Phillip C. Stancil

Department of Physics and AstronomyCenter for Simulational Physics

University of Georgia

Fine-structure Excitation of Atomic Ions:Astrophysical applications,status, andneeds

UGA Theoretical Atomic and Molecular Astrophysics Group

Charge exchangeMolecular rovibrational excitationPhotodissociationRadiative associationMolecular (diatomics) line listsApplications: astrophysics, cold collisions, quantum simulation

Fine-Structure Excitation

Collision of an ion X of charge q (or a neutral) with e- or heavy particle Important in a variety of cool and molecular environments:

Level populations (non-LTE emission spectra)Radiative cooling Temperature, density, radiation diagnosticsMany lines observable by Spitzer, SOFIA, Herschel

No comprehensive data compilation

X(nlLJ) + Y → X(nlLJ �) + Y

Energy Diagram

Atomic Diagnostics of X-ray Irradiated Protoplanetary Disks 7

TABLE 2References for excitation coefficients*

Ion referencesNe II fine structure Gri!n et al. (2001) (e)Ne III fine structure Butler & Zeippen (1994) (e)O I fine structure Launay & Roue" (1977a); Abrahamsson et al. (2007) (H); Jaquet et al. (1992) (H2);

Monteiro & Flower (1987) (He); Chambaud et al. (1980) (p); Pequignot (1990) (e)O I forbidden Krems et al. (2006) (H); Zatsarinny & Tayal (2003) (e)S I fine structure Tayal (2004) (e)S II forbidden Tayal (1997) (e)C I fine structure Launay & Roue" (1977a); Abrahamsson et al. (2007) (H);

Monteiro & Flower (1987) (He, H2)C I forbidden Pequignot & Aldrovandi (1976); Zatsarinny et al. (2005) (e)C II fine structure Barinovs et al. (2005) (H); Launay & Roue" (1977b) (H2); Wilson & Bell (2002) (e)*The collision partners are specified inside the brackets.

Fig. 7.— The spatial variation of the Ne II 12.81 µm fine structureemissivity, defined by equation (1), in units of erg cm!3s!1.

have calculated the 12.81µm emissivity (including linetrapping) with this value for the H de-excitation rate,and find that it increases the flux by about a factor oftwo. This increase arises mainly at large radial distances,where the deviations from a thermal population are thegreatest. We conclude that our calculations based onelectronic excitation alone underestimate the Ne II 12.81µm flux, but possibly not by a large factor. Actual cal-culations of the atomic hydrogen excitation rate wouldbe most welcome.

4.2. Oxygen

Figure 8 gives the 5 levels that we consider in cal-culating fine-structure and forbidden line emission fromatomic oxygen and sulfur. According to Table 2, col-lisional excitation rates for the fine-structure levels ofO I are available for collision partners e, p, H, H2 andHe. Atomic hydrogen tends to dominate and, even with-out line trapping, the critical densities are modest, typi-cally ncr(H) < 106cm!3. At radii (> 25AU), densities ofthis order are not reached until vertical column densitiesNH ! 1020

" 1021 cm!2. The full emissivity calculationis particularly relevant at large radii.

Figure 9 shows the spatial distribution of the emissivityof the O I 63µm fine-structure line. The results for theO I 146µm line (not shown) are similar. The spatial pro-file is particularly complicated at small radii (< 10AU).The emissivity has two peaks going down into the disk(increasing vertical column at fixed radius) that resultfrom a combination of e!ects: density and temperaturevariations, abundance changes and line-trapping. Thedeclining temperature, the conversion of atomic oxygen

Fig. 8.— Energy level diagrams for O and S showing the low-lying levels that generate the fine-structure and forbidden lines.Note that the energy scale is in Kelvin.

Fig. 9.— Spatial distribution of the O I 63 µm fine-structure lineemissivity, defined by equation (1), in units of erg cm!3 s!1.

into molecules (CO, H2O and O2), and trapping at largecolumns all tend to decrease the emissivity. These ef-fects are illustrated by Figure 10, which shows the specificemissivity (per O atom). For small column densities, itis constant when the population is thermalized (at radiiR < 10 AU, where the critical densities are reached), orit increases toward a maximum when the critical den-sities are only attained at larger vertical column densi-

Example Spectra

Observed spectra of planetary nebula NGC 7027

Osterbrock & Ferland (2006)

Example Spectra

Model spectra of a photo-dissociation region (Orion Nebula)

Osterbrock & Ferland (2006)

Example Spectra

Observed spectra of a galaxy

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Harwit et al., white paper (2010)

Key Fine-structure Diagnostics

Diagnostic lines versus ionization potential for various environments

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Harwit et al., white paper (2010)

Temperature-Density Diagnostic

O2+ line ratios combined with forbidden linesComparison to planetary nebulae observations

Osterbrock & Ferland (2006)

log ne

YSO Disks

HH 46 [O I]

Most of [O I] along outflow is in high-velocity jetvan Dishoeck (2009)

Which physical component dominates PACS lines?

