Hydrogen collisions in planetary atmospheres, ionospheres, and magnetospheres

ARTICLE IN PRESS

Planetary and Space Science 56 (2008) 1733–1743 www.elsevier.com/locate/pss

Hydrogen collisions in planetary atmospheres, ionospheres, and magnetospheres David L. Huestis Molecular Physics Laboratory, SRI International, Menlo Park, CA 94025, USA Received 22 July 2008; accepted 24 July 2008 Available online 29 July 2008

Abstract Hydrogen is the most abundant element in the universe. Molecular hydrogen is the dominant chemical species in the atmospheres of the giant planets. Because of their low masses, neutral and ionized hydrogen atoms are the dominant species in the high atmospheres of many planets. Finally, protons are the principal heavy component of the solar wind. Here we present a critical evaluation of the current state of understanding of the chemical reaction rates and collision cross sections for several important hydrogen collision processes in planetary atmospheres, ionospheres, and magnetospheres. Accurate ab initio quantum theory will play an important role. The collision processes are grouped as follows: (a) H++H charge transfer, (b) H++H2(v) charge transfer and vibrational relaxation, and (c) H2(v,J)+H2 vibrational, rotational, and ortho–para relaxation. In each case we provide explicit representations as tabulations or compact formulas. Particularly important conclusions are that H++H2(v) collisions are more likely to result in vibrational relaxation than charge transfer and H2 ortho–para conversion is at least an order-of-magnitude faster than previously assumed. r 2008 Elsevier Ltd. All rights reserved. Keywords: Critical evaluation; Collision cross sections and chemical reaction rates; Hydrogen; Planetary atmospheres, ionospheres, and magnetospheres; Charge exchange; Ortho–para conversion

1. Introduction Collisions of hydrogen atoms, ions, and molecules are key components of models of the atmospheres of the outer planets. The reliability of these models depends on the extent to which the basic collision parameters, i.e. collision cross sections and reaction rate coefficients, are accurately known. Here we illustrate examples of ‘‘critical evaluation’’ of rates and cross sections for three classes of hydrogen collisions: (a) H++H charge transfer, Tel.: +1 650 859 2611; fax: +1 650 859 6196.

E-mail address: [email protected] 0032-0633/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2008.07.012

(b) H++H2(v) charge transfer and vibrational relaxation, and (c) H2(v,J)+H2 vibrational, rotational, and ortho–para relaxation. The primary outputs are ‘‘recommended’’ values, documentation of the sources, how selections were made, and expected uncertainties. Secondary outputs include identifying past erroneous interpretations and needs for future laboratory or theoretical investigations. A key objective is to facilitate interpretation and modeling of atmospheric data without requiring detailed study of the relevant laboratory/theory inputs. Corresponding ‘‘critical evaluations’’ of jovian ion–molecule chemistry (Huntress, 1974) and of fundamental processes of importance at higher collision energies in fusion plasmas (Janev et al., 2003)

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assist in the evaluations reported here. The basic steps could be outlined as follows:

2.2. Introduction to laboratory experimental and theoretical investigations of H++H charge exchange

1. Read the atmospheric and astrophysical literature to identify critical atomic, molecular, and optical processes. 2. Read the laboratory & theory literature. 3. Select the best sources. 4. Produce a synthesis and recommendation. 5. Document the results in formats that are easy to access and use.

The first experimental issue is the fact that the target hydrogen atoms must be generated by dissociation of H2 molecular sources, requiring subtraction of the contributions of collisions with the remaining fraction of undissociated diatomics. This is reasonably straight forward, but some of the early experiments were compromised by lack of sufficient care. In addition, some of the high energy studies have used the diatomic molecular gases as surrogates for the atoms, assuming that the cross sections for the atoms are adequately approximated as half those of the diatomics. Finally, since the cross sections for charge exchange are usually measured in beam studies only by detecting the production of fast neutrals, they provide no information about the final state of the projectile or target atoms, e.g. see Reaction (2),

The evaluator needs years of laboratory, theory, atmospheric, and astrophysical research experience. 2. Charge exchange collisions of H+ with H The energetically degenerate charge transfer collision þ

