Identities and approximations are developed for the first and second moments of the spectral density of center-of-mass oscillations of a commensurate quantum monolayer solid. This corresponds to the one-phonon approximation for inelastic scattering at a reciprocal lattice vector. The results are applied with parameters for monolayers of helium/graphite and hydrogen/graphite. Estimates of the zone-center gap with averages in an optimized Jastrow trial function are internally consistent to better than 10%. There is a marginal level of agreement between the zone-center gap for helium/graphite observed with inelastic neutron scattering and the gap calculated with the Fourier component of the helium/graphite potential derived from atom-surface scattering. The application for hydrogen/graphite amounts to a reanalysis of previous calculations and is in good agreement with them.

• Received 4 October 1993

DOI:https://doi.org/10.1103/PhysRevB.49.7654