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Charge transfer between Si3+ and helium at thermal and low energies
Chemical Physics 238 Ž1998. 401–406
Charge transfer between Si 3q and helium at thermal and low energies P. Honvault a
a,b
, M.C. Bacchus-Montabonel b, M. Gargaud c , R. McCarroll
a
Laboratoire de Dynamique Moleculaire et Atomique (URA 774 du CNRS), UniÕersite´ Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris ´ Cedex 05, France b (UMR 5579 du CNRS), UniÕersite´ Lyon I, 43, Bd du 11 NoÕembre 1918, 69622 Laboratoire de Spectrometrie ´ Ionique et Moleculaire ´ Villeurbanne Cedex, France c ObserÕatoire de l’UniÕersite´ de Bordeaux I (INSU r CNRS URA 352), B.P. 89, 33270 Floirac, France Received 8 July 1998
Abstract Cross sections and rate coefficients for selective electron capture and excitation in Si 3q–He collisions in the thermal-eV energy range are calculated using a quantum mechanical method. It is confirmed that electron capture to the ground state of Si 2q is the dominant charge transfer mechanism and that the inverse process in which Heq captures an electron in a collision with ground state Si 2q can occur easily. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction A quantitative analysis of the emission-line spectra of ionized astronomical objects Žsuch as planetary nebulae, nova shells, active galactic nuclei, etc.. requires reliable data on the microscopic ionization and recombination processes involved. Recombination may occur either by radiative Žor dielectronic. capture or by charge transfer from neutral species. The charge transfer recombination process with atomic hydrogen or helium is particularly important in astrophysical plasmas for many doubly or triply charged ions, whose emission lines are used to furnish direct information on the ionization structure of astronomical objects. Emission and absorption lines of Siq, Si 2q, and Si 3q are observed in many spectra w1–5x. The intensity line ratios of Si 2q are used to deduce electron temperature and density in emissive regions w6–8x.
As a general rule, charge transfer recombination of an ion with charge q leads to the formation of an excited state with charge q y 1. In a dilute medium Žsuch as the astronomical emission-line objects., these states decay radiatively to the ground state before a collision occurs. In that case, the inverse charge transfer ionization is ineffective. On the other hand, there are some doubly and triply charged ions for which electron capture can lead directly to the formation of ground states. When this occurs, ionization may also occur rapidly via the inverse charge transfer process. Two important reactions of this kind are the following Si 2q Ž 3s 2 . 1 S q H Ž 1s . 2 S l Siq Ž 3s 2 3p . 2 P q Hq
Ž 1. Si
3q
2
2 1
Ž 3s . S q He Ž 1s . S l Si
0301-0104r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 3 3 1 - 0
2q
2 1
Ž 3s . S q Heq Ž 1s . 2 S
Ž 2.
402
P. HonÕault et al.r Chemical Physics 238 (1998) 401–406
These reactions are critical in determining the fractional abundances of the Si qq ions for q s 1, . . . , 4. Baliunas and Butler w9x show that inclusion of these two charge transfer reactions change significantly the Si 2q and Si 3q abundance ratios in coronal-type plasmas for temperatures in the range 10 4 - T - 10 5 K. Because of the inverse reaction Ž2., the Si 3q abundance is increased and is dominant over a wide temperature range. Reaction Ž1. has been investigated theoretically w10,11x using model potential methods to represent the Si 2q core electrons. As for reaction Ž2., there have only been empirical estimates w12x of the cross section. In this work we shall not consider the Si 3qrH charge transfer reaction, since it takes place via capture to an excited state of Si 2q. For that reason charge transfer ionization of Si 2q by Hq is ineffective. The Si 3qrHe reaction Ž2. is also of interest as a bench-mark for low energy experiments. At present, there are serious disagreements between some of the new ion-trap measurements w13x and theoretical predictions w14x, which are difficult to understand. For example, no reasonable conjecture seems able to explain why the measured reaction rate for O 2qrHe charge exchange at thermal energies is a factor of 1000 smaller than the theoretical results w14,15x. The Si 3qrHe system would appear to be a better candidate for comparing experiment w13x and theory. As in the case for the O 2qrHe system, theory predicts a large cross section at thermal energies, but the Si 3qrHe system is less likely to be affected by contamination from metastable states.
2. Theory The methods used in this work are similar to those used for recent work on the O 2qrH and O 2qrHe systems w14,16,17x. Only the basic essentials of the theoretical model are recalled here. Although we shall mainly be concerned with reaction Ž2., we consider the complete network of asymptotic channels interacting with both ground Ž3s.2 S and excited Ž3p.2 P states of Si 3q ions. The reason is that while at astrophysical energies Ž0.1–10 eV. there is little interference from adiabatic states to the Ž3p.2 P state of Si 3q, such is not the case at
higher energies where coupling to the Ž3p.2 P channels can become appreciable. We shall assume spinorbit coupling to be of negligible importance in the electron capture process so that electron spin is conserved in the collision process. We need therefore consider only the network of doublet states correlated to the entry channel. The possible reaction channels Žand the asymptotic states to which they are correlated. are designated as follows: Si 2q Ž 3s 2 . 1 S q Heq Ž 1s . 2 S 3
2
Sq
Ž 3.
