  The accurate characterization of eigenmodes and eigenfrequencies of
two-dimensional ion crystals provides the foundation for the use of such
structures for quantum simulation purposes. We present a combined experimental
and theoretical study of two-dimensional ion crystals. We demonstrate that
standard pseudopotential theory accurately predicts the positions of the ions
and the location of structural transitions between different crystal
configurations. However, pseudopotential theory is insufficient to determine
eigenfrequencies of the two-dimensional ion crystals accurately but shows
significant deviations from the experimental data obtained from resolved
sideband spectroscopy. Agreement at the level of 2.5 x 10^(-3) is found with
the full time-dependent Coulomb theory using the Floquet-Lyapunov approach and
the effect is understood from the dynamics of two-dimensional ion crystals in
the Paul trap. The results represent initial steps towards an exploitation of
these structures for quantum simulation schemes.
