Energy distribution of fragments from H2 and H2 + dissociation by electron impact for the use in numerical models applied to negative ion sources

In negative hydrogen ion sources, the kinetic energy of the atoms is directly related to the negative ion yield at the caesiated converter, with a larger contribution from hot atoms. The energy distribution of hydrogen atoms is related to the formation process: either the kinetic energy release, resulting from dissociation of the hydrogen molecules or molecular ions, or the proton neutralization either in the volume or during reflection at walls. The interpretation of recent experimental measurements related to the translational energy distribution of atoms or positive ions could profit from accurate inclusion of the initial energy distribution in numerical models. In this work, we focus on the calculation of the kinetic energy release for the various dissociation channels due to electron impact on H2 and H2 +, in the Franck-Condon and delta approximation. Since in negative ion sources non-equilibrium vibrational distributions of H2 are found, the energy distribution of fragments is calculated for all vibrational levels. The inverse cumulative distribution functions related to the main dissociation processes are given, as well as the cumulative distributions for all dissociation channels by electron impact, for simple implementation in Monte Carlo numerical simulations. Finally, the application of the method to few cases of interest for negative ion sources is discussed.

5 Discussion of results for negative ion sources 15 5.1 Collisional processes generating H involving negative ions 15 5.2 Kinetic energy of the fragments in case of non-negligible initial velocity 15 5.3 Relevance of the initial atom velocity with respect to the atom mean free path 15 5.4 Contributions to hot atom fluxes on the negative-ion converter surface 16

Conclusion 17 1 Role of hot atoms in negative ion sources
In surface-plasma negative ion sources, the production of negative hydrogen ions is enhanced by surface conversion mechanism [1] (i.e.scattering of atoms and protons from a low-work function materials, commonly obtained by the introduction of caesium in the discharge chamber).In filament-arc sources, the direct correlation between the density ratio of atomic to molecular hydrogen n  /n 2 and the negative ion current was derived experimentally [2].This can be justified by the relatively low ionization degree found in negative ion sources, that compensate the lower translational temperature of atoms with respect to ion species (H + , H + 2 , H + 3 ), so that their flux onto the converter surface can be an order of magnitude larger.However, this picture is only valid if the translational energy of the atoms is large enough to guarantee a significant conversion yield in the scattering at the wall, as the scattering models predict a dependence on either the energy [3] or velocity of scattered particles [4].For this reason the characterization of atomic population has received considerable attention along the years.The density ratio n  /n 2 in RF driven sources is found to be about 0.3, with an increase from the driver to extraction region [5], somewhat in line with previous estimates [6]; earlier measurements in filament arc sources indicated a ratio from 0.6-0.7 [6,7].If the translational temperature of the H atom can be associated to the temperature of excited atoms, it can be measured by the Doppler broadening of Balmer emission lines via optical emission spectroscopy: in RF sources, a two temperature distribution of n = 3 excited atoms is found [8], with cold atoms at 2200 K and hot atoms around 2.5 eV, with -1 -density fractional ratios from 1:3 to 1:1 (depending on the operational parameters).Laser-induced excitations from n = 2 to n = 3 were also exploited [9], finding a relatively low temperature of 0.3 eV.Measurement of the energy of atoms is so important for negative ion production that two-photons laser absorption spectroscopy [10] is being applied, allowing the study of the H[1s] population.Negative ion sources are commonly designed following a "tandem" concept, with a driver region where a relatively high electron temperature sustains the plasma discharge and the gas dissociation, and an extraction region protected by a transverse magnetic field [11] that filters out hot electrons thus creating a protected region where cold electrons and positive ions can diffuse, and negative ions survive for a sufficiently long time to be extracted in the beam with minimal electron co-extraction.In such design, the properties of plasma particles -including neutrals -vary strongly depending on the observed region.
For the fundamental role of neutrals in negative ion sources, dedicated Monte Carlo simulation codes were applied to negative ion sources of both types (see for instance [12,13]).Simulations of the RF driven sources indicate that, in terms of production rate, a roughly equal share is ascribed to atom produced via dissociation processes in the volume and via surface processes [14], however the contribution to the atom flux as a function of the impinging energy on the converter surface is not discussed.In the presence of a transverse magnetic field to cool down electrons and minimise the electron detachment of negative ions by electron impact, plasma drifts occur: the correlation between non-uniform beam current densities over the extraction apertures and the neutral population was shown in filament-arc sources [15].In that analysis, the key role of high-energy tails in the energy distribution was highlighted, showing that the dissociation atoms that did not bounce yet on the source walls, sharing less than 10% of the flux (and necessarily less in terms of density), can have twice the effectiveness in producing negative ions than the remaining 90% thermalised atoms.As said, the initial kinetic energy of the product of the dissociation (i.e.H and H/H + ) is a key ingredient for the operation of caesium-enhanced negative ion hydrogen sources; a more accurate description of the production mechanisms of fast atoms, and its inclusion in numerical models such as the ones discussed before, may largely benefit to the comprehension and optimization of surface-plasma sources for negative ion beams.
The kinetic energy release of fragments from electron-impact dissociation of the hydrogen molecule and molecular ion was studied by various authors.The studies by Celiberto [16,17] in the framework of semiclassical approximation highlighted the influence of vibrationally excited molecules, commonly found in negative ion sources, in shaping the translational energy of the fragments.Recently, the convergent close-coupling method was applied to selected dissociation channels of H 2 and H + 2 by Scarlett [18,19].However, a comprehensive study of the kinetic energy release for the various dissociation channels and its dependence on the electron temperature in a plasma discharge is not available yet.
In this paper, we derive the kinetic energy release of a dissociation channel due to electron impact on molecules and molecular ions (section 2).We review the fundamental dissociation processes, and provide supplementary tables in a convenient form for sampling the KER differential cross sections, in form of inverted cumulative energy distribution for the various processes (section 3).The provided tables can be easily implemented in kinetic simulations of plasma discharges (particle in cell models as well as test particle models), allowing a more accurate simulation of the atomic population.Effective differential cross sections are then derived for Maxwellian plasmas in section 4; under simplifying approximations concerning the vibrational population of the H 2 gas, the KER cross-sections are used to discuss the most relevant channels in the production of energetic atoms for selected electron -2 -temperatures (few cases relevant for different regions of RF driven and filament-arc negative ion sources for fusion).Section 5 provides important notes concerning the application of these results to the modeling of negative ion sources, before proposing concluding remarks in section 6.