2
Ž 4.
2
Si 2q Ž 3s3p . P q Heq Ž 1s . S Si
3q
2
2 1
2
Ž 3s . S q He Ž 1s . S
q
Ž 5.
2
Ž 6.
S
Si 2q Ž 3s3p . 1 P q Heq Ž 1s . 2 S 2
2 1
2
Si 3q Ž 3p . P q He Ž 1s . S
Sq
Sq
Ž 3s3p. P q He Ž 1s . S
P
"
Ž 8.
Si 2q Ž 3s3p . 1 P q Heq Ž 1s . 2 S
2
P"
Ž 9.
Si
2
Ž 7.
2
2q
3
Sq
q
Si 3q Ž 3p . 2 P q He Ž 1s 2 . 1 S Si 2q Ž 3p
2 1
.
2
D q Heq Ž 1s . 2 S
P" 2
1
Si 2q Ž 3s3d . D q Heq Ž 1s . 2 S Si 2q Ž 3p
2 3
.
2
P q Heq Ž 1s . S
2
Ž 10 .
Sq,2 P ,2D
Ž 11 .
2
Sq,2 P ,2D
Ž 12 .
y 2
Ž 13 .
S , P
The potential energy curves of the manifold of the Sq and 2 P states, correlated to channels 3–13 have been calculated for interatomic distances in the 2–25 a 0 range by an ab-initio configuration interaction method based on the CIPSI algorithm w18,19x. A non-local pseudo-potential w20x has been used to represent the core electrons of the Si atom. The calculations have been performed with CI spaces of about 300–400 determinants for the zeroth-order diagonalization with a threshold h s 0.01 for the contribution to the perturbation. The basis of atomic functions is presented in Table 1. For Si, we have constructed a 9s7p2d basis of Gaussian functions, contracted to 5s4p2d, starting from the basis sets of McLean and Chandler w21x. The exponents and contraction coefficients have been optimized on the Si 3q Ž3s.2 S and Si 2q Ž3s3p.3 P. For He, we have chosen the 4s1p basis already used in previous applications w22x. A reasonably good agreement with the experimental values of Bashkin and Stoner w23x is observed for a large number of Si 2q 2
P. HonÕault et al.r Chemical Physics 238 (1998) 401–406
403
Table 1 Exponents and contraction coefficients of the Gaussian basis set of Si and He S
Si
He
P
D
Exp.
Cont. coeff.
Exp.
Cont. coeff.
Exp.
Cont. coeff.
657.47 214.00 77.606 30.640 12.816
y0.000024 0.000016 y0.000377 0.000229 y0.002738
95.352 30.334 10.944
y0.000175 y0.000102 y0.004648
0.5 0.2
1.0 1.0
4.042 1.908
0.011817 y0.037812
3.027 1.609 0.274 0.128 97.709 14.857 3.373 0.897 0.251 0.074
1.0 1.0 1.0 1.0 0.00759 0.05413 0.21595 1.0 1.0 1.0
0.347 0.139
1.0 1.0
0.27
1.0
and Si 3q energy levels ŽTable 2.. This accuracy, which is better than 1% for the molecular energies in the dissociation limit, should be sufficient for a satisfactory description of the electron capture dynamics. The radial coupling matrix elements between all pairs of states of the same symmetry have been calculated by means of the finite difference technique: g K LŽ R . s ²CK < E rE R
with the parameter D s 0.0012 au as previously tested and using the silicon nucleus as origin of Table 2 Atomic levels ŽeV.: comparison with Bashkin and Stoner experimental levels Levels
Calculation
Experiment
Si 3q Ž3p. 2 P Si 3q Ž3s. 2 S Si 2q Ž3p 2 . 3 P Si 2q Ž3s3d. 1 D Si 2q Ž3p 2 . 1 D Si 2q Ž3s3p. 1 P Si 2q Ž3s3p. 3 P Si 2q Ž3s 2 . 1 S
41.972 33.191 15.896 15.066 15.072 10.440 6.392 0.0
42.367 33.491 16.113 15.152 15.152 10.276 6.568 0.0
electronic coordinates. The rotational coupling matrix elements ²CK < iL y
404
P. HonÕault et al.r Chemical Physics 238 (1998) 401–406
Fig. 1. Adiabatic potential energy curves for the 2 Sq Ža. and 2 P Žb. states of SiHe 3q. The different channels are labeled as in the main text.