H þ H2H þ Hþ

(1)

has exceptionally large reaction cross sections throughout an extended range of collision energies, from sE1013 cm2 from at 104 eV collision energy, to 1015 cm2 at 10+4 eV. Below we provide a new parameterization that quantitatively summarizes the total charge transfer cross section with an accuracy of 79% from below 103 to above 5 10 eV. As the simplest atomic collision, ab initio theory provides the bulk of the reliable information. Theory is quantitatively accurate below 10 eV and quite good to beyond 100 keV. In contrast, laboratory experiments are very difficult, and provide few usable data below 10 keV. 2.1. Astrophysical and atmospheric H++H charge exchange Reaction (1) has general astrophysical importance because it modifies the intensity of the hydrogen 21 cm hyperfine emission line (Furlanetto and Furlanetto, 2007). The simple interpretation is that electron exchange scrambles the correlation between electron-spin and proton-spin (Smith, 1966). Because the H++H charge transfer at very low energies is dominated by the R4 charge-induced-dipole polarization attraction at large internuclear distances, it is predicted to be the most important source of 21 cm emission at temperatures below about 5 K (Furlanetto and Furlanetto, 2007). In planetary atmospheres, ionospheres, exospheres, and magnetospheres (Smith and Bewtra, 1978; Tinsley, 1979, Hodges and Breig, 1991; Hodges, 1993; Chen et al., 2001; Miller et al., 2004), Reaction (1) (a) limits the lifetime of magnetospheric ring-current H+ ions through collisions with hydrogen atoms in the geocorona; (b) facilitates exospheric hydrogen escape by generating fast H atoms, including isotopic fractionation; and (c) mediates energy exchange between the ionosphere, exosphere, magnetosphere, and solar wind.

Hþ ðfastÞ þ Hð1s; slowÞ ! Hðn‘; fastÞ þ Hþ

ðslowÞ

(2)

Production of more energetic final states becomes more probable at higher collision energies. For example, at an energy of 50 keV, Reaction (1) with ground state products [i.e. Reaction (3)], Hþ ðfastÞ þ Hð1s; slowÞ ! Hð1s; fastÞ þ Hþ

ðslowÞ;

(3)

corresponds to only about 20% of inelastic proton collisions with hydrogen atoms (Shakeshaft, 1978), while excited states are produced 30% of the collisions (Park et al., 1976), and ionization results in the remaining 50% (Park et al., 1977), see Reaction (4). Hþ ðfastÞ þ Hð1s; slowÞ ! Hþ ðfastÞ þ Hþ

ðslowÞ þ e

(4)

A number of methods have been used in the theoretical investigations, spanning a wide range of documented reliability. At low energies, below about 100 eV, the most reliable approaches follow nonadiabatic couplings between the low-lying electronic states of the transient diatomic molecule, Hþ 2 . At higher collision energies, the adiabatic potential-curve model breaks down as the heavy particle velocities approach and exceed the effective velocity of the bound electrons, at about 25 keV ion projectile energy for H+ collisions with H. At energies above about 10 MeV, a new mechanism appears (Bransden, 1970). In this case the projectile proton is traveling so much faster than the target electron that the dominant charge transfer process becomes a head-on (‘‘knock-on’’) backscattering collision in which target proton is ejected with the full projectile energy, stopping the projectile proton, which then captures the ‘‘stationary’’ electron. 2.3. Critical evaluation of cross sections for H++H charge transfer collisions The collision protons with atomic hydrogen are the simplest of all atomic collisions. Reaction (1) has received

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extensive theoretical and experimental investigation, only a portion of which we can review here. We will restrict our discussion to the collision energy dependence of the total cross section for production of fast neutral H, that is, specializing to Reaction (1). With this restriction, we can construct a piece-wise polynomial representation that adequately summarizes the results from theoretical and experimental investigations. Elastic, final-state-resolved and angle-resolved (differential) scattering have also been well studied and can provide useful insights to the details of the collision mechanisms, as well as providing data for charge-neutral energy and momentum coupling in ionospheres, but are not evaluated here. For proton energies less than about 100 eV, Reactions (1) and (2) are dominated by the single channel (3) above, which implies that reliable charge transfer cross sections should be calculable given accurate potential energy curves þ for the Hþ 2 (1ssg) and H2 (2psu) electronic states (Bates et al., 1953; Wind, 1965; Peek, 1965a, b; Madsen and Peek, 1971; Power, 1973). The advances in the quality of the potential curves, sophistication of scattering theory, and computer technology have led to ever more reliable estimates of the charge transfer cross sections (Dalgarno and Yadav, 1953; Bates and Boyd, 1962; Smith, 1966, 1967; Hunter and Kuriyan, 1977; Davis and Thorson, 1978; Hodges and Breig, 1991; Krstic and Schultz, 1999; Krstic et al., 2004; Furlanetto and Furlanetto, 2007). The recent works are consistent with each other, extend to very low collision energies (104 eV or lower), and have sufficiently narrow energy spacing to resolve sharp scattering resonances. They illustrate the Stueckelberg oscillations (Stueckelberg, 1932; Nikitin and Umanskii, 1984; Macek et al., 2004) that are expected when the nominal charge transfer probability exceeds 50%. They also illustrate the importance of the R4 charge-induced-dipole attraction at energies below 0.01 eV. As suggested above, collisions at higher energies are so violent that the full range to bound and continuum electronic states of Hþ 2 becomes accessible; as a result, theoretical methods tend to be less reliable. We include two works that give reasonable agreement with experiments in the 25–200 keV range. Shakeshaft (1978) used a pseudostate coupled channels approach, while Olson (1983) used a classical-trajectory Monte Carlo method. The Born approximation (Openheimer, 1928; Brinkman and Kramers, 1930; Jackson and Schiff, 1953; Toshima, 1999) has also been used with some success in the same energy region. From the numerous laboratory experimental studies in the literature, we have selected data for subsequent analysis from only two of the most recent studies (McClure, 1966; Wittkower et al., 1966) as the only that appear to be quantitatively reliable because of experiment-to-experiment consistency and agreement with theory. They cover the range from 2 to 250 keV. We lack high-quality laboratory data at lower energies. From the various theoretical investigations at low collision energy, we have also selected only a subset

