Astron. Astrophys. 362, 774-779 (2000)

2. H₂ formation

The present proposal introduces a reservoir of electrons stored in negative ions, primarily H^-. We show in Sect. 2.1 that this reservoir represents a negligible perturbation to the distribution of charge within the gas. We also consider other processes, such as photodetachment, which may compete with associative detachment in removing H^-. In Sect. 2.2, the mobility of electrons on a grain surface is discussed. In the concluding remarks in Sect. 3, experimental, theoretical and observational aspects of the model are mentioned.

The surface formation of H^- may take place through either a Rideal-Eley mechanism, a Hinshelwood mechanism or a combination. In a Rideal-Eley mechanism, a gas phase H atom removes an electron directly from a grain surface. This is appropriate if there are abundant electrons on the grain surface (see Sect. 2.2). A mechanism may also be envisaged whereby an electron encounters the grain and attaches to an adsorbed H atom via a Rideal-Eley process. This may operate if there is extensive coverage of H, for example on a cold grain. In a Hinshelwood mechanism, a gas phase H atom adsorbs on the surface and then forms H^- through diffusive exploration of the surface by H and electrons. The Hinshelwood mechanism is very similar to the standard H + H formation process, but here one adsorbed H atom is replaced by an electron on the surface.

It is not evident that a surface adsorbed H atom reacting with a surface adsorbed electron would necessarily lead to evaporation of the resultant H^-. If the assumption of a negligible image force, as suggested above for electrons, may be extended to H^-, then H^- would not be retained on the surface electrostatically. However H^- may become bound to the surface by chemical forces. H^- on the surface (however bound) could for example react with a surface-mobile H, perhaps enhancing the efficiency of H+H recombination on a grain surface. In addition if a grain is overall positively charged, for example in a diffuse region, then H^- would require some kinetic energy to escape the surface. For example an energy of 14 meV (170 K) would be required to escape a grain of radius 0.1 µm with a single positive charge. Retention of H^- on the surface is not considered further in the present work for lack of appropriate data.

2.1. Abundances of H^-

The formation rate of molecular hydrogen in the standard model is given by a rate of hydrogen atoms adsorbing on grains, multiplied by an efficiency for conversion of adsorbed H atoms into H₂ via a surface chemical reaction. The formation rate in the present case is given in a very similar manner, again by a rate of H atoms striking grains, but now multiplied by an efficiency for conversion of H into gas phase H^- through attachment of an electron at the surface. The rate of reaction of H^- in the gas phase with H atoms to form H₂ is effectively instantaneous on the timescale of production of H^- by collision with grains. The relative efficiency of the standard surface and negative ion mechanisms is given simply by the relative efficiency of conversion of H atoms into H₂ at the surface and the efficiency of H^- production at the surface.

The rate of formation of H₂ per cm³ may be expressed as , where N is the total number density (= n(H) + 2n(H₂)) in the medium, n(H) is the number density of H atoms and k is an effective second order rate coefficient. If a typical average grain population is assumed with radius 0.1  µm, density 3 g cm^-3, grain to gas mass ratio = 10^-2, then at temperature T, k may be estimated to be  6.6 x 10^-18 cm³ s^-1 for unit electron attachment efficiency of H at the surface.

The effective second order rate coefficient k has a value, given unit surface attachment efficiency, which is the same as that associated with the standard mechanism of H₂ formation, which also assumes unit efficiency at the surface. This arises since the rate of H₂ formation is effectively first order in the H atom concentration in both cases and the rate determining step is the flux of H to the surface. Since k  1-3x10^{- 17} cm³ s^-1 is the value required to match observations (Jura 1975), the present proposal suffers the same constraint as the standard model, that the efficiency of the surface process be high.

The steady-state gas phase abundance of H^- implied by the model may be estimated by considering the formation and destruction pathways and is given by

where is the rate coefficient for the reaction of H with H^-, reaction (2), with a value of 1.3 x 10^-9 cm³ s^-1 (Schmeltekopf et al. 1967). k^ph is the rate coefficient for photodetachment, integrated over all wavelengths, with a value of 3.4 x 10^-8 s^-1 for the unshielded ISM (Rawlings et al. 1988). is the rate coefficient for removal of H^- through reaction with all positive ion species, number density n(I⁺), with a value of 4x10^-6 cm³ s^-1, assumed insensitive to the nature of the positive ion (Dalgarno & McCray 1973). k^add refers to additional processes which are considered separately below and are found to contribute negligibly to H^- destruction under the various conditions under which H₂ formation takes place.

