Vibrationally resolved processes in slow ion-molecule and atom-molecular ion
6. Three-body, diatomic association
We consider also the process of three-body, diatomic association, a process that may also be referred to using the quite general term three-body recombination. For cold hydrogen plasmas in particular, three-body, diatomic
diatom ( ), or a diatomic molecular ion ( ), in the presence of a third heavy particle. In this non-radiative process, the heavy particle, here a proton or hydrogen atom, scatters on the associating system and plays the role of catalyzer for the reaction, enabling energy and momentum conservation between reactant and the product channels.
Specifically, this process includes the reactions in Eq. (1.g) and (1.h).where any of the particles in the entrance channel can play the role of the scattered, catalyzing particle and denotes the vibrational quantum number of the associated diatomic. As discussed in connection with the study of collisional dissociation in and systems, the dissociation during the incoming and outgoing parts of the collision takes place due to series of consecutive Landau-Zener (LZ) nonadiabatic couplings (at large R distances), and series of hidden crossings at smaller R. The same series are operative under time reversal, and constitute the mechanisms for the associative reactions (1.g) and (1.h).
The difference to the dissociation spectra calculations is in the boundary conditions, describing initial population of the continuum states, normalized here to the unit flux.
It should be emphasized that each of the reactions (1.g) and (1.h) contain two channels: direct association (DA) and charge transfer association (CTA),
(DA) (19)
(CTA) (20) (DA) (21)
(CTA) (22)
The rate coefficients for each of these channels are different. For a given projectile ( or H) the CTA processes can be distinguished from the DA processes and are a consequence of charge transfer and particle exchange processes in the continuous and discrete vibronic spectrum.
The CTA rates are about a factor two larger than those for the DA channel in the entire temperature range considered.
As illustrated in Fig. 15 for T = 1,000 K, this is a consequence of the fact that higher vibrational states give the dominant contribution to the association rate, and that the density of these states in is higher than in . The temperature dependences of the DA and CTA total rates, and their sum, can be fitted to the expressions
(23)
where P stands for the reaction product (DA or CTA in
Eqs. (19,20). Thus, , and
, , while for the total rate for
association , The total
and partial rates for the DA and CTA channels in ( ) ( ){cos ( ) ( ) sin( )}
Figure 14. Angular distributions of dissociation fragments in H++ H ) collisions, for various CM collision energies and
(from Ref. [9]).
, except that now the DA process, association into , dominates for the same reasons that CTA is the dominant process for proton projectiles. In this case,
, , , , and
, The total recombination rate in reaction is about a factor two larger than for the case, for all temperatures considered, which could be expected from the stronger polarization interaction between the associating particles and the charged projectile in the former case.
The division of the three-particle system into
sub-systems, and in the two
basic collision arrangements is physically meaningful, and useful, only when the “projectile” can be clearly identified, e.g. by its energy or momentum. If this is not the case, that is, if all the particles in the system are characterized by the same temperature, then the association rates in the three-particle system for producing and are given by sums of the rates for DA and CTA reactions. These sums are displayed in Fig. 16 as well as the total association rate into either or . In a thermal hydrogen plasma of approxi-mately equal densities of neutral and ionized atoms the total three-body, diatomic association rate for production of is 1.8-2.5 times larger than the rate for association of , for the considered range of temperatures. Also shown in Fig. 16 is the three-body recombination rate for neutral hydrogen, obtained by Orel [29] using a calculational method quite different than the present one, as well as experimental data of Jacobs for H2 association in the collision system [30].
In summary, the calculated total association rates in the H+ + H + H system are about 5×10-32 – 6×10-34 cm6/s for temperatures in the interval 300-20,000 K, and decrease as (approximately) as temperature increases. This indicates that the three-body, diatomic association in H++H+H may be an important mechanism for and/or formation in astrophysical environments containing significant
populations of protons and hydrogen atoms. In fusion divertor plasmas, however, even with temperatures as low as
~ 1 eV and typical plasma and gas densities of 1014 cm–3, the three-body, diatomic association process yields total rate coefficient of cm6/s and the probability rate of s–1. Apparently, this will not play a significant role in volume plasma recombination in the presence of more powerful competing processes (such as the three-body recombination in , for instance).