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(Brinkman and Kramers, 1930; Dalgarno and Yadav, 1953; Jackson and Schiff, 1953; Bates and Boyd, 1962; Smith, 1967; Hunter and Kuriyan, 1977; Shakeshaft, 1978; Olson, 1983; Hodges and Breig, 1991) for analysis. In this case, we have excluded some of the most reliable recent information (Davis and Thorson, 1978; Krstic and Schultz, 1999; Krstic et al., 2004; Furlanetto and Furlanetto, 2007) because it presents more fine details than our analysis can reproduce, and because it agrees quantitatively with the best low-energy data that was included, when degraded to the same energy resolution. Fig. 1 shows the selected data and a piece-wise polynomial fit defined by the FORTRAN code fragment listing in Table 1. The onset of rapid decline in the cross section above 25 keV is indicted (x ¼ 4.38). Fig. 2 shows an enlarged representation of the charge transfer cross section above 10 keV. We can see that the ‘‘accurate Born approximation’’ (Jackson and Schiff, 1953) is quite reasonable at high energy, while the Openheimer (1928) or Brinkman and Kramers (1930) formulation is a significant overestimate, as has been observed by a number of previous investigations (Bransden, 1970). Fig. 3 shows the deviation between the data and fit lines in Figs. 1 and 2. The overall quality of fit of 79% is indicated. At energies Epo25 keV, the theoretical results are clearly mutually consistent, with large variations from the fit formula due to actual narrow scattering resonances. At energies above 10 keV the divergence appears to be due to inaccuracies in and differences between the available theoretical calculations and experimental measurements.

Fig. 1. Fitting the H++H charge transfer cross section. The formula for the piece-wise polynomial fit is given in Table 1. DY53 ¼ Dalgarno and Yadav (1953); BB62 ¼ Bates and Boyd (1962); McC66 ¼ McClure (1966); WRG66 ¼ Wittkower et al. (1966); Sm67 ¼ Smith (1967); HK77 ¼ Hunter and Kuriyan (1977); Sh78 ¼ Shakeshaft (1978); Ol83 ¼ Olson (1983); HB91 ¼ Hodges and Breig (1991).

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Table 1 FORTRAN code fragment that returns values of the piece-wise linear fit for the cross section shown in Fig. 1 for the process H++H(1s)-H(n‘)+H+ Given the proton initial energy, Ep, in units of eV, sigma(Ep) returns the charge transfer cross section in units of cm2 real functionsigma(Ep) x ¼ alog10(Ep) if(x.le.3.3396) then y ¼ 12.9451 else if(x.le.2.2227) then y ¼ -15.8262–0.86272*x else if(x.le.1.202) then y ¼ -14.4664–0.25094*x else if(x.le.2.4513) then y ¼ 14.3008+(0.124550.0094356*x)*x else if(x.le.4.1059) then y ¼ 14.9562+(+0.368210.10138*x)*x else y ¼ 43.9887+(+14.41201.79965*x)*x end if sigma ¼ 10.0**y return end

Fig. 3. Fitting error for H++H charge transfer cross section. DY53 ¼ Dalgarno and Yadav (1953); BB62 ¼ Bates and Boyd (1962); McC66 ¼ McClure (1966); WRG66 ¼ Wittkower et al. (1966); Sm67 ¼ Smith (1967); HK77 ¼ Hunter and Kuriyan (1977); Sh78 ¼ Shakeshaft (1978); Ol83 ¼ Olson (1983); HB91 ¼ Hodges and Breig (1991).

