We generalize the analysis of the normal modes for a rotating ionic Coulomb crystal in a Penning trap to allow for inhomogenieites 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.

The Coulomb repulsion between ions in a linear Paul trap gives rise to anharmonic terms in the potential energy when expanded about the equilibrium positions. We examine the effect of these anharmonic terms on the accuracy of a quantum simulator made from trapped ions. To be concrete, we consider a linear chain of Yb 171+ ions stabilized close to the zigzag transition. We find that for typical experimental temperatures, frequencies change by no more than a factor of 0.01% due to the anharmonic couplings. Furthermore, shifts in the effective spin-spin interactions (driven by a spin-dependent optical dipole force) are also, in general, less than 0.01% for detunings to the blue of the transverse center-of-mass frequency. However, detuning the spin interactions near other frequencies can lead to non-negligible anharmonic contributions to the effective spin-spin interactions. We also examine an odd behavior exhibited by the harmonic spin-spin interactions for a range of intermediate detunings, where nearest-neighbor spins with a larger spatial separation on the ion chain interact more strongly than nearest neighbors with a smaller spatial separation.