  Electrostatic-elastic coupling in colloidal crystals, composed of a mobile
Coulomb gas permeating a fixed background crystalline lattice of charged
colloids, is studied on the continuum level in order to analyze the
lattice-mediated interactions between mobile charges. The linearized, Debye-H\"
uckel-like mean-field equations allow for an analytic solution incorporating
the minimal coupling between electrostatic and displacement fields leading to
an additional effective attractive interaction between mobile charges. For
small screening lengths, the interactions between mobile like charges exhibit a
coupling-strength-dependent behavior, with lattice mediated effective
interaction ranging from weak to strong attraction, while for large screening
lengths the lattice mediated interaction remains purely attractive. By
analyzing Gaussian fluctuations around the mean-field solution we identify its
region of stability in terms of the electrostatic-elastic screening length. A
detailed continuum theory incorporating the standard lattice elasticity and
electrostatics of mobile charges augmented by the electrostatic-elastic
coupling terms can serve as a baseline for more detailed microscopic models.
