Nuclear reactions in a lattice are described using an extension of the resonating group method to include the lattice explicitly. Phonon exchange during fusion and dissociation reactions is predicted Second-order site-other-site reactions are predicted under conditions where the reactions at each site exchange phonons with a common phonon mode. The null reaction in which a dd-fusion at one site is coupled to He dissociation at another site is modeled. Coupled-channel equations are developed for the twosite problem, as an illustration of the application of the lattice resonating group method. We have proposed previously that there should exist compact state solutions of the coupled-channel equations, as the associated exchange potential can be attractive. Such states have been proposed to account for the Kasagi effect, and to provide a foundation for many of the anomalies that are seen in metal deuterides. Our analysis of the twosite system suggests so far has not yielded compact state solutions. Preliminary results for the many-site problem are presented, which leads to evidence in support of the stability of the compact states, and which leads to a new overall picture for the anomalies.