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Astron. Astrophys. 334, 363-375 (1998)

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1. Introduction

1.1. General

The detailed mechanism for the formation of the H2 molecule in the interstellar medium (ISM) really appears to stand up to now as an open question for astrophysicists. Indeed the strong abundance of H2 in the ISM cannot be explained by the classical three-body collisions due to the very low density of the interstellar gas (about 102 - 104 atoms/cm3 for a typical temperature of about 10 [FORMULA] K) nor by the appropriate mechanisms usually at work in space, like radiative association. Consequently astrophysicists have been interested for a long time in the dynamical processes which could induce a high efficiency of the H2 formation in such a rarefied medium.

One another fundamental point for the understanding of the chemical evolution of the ISM is the redistribution of available energy in the nascent H2 molecules. Indeed part of this energy could go into H2 vibration or/and rotation. On the other hand UV radiation from the stars induces electronic transitions to the B [FORMULA] (Lyman band) and C [FORMULA] (Werner band) electronic states. These two electronic states lead to the destruction of the H2 molecule by spontaneous radiative dissociation (coupling to the nuclear continuum) with an efficiency of about 10 [FORMULA] (Stephens & Dalgarno 1972; Schmoranzer et al. 1990). But they also radiatively de-excite to bound rovibrational states of the ground electronic state X [FORMULA]. This process, known as the UV pumping, prepares population in highly excited rovibrational levels in the ground X [FORMULA] which then cascade down by infrared (IR) emission. The spectrum and intensity of the IR radiation, now currently observed from both ground-based (Burton et al. 1992; Brand 1993; Field et al. 1994; Lemaire et al. 1996; Rouan et al. 1997; Sugai et al. 1997) and space-based ISO telescopes, will naturally depend on the UV radiation flux but also on the initial rovibrational population of the nascent H2 molecule, as recently demonstrated by Le Bourlot et al. (1995). Burton et al. (1992) have proposed that the formation mechanism could be at the origin of the excess IR emission observed in the NGC2023 reflection nebula. Futhermore information about the energy released into translation of H2 is important since it can contribute to the heating of the gas in diffuse clouds (Jura 1976).

As the presence of small particles (interstellar dust or grains) in the ISM is now well established (mainly through the spectral analysis of the extinction curve and by their typical infrared signature), reactive processes at the gas-solid interface have been proposed as an altervative solution to explain the H2 recombination (Hollenbach & Salpeter 1970, 1971; Jura 1975, Watson 1976; Goodman 1978). Astrophysical implications of these gas-surface interactions have been widely discussed through different models (Black & Dalgarno 1976; Hunter & Watson 1978; Duley & Williams 1986, 1993; Dalgarno 1993). Unfortunately the chemical nature, the size and the structure of these pieces of solid and the density of such material are not well characterized yet. Most recent results on the exctinction of dense regions in the infrared part of the spectrum, obtained thanks to ISO satellite (Abergel et al., private communication), show that the grain model proposed by Draine & Lee (1984) accounts well for the main features: absorption near 10 µm by the silicates component of the grain distribution, and gradual rise towards shorter wavelengths due to the graphite component. Indeed the graphite character of a significant proportion of the grains is also attested by the so-called "UV-bump" at 217 nm, as first suggested by Mathis et al. (1977) and later quantified in various ways by many authors (see in particular Fitzpatrick et al. (1990)). This presence of graphite has been our motivation for the present study: a microscopic description of the [FORMULA] formation near a "perfect" graphite surface. We have to keep in mind that the chemical reaction studied in this work (graphite+H +H - [FORMULA] graphite + H2) has to be considered as a model system for the ISM to extract general trends. In the ISM, the reality is obviously more complex due to different factors which can strongly affect the cross-sections associated to the reactive processes. For example the presence of condensable molecules ([FORMULA], CO, [FORMULA]...) on the surface of the dust particles in the molecular clouds (characterized by low temperature and low radiation flux) can induce strong changes in the electronic structure and consequently in the reactivity on these surfaces. Another important point to mention is that the structure (planar, spherical,...) of the grains and the eventual presence of defects on the surface could also play a crucial role in such reactive processes.

As the microscopic detailed mechanism of surface recombination is concerned, two schematic descriptions are generally proposed to explain the formation of a chemically bound system, here H2 near a surface. The first one, called the Langmuir-Hinschelwood (LH) mechanism assumes that the atomic reactants are already adsorbed on the surface. The adsorbates then migrate from site to site until they react upon encounter. The final product is then desorbed to the gaseous phase. In this process, reactivity occurs between two atoms already adsorbed and thermalized by the surface. Consequently, the reactivity will be strongly dependent on the surface temperature and coverage.

The second one, called the Eley-Rideal (ER) mechanism, corresponds to the process studied in this work. It is a direct interaction in which a reactive collision occurs between a gas phase atom and an adsorbed atom. If this adsorbed H atom is rather localized on the surface, we can expect that the reactivity will be less dependent on the surface temperature than in the LH mechanism. It is important to note that the energetics of the reaction will be different in the two processes due to the difference in the initial state of one of the reactants. Indeed the formation energy ([FORMULA]) is equal to [FORMULA] - [FORMULA] in the ER process while this energy is smaller and equal to [FORMULA] - 2 [FORMULA] in the LH process. [FORMULA] and [FORMULA] are respectively the binding energies of H and H2 on the graphite surface.