Three different environments are considered, the interior of a dark cloud, the diffuse region at the borders of such a cloud and a dense photodissociation region (PDR). The number density of H and of positive ions may be specified as follows. Turning first to a dark cloud, the equilibrium density of H atoms is given approximately by the cosmic ray ionizing flux, expressed as a frequency, /10^-17 cm^-3 (e.g. Le Bourlot et al. 1995) and is thus independent of total number density. A figure of between 1 and 5 cm^-3 may therefore be appropriate. Measurements of n(H) have however been made in the dark cloud L134 (van der Werf et al. 1988) which show a considerably larger value lying between 30 and 40 cm^-3. The authors suggested that this may indicate incomplete H to H₂ conversion and that this may in fact be typical of a dark cloud. We adopt the observational figure of (say) 30 cm^-3 and a temperature of 10 K. The number density of positive ions may be estimated, following Oppenheimer & Dalgarno 1974, from [I⁺]3.4x10^-5, where the prevalent ion may be assumed to be Na⁺. is the factor of elemental depletion, assumed to be 0.1. Using L134 as an example, the core has a density of 1.2 x 10⁴ cm^-3 (van der Werf et al. 1988) and the degree of ionization is correspondingly  7 x 10^-7, suggesting a total ion concentration of  8x10^-3 cm^-3. In a dark cloud, the photo-detachment rate coefficient in Eq. 3 is unimportant since the flux of radiation generated within a dark cloud through cosmic ray events yields a negligible radiation field for photodetachment (Le Bourlot et al. 1995). Ignoring for the present k^add, Eq. 3 yields a number density of H^- of  1.5 x 10^-4 cm^-3. Thus the charge carried by H^- in these regions is  2% of the total.

Moving to diffuse photodissociation regions at the borders between a dark molecular cloud and the intercloud medium, there may for example be a total number density of 500 cm^-3, with about 20% of H-nuclei in the form of H₂ and 80% as H atoms, with a temperature of (say) 300 K. Ionization in this region is dominated by C⁺ and may typically be as high as 10^-4 or more (van Dishoeck 1998). Assuming that the interstellar radiation field is unshielded, Eq. 3 yields a number density of H^- of  4 x 10^-5 cm^-3. The charge carried by H^- in these regions is therefore less than 0.1% of the total. In addition, in the diffuse medium specified here the removal of H^- by reaction with H to form H₂ is  4 times more rapid than the removal by positive ions, represented by n(I⁺) in Eq. 3.

With regard to the relative importance of reaction of H^- with H and photodetachment of H^- in diffuse regions, the photodetachment rate of H^- and the H₂ formation rate are equal for an H atom concentration of 25 cm^-3. Passing from the intercloud medium into a molecular cloud, H:H₂ ratios would reflect the balance between associative detachment, photodetachment of H^- and photodissociation of H₂ (Jura 1975), with reactions between H^- and positive ions also playing a role, as indicated above. Visual extinction eventually suppresses both photodissociation, at A_v of unity, and photodetachment at A_v of (say) 10 magnitudes. Accurate estimates of the H/H₂ ratio require detailed modeling taking account of density profiles, but it is evident that diffuse clouds are predicted to be surrounded by a halo containing H atoms to a depth of a few tenths to  1 pc. Substantial column densities of H are observed in the direction of clouds such as Oph, Per and Per (Bohlin et al. 1978; Wagenblast 1992) consistent with both the present and the standard models of H₂ formation.

Turning to dense PDRs associated with reflection nebulae, the proximity of a hot star yields a radiation field which may be several thousand times the average interstellar field. For example in NGC2023, there is a region of vibrationally excited H₂ emission close to the exciting B-star, where the radiation field is  5000 times the average field and the number density has been estimated to be  10⁵ cm^-3 (Field et al. 1998). The gas has a high proportion of H atoms in the emitting region. The photodetachment rate coefficient is 1.7 x 10^-4 s^-1 and the pseudo-first order rate coefficient for the formation of H₂ is 1.3 x 10^-4 s^-1 for n(H)  N. Thus the processes of reaction with H atoms to form H₂ or to photodetach compete strongly in such a region. In this connection, the value of A_v in NGC2023 is only  10^-2 at a distance of 10^-4 pc into the illuminated gas, where most of the H₂ emission is formed. Assuming that the degree of ionization does not exceed a few x 10^-3, the influence of H^- removal by positive ions is not important in these regions, noting that the temperature is  500 K (Lemaire et al. 1999).

Collisional detachment of H^- with H atoms and with gas phase electrons

are additional mechanisms that may in principle compete with reaction of H^- with H to form H₂. Reactions (4) and (5) are represented by k^add in Eq. 3. Reaction (4) however takes place only at higher kinetic energy and can be ignored (Esaulov 1986). Reaction (5) has been studied in detail by Vejby-Christensen et al. 1996 and has a threshold energy of 1.5 eV. Reaction (5) is therefore unlikely to influence the chemistry of H^- in the present context.

