Electron Energy Transfer Rate Coefficients of Carbon Dioxide - The

An extensive set of integral cross sections (ICSs) for electron impact vibrational excitation of the CO2 molecule has been used to calculate electron ...
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Electron Energy Transfer Rate Coefficients of Carbon Dioxide G. B. Poparic´,*,† M. M. Ristic´,‡ and D. S. Belic´† Faculty of Physics, UniVersity of Belgrade, Studentski trg 12, P.O. Box 368, 11000 Belgrade, Serbia, and Faculty of Physical Chemistry, UniVersity of Belgrade, Studentski trg 12, P.O. Box 137, 11000 Belgrade, Serbia ReceiVed: September 5, 2009; ReVised Manuscript ReceiVed: October 28, 2009

An extensive set of integral cross sections (ICSs) for electron impact vibrational excitation of the CO2 molecule has been used to calculate electron energy transfer rate coefficients. The ICSs for electron impact symmetric stretch vibrational excitation are measured by using a high resolution double trochoidal electron spectrometer, while ICSs for the bending and asymmetric vibrations have been adopted from previous publications. Calculations of the energy transfer rate coefficients are performed for the equilibrium conditions in the mean electron energy range from 0 to 11 eV. By use of extended Monte Carlo simulations, electron energy distribution functions (EEDFs) and electron energy transfer rate coefficients are determined in the nonequilibrium conditions, for low and moderate values of the electric field over gas number density ratios, E/N, up to 150 Td. Contributions of higher vibrational levels are emphasized. The results are compared with the data available in the literature. Introduction Electron collision processes with the carbon-dioxide molecule are important in many naturally occurring phenomena and also in plasma devices and laser technology. In particular, CO2 has been intensively studied as an important constituent of the Earth’s atmosphere, responsible in part for the greenhouse effects. It is also well-known that electron cooling in collisions with CO2 molecules represents an important energy transfer process in the atmospheres of Mars and Venus.1,2 For modeling all of these phenomena, one needs to know cross sections and rate coefficients for the various involved processes. At low electron energies, vibrational excitation is the dominant process of energy transfer. It has been studied in detail by Bruche,3 Cadez et al.,4 Johnstone et al.,5 Kitajima et al.6 and by Allan.7 Low energy vibrational excitation of the CO2 molecule has two significant resonant contributions: a 2Πu shape resonance, observed by Bonnes and Schulz,8 with the maximum between 3 and 4 eV and a virtual state below 2 eV. The last resonance causes ICSs for low quantum number vibrational transitions to rise steeply above their threshold energy. It has been studied by Morrison,9 Herzenberg,10 Kochem et al.,11 Estrada and Domcke,12 Morgan et al.,13 Morgan,14 Rescigno et al.,15 Mazevet et al.,16 Field et al.17 and by Allan.7,18 In this paper we will present our experimental results for ICSs of the first eight members of the symmetric stretch vibrational series. They are used, together with the extended set of data from other sources, to calculate electron impact energy transfer rate coefficients in the low electron energy region. Calculations are performed for the equilibrium conditions by using a Maxwellian electron energy distribution. An advanced Monte Carlo simulation program has been developed to calculate electron energy distribution functions (EEDFs) and electron energy transfer rate coefficients in nonequilibrium conditions in the presence of a homogeneous electric field for typical, moderate values of the electric field over gas number density * To whom correspondence should be addressed: E-mail: Goran_Poparic@ ff.bg.ac.rs. † Faculty of Physics. ‡ Faculty of Physical Chemistry.