Calculation of the kinetic energy release
The method for calculating the kinetic energy release follows the approach for calculating the cross sections for dissociation processes originally described in ref. [20], and applied also in [16] and [17].In the δ-approximation (see for instance [21]), the wave function of the free particle at the turning point is approximated so that the overlap integral of the vibrational wave function   () of the bound state, and the wave function of repulsive states (i.e. the Frank-Condon density), reads: The vibrational wave functions   () in steady state is obtained from the solution of the Schrödinger equation: where   is the reduced mass (roughly 0.5 amu in this case), and the solution is obtained following the approach described in [22].Figure 1 shows the potential energies  () that were considered in this work.The potentials of the ground or excited states of H 2 are taken from [23][24][25][26] and the excited states of H + 2 from [27].For the resonant states, we considered ref.[28] and references therein.The agreement of calculated eigenstates   with results available in literature was verified (e.g.[22,23]).
Differentiating the dissociation cross section  (, ) for the -th vibrational state of the initial state, yields the KER differential cross section: where the total kinetic energy of released fragments coincides with the energy of continuum  ′ () of the repulsive state (or the repulsive portion of bound potential), depending on the internuclear distance .In this approximation the influence of  on  (, ) is not considered.Since  () for repulsive potentials (or the repulsive portion of bound potentials) is an invertible function, so is  ′ (), and   (, ) can be written in terms of   ( ′ , ).In this sense, as shown in eq.(2.3), the functions   ( ′ , ) are independent on the electron impact energy .For biatomic isotopic molecules, the kinetic energy  of each fragment is identical, so that  ′ = 2.Therefore, the energy distribution function   (, ) of the fragments (i.e. the probability to generate a fragment of energy ) is determined by the function   ( ′ , ), with the only constraint from the electron energy given by the energy conservation  It is known that the δ-approximation it is best suited for atom production from low vibrational levels.Figure 2 provides an example of calculated normalized KER cross-sections, for electron-induced dissociation of H + 2 ; at least for low vibrational states, it is in good agreement the literature.In the following section 3, a cumulative distribution function is calculated for the various -th processes (, ), and it is given in form of inverse normalized function in figures and supplementary tables.On the other hand, in eq.(2.3) the dependency on the electron energy of a dissociation event to happen is given by the proper cross sections  (, ), thus composing the KER differential cross section.

Use of inverse cumulative distributions
In section 3, the cumulative distribution function   (, ) is given in form of inverse normalized functions.For an electron impact with electron energy , the kinetic energy  of atom fragments can be obtained from the inverse cumulative distribution as follows.For the -th process and the -th initial vibrational level, use a random number  in the interval [0:1] to access the table  ,,  at column index  = floor( •  col ).The tables were constructed with uniform steps between zero and one, therefore it is possible to use the column indexes.Repeat until 2 ,,  <  −  diss .Interpolate linearly If such an accept-reject method is not considered computationally efficient comparing to accessing a further table, a table with the maximum  max providing fragment energies consistent with the impact energy can be prepared, so that  ∈ [0 :  max ] does not require the accept criteria given by the energy conservation.
It is important to stress the fact that, as discussed, the provided inverse cumulative distribution are not differential cross sections.The probability of a collision event to happen shall be calculated using the proper cross sections  (, ), for which a convenient technique for the Monte-Carlo approach, used also for the calculation provided in the following section, is the well known null-collision method [29].

