our
formal developments are completely general, but we choose to examine a crystal
of Be+ ions with BeH+ defects to compare with current experimental efforts. We
examine the classical phonon modes (both transverse and planar) and we
determine the effective spin-spin interactions when the system is driven by an
axial spin-dependent optical dipole force. We examine situations with up to
approximately 15% defects. We find that most properties are not strongly
influenced by the defects, indicating that the presence of a small number of
defects will not significantly affect experiments.
