  We extend the analytic theory presented by Sternberg et al. (2014) and Bialy
& Sternberg (2016) for the production of atomic hydrogen (HI) via FUV
photodissociation at the boundaries of dense interstellar molecular (H$_2$)
clouds, to also include the effects of penetrating (low-energy) cosmic-rays for
the growth of the total HI column densities. We compute the steady-state
abundances of the HI and H$_2$ in one-dimensional gas slabs in which the FUV
photodissociation rates are reduced by depth-dependent H$_2$ self-shielding and
dust absorption, and in which the cosmic-ray ionization rates are either
constant or reduced by transport effects. The solutions for the HI and H$_2$
density profiles and the integrated HI columns, depend primarily on the ratios
$I_{\rm UV}/Rn$ and $\zeta/Rn$, where $I_{\rm UV}$ is the intensity of the
photodissociating FUV field, $\zeta$ is the H$_2$ cosmic-ray ionization rate,
$n$ is the hydrogen gas density, and $R$ is the dust-surface H$_2$ formation
rate coefficient. We present computations for a wide range of FUV field
strengths, cosmic-ray ionization rates, and dust-to-gas ratios. We develop
analytic expressions for the growth of the HI column densities. For Galactic
giant molecular clouds (GMCs) with multiphased (warm/cold) HI envelopes, the
interior cosmic-ray zones will dominate the production of the HI only if $\zeta
\gtrsim 4.5\times 10^{-16} \times (M_{\rm GMC}/10^6 \
M_{\odot})^{-1/2}$~s$^{-1}$, where $M_{\rm GMC}$ is the GMC mass, and including
attenuation of the cosmic-ray fluxes. For most Galactic GMCs and conditions,
FUV photodissociation dominates over cosmic-ray ionization for the production
of the HI column densities. Furthermore, the cosmic-rays do not affect the
HI-to-H$_2$ transition points.

Astrophysics of Galaxies