KER in electron-induced dissociation processes
The channels leading to dissociation that were considered in this work are presented in the following, listed by increasing threshold energy; this categorization is convenient for the studies related to negative ion sources, in which -due to the presence of a filter field -different regions of the plasma exhibit different electron temperatures.Dissociation channels for the hydrogen molecule are discussed in section 3.1, for the hydrogen molecular ion in section 3.2.For all the processes discussed here, the inverse cumulative distribution is provided, and the method to sample the fragment energies was presented in section 2.2.Table 1 lists the supplementary data tables.It shall be noted that the inverse cumulative distributions given here can be used to sample the kinetic energy of either atoms or protons produced in the dissociation.

e-H 2 dissociation processes by threshold energy
At low energy, the shape resonances of H − 2 dominates the mechanism of H 2 dissociation.The dissociation via B 2 Σ + g resonance is the dominant channel, rather than via the X 2 Σ + u resonance, at low and intermediate vibrational levels [30].Within our approximations, the KER of the former channel is hardly distinguishable from the case of the repulsive state b 3 Σ + u (the two potentials almost overlap), and they were treated identically in section 4. The KER of the dissociation via X 2 Σ + u resonance was not calculated and the process was neglected; the H − 2 X 2 Σ + u resonance is known for being the only possible channel of vibrational excitation at low electron temperature.
The H − 2 shape resonance contributes also via dissociative attachment to the H population; the KER is calculated directly from the available cross-sections in this case, by subtracting the energy threshold and equally dividing the excess energy among the H and H − .Dissociative attachment via X 2 Σ + u [30], B 2 Σ + g [31] resonances were considered.Note that when the electron temperature is sufficiently large, the negative ion rapidly undergoes electron detachment by electron impact, doubling the influence of these channels (in section 4, only the initial H production is considered).
-5 - In the Franck-Condon region, the electron energy threshold is of the order of ∼10 eV, and dissociation fragments have energy in the range of 2-4 eV for initial  = 0.The vibrational state-resolved cross sections were calculated several years ago [32] by using the Gryzinski-Bauer-Bratky (GBB) approximation [33,34]; recently obtained MCCC results [18] are in good agreement for low , while they indicate a larger cross section for high .The latter cross-sections were included in the calculation of section 4.
At intermediate energies, where the corresponding cross section for the b 3 Σ + u state loses its importance dissociation can occur through excitation of the repulsive portion of singlet states: for this reason, while the energy threshold is relatively high, the KER is generally low for all these processes.These processes are anyway important for creating excited atomic hydrogen atoms ( ≥ 2).We calculated the energy release for excitation involving the repulsive portion of singlet states u for all vibrational states of X 1 Σ + g .In the calculation of section 4, we adopted the cross-sections from [35]; more recent calculations, also based on the GBB, are available as well [36].Another possible channel is the radiative decay from these excited states onto the repulsive portion of the ground state; cross sections including the most relevant contributions can also be found in [35], however the KER for this two-steps process was not calculated here.
Another possible channel in the intermediate electron-energy range is the dissociation via excitation of the triplet state a 3 Σ + g , which is radiatively coupled to the repulsive state b 3 Σ + u .The radiative lifetime is of the order of 1.1×10 −8 s [37], so that the vibrational populations in the a 3 Σ + g state shall be considered for the calculation of the KER (the depopulation via collisional processes is less frequent, with characteristic times of the order of 10 −4 s for ionization, considering for instance ionization cross-sections from ref. [36]).Following the method of ref. [20], the state-to-state resolved cross section -6 -was calculated in the GBB approximation to determine the vibrational population in the a 3 Σ + g state; we renormalized each   (,  ′ ) imposing that the total cross section  ,=0 =  ′′   ( = 0,  ′ ) had a peak value of cross-sections calculated with state-of-the-art calculation methods (see [38] and references therein), finally yielding a set of cross-sections in sagreement with the classical treatment [39].The vibrational population of the a 3 Σ + g state, obtained for the vibrational population of H 2 given in ref. [12], is presented in table 2. Clearly, this excited state can be populated also from triplet states at higher energies (yielding a different vibrational population).
The behavior of the c 3 Π u state is less clear; metastable to X 1 Σ + g for  = 0 with characteristic time of 10 −3 s, its higher levels are optically coupled to a 3 Σ + g or, in general, have relatively large cross sections for the excitation to a 3 Σ + g , >1 levels.For simplicity, given the similarity of the potentials and the vicinity of the vibrational energies, in the section 4 we do not distinguish between the a 3 Σ + g and the c 3 Π u channels, and applied the results for a 3 Σ + g state decay also to the c 3 Π u state.Also the dissociative attachment via Rydberg resonance of C 2 Σ + g [40] can be considered among the collisions at intermediate electron energies.The KER can be treated as discussed previously for DA in low-lying shape resonances.
At higher electron energies, dissociative excitation via repulsive portion of the D 1 Π u singlet state can be considered, for which the KER for all initial vibrational states of X 1 Σ + g are calculated.The process contributes to excited atomic hydrogen atoms (n = 3).
At around the same electron impact energy, a significant excitation cross section is found for excitation to the d 3 Π u triplet state, accessible at relatively higher electron energies.Its radiative decay proceeds via a 3 Σ + g state, which later dissociate to two ground state atoms via b 3 Σ + u , in a three-step process.The cascade process leading to dissociation, relevant also for astrophysical plasmas, is also investigated in ref. [41].We calculated the ,  ′ vibrationally-resolved cross sections for the excitation to d 3 Π u state again in the GBB approximation; the total cross-section from  = 0 integrated over all  ′ has a comparable peak and is within a factor two from available recommended data [38,42].The mixing of the vibrational states in the d 3 Π u -a 3 Σ + g , transition can be calculated according to the transition probabilities [22]; the vibrational population is somewhat preserved, at least for the low vibrational states.In addition to the vibrational population coefficients calculated for the rates of excitation from X 1 Σ + g ( = 0) to the a 3 Σ + g , table 2 exemplifies also the populations coefficients for the a 3 Σ + g via radiative decay of d 3 Π u , together with those of the d 3 Π u state, for Maxwellian electrons with T  = 2.5 eV.The contribution to dissociation of higher triplet states, and of e 3 Σ + u , are neglected; despite its lower lying energy level, excitation to e 3 Σ + u has a factor two lower total cross section with respect to d 3 Π u , with only slightly lower threshold (as conveniently presented in [43]).All triplet states are assumed to cause dissociation to ground state atoms (i.e. the contribution from the repulsive portion of potential can be neglected).Finally, the vibrational population of a 3 Σ + g can be used to calculate the KER of dissociation fragments caused by the excitation to d 3 Π u .
With slightly higher energy threshold, the dissociative ionization channel can occur via the repulsive portion of the H + 2 (X 2 Σ + g ) potential.The cross sections for this process were reviewed in [36], and vibrationally-resolved cross sections are available (see for instance [35]).We used vibrationally-resolved cross sections from [35] corrected to match the experimental results [44].
At higher energies, dissociative ionization can occur via the repulsive X 2 Σ + u state.Being a repulsive potential, the dissociation fragments can have relatively large energies (similarly to the dissociation via b 3 Σ + u ).Dissociation via doubly-excited states is also accessible at high electron energies, yielding to highly-excited atoms ( ≥ 2).
-7 - The inverse normalized cumulative distribution for all the electron-induced processes discussed in this section are presented in figure 3.In general, a wavy inverse function is visible for increasing vibrational states.It can also be seen that, for high vibrational states, lower kinetic energies for the fragments are accessible compared to low vibrational states; clearly, this is associated to a lower threshold for the cross-section in terms of electron impact energy.

