  We theoretically and experimentally examine the effects of anharmonic terms
in the trapping potential for linear chains of trapped ions. We concentrate on
two different effects that become significant at different levels of
anharmonicity. The first is a modification of the oscillation frequencies and
amplitudes of the ions' normal modes of vibration for multi-ion crystals,
resulting from each ion experiencing a different curvature in the potential. In
the second effect, which occurs with increased anharmonicity or higher
excitation amplitude, amplitude-dependent shifts of the normal-mode frequencies
become important. We evaluate normal-mode frequency and amplitude shifts, and
comment on the implications for quantum information processing and quantum
state engineering. Since the ratio of the anharmonic to harmonic terms
typically increases as the ion--electrode distance decreases, anharmonic
effects will become more significant as ion trap sizes are reduced. To avoid
unwanted problems, anharmonicities should therefore be taken into account at
the design stage of trap development.