The technical problem is illustrated by the following simple model: þ H2 þ hv ! e þ Hþ 2 or e þ H þ H

(5)

þ Hþ 2 þ H2 ! H3 þ H ðfastÞ

(6)

Hþ 3 þ e ! H2 þ H or 3H ðfastÞ

(7)

Hþ þ 2H2 ! Hþ 3 þ H2

(8)

Hþ þ e ! H þ hv

ðslowÞ

(9)

ðslowÞ +

Fig. 2. High-energy H++H charge transfer cross section. The formula for the piece-wise polynomial fit is given in Table 1. BK30 ¼ Brinkman and Kramers (1930); JS53 ¼ Jackson and Schiff (1953); McC66 ¼ McClure (1966); WRG66 ¼ Wittkower et al. (1966); Sh78 ¼ Shakeshaft (1978); Ol83 ¼ Olson (1983).

3. H++H2(v) ion–molecule reactions in giant planet ionospheres The Pioneer and Voyager radio occultation experiments found electron densities in the ionospheres of the giant planets that were about an order-of-magnitude smaller than expected. Subsequent measurements from the Galileo and Cassini spacecraft have confirmed these observations. This has been one of the major puzzles in understanding planetary ionospheres.

Unless some new reaction is found to convert H into þ Hþ 2 , H3 , or some other species that rapidly leads to electron–ion recombination, models predict that the protons (and thus the electrons) will reach greater densities than is consistent with observations. As a result, modelers seized on the suggestion by McElroy (1973) that the endothermic charge transfer reaction Hþ þ H2 ðvÞ ! H þ Hþ 2

(10)

becomes exothermic for vibrational levels vX4, and therefore might be expected to be fast. Modelers also cite Huntress (1974) for a rate coefficient for Reaction (10). In fact, Huntress does not give any number for reactions of vibrationally excited hydrogen. Rather, he reports only a calculated Langevin upper limit (Johnson, 1982) of 2.8  109 cm3/s for the rate coefficient for all products of H++H2(v) along with the estimated rate coefficients for endothermic charge transfer starting with v ¼ 0 of 1.9–2.8  1010 cm3/s derived from atomic beam collision experiments in the energy range 6–8 eV (which are not relevant for conditions in giant planet ionospheres).

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A number of modeling studies (Cravens, 1974; Atreya et al., 1979; McConnell et al., 1982; Cravens, 1987; Moses and Bass, 2000; Hallett et al., 2004; Moore et al., 2004; Majeed et al., 2004; Hallett et al., 2005a, b) have followed the McElroy suggestion. Given a rate coefficient for Reaction (10), the key piece of missing information would be the H2 vibrational distribution. A few studies explored reactions producing vibrationally excited hydrogen. Others used parameterized models of the vibrational distributions in the giant planet ionospheres, represented as altitude dependent rates for Reactions (10) or (9). 3.1. H++H2(v) ion–molecule reactions in plasma fusion reactors The magnetic confinement fusion research community has been trying to create a dense energetic plasma of protons. Thus, any process that converts protons into neutral hydrogen atoms can become a serious loss of efficiency. As a result there has been a strong interest in the charge transfer Reaction (10) and its reverse, as well the vibrational relaxation Reaction (11) Hþ þ H2 ðvi Þ ! Hþ þ H2 ðvf ovi Þ

(11)

that had not been considered in previous ionospheric modeling studies. The publications from the plasma fusion community can be of considerable value to the aeronomy/ atmospheric-science community, if we can extrapolate down to ionospheric temperatures. In two recent works (Ichihara et al., 2000; Krstic, 2002), the relevant collision energies are between 0.3 and 3 eV. The lowest energy at which calculations were performed was 0.1 eV, corresponding to a temperature of 1200 K, higher than the maximum temperature in the ionospheres of the gas giant planets. In our critical evaluation below, we will discuss the qualitative implications of the plasma fusion studies and present recommended values for rates of Reactions (10) and (11) at ionospheric temperatures. 3.2. Introduction to the quantum chemistry of H++H2 and H þ Hþ 2 collisions One reason why the H++H2 system has attracted so much interest from theoreticians (Preston and Tully, 1971; Tully and Preston, 1971, Giese and Gentry, 1974; Gentry and Giese, 1975; Schlier et al., 1987; Ichihara et al., 1996, 2000; Krstic, 2002; Krstic et al., 2002; Krstic and Janev, 2003; Kusakabe et al., 2004) is that the two asymptotic channels H++H2 and H þ Hþ 2 have the same energy at one value of the H2 or Hþ internuclear distance, 2 RvibE2.5 bohr or 1.32 A˚, as illustrated in Fig. 4. A second reason for early focus was the development of a simple semiquantitative model for the three low-lying potential energy surfaces of Hþ 3 as well as the nonadiabatic collisional coupling matrix elements between them. This model, called Diatomics in Molecules (DIM) (Ellison et al., 1963), requires knowledge of only the potential energy

