A multi-term solution of the space–time Boltzmann equation for electrons in gases and liquids

In this study we have developed a full multi-term space–time solution of Boltzmann's equation for electron transport in gases and liquids. A Green's function formalism is used that enables flexible adaptation to various experimental systems. The spatio-temporal evolution of electrons in liquids in the non-hydrodynamic regime is benchmarked for a model Percus–Yevick (PY) liquid against an independent Monte Carlo simulation, and then applied to liquid argon. The temporal evolution of Franck–Hertz oscillations in configuration and energy space are observed for the model liquid with large differences apparent when compared to the dilute gas case, for both the velocity distribution function components and the transport quantities. The packing density in the PY liquid is shown to influence both the magnitude and wavelength of Franck–Hertz oscillations of the steady-state Townsend (SST) simulation. Transport properties are calculated from the non-hydrodynamic theory in the long time limit under SST conditions which are benchmarked against hydrodynamic transport coefficients. Finally, the spatio-temporal relaxation of low-energy electrons in liquid argon was investigated, with striking differences evident in the spatio-temporal development of the velocity distribution function components between the uncorrelated gas and true liquid approximations, due largely to the presence of a Ramsauer minimum in the former and not in the latter. (paper)