Inferring elastic properties of an fcc crystal from displacement correlations: sub-space projection and statistical artifacts

We compute the effective dispersion and vibrational density of states (DOS) of two-dimensional sub-regions of three dimensional face centered cubic (FCC) crystals using both a direct projection-inversion technique and a Monte Carlo simulation based on a common underlying Hamiltonian. We study both a (111) and (100) plane. We show that for any given direction of wavevector, both (111) and (100) show an anomalous $\omega^2\sim q$ regime at low $q$ where $\omega^2$ is the energy associated with the given mode and $q$ is its wavenumber. The $\omega^2\sim q$ scaling should be expected to give rise to an anomalous DOS, $D_\omega$, at low $\omega$: $D_\omega \sim \omega^3$ rather than the conventional Debye result: $D_\omega\sim \omega^2$. The DOS for (100) looks to be consistent with $D_\omega \sim \omega^3$, while (111) shows something closer to the conventional Debye result at the smallest frequencies. In addition to the direct projection-inversion calculation, we perform Monte Carlo simulations to study the effects of finite sampling statistics. We show that \emph{finite sampling} artifacts act as an effective disorder and bias $D_\omega$, giving a behavior closer to $D_\omega \sim \omega^2$ than $D_\omega \sim \omega^3$. These results should have an important impact on the interpretation of recent studies of colloidal solids where the two-point displacement correlations can be obtained directly in real-space via microscopy.