e-H +
2 dissociation processes by threshold energy The dissociative recombination (DR) mechanism has large cross section (10 −18 m 2 ) at low electron energies, and yields one of the two atoms in excited state; it is exothermic for H, H[n = 2] fragments (and for high vibrational states also with respect to H, H[n = 3]) and proceeds via a manifold of two-electron excited states [45].The process conserves the total energy, so that the kinetic energy of the two dissociation atoms is  = ( +  diss () +  ion ())/2, with  diss () the dissociation energy of the initial vibrational level of the H + 2 and  ion () the ionization energy of the excited atom.In the calculation of section 4, we used the convenient expressions of the cross sections  DR (, , ) for all v, n states proposed in [43] (obtained fitting the cross sections of reference [46]).
At higher energies, dissociative excitation (DE) may proceed via direct mechanism, involving the two lowest dissociative excited states 2pσ u and 2pπ u (the latter yielding excited atoms H[n = 2]), or via indirect mechanism involving doubly excited auto-ionizing H * * 2 states lying below the 2pσ u state (with lower threshold energies).Results of experimental measurements in the low-energy region showed a different behavior, either with a high cross-section at impact energy as low as 0.01 eV [47], or with a -8 - threshold [48].Recent theoretical studies [49] (that also included DR and H + 2 vibrational excitation), as well as the older study [46] indicated the presence of thresholds, of decreasing energy with increasing initial vibrational level (see also the discussion on this topic in ref. [43]).Due to the uncertainty on the mechanism yielding DE at very low energies, and therefore on the kinetic energy distribution of the fragments, this contribution was not considered in section 4. The vibrational state-resolved cross sections of [46] were considered, for the process yielding H + , H[n = 1] via repulsive potential 2pσ u , and from [42] for the H + , H[n = 2] channel via 2pπ u .For both channels the KER was calculated and inverted.In the ambiguity related to impact energies low-energy, we calculated the influence of including the experimental cross-sections in the 0.01-0.1 eV impact energies from [47]: the rate coefficient for DE would be larger by a factor 2 at T  = 1.5 eV, and by only 15% at T  = 5 eV.
For higher electron impact energy, excitation to higher states may occur (with relatively high thresholds) or ionization to H + , H + ; these former processes are neglected, while the latter are outside the scope of this work as they yield a proton pair.
The inverse normalized cumulative distribution for all the electron-induced processes discussed in this section apart from the dissociative recombination are presented in figure 4; the features of the distributions are similar to those of figure 3. channels is presented for three selected electron temperatures.In section 4.2, the dependence on the electron temperature and the relevant inverse distributions, as well as the resulting rate coefficients for atom production, are finally given.For these calculations, it was necessary to assume vibrational populations of H 2 and H + 2 .In negative ion sources, the molecular vibrational distribution probably varies depending on the source type and also within the ion source; for this study, we selected the -state populations from [12].The chosen distribution includes non-maxwellian high vibrational states, a feature typical of negative ion plasmas and somewhat dependent on the plasma-wall interaction.For the H + 2 ions we used the Von Bush-Dunn distribution [50].Maxwellian electrons are considered.

Contribution from individual collisional processes
Figure 5 presents the cumulative distribution functions of the atom kinetic energies for three electron temperatures, decreasing from 15 to 1 eV.In the figure, also the energy distribution of the excited states n = 2, 3 are presented.In a sense, the case at low temperature and low density shown in figure 5(d), (g) well represents the conditions at the extraction region of a negative ion source.The high temperature case reported in figure 5(a) is closer to the conditions that could be found in a driver region, the region sustaining the plasma, for instance in RF discharge.
