  A model of the collisional kinetics of energetic hydrogen atoms, molecules,
and ions in pure H$_2$ discharges is used to predict H$_\alpha$ emission
profiles and spatial distributions of emission from the cathode regions of
low-pressure, weakly-ionized discharges for comparison with a wide variety of
experiments. Positive and negative ion energy distributions are also predicted.
The model developed for spatially uniform electric fields and current densities
less than $10^{-3}$ A/m$^2$ is extended to non-uniform electric fields, current
densities of $10^{3}$ A/m$^2$, and electric field to gas density ratios $E/N =
1.3$ MTd at 0.002 to 5 Torr pressure. (1 Td = $10^{-21}$ V m$^2$ and 1 Torr =
133 Pa) The observed far-wing Doppler broadening and spatial distribution of
the H$_\alpha$ emission is consistent with reactions among H$^+$, H$_2^+$,
H$_3^+$, and H$^-H$ ions, fast H atoms, and fast H$_2$ molecules, and with
reflection, excitation, and attachment to fast H atoms at surfaces. The
H$_\alpha$ excitation and H$^-$ formation occur principally by collisions of
fast H, fast H$_2$, and H$^+$ with H$_2$. Simplifications include using a
one-dimensional geometry, a multi-beam transport model, and the average
cathode-fall electric field. The H$_\alpha$ emission is linear with current
density over eight orders of magnitude. The calculated ion energy distributions
agree satisfactorily with experiment for H$_2^+$ and H$_3^+$, but are only in
qualitative agreement for H$^+$ and H$^-$. The experiments successfully modeled
range from short-gap, parallel-plane glow discharges to beam-like,
electrostatic-confinement discharges.
