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

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4. Dynamical results and discussion

4.1. Rate of recombination of the H2 molecule

4.1.1. Results of the simulation

The H2 recombination probability during the collision is plotted as a function of the collision energy ([FORMULA]) in Fig. 3 (the direction of the velocity of the incoming H atom was taken perpendicular to the rigid surface, and L = 5 Å). For the two potential models studied, similar behaviour was obtained. A clear increase of the probability with the collision energy appears from Fig. 3a. At low collision energy ([FORMULA]), no hydrogen molecule is formed along the classical trajectories in the phase space, the incoming H atom being reflected back to the gas phase at relatively large z value. The values of [FORMULA] are slightly dependent on the potential model. We obtained [FORMULA] [FORMULA] 3 meV for model (I) and [FORMULA] [FORMULA] 10 meV for model (II) (see Fig. 3b). The effective activation barrier for reactivity is then slightly higher in the case of model (II) although that the H -graphite barrier is lower. This difference is the consequence of the localization of the barrier. Indeed the activation barrier is found at a smaller value of z for model (I) so that the [FORMULA] attractive interaction is then larger than in the case of model (II). Consequently the global barrier in the entrance channel is lower for potential model (I) as compared to model (II).

[FIGURE] Fig. 3a and b. Plot of the H2 recombination probability versus the collision energy ([FORMULA]) obtained in the case of a single H adatom in a cell of 100 Å2, with the two potential models; a corresponds to the full energy range; b is a zoom in the low energy range.

From a classical point of view, no trajectory can go through this potential barrier. However a quantum description of this reactive pathway could give a non-zero probability for the H atom to tunnel through this barrier. Consequently one could presume that the rate of H2 formation at low collision energy in the classical approach is certainly slightly lower than in a complete quantum treatment.

When [FORMULA] becomes larger than [FORMULA], the H2 formation is now energetically allowed for an increasing number of trajectories. Consequently the efficiency of the formation process increases as a function of [FORMULA]. The probability to form a molecule reaches a maximum at [FORMULA] [FORMULA] 0.15 eV. At this energy, almost 65 [FORMULA] of the trajectories are reactive with model (I) and [FORMULA] 45 [FORMULA] with model (II). The recombination is then highly efficient in this range of energy. When the collision energy becomes larger than [FORMULA] 0.15 eV, the efficiency of the recombination process remains almost constant.

Up to now, we have only considered the situation in which the initial velocity of the atomic hydrogen is normal to the graphite surface. In Fig. 4, the formation probability has been plotted as a function of [FORMULA] at a collision energy [FORMULA] = 0.132 eV. When [FORMULA] increases, a net lowering of the recombination rate appears. This reflects the decrease of the normal component of the incoming H linear momentum. Consequently, when [FORMULA] decreases, more and more atoms are reflected on the activation barrier along the z coordinate. If only the normal component of the linear momentum of the incoming particule is to consider in the reactive collision, we expect to find that the recombination probability at [FORMULA] = [FORMULA] for [FORMULA] = 90 [FORMULA] will be equal to the recombination probability at [FORMULA] = [FORMULA] for [FORMULA] = 0 [FORMULA]. In Fig. 4, if we extrapolate the curve up to [FORMULA] = 90 [FORMULA], we obtain [FORMULA] [FORMULA] 0.36 with E0 = 0.132 eV. This value is in very good agreement with that obtained at [FORMULA] = 0 [FORMULA] at the collision energy [FORMULA] = 0.044 eV, i.e. [FORMULA] [FORMULA] 0.38 (see Fig. 3a). Consequently, if we want to transform the collision energy into a gas phase temperature [FORMULA], we have to use the relationship [FORMULA] = [FORMULA] with [FORMULA] = 0 [FORMULA]. Consequently, the gas temperatures, associated to the appearance of the recombination process, are respectively 70 K and 230 K for the potential models (I) and (II).

[FIGURE] Fig. 4. Plot of the H2 recombination probability as a function of the maximum angle [FORMULA] (see text) of the initial velocity from the normal to the surface at [FORMULA] = 0.132 eV in the case of potential model (I) from a set of 4000 trajectories.