7. Conclusions
Since our calculation of all inelastic and elastic processes has been done at the “same footing”, it is of interest to show the interplay of these processes in function of collision energy. Fig. 17 shows the momentum transfer and viscosity cross sections along with the charge transfer (ct), dissociation (dis), and total excitation-de-excitation cross sections ( vib), for a four initial states of . For , charge transfer and dissociation cross sections stay well below the momentum transfer and viscosity cross sections. In the energy range considered, the main contributions to the total vibration cross section comes from the lower vibrational states.
This picture changes for for which the charge transfer cross section dominates (being larger than the one for total vibrational excitation). At the higher end of the collision energies even dissociation becomes a strong process. Thus, for these higher , calculation of the elastic cross sections without taking into account inelastic processes could lead to serious inaccuracies. For the cross sections for all inelastic processes are larger than the momentum transfer and viscosity cross sections and, for , they are larger by more than an order of magnitude.
In the case of the system displayed in Fig. 18, cross sections for the charge transfer processes dominate over the mometum transfer and viscosity cross sections
( )
Figure 15. Final state resolved partial rate coefficients for three-body recombination of (DA) and (CTA) in the presence of at 1000 K (from Ref. [10]).
Figure 16. Total rate coefficients for creation of and in a hydrogen plasma (from Ref. [10]). 2
H H2+
Figure 17. Comparison of momentum transfer and viscosity cross sections for representative excited vibrational states in H++H2(ν) collision with inelastic processes from the same vibrational state (from Ref. [14]).
already for the ground state. For higher the total excitation cross section as well as dissociation become dominating processes, suppressing the elastic scattering channel and its transport-relevant moments. It is interesting to note that for 4, the total excitation–de-excitation and charge transfer cross sections are similar in magnitude, since both are dominantly manifested from exoergic, almost resonant transitions. For the highest vibrational states, charge transfer is suppressed by a large probability of dissociation due to the closeness of the vibrational energy levels to the continuum edge.
The database produced contains all mentioned elastic and inelastic processes for the and that for collision systems (0.5-100 eV) in form of partial and total, initial and final vibrational state resolved cross sections, all obtained at the “same footing”. This currently represents the most comprehensive quantum-mechanically obtained set of inelastic data for collisions that involve hydrogen atoms, ions and molecules. These are available in tabular form at the Controlled Fusion Atomic Data Center (CFADC) web site (www-cfadc.phy.ornl.gov). While the cross sections below 10 eV of CM collision energy are calculated accurately, using only IOSA approximation, the results obtained in the semiclassical framework (above 20 eV) show drawbacks which is characteristics to the straight-line trajectory approximation. In order to study rovibrationally resolved processes at low collision energies, with various isotopic combinations of H and, with resolution of particle rearrangement channels consistently, rotational dynamics has to be fully treated.
Acknowledgments
I acknowledge support from the US Department of Energy, Office of Fusion Energy Sciences, through Oak Ridge National Laboratory, managed by UT-Battelle, LLC under contract DE-AC05-00OR22725.
References
[1] KRASHENINNIKOV, S., Phys. Scr. T96 (2002) 7.
[2] PIGAROV, A.YU., Phys. Scr. T96 (2002) 16.
[3] ICHIHARA, A., IWAMOTO, O., JANEV, R.K., J. Phys. B 33 (2000) 4747.
[4] KRSTIĆ, P.S., Phys. Rev. A 66 (2002) 042717.
[5] KRSTIĆ, P.S., SCHULTZ, D.R., J. Phys. B 32 (1999) 2415.
[6] KRSTIĆ, P.S., SCHULTZ, D.R., Atomic and Plasma-Material Interaction Data for Fusion 8 (1999) 1.