Both processes pre-suppose the existence of a non negligible atomic coverage on the grain surface to take place. It is then of interest to address the question of the nature of the adsorption sites which are occupied in relation to astrophysical conditions. The binding energy for an H atom on graphite can be estimated to about 1.2 eV in a chemisorption site (Fromherz et al. 1993) and to about 50 meV in a physisorption site (Mattera et al. 1980). If we don't consider the chemical reaction induced desorption, the atomic coverage is the result of the competition between the sticking of H -atoms coming from the gas phase and the thermal desorption. The efficiency of this process has a very strong temperature dependence. For typical conditions of the gas phase in interstellar clouds (density [FORMULA] [FORMULA] 1000 cm-3, gas temperature Tg [FORMULA] 100 K) it can be estimated that desorption will overcome sticking as soon as the grain temperature [FORMULA] is larger than 12 K for physisorption and 300 K for chemisorption. Thus for moderately dense interstellar clouds with [FORMULA] = 15-40 K we expect a significant chemisorption coverage and negligible physisorption. The typical time scale for the construction of the coverage is rather short ([FORMULA] 1 year).

1.2. The Eley-Rideal process

The LH mechanism is not the goal of this work. It will only be considered in the context of the discussion of astrophysical conditions.

In the ER process, studied here, three different exit channels are possible. The first one corresponds to the formation of the H2 molecule and its desorption to the gas phase. In this case, we will speak about reactive trajectories. This process can be summarized as:

[EQUATION]

It is important to note that this process can be direct or not. Indeed, during its trajectrory, the incoming H atom can be trapped on the surface, then diffuses more or less easily (depending on its translational kinetic energy) and eventually reacts after a long time. This indirect process is taken into account in the present simulation because the H adsorbed atom is initially located in a given (x,y) cell with periodic boundary conditions. It means that the simulated process corresponds in fact to the reactive collision between a H atom and a hydrogenated surface defined by a density of adsorbates per unit area, i.e. a coverage.

The second one corresponds to the collision of the H atom without involving chemical reaction. In this case, the H atom comes back to the gas phase after elastic and/or inelastic scattering. Such trajectories, which can be summarized as:

[EQUATION]

will be referred to as non-reactive.

The third one corresponds to the sticking of the incoming H atom:

[EQUATION]

This last process involves a coupling between the hydrogen atoms and the surface, which implies that energy transfer towards the bulk solid is possible. It will not be considered in the present work, for which the surface is assumed to be rigid.

From the experimental point of view, it is not easy to distinguish between the different mechanisms for various reasons. The first difficulty is for discreminating the nascent molecules from undissociated hydrogen molecules in the incoming beam or background gas. The second one is related to the actual state of the surface, in particular with respect to H and/or H2 adsorption. The smaller or larger coverage may affect the relative importance of LH versus ER mechanisms. These mechanisms have been investigated mainly on metallic surfaces (Rettner & Auerbach 1994; Rettner 1994; Schermann et al. 1994; Rettner & Auerbach 1996). Only very few experiments have been done on H2 formation from a gas-surface interaction in which the surface could be representative of an interstellar dust grain. In the recent experiments by Pirronello et al. (1997) the discrimination of products from reactants was achieved by detecting HD scattered from an olivine slab at very low temperature irradiated by two independent H and D gas lines. Evidence for the LH mechanism was found. A major interest of the experiments performed by Gough et al. (1996) on carbonaceous materials is to give access to the internal state distribution of the H2 product by means of an original technique. But the surface structure and exact content of the reactant gas is less securely characterized.

On the theoretical side, time-dependent fully-quantum calculations have been performed to describe the ER mechanism leading to the H2 molecule formation on a metallic copper surface Cu(111) in a collinear collision (Jackson & Persson 1992a, 1992b). More recently Jackson & Persson (1995) have done quantum calculations describing the reactive collision near a flat rigid surface, case in which the dynamics can be described with only three degrees of freedom. For the metallic surface, the flat approximation for the surface seems valid because the diffusion barrier is relatively low compared to the chemisorption energy.

In the present work, the corrugation of the graphite surface has been explicitely taken into account. Consequently, a classical and a quasi-classical (QCT) approach have been used. The collision has been analysed from a molecular dynamics (MD) simulation which involves the propagation of classical trajectories in the 12-D phase space. Recently Kratzer (1997) has followed the same theoretical scheme to describe the H2 formation on a silicon surface Si(001). It is interesting to note that Persson & Jackson (1995a, 1995b) have compared their fully-quantum calculations to the QCT method and concluded that the gross features of the dynamics are well described by the QCT method.

Sect. 2 of this paper describes the hamiltonian of the system and the empirical potential model which was used. Sect. 3 presents the computational aspect of the dynamics and the means used to characterize the system. Sect. 4 is devoted to the presentation and the discussion of the results obtained from the MD simulations.

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© European Southern Observatory (ESO) 1998

Online publication: May 12, 1998

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