As said in the previous section, the H adsorbed atom was prepared at a given equilibrium temperature [FORMULA]. All the results shown up to now have been obtained at [FORMULA] [FORMULA] 70 K. The recombination rate has also been calculated for increasing values of [FORMULA] in the 10-250 K range. It was found that the formation rate was almost unsensitive to the initial vibrational energy of the adsorbed H atom.

In Table 4, the formation probability obtained from both CT and QCT calculations at three values of the collision energies are reported. The sensitivity of the process efficiency with respect to the method is weak. The formation probability was found slightly larger in the QCT calculation than in the CT calculation, which can be explained by the zero-point energy ([FORMULA] 0.23 eV) deposited in the H -graphite system at the beginning of each trajectory. It is clear again that the increase of the initial H -graphite vibrational energy does not affect much the efficiency of the process. On the contrary, a 0.20 eV increment in the collision energy completely affects the formation probability. This strong difference reveals that the process is mainly governed by an activation barrier in the entrance channel.


Table 4. Comparison of the results obtained from the CT and QCT methods. [FORMULA] is the recombination probability; [FORMULA], [FORMULA], [FORMULA] are the averaged energies in various degrees of freedom of the nascent H2. Calculations were made with potential model (I). All energies are given in eV.

It is now of interest to analyse the recombination probability as a function of the impact parameter b. It appears that a large proportion of molecules are formed when the impact parameter b is less than 2 Å . For each set of trajectories, we have calculated the opacity function [FORMULA], defined as the ratio of the number of reactive trajectories to the total number of trajectories which are characterized by an impact parameter in the interval [FORMULA]. The opacity function derived from 8000 trajectories at [FORMULA] = 0.026 eV in the case of model (I) is plotted in Fig. 5. From the opacity function, the cross-section can be easily derived as:

[FIGURE] Fig. 5. Plot of the H2 recombination probability as a function of the impact parameter b at [FORMULA] = 0.026 eV. These results have been obtained with potential model (I) from a set of 8000 independent trajectories with L = 5 Å .


The values of [FORMULA], reported in Table 5, are relatively large. However we have to keep in mind that this cross-section reflects not only the direct ER process between the incoming H atom and one isolated hydrogen atom on an infinite surface but also the indirect process. This last one involves trajectories during which the incoming H atom is trapped and migrates on the surface until it reacts with one of the other adsorbed H atoms. In the simulation this process is taken into account thanks to the periodic boundary conditions in the (x,y) plane. To estimate the relative importance of such a process, a process was considered as indirect if the incoming H atom, isolated from the other one, left the cell during the trajectory. In Table 5, the cross-section associated to the direct ER process ([FORMULA]) has been tabulated for three collision energies. It clearly appears that the ratio between direct and indirect processes is strongly dependent on the collision energy. In the low energy regime, all the recombination events are direct. The lower is the collision energy, the more the incoming H atoms are sensitive to the adsorbed H atom which acts as an attractor. This explains why the indirect process becomes negligible when the collision energy decreases. On the other hand, at [FORMULA] = 0.132 eV, a substantial proportion of indirect processes are observed. In this range of collision energy, the linear momentum orientation transfer is dominated by the surface corrugation rather than by the influence of the other H atom and the proportion of indirect trajectories increases. We have also analysed the probability to form H2 in such indirect trajectories: it was found equal to 66 [FORMULA] at [FORMULA] = 0.026 eV and 55 [FORMULA] at [FORMULA] = 0.132 eV respectively. This lowering of the efficiency of the recombination process as a function of the incident kinetic energy corroborates that the migrating H atoms are less and less sensitive to the adsorbed H atom when [FORMULA] becomes larger and larger. This explains the slight decrease of the total recombination efficiency as visible in Fig. 3a.


Table 5. Comparison of the total cross-section ([FORMULA]) and the Eley-Rideal cross-section ([FORMULA]) for potential model (I). The two cross-sections, calculated with L = 5 Å , are defined in the text.

