Inferring elastic properties in colloidal solids: artifacts of a restricted observation window

Recently, it has been shown how to extract information about the effective elasticity in colloidal solids, granular packings, {\it etc.}, using two point displacement correlations as obtained in, \emph{e.g.}, optical microscopy experiments or computer simulations. At its core, this technique relies on the observation that, within the harmonic approximation, the Hamiltonian, $H$, is the inverse of the elastic response function, $G$, \emph{defined over the whole domain of the elastic body}. However, most experiments (and even most simulations) have access to G only over some restricted sub-domain of the experimental system. Here, we study restricted observation domains of various size and dimensionality in face centered cubic (fcc) crystals of various size using a pseudo-analytic approach in which $G$ is obtained analytically and is inverted numerically \emph{on a compact sub-domain} to obtain the projected Hamiltonian, $\tilde{H}$. We show that the effective plane-wave energy, $E_k=\langle \psi_k | \tilde{H}| \psi_k \rangle$, for either a [111] or [100] planar subdomain has an unusual dispersion, $E\sim k$, rather than the familiar $E\sim k^2$ and motivate this observation from continuum considerations. We also show how this leads to an anomaly in the density of states of $\tilde{H}$.