Fig. 4. H2 and Hþ 2 potential energy curves and vibrational energies.

curves for the lowest two electronic states of H2 and of Hþ 2. Preston and Tully (Preston and Tully, 1971; Tully and Preston, 1971; Tully, 1976) pioneered the Trajectory Surface Hopping (TSH) approach that allows us to understand qualitatively the charge exchange mechanism. In the present context we are interested in predicting the rates and products of collisions that begin on the lowest potential energy surface, i.e. the curve labeled H++H2(v) in Fig. 4, with vX4. From a quantum mechanical perspective, the near degeneracy of H2(v ¼ 4) and Hþ 2 ðv ¼ 0Þ suggests rapid interconversion between the two vibronic states. Similarly, from a classical trajectory perspective, as H+ approaches H2(vX5), the atoms in the H2 molecule vibrate back and forth repeatedly through the crossing seam. As the separation between H+ and H2 decreases, the magnitude of the matrix element with the other electronic state, H þ Hþ 2 , increases, and charge transfer becomes more likely. Charge transfer can occur back and forth many times during the collision. To determine the actual distribution of products requires detailed dynamics calculations, but we can argue qualitatively that net charge transfer should occur for roughly half the collisions. 3.3. Critical evaluation of H++H2(v) ion–molecule reactions For the foreseeable future, ab initio quantum theory will be the only reliable source of information about collisions of protons with vibrationally excited hydrogen molecules. This is because it is very difficult to prepare well-defined beams of vibrationally excited hydrogen molecules in the laboratory. For our critical evaluation, we will depend on recent work from the plasma fusion quantum theory community (Ichihara et al., 2000; Krstic, 2002; Krstic et al., 2002; Krstic and Schultz, 2003; Janev et al., 2003). Two studies

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Table 2 Recommended rate coefficients at 600 K for H++H2(v) charge transfer (kCT) and vibrational relaxation (kVR) (in units of 109 cm3/s) V

kCT kVR

0

1

2

3

4

5

6

7

8

0 0

0 1.2

0 1.8

0 1.8

0.6 1.5

1.3 1.2

1.3 1.2

1.3 1.2

1.3 1.2

(Ichihara et al., 2000; Krstic et al., 2002) investigated the charge transfer Reaction (10). By extrapolating to lower temperatures the results from the earlier study (Ichihara et al., 2000, Table 2), we estimate that Reaction (10) has a rate coefficient at 600 K of approximately 1.3  109 cm3/s, for vX4, consistent with numbers in current models. However, the later study (Krstic et al., 2002) (using more capable theory) found collision cross sections at thermal energies for v ¼ 4 that are much smaller than those for vX5. As a result of this disagreement, we have arbitrarily reduced by 50% the ‘‘recommended’’ rate coefficient for v ¼ 4 shown in Table 2. Another fusion-motivated investigation (Krstic, 2002; Fig. 11) suggested that the vibrational relaxation Reaction (11) should be fast. From graphical analysis of this work, we recommend thermalenergy rate coefficients of between 1.2  109 and 1.8  109 cm3/s as indicated in Table 2. New quantum theory calculations are under way to reduce the uncertainties in these recommendations at low energy (Quemener et al., 2008). Inclusion of Reaction (11) will significantly reduce calculated vibrational temperatures in ionospheric models. Protons are less abundant than neutral hydrogen atoms by a factor of about 10,000 and the proton rate coefficients for vibrational relaxation are about a factor of 10,000 larger. Just as important, Reaction (11) depletes excited vibrational population in vibrational levels 1, 2, and 3, which will contain the vast majority of vibrational energy for plausible vibrational distributions. The results of our earlier analysis (Huestis, 2005) have been adopted in a recent ionospheric modeling study (Moore et al., 2006), which suggests that the high influx of water indicated by Cassini observations will lead to a new mechanism for reducing the ionospheric electron density on Saturn (Connerney and Waite, 1984; Waite et al., 1997, Maurellis and Cravens, 2001) Hþ þ H2 O ! H þ H2 Oþ

fact that the two protons are identical causes the excited rotational and vibrational levels to be metastable against relaxation by infrared emission. Nuclear identity also separates the rotational levels into two classes: the even-J (para) and the odd-J (ortho) levels act as if they were different molecules. In order to reliably model, interpret, and explain the observed population distributions, it essential that we have a quantitative understanding of the collisional processes that excite and relax these excited rotational and vibrational levels. Here we will review and evaluate two aspects of this problem. (a) Vibrational and rotational excitation and relaxation in collisions of H2 with H2. (b) Ortho/para conversion in collisions of H2 with H2. 4.1. Critical evaluation of vibrational excitation and relaxation of H2(v,J) in H2+H2 collisions The analysis above about H++H2(v) ion–molecule reactions emphasizes the need for more careful review of all collision processes involving H2(v,J). Additional motivation comes from interstellar shocks (Danby et al., 1987). The simplest to measure is the vibrational relaxation reaction H2 ðv ¼ 1Þ þ H2 ðv ¼ 0Þ ! H2 ðv ¼ 0Þ þ H2 ðv ¼ 0Þ

