The presence of small amounts of atomic hydrogen, detected as absorption dips in the 21 cm line spectrum, is a well-known characteristic of dark clouds. The abundance of hydrogen atoms measured in the densest regions of molecular clouds can be only explained by the dissociation of H$_2$ due to cosmic rays. We want to assess the role of Galactic cosmic rays in the formation of atomic hydrogen, by using recent developments in the characterisation of the low-energy spectra of cosmic rays and advances in the modelling of their propagation in molecular clouds. We model the attenuation of the interstellar cosmic rays entering a cloud and compute the dissociation rate of molecular hydrogen due to collisions with cosmic-ray protons and electrons as well as fast hydrogen atoms. We compare our results with the available observations. The cosmic-ray dissociation rate is entirely determined by secondary electrons produced in primary ionisation collisions. These secondary particles constitute the only source of atomic hydrogen at column densities above $sim10^{21}$ cm$^{-2}$. We also find that the dissociation rate decreases with column density, while the ratio between the dissociation and ionisation rates varies between about 0.6 and 0.7. From comparison with observations we conclude that a relatively flat spectrum of interstellar cosmic-ray protons, as the one suggested by the most recent Voyager 1 data, can only provide a lower bound for the observed atomic hydrogen fraction. An enhanced spectrum of low-energy protons is needed to explain most of the observations. Our findings show that a careful description of molecular hydrogen dissociation by cosmic rays can explain the abundance of atomic hydrogen in dark clouds. An accurate characterisation of this process at high densities is crucial for understanding the chemical evolution of star-forming regions.
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