[7] KRSTIĆ, P.S., “Vibrationally resolved collisions in cold hydrogen plasma”, Nuclear Instruments and Methods 2005 (Proc. CAARI2004, Fort Worth, Texas).
[8] KRSTIĆ, P.S., SCHULTZ, D.R., JANEV, R.K., Phys. Scr.
T96 (2002) 61-71.
[9] KRSTIĆ, P.S., JANEV, R.K., Phys. Rev. A 67 (2003) 022708.
[10] KRSTIĆ, P.S., JANEV, R.K., SCHULTZ, D.R., J. Phys. B 36 (2003) L249.
[11] KRSTIĆ, P.S., SCHULTZ, D.R., “Atomic and molecular transport processes in a cool hydrogen plasma,” in Atomic Processes in Plasmas, R.C. Mancini and R.A. Phaneuf (Eds.), AIP Conference Proceedings 547 (2000) 53.
[12] KRSTIĆ, P.S., SCHULTZ, D.R., “Atomic and Molecular Databases for Fusion Divertor Plasma,” Proc. 3rd ICAMDATA, Gatlinburg, TN, edited by D.R. Schultz et al, (AIP, New York, 2002).
[13] KRSTIĆ, P.S., “Nearly adiabatic atomic physics: Methods-Data-Devices,” Proceedings, 21st Symposium on Physics of Ionized Gases (SPIG), edited by N. Bibic et al. (Nis, Yug., 2002).
[14] KRSTIĆ, P.S., SCHULTZ, D.R., J. Phys. B. 36 (2003) 385.
[15] BAER, M., NAKAMURA, H., J. Chem Phys. 66 (1987) 1363, and references therein.
[16] ELLISON, F.O., J. Am. Chem. Soc. 85 (1963) 3540.
[17] PRESTON, R.K., TULLY, J.C., J. Chem. Phys. 54 (1971) 4297.
[19] SCHMIDT, M.W., BALDRIDGE, K.K., BOATZ, J.A., ELBERT, S.T., GORDON, M.S., JENSEN, J.H., KOSEKI, S., MATSUNAGA, N., NGUYEN, K.A., SU, S., WINDUS, T.L., DUPUIS, M., MONTGOMERY, J.A., "General atomic and molecular electronic structure system,” J.
Comput. Chem. 14 (1993) 1347.
[20] JOHNSON, B., J. Comp. Phys. 13 (1973) 445.
[21] SOLOV’EV, E.A., Usp. Fiz. Nauk, 157 (1989) 437 [Sov.
Phys. Usp., 32 (1989) 228].
[22] KRSTIĆ, P.S., SCHULTZ, D.R., JANEV, R.K., J. Phys. B, (1996) 1941.
[23] HOLLIDAY, M.G., MUCKERMAN, J.T., FRIEDMAN, L., J. Chem. Phys. 54 (1971) 1058.
[24] LINDER, F., JANEV, R.K., BOTERO, J., in “Atomic and Molecular Processes in Fusion Edge Plasmas,” Janev, R.K.
(Ed.), Plenum Press, New York, 1995, p. 397.
[25] SCHINKE, R., MCGUIRE, P., Chem. Phys 31 (1978) 391.
[26] GIESE, C.F., GENTRY, W.R., Phys.Rev.A 10 (1974) 2156.
[27] SCHINKE, R., Chem. Physics 24 (1977) 379.
[28] JANEV, R.K., LANGER, W.D., EVANS, JR, K., POST, JR, D.E., in “Elementary Processes in Hydrogen-Helium Plasmas”, Springer-Verlag, New York (1987).
[29] OREL, A.E., J. Chem. Phys. 87 (1987) 314.
[30] JACOBS, T.A., GIEDT, R.R., COHEN, N., J. Chem. Phys.
47 (1967) 54.
vi
ν
i≥H H+ 2+ H++H2