ratios, E/N. These results are compared with each other and with the results published by other authors. Vibrational Excitation Cross Sections of CO2 Low energy electron impact cross sections for vibrational excitation of the CO2 molecule have been recently summarized by Campbell et al.2 Some lower level excitations for all three vibrational modes of CO2 are investigated both experimentally and theoretically; see Brunger et al.19 Kitajima et al.,6 have measured differential cross sections for electron impact excitation of the (010), (100), (001), and (020) vibrational levels in a crossed-beam experiment for energies from 1.5 to 30 eV. The absolute differential cross sections (DCSs) at 135o for the Fermi dyad [(100), (020)] and the (000), (001), and (101) levels were measured by Allan7,18 by using a hemispherical electron spectrometer that allows the threshold peaks at low electron energies to be seen. Relative DCSs at 90° for the Fermi triad [(200) + (120) + (040)] were also measured by Allan.7 Theoretically, integral cross sections for electron impact vibrational excitation of CO2 are calculated by McCurdy et al.20 They used the complex Kohn variational method in a fully ab initio study. ICSs for excitation of the components of the Fermi dyad [levels (100) and (020)] and of the Fermi triad [(200), (120) and (040)] were calculated. Measurement of the Symmetric Stretch (v00) Vibrational Excitation. Low energy electron impact vibrational excitation cross sections of CO2 are measured in our laboratory by use of a high resolution crossed-beam double trochoidal electron spectrometer. Our experimental procedure has been described in detail by Vicic et al.21 and so only a brief outlook will be given here. The electron beam is prepared by using a trochoidal electron monochromator (TEM) and is crossed at right angles with the gas beam. After the collision, inelastically scattered electrons are analyzed by use of a double TEM device. Due to the presence of a longitudinal magnetic field, needed for the trochoidal electron selectors operation, in our originally designed apparatus the detected signal consisted of the sum of electrons inelastically scattered at 0 and 180 degrees. Selected electrons are detected by use of a channel electron multiplier, counted

10.1021/jp908593e  2010 American Chemical Society Published on Web 12/31/2009

Electron Energy Transfer Rate Coefficients of CO2

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Figure 1. ICSs for symmetric stretch vibrational excitation of CO2: (full line) present results; (dashed line) Campbell et al.2

and the results are stored in an on line-computer. Due to the experimental geometry and the trochoidal movement of scattered electrons perpendicular to the magnetic field, the transmission function of this experiment is scaled as 1/Er,21 where Er is the residual electron energy. Obtained results are corrected for this effect. We have focused our attention to measure the excitation functions from the ground level of CO2 molecule to the first 8 vibrationally excited symmetric stretch levels via the 2Πu shape resonance. As already pointed out, there are Fermi resonances between the vibrational levels of CO2. For example, the (100) and (020) modes and the (200) and (040) ones are coupled, respectively. We have estimated different contributions by comparison of our measurements with the electron energy loss spectra of Johnstone et al.5 and of Kitajima et al.6 Their deconvolution procedure and angular distributions, extrapolated to the low scattering angles, are employed. In such a way the couplings are separated and the (100) and (200) contributions are obtained. For higher levels possible couplings are not considered. We have found that the cross sections for higher members of the bending and asymmetric stretch vibrations decrease more rapidly in comparison to the symmetric stretch ones, and thus their contributions are neglected. Our measurements are in fact relative differential cross sections, and to normalize our results to the absolute scale, we have used recent measurements of Kitajima et al.6 The absolute DCSs of Kitajima et al.6 for the (100) excitation channel at 3.8 eV are extrapolated by a fourth-order Legendre polynomials and integrated. Integral cross section value of 1.33 × 10-16 cm2 is found and our result for the V ) 1 symmetric stretch excitation level is normalized to this value. For the higher levels our relative DCSs are scaled by using the number of counts for each particular vibrational channel under the same experimental conditions. This normalization is performed under the assumption of the angular distributions being the same for all vibrational levels and independent of the electron energy, which is of constant ratios of differential and integral cross sections. This assumption should be valid for a resonant process and can be adopted here since in the considered energy region resonant contribution dominates to a high extent over the direct process. Our normalized results for the symmetric stretch vibrational excitation of the first eight levels, for electron energies from 2.5 to 6 eV, are shown in Figure 1. It should be noted that the cross sections for the higher levels are more than 3 times lower than for the V ) 1 transition. Our results are compared for the first two members of the series, (100) and (200), with the results