All the results discussed below have been obtained with the area of the square cell equal to 100 Å 2. As the cross-sections are certainly dependent on the density of adsorbed H atoms per unit area, or equivalently on the coverage [FORMULA] (in units of a monolayer, i.e. when all the top sites are occupied), we have analysed the evolution of the formation probability as a function of [FORMULA] by changing the area of the cell (a cell of 100 Å2 corresponds to [FORMULA] [FORMULA] 0.05). This procedure allowed to obtain informations in the range of low coverage ([FORMULA]). These results have been reported in Fig. 6a for three different collision energies. It appears that, for the 2 lowest values of [FORMULA], the recombination efficiency is linear with respect to the coverage. On the other hand a slight saturation of the recombination probability appears at [FORMULA] [FORMULA] 0.09 at [FORMULA] = 0.132 eV. In fact the non-linearity appears for a critical H density [FORMULA] = [FORMULA] in which [FORMULA] corresponds to the characteristic value of the impact parameter for which the opacity function vanishes. As [FORMULA] increases as a function of the collision energy, it explains that the non-linearity is observed in the MD calculations for the lowest [FORMULA].

[FIGURE] Fig. 6a and b. Plot of the H2 recombination probability versus the coverage ([FORMULA]) at three different collision energies with potential model (I). a Results obtained for [FORMULA] 0.1 by changing the area of the cell in classical trajectories. b Results obtained for larger coverage (up to 0.4) from the distribution of impact parameters and the data in Fig. 5.

To obtain information relative to larger coverage, an alternative procedure has been used. The opacity functions P(b) have been calculated as explained above for different collision energies. For a given coverage ([FORMULA]), the distribution of the impact parameter ([FORMULA] (b)) was calculated from a set of randomly chosen initial conditions in a cell whose size is adapted to the coverage. From P(b) and [FORMULA] (b), the recombination probability was easily deduced from:


The [FORMULA] function is plotted in Fig. 6b for the same three collision energies up to [FORMULA] [FORMULA] 0.4.

4.1.2. Discussion of the astrophysical consequences

This H2 recombination probability at given coverage [FORMULA] is crucial to extract useful information for astrophysics. Indeed, the most important quantity for interstellar hydrogen chemistry is the H2 formation rate (in [FORMULA]) which is given by:


In this last expression, [FORMULA] and [FORMULA] corresponds respectively to the atomic hydrogen density in the gas phase and to the mean velocity of these H atoms (which is related to the gas temperature if this phase is considered to be at thermodynamical equilibrium in the ISM). [FORMULA] and [FORMULA] are respectively the dust grain density in the ISM and the mean surface offered by the average grain (depending in particular on the geometry of the dust particle).

In standard astrophysical models of H2 formation in the ISM, such as the one proposed by Hollenbach & Salpeter (1970, 1971), the rate is estimated within the hypothesis of a very high recombination probability ([FORMULA] 0.3), supposed to be the product of the sticking probability [FORMULA] by a factor [FORMULA] (close to 1) implicitely assumed to account for the efficiency of the LH mechanism, independently of any explicit reference to the amount of adsorbed hydrogen as measured by the coverage. Indeed this formulation is equivalent to consider that the reaction probability is never limited by the diffusion of the adatoms. This assumption would need a specific study in order to check its validity in the low [FORMULA] range. We estimate nonetheless that diffusion coefficient large enough to permit an average diffusion velocity of the order of 1 Å/year or more is necessary for the reactivity to be dominated by the LH mechanism. Note that these diffusion processes are expected to be very sensitive to the chemical composition and physical structure of the grains surfaces.

On the contrary if we neglect thermal desorption and the LH process, which will tends to be valid in dense and cold regions of the ISM, the steady-state coverage, [FORMULA], resulting from the equilibrium between sticking and H2 formation by the ER mechanism will result from:


[FORMULA] corresponds to the sticking probability for the H atom impinging onto the grain. It depends parametrically on the coverage [FORMULA] and on the mean collision energy. With the very simple assumption that this sticking probability is proportionnal to the number of available sites, we have [FORMULA] = ([FORMULA]) [FORMULA] [FORMULA]. The reaction probability has been found to be an increasing function of the coverage (see Fig. 6). The two probability functions will then cross over at the steady state value [FORMULA] of the coverage. assuming [FORMULA] = [FORMULA] as an order of magnitude, one obtains [FORMULA] = 0.3 at [FORMULA] = 0.013 eV (i.e. [FORMULA] [FORMULA] 300 K). The corresponding recombination probability is [FORMULA] 0.35, which is very much the same as the commonly admitted value (Hollenbach & Salpeter 1970, 1971, Jura 1975). The average gas temperature in the ISM is smaller than 300 K but the above value can be reached in shocked regions or heated regions, such as photon dominated regions where H2 is more readily observed. As a matter of fact the grain to gas relative velocity is the quantity which governs the value of [FORMULA] and it may be affected by grain acceleration and/or friction in stellar winds. Finally one should keep in mind that these numbers depend rather crucially of physical quantities not accurately known such as the barrier in the entrance channel.

4.2. Distribution of excess energy in the nascent H2 molecule

Besides the direct astrophysical consequences, the interest to examine the distribution of excess energy in the nascent H2 molecules has been strengthened by the recent experimental results obtained by Gough et al. (1996) on the internal states distribution of H2 molecules desorbed from carbonaceous surfaces exposed to thermal H atoms. H2 molecules in highly excited vibrational states up to v = 7 were observed. On the other hand no significant rotational excitation was observed. This experimental study has been realized for different values of the surface temperature.

In the MD simulations, we have been interested in the repartition of the excess energy available in the chemical reaction induced in the vicinity of the graphite surface among the various degrees of freedom of the product, i.e. H2. With the two potential models used in this work, the reaction is highly exoenergetic ([FORMULA] E [FORMULA] 3.4 eV). In Fig. 7, the ensemble averages of the translational kinetic energy, rotational energy and vibrational energy of the newly formed H2 molecules have been plotted for 9 different values of [FORMULA] in the case of model (I). As it can be seen in this plot, the partitioning of the energy in the different degrees of freedom is almost independent of the collision energy. This result can be easily understood by noting that the increase of the collision energy represents only a small amount with respect to the reaction exothermicity. A precise analysis of the results given in Table 4 shows that the translational and rotational energies of H2 increase slowly as a function of the collision energy. On the other hand the vibrational energy tends to decrease slightly. But the ordering [FORMULA] [FORMULA] [FORMULA] [FORMULA] [FORMULA] remains unchanged.

[FIGURE] Fig. 7. Distribution of excess energy in the reaction among various degrees of freedom: ensemble averages of translational, rotational and vibrational energies of the newly formed H2 molecules as a function of the collision energy of the impinging H atom. These results have been obtained with potential model (I).

A large amount of the excess energy ([FORMULA] 50 [FORMULA]) is then devoted to the overall translation energy of the H2 molecule. This large kinetic energy associated to the recoil of the H2 molecules following the recombination mechanism at the gas-solid interface could have important consequences for the physics and chemistry of the ISM. Indeed these newly formed H2 molecules could induce an efficient heating of the interstellar gas.

The direction of the desorbed H2 molecule following the collision has also been analysed. This information could be useful for experiments in which the detection of the desorbed H2 molecules could be angularly resolved. The angular distribution of the desorbed H2 molecules is diplayed in Fig. 8 (solid line). It presents a maximum at [FORMULA] [FORMULA] 35 degrees, and is almost unsensitive to the initial collision energy. The angular distribution for the back-scattered H atom (dashed curve) is maximum at [FORMULA] = 0 but is broadened due to the surface corrugation.

[FIGURE] Fig. 8. Angular distribution of the scattered H atoms and of the newly formed H2 molecules at [FORMULA] = 0.132 eV with potential model (I). Angles are referred to the surface normal.

The rotational degrees of freedom are the less favored. Only [FORMULA] 19 [FORMULA] of the excess energy is going into the rotation of the diatomic molecule. The distribution of the rotational populations in the nascent H2 has been obtained from the analysis of the final angular momentum of the diatomics, J. It is plotted in Fig. 9. The mean value of J is [FORMULA] [FORMULA] 10 which corresponds to a mean rotational energy [FORMULA] [FORMULA] 0.70 eV. The broad distribution of the angular momentum is in fact the signature of the anticorrelation between translational and internal energies for the desorbed molecules. Since the simulation is classical, no information about the ortho/para population can be extracted. We can only note that, in the work of Persson & Jackson (1995b), quantum distributions P(J) were generally well reproduced by the QCT simulations. Such a low rotational excitation is consistent with experimental observations by Gough et al. (1996).