Protostellarenvelopewith hot core:Low-J CO

UV irradiatedcavity walls, disksurface:Mid-J CO? Quiescent O I?

Outflow shocks:High-J CO,Hot waterHigh velocity O I

PDR

Disk

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1/5/11 2:04 PMLeiden Atomic and Molecular Database

Page 1 of 1http://www.strw.leidenuniv.nl/~moldata/

LAMDALeiden Atomic and Molecular Database

Atomic datafiles | Molecular datafiles | Data format | RADEXAtomic datafiles

CI CII OIMolecular datafiles

CO CS HCl

OCS SO SO2

SiO SiS SiC2

HCO+ N2H+ HCS+

HC3N HCN HNC

C3H2 H2O H2CO

OH CH3OH NH3

HDO H3O+ HNCO

NO CN CH3CN

O2 HFRadiative transfer

RADEX Benchmarking

Development Update history Future updates

The aim of this project is to provide users of radiative transfer codeswith the basic atomic and molecular data needed for the excitationcalculation. Line data of a number of astrophysically interesting speciesare summarized, including energy levels, statistical weights, Einstein A-coefficients and collisional rate coefficients. Available collisional datafrom quantum chemical calculations and experiments are in some casesextrapolated to higher energies.

Currently the database contains atomic data for 3 species and moleculardata for 29 different species. In addition, several isotopomers anddeuterated versions are available. Work is currently underway to addmore datafiles. We encourage comments from the users in order toimprove and extend the database.

This database should form an important tool in analyzing observationsfrom current and future infrared and (sub)millimetre telescopes.Databases such as these rely heavily on the efforts by the chemicalphysics community to provide the relevant atomic and molecular data.We strongly encourage further efforts in this direction, so that thecurrent extrapolations of collisional rate coefficients can be replaced byactual calculations in future releases.

RADEX, a computer program for performing statistical equilibriumcalculations is made publically available as part of the data base.

NEWS (4 January 2011): Minor update of OH datafile.

If you use the data files in your work please refer to the publication bySchöier, F.L., van der Tak, F.F.S., van Dishoeck E.F., Black, J.H.2005, A&A 432, 369-379 introducing this data base. When individualmolecules are considered, references to the original papers providing thespectroscopic and collisional data are encouraged.

Fredrik Schöier, Floris van der Tak, Ewine van Dishoeck, John Black

This research is supported by the Netherlands Organization forScientific Research (NWO) , the Netherlands Research School forAstronomy (NOVA) and the Swedish Research Council.

Last update on Tue, 04 Jan 2011 10:51:26 GMT. Send comments to moldata@strw.leidenuniv.nl

1/5/11 2:06 PMhttp://www.strw.leidenuniv.nl/~moldata/datafiles/c+.dat

Page 1 of 2

!MOLECULEC+ (atomic ion)!MOLECULAR WEIGHT12.0!NUMBER OF ENERGY LEVELS8!LEVEL + ENERGIES(cm^-1) + WEIGHT + J 1 0.000000000 2.0 0.5 ! 2P 2 63.395087 4.0 1.5 ! 2P 3 43003.291 2.0 0.5 ! 4P 4 43025.285 4.0 1.5 ! 4P 5 43053.568 6.0 2.5 ! 4P 6 74930.074 6.0 2.5 ! 2D 7 74932.608 4.0 1.5 ! 2D 8 96493.727 2.0 0.5 ! 2S !NUMBER OF RADIATIVE TRANSITIONS14!TRANS + UP + LOW + EINSTEINA(s^-1) + FREQ(GHz) + E_u(K) 1 2 1 2.300E-06 1900.5369 91.21 2 3 1 5.530E+01 1289206.2 61871.8 3 3 2 6.550E+01 1287305.7 61871.8 4 4 1 1.710E+00 1289865.6 61903.4 5 4 2 5.240E+00 1287965.1 61903.4 6 4 3 2.390E-07 659.364 61903.4 7 5 2 4.320E+01 1288813.0 61944.1 8 5 4 3.670E-07 847.903 61944.1 9 5 3 3.490E-14 1507.267 61944.1 10 7 1 2.393E+08 2246423.1 107810.6 11 7 2 4.773E+07 2244522.5 107810.6 12 6 2 2.864E+08 2244476.5 107808.4 13 8 1 7.643E+08 2892809.2 138832.1 14 8 2 1.526E+09 2890908.6 138832.1 !NUMBER OF COLL PARTNERS4!COLLISIONS BETWEEN2 C+ + pH2 ! Flower & Launay (1977, JPB, 10, 3673); no corr. for Flower (1988)!NUMBER OF COLL TRANS1!NUMBER OF COLL TEMPS6!COLL TEMPS 10.0 20.0 50.0 100.0 200.0 250.0 !TRANS + UP + LOW + COLLRATES(cm^3 s^-1) 1 2 1 3.0e-10 3.4e-10 3.9e-10 4.3e-10 4.6e-10 4.7e-10!COLLISIONS BETWEEN3 C+ + oH2 ! Flower & Launay (1977, JPB, 10, 3673)!NUMBER OF COLL TRANS1!NUMBER OF COLL TEMPS6!COLL TEMPS 10.0 20.0 50.0 100.0 200.0 250.0 !TRANS + UP + LOW + COLLRATES(cm^3 s^-1) 1 2 1 4.4e-10 4.6e-10 4.9e-10 5.1e-10 5.6e-10 5.7e-10!COLLISIONS BETWEEN5 C+ + H ! Launay & Roueff, 1977, JPB, 10, 879

