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

We compute the effective dispersion and density of states (DOS) of two-dimensional sub-regions of three dimensional face centered cubic (FCC) crystals with both a direct projection-inversion technique and a Monte Carlo simulation based on a common Hamiltonian. We study sub-regions of both (111) and (100) planes. For any direction of wavevector, we show an anomalous $\omega^2\sim q$ scaling regime at low $q$ where $\omega^2$ is the energy associated with a mode of wavenumber $q$. This scaling should 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 the (100) sub-region looks to be consistent with $D_\omega \sim \omega^3$, while the (111) shows something closer to the Debye result at the smallest frequencies. Our Monte Carlo simulation shows that \emph{finite sampling} artifacts act as an effective disorder and bias the $D_\omega$ in the same way as the \emph{finite size} artifacts, giving a behavior closer to $D_\omega \sim \omega^2$ than $D_\omega \sim \omega^3$. These results should have an important impact on interpretation of recent studies of colloidal solids where two-point displacement correlations can be obtained in real-space via microscopy.