Let us focus on the case at T  = 15 eV,  + 2 = 3 × 10 18 m −3 of figure 5(a).Such a high H + 2 density was selected to increase the readability of the distribution related to dissociative excitation and dissociative recombination, compared to e, H 2 processes, and the contribution from these channels can be easily rescaled to different  + 2 .The low energy fragments (i.e.below 0.5 eV) are all generated by excitation to the repulsive portion of the bound singlet states (i.e.dissociative ionization via B 1 Σ + u , C 1 Π u , B ′1 Σ + u ).Dissociation via B ′1 Σ + u singlet state appears one of the dominant channel producing low energy atoms, for the high temperature case (cross section peaks near threshold [35], but this is not the case of recent results [51]).Dissociation via a 3 Σ + g and c 3 Π u are the main contribution for fragments at around 1 eV: their contribution reaches low energies all the way to ∼0 eV, due to the mixing of the vibrational levels towards higher .This is even more the case for the multistep d 3 Π u -a 3 Σ + g channel, which can be identified especially between 1 and 3 eV; however its contribution with respect to the other channels is relatively low (somewhat supporting the approximation of neglecting higher triplet states and the e 3 Σ + u state).All highly-excited triplet states finally dissociating via the b 3 Σ + u state should exhibit a similar spreading.The key role of dissociation via b 3 Σ + u is clearly identifiable, peaking around 2.5 eV, even though its importance with respect to other processes will start decreasing for higher electron temperatures (e.g.T  = 20 eV, since the cross section peaks at around 13 eV).Also the resonance B2 Σ + g gives a contribution to atoms generated with ∼2.5 eV, while the contribution of Rydberg resonance (dissociative attachment) is marginally identifiable at around 3.5 eV.Finally, products of dissociative ionization via H + 2 2 Σ + u are identifiable at around 8 eV.Let's focus on the e, H + 2 dissociation processes.The role of electron-induced H + 2 dissociation is to provide a wide contribution, between 1 and 8 eV, to the energy distribution of atoms; the impact energy of the electron limits the possible KER (as shown later in figure 6(b)), but it cannot exceed about 10 eV.
Let's now discuss figure 5(g)-(h)-(i), that presents the low temperature case with T  = 1 eV, Dissociation via b 3 Σ + u occurs efficiently with the v>9 population in the H 2 target, so that the dissociation contributes with low-energy fragments (not with the characteristic distribution originated by the  = 0 state peaked at around 2.6 eV).At very low energy, the contribution of the dissociative attachment channel is visible; being active for high vibrational levels, the repulsive H − resonance B 2 Σ + g plays a role similar to the b 3 Σ + u state.The larger contribution is due to dissociative recombination: this channel produces ground-state as well as excited atoms, with low Rydberg states having larger energies (i.e.exothermic process with greater gain for n = 2).As one of the two atoms produced by dissociation is always in n = 1 state, in the fragmentation it mirrors the distribution of kinetic energy of the paired excited atom.On the overall, integrating the spectra yields a generation frequency of atoms ν  = 40 × 10 3 1/ (ν  = 23.4 × 10 3 1/s due to collisions with H + 2 alone), with average kinetic energy of 0.65 eV (T = 0.42 eV).About 1/20 of these are n = 2 atoms (average energy of 1.35 eV) and 1/8 are n = 3 atoms with 0.73 eV.
Figure 5(d)-(e)-(f) presents the intermediate electron temperature of T  = 2.5 eV, with the same H + 2 density.The overall production of atoms is higher compared to T  = 1 eV, with ν  = 74 × 10 3 1/.The contribution of collisions with H + 2 alone is ν  = 24 × 10 3 1/, practically identical to the previous case (due to the lower cross sections of the DR channel), with the differences that recombination channel produces n = 4, 5 atoms and dissociative excitation starting to play a more important role.
Although without pretension of being exhaustive, we showed in this discussion that the electron temperature determines very different energy distributions of the dissociation atoms; this is an important result for negative ion sources, which as said exhibit a non-uniform T  , and the accuracy of ion energy distribution calculated from plasma simulation codes may largely profit from implementing these results.In particular, at Te of 2.5 eV and below, the atom energy distribution can quite differ from common approximation [53] to the Franck-Condon energy of ∼2.5 eV resulting from H 2 [ = 0] dissociation via b 3 Σ + u .

