A multi-beam model for low-current, very high E/N discharges in hydrogen.

The multi-beam model of Helm and St\"{o}ri\footnote{D. Helm and H. St\"{o}ri, J. Appl. Phys. {\bf 72}, 3330 (1992).} has been applied to the motion and reactions of H$^+$, H$_2^+$, H$_3^+$, fast H$_2$, and fast H in H$_2$ for a uniform electric field. Plots of most of our analytic expressions for the cross sections are available\footnote{A. Bogaerts and R. Gijbels, Spectrochim. Acta Part B {\bf 57}, 1071 (2002).}. Elastic scattering of non-identical particles is modelled with an energy loss for backward scattering in center-of-mass and the elastic momentum transfer cross section. For identical particles, we use the energy loss for 90$^\circ$ scattering and the elastic viscosity cross section. Calculated rates of excitation of H$_\alpha$ and the uv continuum versus distance and pressure are compared with experiments\footnote{H.A.M. Blasberg and F.J. de Hoog, Physica {\bf54}, 468 (1971).}$^,$\footnote{Z.Lj. Petrovi\'c, B.M. Jelenkovi\'c, and A.V. Phelps, Bull. Am. Phys. Soc. {\bf 37}, 1951 (1992).}. Calculated ion and fast neutral fluxes at the cathode will be compared with calculations using Monte Carlo methods\footnote{T. \v{S}imko, Z. Donk\'o, and K. R\'ozsa, 23rd Intl'l. Conf. on Ionization Phenomena in Gases, (Univ. Paul Sabatier, Toulouse, 1997), p. II-64.}.