(13)

A summary of the experimental measurements from the 1960s and 1970s is presented in Fig. 5. Note that at low temperatures collisions involving ortho-H2 are significantly faster than just pure para-H2. A plausible explanation is that the ortho-H2 molecules are always rotating, even at 0 K, so the collision frequency never goes to zero. The fit formula for the temperature dependence shown in Fig. 5

(12)

4. H2(v,J) Vibrational, rotational, and ortho–para relaxation The excited rotational and vibrational levels of molecular hydrogen are important energy reservoirs in the atmospheres of the giant planets. The small atomic mass produces large rotational and vibrational energy spacings and correspondingly slow collisional relaxation rates. The

Fig. 5. Vibrational relaxation in hydrogen. Shock tube experiments: & Kiefer and Lutz (1966); } Dove and Teitelbaum (1974). Laser Raman excitation and detection experiments: D normal-H2 Audibert et al. (1974); K ortho-H2, J para-H2 and r normal-H2 Audibert et al. (1975).

ARTICLE IN PRESS D.L. Huestis / Planetary and Space Science 56 (2008) 1733–1743 Table 3 Recommended formulas for rate coefficients k(T) in units of cm3/s, for hydrogen relaxation/equilibration collisions, given the temperature (T) in units of Kelvin Vibrational relaxation: H2(v ¼ 1)+H2(v ¼ 0)-H2(v ¼ 0)+H2(v ¼ 0) k1-0(T) ¼ 3.8  109 exp(121.5/T1/3)+4  1013 exp(59/T1/3) +3.5  1018 [for n-H2] Rotational equilibration: H2(J ¼ 2)+H2(J ¼ 0)2H2(J ¼ 0)+H2(J ¼ 0) k2-0(T)+k0-2(T) ¼ 1.3  109 exp(52.1/T1/3) +5  1013 exp(0.08/T1/3) Ortho–para equilibration: H2(ortho)+H2(ortho)2H2(ortho)+H2(para) kop(T)+kpo(T) ¼ [1.56+12.3 exp(173/T)]  1028

and in Table 3 illustrates the benefit for applications of a compact functional form that is based on established physics (Landau and Teller, 1936; Huestis, 2006). There is also some experimental information (Teitelbaum, 1984; Farrow and Chandler, 1988; Kreutz et al., 1988; Ahn et al., 2005) about collisions involving higher vibrational levels, such as H2 ðv ¼ 1Þ þ H2 ðv ¼ 1Þ2 H2 ðv ¼ 2Þ þ H2 ðv ¼ 0Þ

(14)

4.2. Critical evaluation of rotational excitation and relaxation of H2(v,J) in H2+H2 collisions Before tackling the difficult problem of ortho–para conversion, we consider the much simpler case of nuclear-spin conserving rotational relaxation, with DJ ¼ 72. There have been relatively few experimental investigations (Stewart, 1946; Sluijter et al., 1964, 1965; Narayana and Woods, 1970; Mate et al., 2005) of pure rotational excitation and relaxation such as H2 ðJ ¼ 2Þ þ H2 ðJ ¼ 0Þ2H2 ðJ ¼ 0Þ þ H2 ðJ ¼ 0Þ

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backward reactions: kequil ðTÞ ¼ k0!2 ðTÞ þ k2!0 ðTÞ Furthermore, the forward and backward rates are related by detailed balance: k0!2 ðTÞ ¼ k2!0 ðTÞ  g2 =g0 exp½ðE 2  E 0 Þ=ðkTÞ ¼ k2!0 ðTÞ  5 exp½509:7=T using the degeneracy formula gJ ¼ 2J+1 and the measured E2E0 ¼ 354.3735 cm1 (Mikelson, 1998). A reason for this detailed discussion is that theoreticians find it much simpler to calculate the cross section and rate coefficient in the excitation direction, i.e. k0-2(T), and leave it to the reader to calculate the (generally much larger) k2-0(T) by detailed balance. Fig. 6 presents a summary of the results from the experimental and theoretical literature, along with an empirical formula that recapitulates the individual values. The formula, which is also included in Table 3, was obtained by a nonlinear least-squares fit to the latest theoretical values (Flower, 1998; Mate et al., 2005), while constrained to pass through the twice measured experimental value from sound absorption at 77 K (Sluijter et al., 1964, 1965; Narayana and Woods, 1970). Note that when expressed as kequil(T), the values span less than two orders of magnitude from 50 to 1000 K, while k0-2(T) varies by more than four orders of magnitude over the same temperature range (Lee et al., 2006), masking the differences between different approaches under the point symbols. Fig. 6 shows that while differences still exist, overall internal consistency is near. To apply this knowledge in practical applications we also need rate coefficients for ortho–ortho and ortho–para collisions, and in general for initial or final states having more than one quantum of rotational excitation. Some of