Dependence on electron temperature
Dissociation processes have increasingly higher threshold energies, as discussed in the previous section 3.As expected, the energy distribution of the atoms strongly depends on the electron temperature.This is noticeable in figure 6(a), which reports the differential rate coefficients for the energy distribution of fragments produced by electron collisions with neutral molecules.In general, the marked characteristics of dissociation via b 3 Σ + u , i.e. the fragments with energies around 2.5 eV, becomes evident only for electron temperatures above 2 eV.For a stronger rate of this channel (desirable for negative ion sources, since the surface conversion yield largely profit from the impact energy of the atoms) the electron temperature must be larger: at 5 eV, about one-third of dissociation events yield atoms at energies above 2.5 eV.
The cumulative KER differential rate coefficients of atoms produced by electron collisions with H + 2 ions is given in figure 6(b).Molecular ions have a broad distribution of vibrational states, which enables the production of atoms with higher translational energies.However, for the dissociative excitation to contribute significantly with high-energy fragments above 4 eV, an electron temperature greater than ∼5 eV is necessary, as clearly visible in figure 6(b).For lower temperatures, slow fragments obtained by dissociative recombination are dominant.The reader should note that it is possible that the vibrational population of H 2 differs within RF-drivers (or the arc region of filament arc sources) from the distribution selected for these calculations, which is applicable to the expansion region.
By integrating the KER differential cross-sections, we obtained the rate coefficients shown in the inset of figure 6.It is important to stress that, for T  around 5 eV or higher, the rates for atom production are comparable for e,H 2 and e,H + 2 collisions (blue and magenta solid lines in figure); at low electron temperature, the atom production is dominated by DR.
-12 -  In the inset, the integral Maxwellian rate coefficients calculated for the dissociation of one atom is shown (i.e.twice the dissociation rate for H 2 ).In blue by electron impact on H 2 , in magenta by electron impact on H + 2 ; solid line (thick), total; dashed line, rate for  > 1 eV only; dashed black line, via b 3 channel only.The inverse cumulative energy distributions for the two targets H 2 , H + 2 for the use in simulations is given in figure 7(a), (b).The distribution for intermediate temperatures can be obtained by interpolation.The cumulative rates are shown in the inset of figure 6 (thick solid lines), which shall be used for weighting the two normalized contributions from electron impact with H 2 or H + 2 .

