Exact results for hydrogen recombination on dust grain surfaces

The recombination of hydrogen in the interstellar medium, taking place on surfaces of microscopic dust grains, is an essential process in the evolution of chemical complexity in interstellar clouds. Molecular hydrogen plays an important role in absorbing the heat that emerges during gravitational collapse, thus enabling the formation of structure in the universe. The H-2 formation process has been studied theoretically, and in recent years also by laboratory experiments. The experimental results were analyzed using a rate equation model. The parameters of the surface that are relevant to H-2 formation were obtained and used in order to calculate the recombination rate under interstellar conditions. However, it turned out that, due to the microscopic size of the dust grains and the low density of H atoms, the rate equations may not always apply. A master equation approach that provides a good description of the H-2 formation process was proposed. It takes into account both the discrete nature of the H atoms and the fluctuations in the number of atoms on a grain. In this paper we present a comprehensive analysis of the H-2 formation process, under steady state conditions, using an exact solution of the master equation. This solution provides an exact result for the hydrogen recombination rate and its dependence on the flux, the surface temperature, and the grain size. The results are compared with those obtained from the rate equations. The relevant length scales in the problem are identified and the parameter space is divided into two domains. One domain, characterized by first order kinetics, exhibits high efficiency of H-2 formation. In the other domain, characterized by second order kinetics, the efficiency of H-2 formation is low. In each of these domains we identify the range of parameters in which, due to the small size of the grains, the rate equations do not account correctly for the recombination rate and the master equation is needed.