of Campbell et al.;2 see Figure 1. The agreement is remarkable. For the (100) level this is expected having in mind that both results are normalized by the DCSs of Kitajima et al.6 Cross-Section Database. The present ICSs for symmetric stretch vibrational excitation of CO2 are used to calculate electron energy transfer rate coefficients. To complete the database of ICSs for all significant vibrational levels of the CO2 molecule in the considered energy region, we have used the results of other authors, as well. The low energy contribution, below 2.5 eV, via the virtual state is included for the first two symmetric stretch levels. For the (100) level it is estimated from the DCSs of Kitajima et al.6 For the (200) level, we have adopted recommended values from Campbell et al.,2 which are based on the calculation of McCurdy et al.20 and on the measurements of Allan.7 The ICSs for (010), (020), and (040) bending mode are also adopted from Campbell et al.2 These are the recommended results based on the measurements and calculations of different authors: Kitajima et al.,6 Brunger et al.,19 McCurdy et al.,20 and Allan.7 For the asymmetric stretch mode, ICSs for (001) excitation are determined from differential cross sections published by Kitajima et al.6 The contribution via the virtual state at low energy is also included. The ICSs for the other asymmetric stretch levels are much lower in magnitude and are not taken into account here. The ICS for the combined vibrational excitation mode (120) is also adopted from McCurdy et al.20 The sources of the ICSs for elastic scattering and for the electronic excitation and ionization, needed for the nonequilibrium electron energy distribution functions evaluation, will be presented later. Electron Energy Transfer Rate Coefficients An extended set of ICSs for excitation of significant levels in all vibrational modes of the CO2 molecule is used now to calculate corresponding electron energy transfer rate coefficients. These rates represent electron impact energy loss per unit electron and molecule density. Calculations are performed for both equilibrium and nonequilibrium conditions, with the low electron energy region primarily considered. The rates are determined in the mean electron energy region from 0 to 11 eV for the equilibrium case, and in the E/N interval from 0 to 150 Td for the nonequilibrium case. Maxwellian Rate Coefficients. Electron impact energy transfer rate coefficients, for a specific vibrational level excitation, are given by22

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j el) ) εthres Q(E

∫ε+∞ σV(ε)√(ε) fe(Ej el,ε) dε thres

Poparic´ et al.

(1)

j el is the mean electron energy, σv(ε) is the integral cross Here, E section for the specific vibrational level excitation, εthres is the threshold energy for the considered vibrational excitation j el,ε) is the normalized electron energy distribuchannel and fe(E 23,24 tion function

∫0+∞ fe(Ej el,ε) dε ) 1

(2)

For the equilibrium case, the electron energy distribution function is given by the Maxwellian equation:

j el) j el,ε) ) 2π-1/2(3/2E j el)+3/2√ε exp(-3ε/2E fe(E

Figure 2. Present electron energy transfer rate coefficients for equilibrium conditions.

(3)

For this case, the electron energy transfer rates are therefore determined by direct numerical integration of the vibrational ICSs for the Maxwellian electron energy distribution function, according to eq 1. Numerical integration is performed for the mean electron energies in the region from 0 to 11 eV. Maxwellian electron energy transfer rate coefficients are therefore presented in Figure 2. For each considered vibrational ICS, specific rates are determined. The rates corresponding to the same vibrational mode of the CO2 molecule are grouped together and presented in this figure as partial rates. The sum of all partial contributions included in the calculation represents the total rate coefficient. Note that bending vibration degeneracy is taken into account by appropriate statistical weighting. The total rates are also shown in the figure. The total rates increase steeply above the lower excitation threshold and have a maximum of 4.93 × 10-9 eV cm3 s-1 at 4 eV. In the low energy region, below 1 eV of the mean electron energy, or below 10 000 K, the virtual state has the dominant contribution to the energy transfer rate coefficients. The evidence for this is seen in all modes but is pronounced more clearly in the bending mode. Contributions of symmetric stretch vibrations (included are the first eight vibrational levels), bending vibrations (for the 010, 020, and 040 levels) and asymmetric (001 level) as well as combined vibrational excitation (120 level) are calculated and shown separately. As can be seen from Figure 2, the largest contribution to the energy transfer rate coefficients is due to the symmetric vibrations. Present results for the total vibrational electron energy transfer rates for the equilibrium conditions are compared with the previous results published by Morrison and Greene1 and by Campbell et al.2 This comparison is performed below 11 000 K, in the energy region where the other results are evaluated, and is presented in Figure 3. The three sets of data are seen to be in a reasonable agreement in this low energy region, where the excitations via the virtual state dominates. This is expected, having in mind that similar sets of ICSs are used for these calculations. This is in particular true for the present results and for the more recent results of Campbell et al.2 Above 6000 K, in the region where the 2Πu shape resonance becomes more important, the present results are seen to increase significantly relative to the data of Campbell et al.2 As already pointed out, a larger number of vibrational levels is included in our calculations, in particular for the symmetric stretch vibrations, relative to the previous studies. It is clear from Figure 1 that the ICSs decrease in magnitude with an increase in the symmetric stretch quantum number, as would

Figure 3. Comparison of total energy transfer rates for equilibrium conditions with the results of Morrison and Greene1 and Campbell et al.2