(15)

On the other hand, there have been quite a number of theoretical works (Allison and Dalgarno, 1967; Rabitz and Lam, 1975; Green, 1975; Danby et al., 1987; Flower, 1998; Mate et al., 2005; Lee et al., 2006) with increasingly sophisticated dynamical methods and nominally more accurate H2+H2 potential energy surfaces (Schwenke, 1988; Diep and Johnson, 2000; Boothroyd et al., 2002; Lee et al., 2006). Unlike the case for vibration discussed above, many rotational levels are occupied at room temperature, including the 1:3 mixture of the para (even J) and ortho (odd J) forms. This means that in order to isolate a single kinetic process, such as the equation above, it is essential to carry out experiments in pure para-H2 and at sufficiently low temperatures such that the population in J ¼ 2 is a small fraction of the total. A second point is that a vibrational or a rotational relaxation kinetics experiment measures the overall rate of approach to thermodynamic equilibrium. Thus the ‘‘equilibration’’ rate coefficient is the sum of those of the forward and

Fig. 6. Rotational equilibration in hydrogen. SKB64 ¼ Sluijter et al. (1964); SKB65 ¼ Sluijter et al. (1965); AD67 ¼ Allison and Dalgarno (1967); NW70 ¼ Narayana and Woods (1970); RL75 ¼ Rabitz and Lam (1975); DFM87 ¼ Danby et al. (1987); FLl98 ¼ Flower (1998); MTT05 ¼ Mate et al. (2005).

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the required information has already been calculated (Danby et al., 1987; Flower, 1998). Appropriate J-scaling relationships could be developed to enable kinetics modeling at elevated temperatures. 4.3. Critical evaluation of the H2 ortho/para ratio and ortho/para conversion in giant planet atmospheres In spite of their intrinsic weakness, the forbidden rovibrational H2 radiative transitions play major roles in radiative transport in the dense hydrogen-dominated atmospheres of the giant planets, Jupiter, Saturn, Uranus, and Neptune (Trafton, 1967; Trauger and Bergstralh, 1981). In addition, the relative intensities of even- and oddJ transitions can be used to infer the relative populations of para- and ortho-hydrogen. The pressure-broadened pure-rotation quadrupole transitions S(0) (J ¼ 22J ¼ 0) at 28 mm and S(1) (J ¼ 32J ¼ 1) at 17 mm were observed by the Voyager IRIS spectrometer (Hanel et al., 1979a, b; Conrath and Gierasch, 1983, 1984; Conrath et al., 1998). Narrow-band emissions of the S(0) and S(1) lines have been observed from all four gas giant planets by the Infrared Space Observatory (ISO) (Fouchet et al., 2003). The pressure-broadened vibrational overtone quadruple transitions S4(0) (v ¼ 4, J ¼ 2)2 (v ¼ 0, J ¼ 0), at 6436.70 A˚ and S4(1) (v ¼ 4, J ¼ 3)2 (v ¼ 0, J ¼ 1), at 6369.45 A˚ are observable in the visible using ground-based telescopes (Smith, 1978; Sato and Hanson, 1979; Trauger and Bergstralh, 1981; Baines and Bergstralh, 1986; Baines and Smith, 1990; Baines et al., 1995). Finally, features in the ultraviolet spectrum (2110–2300 A˚) of Jupiter observed from the Hubble Space Telescope have been assigned to Raman-shifted solar Fraunhofer lines resulting from Dv ¼ 1, DJ ¼ 0,2 and Dv ¼ 0, DJ ¼ 2 transitions in para and ortho H2 (Betremieux and Yelle, 1999). The overall conclusion from the previous observations and analyses (references cited above and also Massie and Hunten, 1982; Orton et al., 1986; Carlson et al., 1992; Simon-Miller et al., 2002) is that the H2 ortho/para ratio (usually expressed as the para fraction) in the giant planet atmospheres is not in thermodynamic equilibrium at the local temperature versus altitude and latitude. The corollary conclusion is that the rate of ortho–para conversion is slow compared with the rates of vertical and horizontal transport from hotter or colder regions. This implies that the ortho/para ratio may be a valuable tracer of atmospheric motion or an indicator of chemical processes in active atmospheric regions, e.g., ion chemistry at high latitudes associated with aurora. In addition, the latent heat of ortho–para conversion is a substantial fraction of the enthalpy of H2 in the upper atmospheres of the gas giant planets. Thus the H2 ortho–para ‘‘phase transition’’ may be an important driving force for ‘‘weather’’ and storms corresponding to the phase transition between liquid water and water vapor in the terrestrial atmosphere.