We report in Figs. 2 and 3 the corresponding radial and rotational matrix elements. The shape of these matrix elements is quite consistent with the results of Fig. 1. In particular the radial components reproduce quite well the 0.25 a0 internuclear dis-
Fig. 3. Rotational coupling ŽROC. matrix elements between 2 Sq Žthe first number. and 2 P Žthe second number. states of SiHe 3q. The numerical notation is that used in Fig. 1. Ža. With asymptotic notation, for Si 3q Ž3p.2 PqHe and Si 2q Ž3s3p.3 PqHeq. Žb. With asymptotic notation, for Si 3q Ž3p.2 PqHe and Si 2q Ž3s3p.1 Pq Heq.
tance shift between the 2 Sq and 2 P pseudo-crossing positions. The rotational coupling between the wSi 3q Ž3p.2 P q HeŽ1s 2 .1 Sx and wSi 2q Ž3s 2 .1 S q HeqŽ 1s.2 Sx channels is not shown. The following seven reactions have been investigated simultaneously Žchannels 11, 12, 13 being decoupled from the entry channels.. Si 3q Ž 3s . 2 S q He Ž 1s 2 . 1 S Ž channel 5 . ™ Si 2q Ž 3s 2 . 1 S q Heq Ž 1s . 2 S Ž channel 3 .
Ž 14 .
™ Si 2q Ž 3s3p . 3 P q Heq Ž 1s . 2 S Ž channels 4,8 .
Ž 15 . ™ Si 2q Ž 3s3p . 1 P q Heq Ž 1s . 2 S Ž channels 6,9 .
Ž 16 . Fig. 2. Radial coupling ŽRC. matrix elements between 2 Sq Žsolid curves. and 2 P Ždashed curves. states of SiHe 3q. Same notations as in Fig. 1.
™ Si 3q Ž 3p . 2 P q He Ž 1s 2 . 1 S Ž channels 7,10 .
Ž 17 .
P. HonÕault et al.r Chemical Physics 238 (1998) 401–406
Si 3q Ž 3p . 2 P q He Ž 1s 2 . 1 S Ž channels 7,10 . ™ Si 2q Ž 3s 2 . 1 S q Heq Ž 1s . 2 S Ž channel 3 .
Ž 18 .
™ Si 2q Ž 3s3p . 3 P q Heq Ž 1s . 2 S Ž channels 4,8 .
Ž 19 . ™ Si 2q Ž 3s3p . 1 P q Heq Ž 1s . 2 S Ž channels 6,9 .
Ž 20 . plus the inverse charge transfer reaction: Si
2q
2 1
q
2
Ž 3s . S q He Ž 1s . S Ž channel 3 .
™ Si 3q Ž 3s . 2 S q He Ž 1s 2 . 1 S Ž channel 5 .
Ž 21 .
At low energies, especially at thermal energies, the semi-classical formalism is difficult to apply since trajectory effects need to be accounted for. The collision dynamics have been treated by a quantum mechanical approach using the techniques developed for other applications w25x. Allowance for translation effects have been made by introduction of appropriate reaction coordinates w26,27x. This leads to a modification of the radial and rotational matrix elements similar in form to those resulting from the application of the Common Translation Factor ŽCTF. w28x. Subsequently a diabatic transformation of the
Table 3 Electron capture and excitation cross-sections Ž10y1 6 cm2 . as a function of incident ion energy in eV Žlaboratory system. E ŽeV.
s1
1.09 1.63 2.18 4.35 6.53 10.88 16.33 35.00 70.01 121.99 280.02 630.07 1120.1 1750.5 3499.3 5262.4 7000.7
3.63 3.51 3.83 4.98 6.03 7.76 9.45 12.41 14.39 15.38 14.81 13.47 12.14 11.13 9.87 9.07 8.84
s2
0.36 0.65 0.69 0.64 0.60 0.49 0.41 0.37
s3
0.0001 0.0003 0.0008 0.0013 0.0016 0.0023 0.0040 0.0074
s4
sCT
0.0008 0.03 0.10 0.16 0.20 0.27 0.32 0.34
3.63 3.51 3.83 4.98 6.03 7.76 9.45 12.41 14.39 15.74 15.46 14.16 12.78 11.73 10.36 9.48 9.22
405
Table 4 Rate coefficients Ž10y9 cm3 sy1 . for charge transfer recombinaBu tler . Bu tler . tion Ž k CT , k CT and ionization Ž k ion , k ion T ŽK.
k CT
500 1000 2000 3000 5000 10000 20000 30000 40000 50000 100000
0.08 0.10 0.15 0.21 0.33 0.62 1.14 1.58 1.97 2.31 3.60
tler k Bu CT
0.17 0.39 0.96 2.00
k ion
k Butler ion
y y y y y 0.00002 0.0065 0.05 0.15 0.29 1.28
0.00003 0.07
1.21
basis set is first carried out before solving the coupled differential equations for the determination of the S matrix.