Figure 4. Symmetric stretch and total rates for the first 2 symmetric and for our 8 symmetric vibrational levels in CO2.

be also the case with the vibrational excitation rate coefficients. However, the electron energy transfer rate coefficients do not decrease in value as fast since in eq 1 the respective cross sections are multiplied by the increasing threshold energy. Thus, the contributions of higher vibrational levels remain significant and they should not be neglected, whenever it is possible. This is specifically illustrated for the case of the symmetric stretch vibrations in Figure 4. Symmetric stretch energy transfer rates for the first 8 levels are more than a factor of 2 higher than for the case when only the two lower levels are included. Similarly, the significant influence on the total rate coefficients is also demonstrated in this figure.

Electron Energy Transfer Rate Coefficients of CO2 Nonequilibrium EEDFs and Rate Coefficients. To determine electron energy transfer rate coefficients in the case of nonequilibrium conditions, we have developed an extended Monte Carlo simulation technique.25 We have simulated the movement of electrons through the CO2 gas in the presence of an uniform electric field. All scattering processes, both elastic and inelastic, are included in this modeling by using experimentally measured and adopted data for the integral cross sections as a function of energy. The probability for the possible elastic scattering, ionization, and vibrational or electronic excitation is proportional to the value of the corresponding integral cross sections. The decision of which possible processes will happen in each collision event is left to the pseudorandom generated numbers. The scattering angle of the electrons after the collision is determined by using experimentally measured differential cross sections, i.e., the corresponding angular distributions. In that way, the scattering angle is determined also by using pseudorandom generated numbers, but weighted by the real differential cross sections. Since the sets of integral and differential cross sections are measured for discrete values of energy, the cross section data are dynamically interpolated for the actual values of electron energy during its motion. This type of simulation is similar to those developed earlier by White et al.,26 Stojanovic et al.27 and Stojanovic and Petrovic.28 To test our algorithm, we have applied the Reid ramp model gas simulation test,29 and obtained the same results (within the statistical error bars) for the mean electron energy and for the diffusion coefficients as White et al.26 in their benchmark simulations. The present Monte Carlo simulation is also tested by comparing generated electron energy distribution functions in nitrogen,25 for the mean energy values of 0.96 and 2.35 eV, against those in the paper of Mihailov et al.30 Despite the fact that different sets of cross section data are used for the two Monte Carlo simulations, obtained results25 for EEDFs were essentially the same as those found by Mihailov et al.30 For modeling electron diffusion through the carbon-dioxide gas, for the elastic scattering the data from other authors are used. For electron energies of 0.155 and 1.05 eV, the data of Kochem et al.11 are used, while in the energy range from 1.5 to 100 eV the data of Tanaka et al.31 are employed. In modeling inelastic electron collision processes a complete database of ICSs for all significant vibrational modes, as given in previous section, are used. All other excitation processes of the valence and Rydberg levels of the carbon-dioxide molecule have significantly lower cross sections and lie above 7 eV. Their contribution to electron scattering in the low energy region, below 6 eV, is estimated to be about 5% only. Nevertheless, these processes are included in our modeling. Integral cross sections for electron impact excitation of electronic states of CO2 are included for the 1Σu+ and 1Πu levels from Kawahara et al.32 The ICSs for a not specified level with 7 eV threshold energy, cited in the BOLSIG33 database by Pitchford et al.34 as Lowke’s level from Kieffer’s JILA report No 1335 and from a Shimamura review (1989), are also included. The integral cross sections for electron impact ionization of CO2 from Hudson et al.36 are included in our modeling, as well. The nonequilibrium electron energy distribution functions for different values of the mean electron energy are generated by using Monte Carlo simulation for transport of electrons in the carbon-dioxide gas. In Figure 5, EEDFs for two mean electron energy values are shown, for 1.2 eV and for 3.4 eV. These values correspond to the E/N values of 30 and 75 Td, respectively. To allow comparison, Maxwellian distribution

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Figure 5. EEDFs for the mean electron energy values of 1.2 and 3.4 eV, which correspond to the E/N of 30 and 75 Td, respectively.