Associative detachment forms nascent populations of H₂ preferentially in v=6,7,8,9 states (Black et al. 1981; Launay et al. 1991; Cizek et al. 1998). Vibrationally excited H₂ rapidly undergoes dissociative attachment, the reverse of reaction 2 (see Gauyacq 1985). For example H₂(v=8) in collision with low energy electrons forms H + H^- with a rate coefficient between 10^-8 and 10^-9 cm³ s^-1. Vibrational lifetimes for fluorescent emission however lie between 10⁵ and 10⁶ s (e.g. Black & Dalgarno 1976) and therefore vibrational relaxation through fluorescence takes place very much more rapidly than dissociative attachment.

2.2. The mobility of electrons on grain surfaces

In shielded regions, grains of characteristic size around 0.1  µm will tend to carry one or a few net negative charges (Draine & Sutin 1987 and references therein). If grains are unshielded, for example in diffuse regions, or are subjected to a powerful radiation field, as in dense PDRs, the net charge on grains may become positive rather than negative through photoemission of electrons (Watson 1972; Feuerbacher et al. 1973; Bakes & Tielens 1994). The dynamic equilibrium which maintains the grain charge is essentially unaffected by the efficient circulation of charge between the surface and the gas phase implied by the negative ion route for H₂ formation. The reason is that electrons, removed from the surface as H^-, are rapidly reintroduced into the gas phase, through reaction of H^- with H, and therefore remain available for adsorption at grain surfaces. The charge balance with for example net negative charge on the grain is retained unchanged.

When a grain surface has a net charge of (say) 1 electron, does this imply a single electron on the grain or an excess of one electron over the number of positive ions on the grain? The first model would arise from mobile electrons recombining with ions on the surface, and the second from immobile electrons and ions on the surface with charges distributed over the entire surface of a grain (Umebayashi & Nakano 1980). Electron mobility further determines whether surface electrons will migrate to the deepest sites available or remain on more weakly bound sites. Electron mobility also relates to the Hinshelwood surface mechanism, which would be facilitated if rapid surface electron diffusion is possible.

In order to investigate electron mobility, a model is adopted in which favourable adsorption sites are assumed to be separated by some mean distance which is typically very much greater than the surface atom separation (Smoluchowski 1979, 1981). Motion on the surface is achieved by tunneling through the barrier binding the electron to its current site to a nearby available site. The length of the barrier is taken to be the average distance apart of adsorption sites. Thus we estimate below the requisite mean separation of binding sites such that an electron can tunnel from one site to an adjacent site within a specified time through a single tunneling event. Using standard theory of tunneling, the average separation of sites, d, such that electrons can migrate to an adjacent site within some time, t, may be expressed as

where m_e is the electron mass, E_b is the barrier against tunneling, with a value which is some fraction of the electron surface binding energy, E_th is the thermal energy, given by the grain temperature, and r is the extension of the surface vibration of the bound electron (1Å).

Two values of E_b, 75 meV and 700 meV, are used for illustration, the upper value chosen as the limit for which surface H^- formation is exothermic. Positive ions are less mobile than electrons on the surface and, for simplicity, positive ions are in fact assumed immobile. The time t is the residence time either of electrons or of positive ions on the surface, and values are not known. Neutralization of positive ions such as C⁺ may take place by recombination with valence electrons in the solid (Draine & Sutin 1987) and the residence time t may then be that of atoms on the surface. Such a process would not be available to ions of lower ionization potential, such as Na⁺ on grains in dark clouds. Hence values of residence time, t, are likely to be strongly dependent on the grain environment. Values of d are however rather insensitive to the choice of t, which is characterized here in terms of a surface binding energy of 50 meV (Katz et al. 1999) and two values of grain temperature, T_g, of 15 K and 50 K. The residence time then becomes 6x10⁴ s at 15 K and 10^-7 s at 50 K (Smoluchowski 1979). For E_b = 75 meV, T_g = 15 K and 50 K, d = 15 nm and 11.5 nm respectively. For E_b = 700 meV, T_g = 15 K and 50 K, d = 5 nm and 2 nm respectively. The conclusion that can be drawn is that for a substantial range of conditions an adsorbed electron may be able to interact with an ion (or an atom) if the nearest sites lie between 5 and 10 nm apart. On this basis, a grain of radius 0.1 µm, surface area 1.3x10⁵nm², may be able to accommodate a steady state surface population of the order of a thousand electrons and ions. If the radius of each site is  0.2 nm, there are about 10⁶ sites on a typical grain surface. Thus 1 part in 10³ of surface atoms on the grain may be populated by electrons. This may favour a Hinshelwood mechanism for cold grains. If the distance between adsorption sites is less than the values of d estimated above, a succession of tunneling events may take place during residence lifetimes. The surface will then tend to be further depleted of electrons and positive ions.

© European Southern Observatory (ESO) 2000

Online publication: October 24, 2000
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