Scaling relations

The scaling of the integral elastic cross section and its higher moments for H₂ targets show similar scaling with collision energy and reduced mass as in cases of ion-atom and atom-atom systems, except to account for differences in the vibrational couplings and energies. Surprisingly, after averaging over the diatomic orientations, the elastic cross sections scale well with the system’s reduced mass if only the projectile is varied. For a fixed projectile mass the cross section does not depend significantly on the details of the isotopic composition of the molecular target, and no scaling is needed, especially if the collision energy is above 0.3 eV. Figure 8a illustrates these behaviors, combining various projectiles with all six isotopic combinations of H₂. Thus, for variation of the projectile isotope and for variation of the target isotopomers, respectively, we have

σ_el^A++BC(E) = σ_el^H++H2( µ0

µABCE), A=H

σ_el^A++BC(E) = σ_el^A++H2(E) (17)

where µ0 is the reduced mass of H⁺+H₂. The deviations from the H⁺+H₂ curve do not exceed 20% over the whole energy range. These drop to less than 5% for energies close to 100 eV, when the collision time becomes short in comparison to the characteristic vibration time of the target.

152

10^-1 10⁰ 10¹ 10²

FIG. 8. Scaling of integral a) elastic, b) momentum transfer, and c) viscosity cross sections in collisions of hydrogen ions with hydrogen molecules in the ground vibrational state, varying isotopes in both projectile and target.

All the momentum transfer cross sections almost coincide at energies lower than 2 eV (Figure 8b). The curves deviate up to a factor of 3 for energies about 10 eV, where the vibrational transitions are most active. This deviation decreases toward higher energies, as with the elastic cross sections, when the momentum transfer cross section becomes very small and population of the vibrational excited states high. The dispersion in that range may also be attributed to possible convergence errors caused by the implemented trunca-tion of the sum over vibratrunca-tional states which produces the most pronounced uncertainty for large scattering angles that affect most the momentum transfer cross section. Thus

σ^A++BC(E) =σ^H++H2(E) (18)

10⁰ 10¹ 10²

FIG. 9. Scaling of integral a) elastic, b) momentum transfer, and c) viscosity cross sections in collisions of hydrogen atoms with hydrogen molecules in the ground vibrational state, varying isotopes in both projectile and target.

for all A, B, and C.

Figure 8c shows the viscosity cross sections for all of the isotopic variants. Similar to the momentum transfer case, the curves start to deviate above 2 eV, where the details of vibrational excitations are reflected in the mid-range and backward part of the differential cross sections. The deviation decreases toward higher energies. Thus we have

σ_vi^A++BC(E) =σ_vi^H++H2(E) (19)

for all A, B, and C.

The collisions of neutral atoms with neutral molecules retain the properties similar to cases of ion-molecule systems. Unlike the ion-molecule case, the best scaling is reached for the elastic cross sections if it is performed using the reduced masses for all cases,

154

irrespective of the projectile. This is illustrated in Figure 9a. In this case σ_el^A+BC(E) =σ_el^H+H2( µ0

µABCE) (20)

for all A, B, andC.

The momentum transfer and viscosity cross sections coincide, and no scaling is needed, as shown in Figures 9b and 9c. Thu s

σ_mt^A+BC(E) = σ_mt^H+H2(E)

σ_vi^A+BC(E) = σ_vi^H+H2(E). (21)

We note that the dispersion of the results is smaller than in the ion-molecule case, and does not exceed 30% in the vibrationally active region of collision energies (except at 100 eV) for both momentum transfer and viscosity cross sections. For the elastic cross section, the deviations are quantitatively similar to the ion-molecule cases.

5. Conclusions

We have presented the first comprehensive study of elastic scattering of H on H₂ in the range of center of mass collision energies of 0.1-100 eV, based on fully quantal calculations and with the inclusion of vibrational excitation only, while rotational dynamics is treated within the IOSA. In the absence of pre-existing, benchmarking calculations, we have also treated the elastic and vibrationally inelastic scattering of H⁺ on H₂, favorably comparing with the data available in literature for this system. In addition, due to their utility for plasma modeling, transport cross sections as well as scattering on excited vibrational states have also been calculated. We have found that the vibrational excitation cross sections for the H+H₂ system display features not present for H⁺+H₂ collisions. The dominant scattering mechanism for H+H₂is identified as scattering of the projectile on the closest atom in molecule, in contrast to the H⁺+H₂ system, where both elastic scattering and vibrational excitations are governed by perturbation of the electronic distribution of the molecule by the passing proton.

We note that an analysis and description of the relevant scaling of the integral cross sections for scattering of H⁺and H on H and He in various isotopic combinations is pre-sented elsewhere [25, 24] as are fitting coefficients for all of the differential and inte-gral cross sections for ease of use in plasma modeling applications [7]. The calculated data for differential and integral cross sections discussed in the present work as well as those for the various isotopic combinations can be viewed and downloaded from the web site of the Controlled Fusion Atomic Data Center at Oak Ridge National Laboratory, www-cfadc.phy.ornl.gov.

The establishment of the detached plasma regimes in the divertor, which is essential for minimization of plasma-divertor plate interaction and the associated material erosion and material properties degradation effects, can be faciliated by inclusion of certain fast volume plasma recombination mechanism. One of the reaction schemes proposed is the so-called “ion conversion” scheme[27], which involves capture of electron by proton from a vibrationally excited H₂, followed by dissociative recombination of H⁺₂ with a plasma electron. Lack of cross sections for this and many other processes involving excited molec-ular states does not presently allow one to construct the collision-radiative model for the

H₂/H gas. Therefore, these processes should be a central focus in future research in atomic physics for fusion.

This work has been supported by the U.S. Department of Energy, Office of Fusion Energy Sciences, at Oak Ridge National Laboratory which is managed by UT-Battelle, LLC under contract DE-AC05-00OR22725, and performed in conjunction with the IAEA Co-ordinated Research Project (CRP) on “Atomic and Plasma-Wall Interaction Data for Divertor Modeling”.

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