3. Results 3.1. Cross sections We report in Table 3 the charge transfer cross section results obtained for each reaction channel. Taking account of the statistical factors we obtained the following scheme for transitions originating from the ground state Si 3q ions:
s 1 s s5™ 3 s 2 s s5™ 4 q s5™ 8 s 3 s s5™ 6 q s5™ 9 s4 s s5™ 7 q s5™ 10 We define a total charge transfer cross section from ground state Si 3q ions: sCT s s 1 q s 2 q s 3 . Our calculations show Žas expected. that capture to the ground state of Si 2q Žreaction 14. is dominant, although there is a small contribution for energies higher than 120 eV from reaction Ž15. due to the inner crossing at R X s 3.2 a 0 . The contribution of reaction Ž16. is always negligible whatever the energy involved. The cross section Ž s4 . for excitation
P. HonÕault et al.r Chemical Physics 238 (1998) 401–406
406
of the Si 3q into the Ž3p.2 P state becomes appreciable for energies above 600 eV.
help in assessing the accuracy of ab-initio calculations of the molecular parameters Žadiabatic energies and non-adiabatic couplings..
3.2. Rate coefficients For typical astrophysical temperatures only the ground state of Si 3q is significantly populated. The rate constant k CT for charge transfer can then be easily obtained from the cross sections of Table 3 by averaging over a Maxwellian distribution of the kinetic energies of the collision partners. The results are presented in Table 4. The rate constant k CT exceeds 10y9 cm3 sy1 over a wide temperature range Ž2 = 10 4 to 10 5 K.. As T varies from 500 K to 10 5 K, k CT increases from 0.08 = 10y9 cm3 sy1 to 3.52 = 10y9 cm3 sy1 . Table 4 presents also rate coefficients obtained by Butler and Dalgarno w12x. They take into account only reaction Ž14.. Their results are somewhat higher than ours, but bearing in mind the empirical nature of their theoretical model, the agreement is quite reasonable, nevertheless both theoretical approaches lead to cross sections lower than the ion-trap experimental point w13x. The rate constant k ion for the reaction Ž21. can be obtained very simply from k CT by the relations of micro-reversibility:
ž
k ion s g exp y
DE kT
/
k CT ,
Ž 22 .
where g s 1 is the ratio of the statistical weights of the initial and final states and D E s 8.88 eV is the energy gain of reaction Ž14.. The rate constant k ion is large for temperatures above 3 = 10 4 K. Below, it becomes rapidly negligible because of the exponential factor in Ž22..
4. Conclusion The Si 3q–He system presents several unique features which make it an interesting candidate for experimental investigation in the low energy regime. The charge transfer mechanism is controlled by a single avoided crossing and there is little risk of interference with metastable states. The cross section is fairly large and exhibits an appreciable energy dependence in the eV range. All these factors should
References w1x M.A. Hayes, H. Nussbaumer, Astrophys. J. 161 Ž1986. 287. w2x G.A. Doschek, L. Feldman, J.T. Mariska, J.L. Linsky, Astrophys. J. Lett. 226 Ž1978. L35. w3x P.G. Judge, Mon. Not. Roy. Astr. Soc. 223 Ž1986. 239. w4x W.A. Feibelman, Astrophys. J. 258 Ž1982. 548. w5x G.D. Sandlin, J.D.F. Bartoe, G.F. Baureckner, R. Tousey, M.E. Van Hoosier, Astrophys. J. Suppl. 61 Ž1986. 801. w6x P.L. Dufton, F.P. Keenan, A.E. Kingston, Mon. Not. Roy. Astr. Soc. 209 Ž1984. 1. w7x H. Nussbaumer, Astron. Astrophys. 155 Ž1986. 205. w8x F.P. Keenan, P.L. Dufton, A.E. Kingston, Mon. Not. Roy. Astr. Soc. 225 Ž1987. 859. w9x S.L. Baliunas, S.E. Butler, Astrophys. J. 235 Ž1980. L45. w10x R. McCarroll, P. Valiron, Astron. Astrophys. 53 Ž1976. 83. w11x M. Gargaud, R. McCarroll, P. Valiron, Astron. Astrophys. 106 Ž1982. 197. w12x S.E. Butler, A. Dalgarno, Astrophys. J. 241 Ž1980. 838. w13x Z. Fang, V.H.S. Kwong, Astrophys. J. 483 Ž1997. 527. V.H.S. Kwong, Z. Fang, Phys. Rev. Lett. 71 Ž1993. 4127. w14x M. Gargaud, M.C. Bacchus-Montabonel, R. McCarroll, J. Chem. Phys. 99 Ž1993. 