The principal areas of controversy in the literature are estimation of the rate of ortho–para conversion in the giant planet atmospheres and identification of the controlling chemistry. Leading candidate mechanisms are collisions of H2 with paramagnetic aerosols and with the weak magnetic moment of ortho-H2. In a conference presentation (Huestis, 2003), we described possible conversion mechanisms and outlined quantum mechanical calculations of cross sections and rate coefficients for ortho–para conversion in H2–H2 collisions. Here we will review the current state of knowledge. The most significant ‘‘new’’ development is discovery that experimental measurements of the rate of ortho/para conversion in gas-phase hydrogen had been reported in the literature in 1997 (Milenko et al., 1997). It remained unnoticed because an online electronic version of the publication was not available until the spring of 2006. Analysis showed that radiative conversion is too slow by many orders of magnitude for Jo3. Spin-scrambling by atom exchange with H or H+ is also too slow because either the rate coefficient is known to be too small (H) or the collider density is too low (H+). The chemist’s favorite candidate is the four-center exchange reaction H2 ðaaÞ þ H2 ðbbÞ2H2 ðabÞ þ H2 ðbaÞ

(16)

which would also spin-scramble the ortho and para levels. This reaction, with a calculated activation energy that is comparable to the H2 dissociation energy (CarmonaNovillo et al., 2007), would be much too slow at the relevant planetary temperatures. We did not conduct a detailed analysis of the (speculative) paramagnetic aerosol suggestion because we had insufficient information to produce adequate constraints and because interpretation and extrapolation of the available information suggested that direct conversion by spin–spin magnetic interactions in the collisions H2 ðorthoÞ þ H2 ðorthoÞ2H2 ðorthoÞ þ H2 ðparaÞ ! H2 ðparaÞ þ H2 ðparaÞ

(17)

appeared to be significantly faster than the upper limits inferred from atmospheric observations. Fig. 7 summarizes our assessment of the current state of knowledge. Conrath and Gierasch (1984) constructed an estimate for H2+H2 collisions based on scaling the rates for O2 catalysis from Farkas (1935), which the present analysis suggests is an underestimate. In addition, more recent modeling of observational data (Conrath et al., 1998; Fouchet et al., 2003) appears to require even slower rates of ortho–para conversion. Our scaling (Huestis, 2003) of experimental conversion rates (Farkas, 1935) in liquid water and gaseous oxygen produced estimates that were consistent with measured conversion rates in liquid and solid hydrogen (Farkas, 1935). Our analysis has been quantitatively confirmed by the previously unknown experiments in gaseous hydrogen (Milenko et al., 1997).

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Nevada, Las Vegas, respectively. This work was funded in part by grants from the NSF Aeronomy, NSF Planetary Astronomy, NASA Planetary Atmospheres, and NASA Outer Planets Research programs. References

Fig. 7. Rate coefficients for ortho–para conversion in H2+H2 collisions. Fa35 ¼ Farkas (1935); CG84 ¼ Conrath and Gierasch (1984); MSS97 ¼ Milenko et al. (1997); CGU98 ¼ Conrath et al. (1998); FLF03 ¼ Fouchet et al. (2003).

The overall summary of the current state of knowledge is that while atmospheric modeling preferred time scales for ortho–para conversion in H2+H2 collisions of 30–100 years, the current best estimates from existing laboratory data suggest time scales that are much faster, by factor of 10 or more. This conclusion is significant because it places new constraints on the inference of atmospheric transport based on the observed ortho/para ratio. Additional ab initio quantum chemical calculations are needed to confirm this analysis. 5. Conclusions We have presented critical evaluations of the current state of understanding of the chemical reaction rates and collision cross sections for (a) H++H charge transfer, (b) H++H2(v) charge transfer and vibrational relaxation, and (c) H2(v,J)+H2 vibrational, rotational, and ortho–para relaxation. In each case we provided explicit representations as tabulations or compact formulas. Particularly important conclusions are that H++H2(v) collisions are more likely to result in vibrational relaxation than charge transfer and H2 ortho–para conversion is at least an order-of-magnitude faster than previously assumed. Acknowledgements It is a pleasure to acknowledge collaborations with Professors A. Peet Hickman and Balakrishnan Naduvalath, from Lehigh University and the University of

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