Discussion of results for negative ion sources
In this section we discuss the application of the presented results to the specific case of negative ion sources.In particular, the kinetic energy of atoms produced by mutual neutralization of ions with negative ions is discussed in section 5.1, and the caveat on the conservation of total momentum is discussed in section 5.2.The importance of the KER from dissociation events in the development of an equilibrium translational energy distribution function must be put into context considering the collision frequency for momentum transfer with other species, discussed in section 5.3, as well as the relative importance with respect to neutral atoms of relatively high kinetic energy produced in the neutralization at walls of positive ions, discussed in section 5.4.

Collisional processes generating H involving negative ions
In surface-plasma negative ion sources with Cs-enhanced negative ion production, at the extraction region the local rate for generation of hydrogen atoms due to mutual neutralization can be comparable or exceed the volumetric rate by electron impact, due to the very low electron temperature; however, this is clearly not the case when the whole plasma volume is considered.Let's compare the dissociation of atoms with other possible collisional channels present in at the extraction region of a negative ion source.For convenient discussion we will consider the same plasma properties as in figure 5(g) i.e   = 1 eV,  2+ = 2 × 10 17 m −3 .In the presence of negative ions, mutual neutralization H − ,H + 2 occurs.If we take a case  − =  2+ , the volumetric rate   / would be roughly a half of the contribution from electrons if   =  2+ .Concerning the excited states, mutual neutralization H − , H + would yield both H[n = 2] and H[n = 3] excited atoms, and fragments would take translational energy of 2.65 eV and 0.76 eV respectively [52]; even considering  − =  + , in total   / would be 4 times smaller than the dissociative recombination channel.When only the H[n = 3] product is considered, the mutual neutralization is two times more effective in populating H[n = 3], while n = 2 products are generated more effectively (∼4 times) by the dissociative recombination channel.H γ transition is used to estimate the atom to molecule density ratio; the excitation to n = 5 due to dissociative recombination has a rate of 600 1/ (680 1/ at T  = 2.5 eV); if we take   [1] = 1 × 10 19 m −3 , for comparison at   = 1 eV the atom excitation has a rate of 0.2 1/ while at   = 2.5 eV is 200 1/ (using atom excitation cross sections from [43]).

Kinetic energy of the fragments in case of non-negligible initial velocity
It shall be noted that, in addition to the KER due to the dissociation event, fragments from H + 2 inherit the kinetic energy of the ion, which may not be negligible in one application.Also mutual neutralization of H − with H + and H + 2 yields atoms that, in addition to the energy release, inherit the kinetic energies of charged species.

Relevance of the initial atom velocity with respect to the atom mean free path
Momentum transfer in general with plasma ions (H + , H + 2 , H + 3 ) has a relatively large cross section (of the order of 10 −19 m 2 at 1 eV).However, for the relatively low ion velocity and plasma density in the weakly-ionized plasmas of interest (i.e. + = 10 17 /10 18 m −3 ,  gas > 3 × 10 19 m −3 ) the mean free path of hydrogen atoms for momentum transfer is rather large (>1 m), at least in the extraction region of negative ion sources.Charge exchange with protons in particular is a mechanism that may -15 -determine the total momentum exchange for hydrogen atoms; for the same reason however, its rate is low in comparison to the rate of dissociation mainly caused by the interaction with electrons, so that the influence on the kinetic energy of atoms cannot be dominant.

Contributions to hot atom fluxes on the negative-ion converter surface
The negative ion yield is tightly related to the energy distribution of the atoms reaching the caesiated converter.The energy distributions of atoms developed in this work due to electron-induced dissociation can be compared, at least qualitatively, with the energy distribution expected for atoms generated by ion neutralization at walls.
In particular, for this assessment we will only consider the neutralization occurring on the rear walls of the expansion region of RF-driven sources, while other surfaces are neglected -e.g. the rear wall of the driver itself (where in most designs, cusp magnets are installed and limit the interaction of charges with the surface), rather than the lateral walls enclosing the expansion region.Recent measurements at the rear wall of the expansion region of a RF-driven source [54] indicate in uniform plasma conditions (named "reversed field" in that paper) a plasma density between 2 × 10 17 m −3 with T  of 4 eV (lower limit) and 5 × 10 17 m −3 with T  of 10 eV (upper limit), with a floating potential of about 10 V with respect to the source walls.Let us assume for simplicity that the plasma is composed only by protons with a Bohm flux, taking 0.6 neutralisation probability and an energy accommodation of 0.5 in the impact with the wall [14], and assume a cosine law for the reflection velocity (i.e.average perpendicular velocity of 0.66).In this approximation the flux of atoms directed perpendicularly to the plasma grid would be 2 × 10 21 m −2 s −1 with energies uniformly distributed from the a half of the impact energy down to zero (reaching 9 × 10 21 m −2 s −1 at 9 eV if the upper values are considered).
Let us calculate the atom fluxes produced by dissociation coming from the driver.For this purpose we converted the fragment energy distribution of figure 6(a) and (b) in isotropic velocities, and only atoms with positive velocity along one component (i.e.perpendicular to the plasma grid) will be studied; we take an average electron density of 1.5 × 10 18 m −3 and a depth of the driver plasma of 0.1 m.For this comparison, we will assume that the area of the driver open end and of the rear wall is identical.
The energy distributions for the fluxes of the two families of hot atoms are compared in figure 8.A figure of merit of their effectiveness in producing negative ions is the integral of the distribution above 1 eV, since the atom conversion probability into H − is quite uniform above that impact energy.-16 -If the exit area of the driver equals the area of the rear wall of the source, we find 2.2 × 10 21 m −2 s −1 particle with energy greater than 1 eV coming from the rear walls (1.4 × 10 22 m −2 s −1 in the upper limit, magenta line) and 2.3 × 10 22 m −2 s −1 coming from volume dissociation within the driver.It means the role of volume dissociation form the driver is dominant for the production of negative ions, but the contribution from neutralised ions could be of the same order under certain conditions.However, the latter mechanism could contribute with a much greater flux at impact energies above 5 eV, thus with precursors atoms that may generate more energetic negative ions, with possible detrimental effect on the optics of the extracted negative ion beam.