Figure 6. Electron energy transfer rate coefficients for the nonequilibrium conditions, as a function of E/N.

functions for both mean electron energy values are also shown in Figure 5. Results are shown in a semilogarithmic plot; namely, the electron energy distribution function divided by the square root of the energy is shown as a function of electron energy, because in that way the Maxwellian distribution function becomes a straight line. A significant difference between these two sets of data, for the equilibrium and nonequilibrium cases, can be seen from this figure. Note that the difference is more pronounced for the lower mean electron energy value. For the same E/N values (30 and 75 Td) and the same sets of cross section data, the EEDFs are also generated by numerically solving the Boltzmann equation based on a twoterm Legendre expansion of the velocity distribution function (BOLSIG v1.05).34 These results, also shown in Figure 5, are in very good agreement with the present Monte Carlo simulation data. Electron energy distribution functions are calculated for a large number of mean electron energy values in the energy region from 0 to 6 eV, which correspond to the E/N values in the region from 0.1 to 150 Td. These EEDFs are used together with the vibrational excitation integral cross sections to finally calculate the corresponding nonequilibrium energy transfer rate coefficients. Obtained results for the total and partial vibrational excitation energy transfer rate coefficients are shown in Figure 6, as a function of the ratio of the electric field over gas number density, E/N. The total vibrational excitation energy transfer rate coefficients are also listed in Table 1, for selected mean electron energies for both Maxwellian and nonequilibrium cases, in the region up to 6 eV. Electron energies listed for the nonequilibrium case are the mean electron energies for the corresponding E/N

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Poparic´ et al.

TABLE 1: Comparison of the Total Energy Transfer Rate Coefficients for Maxwellian (QM) and Nonequilibrium (QN) Conditions in 10-9 eV cm3 s-1 Maxwellian conditions j el (eV) QM E 0.05

0.03

0.25 0.45 0.65 0.85 1.05 1.25 1.45

0.80 1.05 1.18 1.39 1.71 2.11 2.55

1.65 1.85 2.05 2.25 2.45 2.65 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.05 5.25 5.45 5.65

2.97 3.36 3.71 4.00 4.24 4.44 4.59 4.71 4.80 4.86 4.90 4.93 4.93 4.92 4.91 4.88 4.85 4.81 4.76 4.72 4.67

5.85 6.05 6.25

4.61 4.56 4.50

nonequilibrium conditions j el (eV) E/N (Td) E QN 0.5 1 3 5 10 15 20

0.05 0.06 0.08 0.10 0.14 0.23 0.48

0.002 0.01 0.06 0.14 0.38 0.65 0.99

25

0.85

1.42

30

1.24

1.98

35

1.56

2.60

40 45 50 55 60 65 70

1.80 2.03 2.25 2.47 2.71 2.93 3.15

3.05 3.58 3.92 4.15 4.41 4.57 4.67

75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

3.41 3.65 3.89 4.08 4.32 4.48 4.70 4.88 5.11 5.20 5.36 5.53 5.72 5.89 6.03 6.21

4.77 4.80 4.77 4.76 4.80 4.75 4.70 4.69 4.63 4.64 4.65 4.56 4.46 4.41 4.34 4.34

values. The comparisons of the total and partial energy transfer rate coefficients for Maxwellian and nonequilibrium cases are also presented in Figure 7. The two sets of data are seen, both from Table 1 and from Figure 7, to be in a very good agreement with each other over the whole considered energy region. This is not quite expected, bearing in mind that the electron energy distribution functions for these two cases are different, as shown in Figure 5.

Figure 7. Comparison of the total and partial energy transfer rates for equilibrium (solid symbols) and nonequilibrium conditions (open symbols).