[FIGURE] Fig. 9. Rotational angular momentum distribution of the nascent H2 molecules, P(J), in a QCT calculation at [FORMULA] = 0.132 eV. J is expressed in [FORMULA] units.

We have analysed the possible alignment of the angular momentum of the H2 desorbed molecule to obtain dynamical information on the formation process. Such alignment effects of the desorbed molecules could be experimentally analysed from the spectroscopy of the H2 molecules with polarized laser beams. An alignment effect in the thermal desorption of H2 from a palladium surface has been recently observed (Wetzig et al. 1996) in this way.

The distribution of [FORMULA] for all the trajectories, and also for the selected direct collisions trajectories, has been analysed. These plots (not shown here) display a maximum around zero which means that the angular momentum orientation is favored in the plane parallel to the graphite surface. Surprisingly, the two distributions (direct and total) are almost the same. It indicates that the desorption induced by a hot migrating atom produces almost the same distribution that a direct ER process. In fact, analysis of the direct trajectories reveals that, after the direct harpooning of the incoming H atom, the H2 molecule may migrate for some time on the surface before being desorbed. This explains why there is no real difference between the two distributions. If we reject these trajectories (i.e. considering only trajectories involving direct formation and direct desorption), the distribution of [FORMULA] exhibit as more pronounced peak around zero, as expected. For these latter trajectories, we have plotted in Fig. 10 the correlation between [FORMULA] and [FORMULA]. A clear maximum appears when ([FORMULA])2 +([FORMULA])2 [FORMULA] 1, corresponding to trajectories with [FORMULA] [FORMULA] 0. This difference then demonstrates that trajectories involving atomic or molecular migrations tend to be associated to helicopter-like motion in which the angular momentum is along the z direction. A more detailed analysis of the [FORMULA] and [FORMULA] distributions also reveals preferential directions peaks which can be correlated to the topology of the graphite-H PES.

[FIGURE] Fig. 10. Plot of the ([FORMULA], [FORMULA]) correlation for the newly formed H2 molecules. This distribution has been obtained at [FORMULA] = 0.195 eV with potential model (I) from a set of 8000 trajectories.

About 31 [FORMULA] of the excess energy is transfered into vibrational excitation. The relative population distribution [FORMULA] in successive vibrational states obtained at [FORMULA] = 0.132 eV is shown in Fig. 11. From this set of 8000 trajectories, we observe high vibrational excitations up to v = 6, which is in quite good agreement with the experimental work of Gough et al. (1996) where a vibrational population up to v = 7 has been observed. The calculated vibrational distribution does not exhibit population inversion, nor is characteristics of a thermal distribution which could be the case in a LH process.

[FIGURE] Fig. 11. Vibrational population distribution of the newly formed H2 molecules, relative to the v=0 population. [FORMULA] = 0.132 eV, potential model (I), averaged over a set of 8000 trajectories.

However the experimental vibrational distribution reported by Gough et al. (1996) is strongly different from the present prediction. In particular, they found that most of the molecules are formed in the ground vibrational state. They have estimated that the ratio [FORMULA] is less than 0.13. In the calculation presented here, we have found [FORMULA] =0.84. This disagreement call for some comments, since there are many reasons for which calculations and measurements can give different results.

First of all, as said by Gough et al. (1996), a large fraction of the newly formed H2 molecules may collide many times with the surface before being detected due to the configuration of the experimental set-up. If this is the case, these collisions obviously induce a de-excitation of the diatomics which will alter the measured vibrational distribution. On the other hand it is not clear at all that the microscopic mechanism which mainly induce H2 desorption in the experiment is the ER mechanism. It may be partly or largely affected by the LH process. Another uncertainty resides in the parametrization of the interaction potential used in the present work, which may not perfectly represent the reactive zone. If we analyse the energy partitionning obtained with potential model (II), the translational contribution is again the most important (41 [FORMULA]) but is however slightly smaller than with the first parametrization. On the other hand, the vibrational contribution (38 [FORMULA]) is now almost equal to the translational one, with a maximum for v = 1. The rotational contribution (21 [FORMULA]) is almost unchanged with respect to model (I). These three contributions are again almost unsensitive to the collision energy over the whole range in which the reactive process has been studied. In summary, potential model (II) tends to increase the vibrational contribution with respect to the translational one.