Conclusion
In this paper we studied the kinetic energy release for the dissociation of hydrogen molecule and molecular ion caused by electron impact, providing inverse cumulative energy distribution for an easy implementation in kinetic simulations of hydrogen plasma discharges.Tables are provided for the single dissociation channels, as well as for the total dissociation of H 2 and H + 2 .It was shown that the energy distribution of the fragments has a great dynamic with the electron temperature.This feature is commonly neglected, as it is often assumed that atoms produced by dissociation all proceeds via the repulsive b 3 state and take the Franck-Condon energy of ∼2.5 eV.
In general, at electron temperature between 1.5 and 5 eV, atoms generated by H 2 dissociation have a higher energy than atoms generated by H + 2 dissociation.However, atoms at relatively high energies can be produced at T  >5 eV by the latter process.In addition at low electron temperature T  ≤1 eV, a condition desired at the extraction region of negative ion sources, it is the dissociation of H + 2 that would provide atoms of higher energy, if H + 2 were available in large number.Rates of mutual neutralisation of negative ions were discussed in relation to dissociation of H + 2 in a condition relevant to the recombining plasma at the extraction region, but the lack of experimental data providing the ion effective mass does not allow a conclusive statement; on the other hand, we noted that for RF-driven sources, the contribution of hot atoms from dissociation processes inside the driver appears dominant with respect to that of neutralisation in the scattering of plasma ions at walls.
Future work shall consider the energy release in the dissociation of the H + 3 ion, for the importance of its role in the negative ion sources.In addition, for a more accurate calculation of the KER differential cross sections from high -states, a different approach than the δ-approximation could be used.

Figure 2 .
Figure 2. Normalized KER cross sections of the 2pπ  state of H + 2 at  = 50 eV, electron scattering on the  = 0, 3 states compared with the calculations of ref. [19] based on a different approximation.

Figure 4 .
Figure 4. Inverse cumulative distribution of the kinetic energy for the fragments of selected dissociation processes, for the interaction of electrons with the hydrogen molecular ion (e, H + 2 ) [As written in the text, this is not a differential cross section].

Figure 6 .
Figure 6.Influence of the electron temperature on the energy distribution of atoms generated by dissociative processes: (a) by electron impact on H 2 (b) by electron impact with H + 2 .In the inset, the integral Maxwellian rate coefficients calculated for the dissociation of one atom is shown (i.e.twice the dissociation rate for H 2 ).In blue by electron impact on H 2 , in magenta by electron impact on H + 2 ; solid line (thick), total; dashed line, rate for  > 1 eV only; dashed black line, via b 3 channel only.

Figure 7 .
Figure 7. Inverse normalized cumulative energy distribution of hydrogen atom fragments produced in the collision with H 2 (on the left) and with H + 2 (on the right).

Figure 8 .
Figure 8. Energy distribution of atom flux on PG due to dissociation in the drivers (blue line) and first scattering of protons at the rear walls (red line), for which an upper limit is also indicated (magenta line).

Table 1 .
List of data tables.
2 7 XIV At low-intermediate electron energy range, H 2 dissociation proceeds via b 3 Σ + u repulsive state.

Table 2 .
Calculated population of vibrational levels in the excitation from X 1 Σ + g ( = 0) by Maxwellian electrons at 2.5 eV.The numbers in parentheses denote powers of 10.