  In this work we investigate the effect a crystalline quark-hadron mixed phase
can have on the neutrino emissivity from the cores of neutron stars. To this
end we use relativistic mean-field equations of state to model hadronic matter
and a nonlocal extension of the three-flavor Nambu-Jona-Lasinio model for quark
matter. Next we determine the extent of the quark-hadron mixed phase and its
crystalline structure using the Glendenning construction, allowing for the
formation of spherical blob, rod, and slab rare phase geometries. Finally we
calculate the neutrino emissivity due to electron-lattice interactions
utilizing the formalism developed for the analogous process in neutron star
crusts. We find that the contribution to the neutrino emissivity due to the
presence of a crystalline quark-hadron mixed phase is substantial compared to
other mechanisms at fairly low temperatures ($\lesssim 10^9$ K) and quark
fractions ($\lesssim 30\%$), and that contributions due to lattice vibrations
are insignificant compared to static-lattice contributions. There are a number
of open issues that need to be addressed in a future study on the neutrino
emission rates caused by electron-quark blob bremsstrahlung. Chiefly among them
are the role of collective oscillations of matter, electron band structures,
and of gaps at the boundaries of the Brillouin zones on bremsstrahlung, as
discussed in the summary section of this paper. We hope this paper will
stimulate studies addressing these issues.