4495. w15x S.E. Butler, T.G. Heil, A. Dalgarno, J. Chem. Phys. 80 Ž1984. 4986. w16x P. Honvault, M.C. Bacchus-Montabonel, R. McCarroll, J. Phys. B: At. Mol. Opt. Phys. 27 Ž1994. 3115. w17x P. Honvault, M. Gargaud, M.C. Bacchus-Montabonel, R. McCarroll, Astron. Astrophys. 302 Ž1995. 931. w18x B. Huron, J.P. Malrieu, P. Rancurel, J. Chem. Phys. 58 Ž1973. 5745. w19x S. Evangelisti, J.P. Daudey, J.P. Malrieu, J. Chem. Phys. 75 Ž1983. 91. w20x M. Pelissier, N. Komiha, J.P. Daudey, J. Comput. Chem. 9 Ž1988. 298. w21x A.D. McLean, G.S. Chandler, J. Chem. Phys. 72 Ž1980. 5639. w22x M.C. Bacchus-Montabonel, Phys. Rev. A 46 Ž1992. 217. w23x S. Bashkin, J.O. Stoner, Atomic Energy Levels and Gotrian Diagrams, North Holland, Amsterdam, 1975. w24x F. Fraija, A.R. Allouche, M.C. Bacchus-Montabonel, Phys. Rev. A 49 Ž1994. 272. w25x M. Gargaud, J. Hanssen, R. McCarroll, P. Valiron, J. Phys. B: At. Mol. Opt. Phys. 14 Ž1981. 2259. w26x M. Gargaud, R. McCarroll, P. Valiron, J. Phys. B: At. Mol. Opt. Phys. 20 Ž1987. 1555. w27x R. McCarroll, D.S.F. Crothers, Advances in Atomic, Molecular and Optical Physics, vol. 32, D.R. Bates, B. Bederson ŽEds.., Academic Press, New York, 1994, p. 253. w28x L.F. Errea, L. Mendez, A. Riera, J. Phys. B: At. Mol. Opt. Phys. 15 Ž1982. 101.
Charge transfer between Si 3q and helium at thermal and low energies P. Honvault a
a,b
, M.C. Bacchus-Montabonel b, M. Gargaud c , R. McCarroll
a
Laboratoire de Dynamique Moleculaire et Atomique (URA 774 du CNRS), UniÕersite´ Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris ´ Cedex 05, France b (UMR 5579 du CNRS), UniÕersite´ Lyon I, 43, Bd du 11 NoÕembre 1918, 69622 Laboratoire de Spectrometrie ´ Ionique et Moleculaire ´ Villeurbanne Cedex, France c ObserÕatoire de l’UniÕersite´ de Bordeaux I (INSU r CNRS URA 352), B.P. 89, 33270 Floirac, France Received 8 July 1998
Abstract Cross sections and rate coefficients for selective electron capture and excitation in Si 3q–He collisions in the thermal-eV energy range are calculated using a quantum mechanical method. It is confirmed that electron capture to the ground state of Si 2q is the dominant charge transfer mechanism and that the inverse process in which Heq captures an electron in a collision with ground state Si 2q can occur easily. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction A quantitative analysis of the emission-line spectra of ionized astronomical objects Žsuch as planetary nebulae, nova shells, active galactic nuclei, etc.. requires reliable data on the microscopic ionization and recombination processes involved. Recombination may occur either by radiative Žor dielectronic. capture or by charge transfer from neutral species. The charge transfer recombination process with atomic hydrogen or helium is particularly important in astrophysical plasmas for many doubly or triply charged ions, whose emission lines are used to furnish direct information on the ionization structure of astronomical objects. Emission and absorption lines of Siq, Si 2q, and Si 3q are observed in many spectra w1–5x. The intensity line ratios of Si 2q are used to deduce electron temperature and density in emissive regions w6–8x.
As a general rule, charge transfer recombination of an ion with charge q leads to the formation of an excited state with charge q y 1. In a dilute medium Žsuch as the astronomical emission-line objects., these states decay radiatively to the ground state before a collision occurs. In that case, the inverse charge transfer ionization is ineffective. On the other hand, there are some doubly and triply charged ions for which electron capture can lead directly to the formation of ground states. When this occurs, ionization may also occur rapidly via the inverse charge transfer process. Two important reactions of this kind are the following Si 2q Ž 3s 2 . 1 S q H Ž 1s . 2 S l Siq Ž 3s 2 3p . 2 P q Hq
Ž 1. Si
3q
2
2 1
Ž 3s . S q He Ž 1s . S l Si
0301-0104r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 3 3 1 - 0
2q
2 1
Ž 3s . S q Heq Ž 1s . 2 S
Ž 2.