Databases?

Ion e- H H2 He H+ λ(µm)

O Bell1998

Abraham.2007

Jaquet1992

Monteiro1987

Chambau 1980 63,145

O2+ McLaugh.1998 288

O3+ Tayal2006 26

C Johnson1987

Abraham.2007

Schroed.1991

Staemml.1991

Roueff1990 370

C+ Tayal2008

Barinovs2005

Launay1977 158

N+ Hudson2004

122, 205

N2+ Blum1992 57

Ion e- H H2 He H+ λ(µm)

Ne+ Griffin2001 12.8

Ne2+ McLaugh.2000 16, 36

Ne4+ Griffin2000 14, 24

Ne5+ Mitnik2001 7.6

Mg3+ Witthoeft(2007) 4.5

Mg4+ Butler1994

5.6, 13.5

Mg6+ Lennon1994 5.5, 9

Ion e- H H2 He H+ λ(µm)

Mg7+ Zhang1994 3.0

Al 89

Al4+ Witthoeft(2006) 2.9

Si 25, 57

Si+ Tayal2008

Barinovs2005 35

Si5+ Witthoeft(2007) 1.96

Si6+ Butler1994 2.5, 6.5

Ion e- H H2 He H+ λ(µm)

Si8+ Lennon1994 2.5, 3.9

Si9+ Zhang1994 1.43

P+ Tayal2003? 33, 61

P2+ Krueger1970 17.9

P6+ Witthoeft(2007) 1.37

S Tayal2004 25, 57

S2+ Tayal1999 19, 34

Ion e- H H2 He H+ λ(µm)

S3+ Tayal2000 10.5

S7+ Witthoeft(2007) 1

Ar+ Pelan1995 6.98

Ar2+ Munoz-2009 9, 22

Ar4+ Ludlow2010 7.9, 13

Ar5+ Ludlow2010 4.53

Ca3+ Pelan1995 3.2

Ion e- H H2 He H+ λ(µm)

Ca4+ Galavis1995 4.1, 11.5

Fe ? 24, 34

Fe+ ? 26, ...

Fe2+ ? 22.9, ...

Fe4+ Ballance(2007?) 70, ...

Fe5+ Ballance(2008?) 20, ...

Fe6+ Witthoeft(2007) 9.5, 7.8

Fine-structure excitation: Needs

Ne+ + H, Ne2+ + H protoplanetary disks

(Meijerink et al. 2007)

Ne++H,H2, Ne2+ + H,H2 in XDRs

(Abel 2008)

S + Hprotoplanetary disks

(Meijerink et al. 2007)

[NeI

II]/[N

eII]

[NeIII]/[NeII] ratio used as a diagnostic of AGNs

Usually only e-collisions considered

e

e, H, H2

[NeIII] and [NeII] lines observed in protoplanetary disks with Spitzer

H collisional rates needed

[SI] observed in proto-planetary disks

H collisional rates needed

Summary - Needs

Fine-structure excitation by electrons:Calculations for O and C need revisitingNeon ions look in good shapeOther possibilities: N2+, Ar2+, Al, and SFurther work on: Fe, Fe+, Fe2+????

