Semi-empirical dielectric descriptions of the Bethe surface of the valence bands of condensed water

The Bethe surface of a material is an essential element in the study of inelastic scattering at low impact energies where the optical approximation fails. In this work we examine various semi-empirical models for the dielectric response function of condensed water towards an improved description of the energy-loss function over the whole energy–momentum plane (i.e. Bethe surface). The experimental “optical” data (i.e. at zero momentum transfer) for the valence bands of liquid and solid water are analytically represented by a sum-rule constrained linear combination of Drude-type functions. The dependence on momentum transfer is introduced through various widespread “extension” schemes which are compared against the available Compton scattering data. It is shown that the widely used Lindhard function along with its “single-pole” (or “δ-oscillator”) approximation used in the Penn and Ashley models, as well as the Ritchie and Howie extended-Drude scheme with a simple quadratic dispersion, predict a sharp Bethe ridge which compares poorly with the experimental profile. In contrast, the Mermin dielectric function provides a more realistic account of the observed broadening with momentum transfer. An improved fully-extended-Drude model is presented which incorporates the momentum broadening and line-shift of the Bethe ridge and distinguishes between the different dispersion of the discrete and continuum spectra of water.