402
P. HonÕault et al.r Chemical Physics 238 (1998) 401–406
These reactions are critical in determining the fractional abundances of the Si qq ions for q s 1, . . . , 4. Baliunas and Butler w9x show that inclusion of these two charge transfer reactions change significantly the Si 2q and Si 3q abundance ratios in coronal-type plasmas for temperatures in the range 10 4 - T - 10 5 K. Because of the inverse reaction Ž2., the Si 3q abundance is increased and is dominant over a wide temperature range. Reaction Ž1. has been investigated theoretically w10,11x using model potential methods to represent the Si 2q core electrons. As for reaction Ž2., there have only been empirical estimates w12x of the cross section. In this work we shall not consider the Si 3qrH charge transfer reaction, since it takes place via capture to an excited state of Si 2q. For that reason charge transfer ionization of Si 2q by Hq is ineffective. The Si 3qrHe reaction Ž2. is also of interest as a bench-mark for low energy experiments. At present, there are serious disagreements between some of the new ion-trap measurements w13x and theoretical predictions w14x, which are difficult to understand. For example, no reasonable conjecture seems able to explain why the measured reaction rate for O 2qrHe charge exchange at thermal energies is a factor of 1000 smaller than the theoretical results w14,15x. The Si 3qrHe system would appear to be a better candidate for comparing experiment w13x and theory. As in the case for the O 2qrHe system, theory predicts a large cross section at thermal energies, but the Si 3qrHe system is less likely to be affected by contamination from metastable states.
2. Theory The methods used in this work are similar to those used for recent work on the O 2qrH and O 2qrHe systems w14,16,17x. Only the basic essentials of the theoretical model are recalled here. Although we shall mainly be concerned with reaction Ž2., we consider the complete network of asymptotic channels interacting with both ground Ž3s.2 S and excited Ž3p.2 P states of Si 3q ions. The reason is that while at astrophysical energies Ž0.1–10 eV. there is little interference from adiabatic states to the Ž3p.2 P state of Si 3q, such is not the case at
higher energies where coupling to the Ž3p.2 P channels can become appreciable. We shall assume spinorbit coupling to be of negligible importance in the electron capture process so that electron spin is conserved in the collision process. We need therefore consider only the network of doublet states correlated to the entry channel. The possible reaction channels Žand the asymptotic states to which they are correlated. are designated as follows: Si 2q Ž 3s 2 . 1 S q Heq Ž 1s . 2 S 3
2
Sq
Ž 3.
2
Ž 4.
2
Si 2q Ž 3s3p . P q Heq Ž 1s . S Si
3q
2
2 1
2
Ž 3s . S q He Ž 1s . S
q
Ž 5.
2
Ž 6.
S
Si 2q Ž 3s3p . 1 P q Heq Ž 1s . 2 S 2
2 1
2
Si 3q Ž 3p . P q He Ž 1s . S
Sq
Sq
Ž 3s3p. P q He Ž 1s . S
P
"
Ž 8.
Si 2q Ž 3s3p . 1 P q Heq Ž 1s . 2 S
2
P"
Ž 9.
Si
2
Ž 7.
2
2q
3
Sq
q
Si 3q Ž 3p . 2 P q He Ž 1s 2 . 1 S Si 2q Ž 3p
2 1
.
2
D q Heq Ž 1s . 2 S
P" 2
1
Si 2q Ž 3s3d . D q Heq Ž 1s . 2 S Si 2q Ž 3p
2 3
.
2
P q Heq Ž 1s . S
2
Ž 10 .
Sq,2 P ,2D
Ž 11 .
2
Sq,2 P ,2D
Ž 12 .
y 2
Ž 13 .
S , P
The potential energy curves of the manifold of the Sq and 2 P states, correlated to channels 3–13 have been calculated for interatomic distances in the 2–25 a 0 range by an ab-initio configuration interaction method based on the CIPSI algorithm w18,19x. A non-local pseudo-potential w20x has been used to represent the core electrons of the Si atom. The calculations have been performed with CI spaces of about 300–400 determinants for the zeroth-order diagonalization with a threshold h s 0.01 for the contribution to the perturbation. The basis of atomic functions is presented in Table 1. For Si, we have constructed a 9s7p2d basis of Gaussian functions, contracted to 5s4p2d, starting from the basis sets of McLean and Chandler w21x. The exponents and contraction coefficients have been optimized on the Si 3q Ž3s.2 S and Si 2q Ž3s3p.3 P. For He, we have chosen the 4s1p basis already used in previous applications w22x. A reasonably good agreement with the experimental values of Bashkin and Stoner w23x is observed for a large number of Si 2q 2
P. HonÕault et al.r Chemical Physics 238 (1998) 401–406
403
Table 1 Exponents and contraction coefficients of the Gaussian basis set of Si and He S
Si
He
P
D
Exp.
Cont. coeff.
Exp.
Cont. coeff.
Exp.
Cont. coeff.