Fine-structure excitation by heavy particlesCalculations for O and C with H2, He, H+ need revisitingO + H+ (later C + H+) in collaboration with David Schultz and Yong WuC+ + H2 should be revisited

Summary - Needs

Fine-structure excitation by heavy particles (cont’d):Calculations of H collisions needed for S, Si, Fe, Al, Ne+, N+, Fe+, Ne2+, O2+ , Fe2+, S2+

Calculations of He and H2 collisions needed for S, Si, Al ,Fe, Ne+, N+, Fe+, Ne2+

Calculations of H+ collisions needed for S and FeConsideration of internuclear-dependent spin-orbit coupling

Compilations neededMeasurements needed

Acknowledgements

Funding from NASA:Astronomy and Physics Research and Analysis (APRA)Origins of Solar Systems Program (OSS)Astrophysics Data Program (ADP)Astrophysics Theory Program (ATP)

Contributions from collaborators and colleagues:Gary Ferland, Mitch Pindzola, Ryan Porter, David Schultz, Yong Wu

Fine-structure excitation: Highlights

O + H, C+H MOCC Calculations

Abrahamsson et al. (2007)

C+ + e R-matrix calculationsTayal (2008)

computed using the conventional basis set of only the 3P states.The couplings to the 1D state appear to be insignificant even athigh collision energies of 10,000 cm!1.

4. SUMMARY

Using accurate interaction potentials of Parlant & Yarkony(1999) and Kalemos et al. (1999), we have recalculated the rate

coefficients for the fine-structure excitations in collisions ofO(3P) and C(3P) with atomic hydrogen. The results are pre-sented in the form of analytical functions representing the ratecoefficients over a wide range of temperatures. To verify theaccuracy of the calculations, we examined the sensitivity of therate coefficients to variations of the interaction potentials atshort range and the couplings to electronically excited statesof oxygen. This is the first calculation of fine-structure excita-tion dynamics of O(3P) with the couplings to the 1D statesincluded. Given the recent progress in interstellar spectros-copy measurements, the refined results presented here may im-prove our understanding of the conditions in the interstellarmedium.

We thank Zhiying Li for verifying some of our results andfor pointing out several misprints. We thank A. Kalemos andD. R. Yarkony for sending us the results of their potential en-ergy calculations, and G. Barinovs for help with interpolatingthe interaction potentials for CH. The work of E. A. and R. K.was supported by NASA grant NNG 06-GJ11G from the As-trophysics Theory Program and the Natural Sciences and En-gineering Research Council (NSERC) of Canada and of A. D.by the Chemical Sciences, Geosciences and Biosciences Divi-sion of the Office of Basic Energy Sciences, Office of Science,US Department of Energy.

REFERENCES

Arthurs, A. M., & Dalgarno, A. 1960, Proc. R. Soc. London, Ser. A, 256, 540Bensch, F. 2006, A&A, 448, 1043Bernard-Salas, J., & Tielens, A. G. G. M. 2005, A&A, 431, 523Cowie, L. L., & Songaila, A. 1986, ARA&A, 24, 499Dalgarno, A., & McCray, R. A. 1972, ARA&A, 10, 375Ge, J., Bechtold, J., & Black, J. H. 1997, ApJ, 474, 67Ho, T.-S., & Rabitz, H. 1996, J. Chem. Phys., 104, 2584Kalemos, A., Mavridis, A., & Metropoulos, A. 1999, J. Chem. Phys., 111, 9536Kramer, C., Jakob, H., Mookerjea, B., Schneider, N., Brull, M., & Stutzki, J.2004, A&A, 424, 887

Krems, R. V., Jamieson, M. J., & Dalgarno, A. 2006, ApJ, 647, 1531

Launay, J. M. 1977, J. Phys. B, 10, 3665Launay, J. M., & Roueff, E. 1977, A&A, 56, 289Oka, T., Iwata, M., Maezawa, H., Ikeda, M., Ito, T., Kamegai, K., Sakai, T., &Yamamoto, S. 2004, ApJ, 602, 803

Papadopoulos, P. P., & Greve, T. R. 2004, ApJ, 615, L29Parlant, G., & Yarkony, D. R. 1999, J. Chem. Phys., 110, 363Pequignot, D. 1990, A&A, 231, 499Quast, R., Baade, R., & Reimers, D. 2002, A&A, 386, 796Shaw, G., Ferland, G. J., Srianand, R., & Abel, N. P. 2006, ApJ, 639, 941Yarkony, D. R. 1992, J. Chem. Phys., 97, 1838

Fig. 2.—Cross sections for the fine-structure excitations in O(3P)-H collisions.Circles: j " 2 ! j " 1 transition; squares: j " 1 ! j " 0 transition; diamonds:j " 2 ! j " 0 transition. Solid curves represent calculations including the 1Dstate of oxygen as described by Krems et al. (2006).

