Phonon exchange with nuclei in the course of fusion reactions that occur in a solid has not been analyzed previously. This problem has become of interest in connection with claims of observations of anomalies in metal deuterides. If the strong force interaction were dependent only on position (and not spin or isospin), then the coupling with phonons can be developed directly. Since a nuclear interaction can change the lattice constituents, the initial and final state lattices can be different, and we must include this in the formulation. For more realistic strong force models with spin and isospin dependence, we can use correlated nuclear wavefunctions which are made up of products of space, spin and isospin components. In this case, the spin and isospin algebra can be done analytically, producing channel-dependent potentials that are only space dependent. The formulation that results can be used for quantitative estimates of phonon exchange.