657.47 214.00 77.606 30.640 12.816
y0.000024 0.000016 y0.000377 0.000229 y0.002738
95.352 30.334 10.944
y0.000175 y0.000102 y0.004648
0.5 0.2
1.0 1.0
4.042 1.908
0.011817 y0.037812
3.027 1.609 0.274 0.128 97.709 14.857 3.373 0.897 0.251 0.074
1.0 1.0 1.0 1.0 0.00759 0.05413 0.21595 1.0 1.0 1.0
0.347 0.139
1.0 1.0
0.27
1.0
and Si 3q energy levels ŽTable 2.. This accuracy, which is better than 1% for the molecular energies in the dissociation limit, should be sufficient for a satisfactory description of the electron capture dynamics. The radial coupling matrix elements between all pairs of states of the same symmetry have been calculated by means of the finite difference technique: g K LŽ R . s ²CK < E rE R
with the parameter D s 0.0012 au as previously tested and using the silicon nucleus as origin of Table 2 Atomic levels ŽeV.: comparison with Bashkin and Stoner experimental levels Levels
Calculation
Experiment
Si 3q Ž3p. 2 P Si 3q Ž3s. 2 S Si 2q Ž3p 2 . 3 P Si 2q Ž3s3d. 1 D Si 2q Ž3p 2 . 1 D Si 2q Ž3s3p. 1 P Si 2q Ž3s3p. 3 P Si 2q Ž3s 2 . 1 S
41.972 33.191 15.896 15.066 15.072 10.440 6.392 0.0
42.367 33.491 16.113 15.152 15.152 10.276 6.568 0.0
electronic coordinates. The rotational coupling matrix elements ²CK < iL y
404
P. HonÕault et al.r Chemical Physics 238 (1998) 401–406
Fig. 1. Adiabatic potential energy curves for the 2 Sq Ža. and 2 P Žb. states of SiHe 3q. The different channels are labeled as in the main text.
We report in Figs. 2 and 3 the corresponding radial and rotational matrix elements. The shape of these matrix elements is quite consistent with the results of Fig. 1. In particular the radial components reproduce quite well the 0.25 a0 internuclear dis-
Fig. 3. Rotational coupling ŽROC. matrix elements between 2 Sq Žthe first number. and 2 P Žthe second number. states of SiHe 3q. The numerical notation is that used in Fig. 1. Ža. With asymptotic notation, for Si 3q Ž3p.2 PqHe and Si 2q Ž3s3p.3 PqHeq. Žb. With asymptotic notation, for Si 3q Ž3p.2 PqHe and Si 2q Ž3s3p.1 Pq Heq.
tance shift between the 2 Sq and 2 P pseudo-crossing positions. The rotational coupling between the wSi 3q Ž3p.2 P q HeŽ1s 2 .1 Sx and wSi 2q Ž3s 2 .1 S q HeqŽ 1s.2 Sx channels is not shown. The following seven reactions have been investigated simultaneously Žchannels 11, 12, 13 being decoupled from the entry channels.. Si 3q Ž 3s . 2 S q He Ž 1s 2 . 1 S Ž channel 5 . ™ Si 2q Ž 3s 2 . 1 S q Heq Ž 1s . 2 S Ž channel 3 .
Ž 14 .
™ Si 2q Ž 3s3p . 3 P q Heq Ž 1s . 2 S Ž channels 4,8 .
Ž 15 . ™ Si 2q Ž 3s3p . 1 P q Heq Ž 1s . 2 S Ž channels 6,9 .
Ž 16 . Fig. 2. Radial coupling ŽRC. matrix elements between 2 Sq Žsolid curves. and 2 P Ždashed curves. states of SiHe 3q. Same notations as in Fig. 1.
™ Si 3q Ž 3p . 2 P q He Ž 1s 2 . 1 S Ž channels 7,10 .
Ž 17 .
P. HonÕault et al.r Chemical Physics 238 (1998) 401–406
Si 3q Ž 3p . 2 P q He Ž 1s 2 . 1 S Ž channels 7,10 . ™ Si 2q Ž 3s 2 . 1 S q Heq Ž 1s . 2 S Ž channel 3 .
Ž 18 .
™ Si 2q Ž 3s3p . 3 P q Heq Ž 1s . 2 S Ž channels 4,8 .
Ž 19 . ™ Si 2q Ž 3s3p . 1 P q Heq Ž 1s . 2 S Ž channels 6,9 .
Ž 20 . plus the inverse charge transfer reaction: Si
2q
2 1
q
2
Ž 3s . S q He Ž 1s . S Ž channel 3 .
™ Si 3q Ž 3s . 2 S q He Ž 1s 2 . 1 S Ž channel 5 .
Ž 21 .
