  We present the study of CI/H$_2$ relative abundance in the diffuse cold
neutral medium. Using the chemical and thermal balance model we calculated the
dependence of CI/H$_2$ on the main parameters of the medium: hydrogen number
density, metallicity, strength of the UV field, and cosmic ray ionization rate
(CRIR). We show that observed relative CI and H$_2$ column densities in damped
Lyman alpha systems (DLAs) at high redshifts can be reproduced within our model
assuming the typically expected conditions in the diffuse cold neutral medium
(CNM). Using the additional observed information the on metallicity, HI column
density, and excitation of CI fine-structure levels, as well as temperature we
estimated for a wide range metallicities in the CNM at high redshifts that
CRIRs to be in the range from $\sim10^{-16}$ to $\rm few \times 10^{-15}\rm
s^{-1}$, hydrogen number densities to be in range $\sim10 - 10^3$cm$^{-3}$, and
UV field in range from $10^{-2}$ to $\rm few \times 10^2$ of Mathis field. We
argue, that since the observed quantities used in this work are quite
homogeneous and much less affected by the radiative transfer effects (in
comparison with e.g. dissociation of HD and UV pumping of H$_2$ rotational
levels) our estimates are quite robust against the assumption of the exact
geometrical model of the cloud and local sources of the UV field.
