  We employ spin-dependent optical dipole forces to characterize the transverse
center-of-mass (COM) motional mode of a two-dimensional Wigner crystal of
hundreds of $^9$Be$^+$. By comparing the measured spin dephasing produced by
the spin-dependent force with the predictions of a semiclassical dephasing
model, we obtain absolute mode temperatures in excellent agreement with both
the Doppler laser cooling limit and measurements obtained from a previously
published technique (B. C. Sawyer et al. Phys. Rev. Lett. \textbf{108}, 213003
(2012)). Furthermore, the structure of the dephasing histograms allows for
discrimination between initial thermal and coherent states of motion. We also
apply the techniques discussed here to measure, for the first time, the ambient
heating rate of the COM mode of a 2D Coulomb crystal in a Penning trap. This
measurement places an upper limit on the anomalous single-ion heating rate due
to electric field noise from the trap electrode surfaces of
$\frac{d\bar{n}}{dt}\sim 5$ s$^{-1}$ for our trap at a frequency of 795 kHz,
where $\bar{n}$ is the mean occupation of quantized COM motion in the axial
harmonic well.