An other possibility could be that a large part of the excess energy is transferred to the excitation of the solid thanks to the coupling with the phonon modes. However preliminary MD results using a non rigid surface model show that this energy transfer is not really efficient in this range of collision energy. Finally a point of major importance is the difference in the structure of the carbonaceous surface involved in the experiment with respect to the perfect graphite surface used in the calculation.

The above discussion concerned the energy distribution obtained from the CT calculations. In Table 4, we present the comparison between the CT and QCT methods. In the QCT calculation, the total energy (translational + internal energies) are larger than in the CT calculation due to the ZPE involved in the simulation. At high collision energy ([FORMULA] = 0.132 eV), this excess energy is equally shared by translation, vibration and rotation. On the other hand, at low energy, the ZPE seems to be preferentially transferred to the translational and rotational degrees of freedom. As a counterpart the mean vibrational energy slightly decreases with respect to the CT calculation. Indeed increasing [FORMULA] or the initial adsorption energy induces the same effect (relative increase of [FORMULA] and decrease of [FORMULA]). This shows that the energy distribution is mainly governed by the transition state region corresponding to the H2 molecule adsorbed on the surface.

4.3. Energy transfer in the non reactive trajectories

We turn now to the analysis of the non reactive trajectories during which the incoming H atoms are back-scattered. We defined a parameter Q which characterizes the degree of inelasticity of the collision:


[FORMULA] and [FORMULA] are respectively the mean values of the initial and final collision energy of the H atoms after the scattering, the averages being calculated over the subset of trajectories which are non reactive only. As the surface is considered as rigid, the value of Q reflects in fact the energy transfer between the two H atoms. The values of Q have been reported in Table 6 at five different collision energies for the CT calculations. In the low collision energy regime, the collision appears as elastic (Q = 0). The incoming particle is reflected at a large z value and the energy transfer between the hydrogen atoms is not efficient. At intermediate collision energy ([FORMULA] [FORMULA] 0.05 eV), the incoming H atom can come close to the surface and consequently can begin to efficiently exchange energy with the other particle. At [FORMULA] =0.132 eV, Q is equal to 0.025. At this collisional energy, analysis of the trajectories reveals that some H -impact induced migration of the previously chemisorbed atom on the surface appears. It means that an energy transfer of about 0.25 eV (energy of the diffusion barrier) becomes possible. For example, at [FORMULA] = 0.262 eV, 8000 trajectories have been run. 2704 trajectories were found non-reactive and 36 trajectories have induced an atomic migration. These diffusing H atoms can play an important role in the reactivity process because, following the first H impact, adsorbed atoms can migrate and thus possibly react with other chemisorbed H atoms to also induce a H2 desorption. In Fig. 12, we have reported the final (x,y) positions of the adsorbed H atom (filled circles) at the end of each trajectory involving the back-scattering of H. At the beginning of the trajectories, the initial position of the adsorbed atom was around (x=1.4 Å ;y=0). The carbon atoms (open circles) are also drawn in this picture to put into evidence that the induced migrations appear along the C-C bonds. An analysis of the trajectories reveal that the induced migrations are almost equally generated by direct ([FORMULA]) and indirect ([FORMULA]) collisions (see Table 6). This induced migration has been recently observed by Eilmsteiner et al. (1996) in experiments where the desorption of the D2 molecule was detected following the impact of H on a deuterated Ni(100) surface.

[FIGURE] Fig. 12. Final (x,y) in-plane localization of the H adsorbed atom in the non reactive trajectories, obtained at [FORMULA] = 0.262 eV with potential model (I) from a set of 8000 trajectories. Each open circle represents a carbon atom.


Table 6. The value of the inelasticity parameter Q (see Eq. (18)) for different collision energies within the CT approach. These results have been obtained with potential model (I). All the energies are given in eV.

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Online publication: May 12, 1998