Despite the very specific form of the nonequilibrium EEDF relative to the Maxwellian distribution function, for understanding the behavior of the rate coefficients one should pay particular attention to the regions of energy in which the ICSs for electron impact vibrational excitation of the CO2 molecule have maxima. These regions are the virtual state region below 1 eV and the 2Πu shape resonance region between 3 and 5 eV, the second one having a dominant contribution. As can be seen from Figure 5, the nonequilibrium EEDF crosses the Maxwellian EEDF generally in the 2Πu resonance region, below 4 eV. Thus, the integration of the product of EEDF and ICS for the two considered cases, left and right from the intersection, obviously can lead to the same value of the rates. As a consequence, it is possible to use the Maxwellian electron energy rate coefficients for modeling CO2 in the nonequilibrium conditions, as well. This is convenient, since they are easier to calculate. Conclusions Electron impact vibrational excitation of the CO2 molecule in the low electron energy region has been investigated. By using a high resolution trochoidal electron spectrometer, we determine integral cross sections for electron impact excitation of the symmetric stretch vibrations. The ICSs for bending and asymmetric vibrations have been adopted from previous publications. Electron energy transfer rate coefficients for the equilibrium electron energy distribution conditions are calculated in the mean electron energy region from 0 to 11 eV. By use of our extended Monte Carlo simulation technique, the electron energy distribution functions and electron energy transfer rate coefficients are determined under nonequilibrium conditions, in the presence of a homogeneous external electric field. Calculations are performed in the mean electron energy region from 0 to 6 eV, or for the E/N values below 150 Td. Comparisons between these two sets of data, as well as with the previous results, have been performed. Acknowledgment. This work was supported in part by the Ministry of Science and Technological Development of the Republic of Serbia by the Project No. 141015. References and Notes (1) Morrison, M. A.; Greene, A. E. J. Geophys. Res. 1978, 83, 1172. (2) Campbell, L.; Brunger, M. J.; Rescigno, T. N. J. Geophys. Res. 2008, 113, E08008. (3) Bruche, E. Ann. Phys. (Leipzig) 1927, 83, 1065. (4) Cadez, I.; Gresteau, F.; Tronc, M.; Hall, R. I. J. Phys. B. 1977, 10, 3821–3834. (5) Johnstone, W. M.; Akther, P.; Newell, W. R. J. Phys. B: At. Mol. Opt. Phys. 1995, 28, 743–753. (6) Kitajima, M.; Watanabe, S.; Tanaka, H.; Takekawa, M.; Kimura, M.; Itikawa, Y. 2001, 34, 1929–1940. (7) Allan, M. J. Phys. B: At. Mol. Opt. Phys. 2002, 35, L387–L395. (8) Bonnes, M. J. W.; Schulz, G. J. Phys. ReV. A 1974, 9, 1969. (9) Morrison, M. A. Phys. ReV. A 1982, 25, 1445. (10) Herzenberg, A. Electron-Molecule Collisions, 1st ed.; Shimamura, I., Takayanagi, K. Eds.; Plenum: New York, 1984. (11) Kochem, K. H.; Sohn, W.; Nebel, N.; Jung, K.; Ehrhardt, H. J. Phys. B: At. Mol. Phys. 1985, 18, 4455–4467. (12) Estrada, H.; Domcke, W. J. Phys. B: At. Mol. Phys. 1985, 18, 4469– 4479. (13) Morgan, L. A.; Gillan, C. J.; Tennyson, J.; Chen, X. J. Phys. B 1997, 30, 4087–4096. (14) Morgan, L. A. Phys. ReV. Lett. 1998, 80, 1873–1875. (15) Rescigno, T. N.; Byrum, D. A.; Isaacs, W. A.; McCurdy, C. W. Phys. ReV. A 1999, 60, 2186–2193. (16) Mazevet, S.; Morrison, M. A.; Morgan, L. A.; Nesbet, R. K. Phys. ReV. A 2001, 64, 040701. (17) Field, D.; Jones, N. C.; Lunt, S. L.; Ziesel, J. P. Phys. ReV. A 2001, 64, 022708.

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J. Phys. Chem. A, Vol. 114, No. 4, 2010 1615 (28) Stojanovic´, V. D.; Petrovic´, Z. L. J. Phys. D: Appl. Phys. 1998, 31, 834–846. (29) Reid, I. D. Aust. J. Phys. 1979, 32, 231–254. (30) Mihajlov, A. A.; Stojanovic´, V. D.; Petrovic´, Z. L. J. Phys. D: Appl. Phys. 1999, 32, 2620–3629. (31) Tanaka, H.; Ishikawa, T.; Masai, T.; Sagara, T.; Boesten, L. Phys. ReV. A 1998, 57, 1798–1808. (32) Kawahara, H.; Kato, H.; Hoshino, M.; Tanaka, H.; Campbell, L.; Brunger, M. J. J. Phys. B: At. Mol. Opt. Phys. 2008, 41, 085203. (33) http://www.siglo-kinema.com/bolsig.htm. (34) Pitchford, L. C.; Oneil, S. V.; Rumble, J. J. R. Phys. ReV. A 1981, 23, 294–304. (35) Kieffer, L. J. JILA report No 13, unpublished ed.; JILA: Boulder, CO, 1973. (36) Hudson, J. E.; Vallance, C.; Harland, P. W. J. Phys. B: At. Mol. Opt. Phys. 2004, 37, 445–455.

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