At low energies, especially at thermal energies, the semi-classical formalism is difficult to apply since trajectory effects need to be accounted for. The collision dynamics have been treated by a quantum mechanical approach using the techniques developed for other applications w25x. Allowance for translation effects have been made by introduction of appropriate reaction coordinates w26,27x. This leads to a modification of the radial and rotational matrix elements similar in form to those resulting from the application of the Common Translation Factor ŽCTF. w28x. Subsequently a diabatic transformation of the
Table 3 Electron capture and excitation cross-sections Ž10y1 6 cm2 . as a function of incident ion energy in eV Žlaboratory system. E ŽeV.
s1
1.09 1.63 2.18 4.35 6.53 10.88 16.33 35.00 70.01 121.99 280.02 630.07 1120.1 1750.5 3499.3 5262.4 7000.7
3.63 3.51 3.83 4.98 6.03 7.76 9.45 12.41 14.39 15.38 14.81 13.47 12.14 11.13 9.87 9.07 8.84
s2
0.36 0.65 0.69 0.64 0.60 0.49 0.41 0.37
s3
0.0001 0.0003 0.0008 0.0013 0.0016 0.0023 0.0040 0.0074
s4
sCT
0.0008 0.03 0.10 0.16 0.20 0.27 0.32 0.34
3.63 3.51 3.83 4.98 6.03 7.76 9.45 12.41 14.39 15.74 15.46 14.16 12.78 11.73 10.36 9.48 9.22
405
Table 4 Rate coefficients Ž10y9 cm3 sy1 . for charge transfer recombinaBu tler . Bu tler . tion Ž k CT , k CT and ionization Ž k ion , k ion T ŽK.
k CT
500 1000 2000 3000 5000 10000 20000 30000 40000 50000 100000
0.08 0.10 0.15 0.21 0.33 0.62 1.14 1.58 1.97 2.31 3.60
tler k Bu CT
0.17 0.39 0.96 2.00
k ion
k Butler ion
y y y y y 0.00002 0.0065 0.05 0.15 0.29 1.28
0.00003 0.07
1.21
basis set is first carried out before solving the coupled differential equations for the determination of the S matrix.
3. Results 3.1. Cross sections We report in Table 3 the charge transfer cross section results obtained for each reaction channel. Taking account of the statistical factors we obtained the following scheme for transitions originating from the ground state Si 3q ions:
s 1 s s5™ 3 s 2 s s5™ 4 q s5™ 8 s 3 s s5™ 6 q s5™ 9 s4 s s5™ 7 q s5™ 10 We define a total charge transfer cross section from ground state Si 3q ions: sCT s s 1 q s 2 q s 3 . Our calculations show Žas expected. that capture to the ground state of Si 2q Žreaction 14. is dominant, although there is a small contribution for energies higher than 120 eV from reaction Ž15. due to the inner crossing at R X s 3.2 a 0 . The contribution of reaction Ž16. is always negligible whatever the energy involved. The cross section Ž s4 . for excitation
P. HonÕault et al.r Chemical Physics 238 (1998) 401–406
406
of the Si 3q into the Ž3p.2 P state becomes appreciable for energies above 600 eV.
help in assessing the accuracy of ab-initio calculations of the molecular parameters Žadiabatic energies and non-adiabatic couplings..
3.2. Rate coefficients For typical astrophysical temperatures only the ground state of Si 3q is significantly populated. The rate constant k CT for charge transfer can then be easily obtained from the cross sections of Table 3 by averaging over a Maxwellian distribution of the kinetic energies of the collision partners. The results are presented in Table 4. The rate constant k CT exceeds 10y9 cm3 sy1 over a wide temperature range Ž2 = 10 4 to 10 5 K.. As T varies from 500 K to 10 5 K, k CT increases from 0.08 = 10y9 cm3 sy1 to 3.52 = 10y9 cm3 sy1 . Table 4 presents also rate coefficients obtained by Butler and Dalgarno w12x. They take into account only reaction Ž14.. Their results are somewhat higher than ours, but bearing in mind the empirical nature of their theoretical model, the agreement is quite reasonable, nevertheless both theoretical approaches lead to cross sections lower than the ion-trap experimental point w13x. The rate constant k ion for the reaction Ž21. can be obtained very simply from k CT by the relations of micro-reversibility:
ž
k ion s g exp y
DE kT
/
k CT ,
Ž 22 .
where g s 1 is the ratio of the statistical weights of the initial and final states and D E s 8.88 eV is the energy gain of reaction Ž14.. The rate constant k ion is large for temperatures above 3 = 10 4 K. Below, it becomes rapidly negligible because of the exponential factor in Ž22..
4. Conclusion The Si 3q–He system presents several unique features which make it an interesting candidate for experimental investigation in the low energy regime. The charge transfer mechanism is controlled by a single avoided crossing and there is little risk of interference with metastable states. The cross section is fairly large and exhibits an appreciable energy dependence in the eV range. All these factors should
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