ABRAHAMSSON, KREMS, & DALGARNO1174

Si+ + e R-matrix calculationsTayal (2008)

632 S. S. Tayal: Electron impact excitation collision strength for transitions in C II

been shown in Table 3 with other previous calculations. Ourcalculation agrees very well with the calculation of Tachiev &Froese Fischer (2000) and Correge & Hibbert (2002) for mostfine-structure transitions. The CIV3 results from the calculationof Correge & Hibbert (2002) include core correlation and arefine-tuned to experimental energies. Our results show some dis-crepancies with the calculation of Galavis et al. (1998) for sev-eral fine-structure transitions.

The length ( fL) and velocity ( fV ) values of oscillatorstrengths and transition probabilities (AL and AV ) for dipole-allowed transitions among fine-structure levels have been tabu-lated in Table 4, and our results have been compared with the cal-culation of Tachiev & Froese Fischer (2000). We have reportedresults for the allowed transitions between the levels of doubletand quartet symmetries. The agreement between the length andvelocity forms of oscillator strengths may to some extent in-dicate the accuracy of wave functions and the convergence ofCI expansions. The convergence of results is an important ac-curacy criterion. There is normally a good agreement betweenthe present length and velocity forms of oscillator strengthsand with the results from the calculation of Tachiev & FroeseFischer (2000). The agreement between our results and Tachiev& Froese Fischer (2000) is normally within 20% for most dipole-allowed transitions. The agreement provides us confidence in theaccuracy of our target wave functions.

Accurate description of target wave functions is an essen-tial part of a reliable scattering calculation. The quality of targetwave functions used in our scattering calculation is very goodas has been assessed by comparing computed excitation ener-gies and oscillator strengths with experiment and other reliablecalculations. We included 42 fine-structure levels arising fromthe 23 LS 2s22p 2P!, 2s2p2 4P, 2D, 2P, 2S, 2s2ns (n = 3–6) 2S,2s2np (n = 3–5) 2P!, 2s2nd (n = 3–5) 2D, 2p3 4S ! 2P!, 2D!,2s2n f (n = 4, 5) 2F!, 2s2p3s 4P!, 2P! and 2s2p3p 2P terms. Wehave plotted resonant collision strengths in the thresholds energyregion and non-resonant background collision strengths abovethe highest excitation threshold energy region up to 2.0 Rydas a function of electron energy for the forbidden 2s22p 2P!1/2–2P!3/2 and allowed 2s22p 2P!3/2–2s2p2 2D5/2 transitions in Figs. 1and 2 respectively. It is clear from these figures that the res-onance structures are complex and make significant enhance-ments in collision strengths. The non-resonant background col-lision strength for the allowed transition is larger than for theforbidden transition. The resonance enhancement in collisionstrengths for the forbidden transitions is normally larger com-pared to allowed transitions. The relativistic e!ects appear tobe small for these transitions. We chose a fine energy mesh forcollision strength calculation in the thresholds energy region todelineate the resonance structures. The collision problem in theexternal region was solved using an energy mesh of 0.0002 Rydin the closed-channels energy regions up to 1.658 Ryd. In the en-ergy region of all open channels where there are no resonancesand collision strengths show smooth variation we used an energymesh of 0.2 Ryd. Resonance structures are quite dense in the en-ergy region up to the 2s23p 2P! threshold around 1.20 Ryd. Ourcalculation properly includes important short-range correlatione!ect to ensure correct position of resonances in the low energyregion.

A good agreement with the measured absolute direct excita-tion cross sections for the intercombination 2s22p 2P!–2s2p2 4Pand resonance 2s22p 2P!–2s2p2 2D, 2S transitions (Smith et al.1996) has been obtained. The fine-structure components of thesemultiplets were not resolved in the experiment because for very

0

1

2

3

4

5

6

7

8

9

10

11

0 0.5 1 1.5 2

CO

LLIS

ION

ST

RE

NG

TH

ELECTRON ENERGY (RYD)

Fig. 1. Collision strength for the forbidden 2s22p 2P!1/2–2P!3/2 transitionas a function of electron energy.

2

3

4

5

6

7

8

0.6 0.8 1 1.2 1.4 1.6 1.8 2

CO

LLIS

ION

ST

RE

NG

TH

ELECTRON ENERGY (RYD)

Fig. 2. Collision strength for the allowed 2s22p 2P!3/2–2s2p2 2D5/2 tran-sition as a function of electron energy.

small separations between fine-structure levels. The uncertaintyin experimental cross sections is about 20% and the energy reso-lution is 0.250 eV. We have convoluted theoretical cross sectionsto the experimental energy spread. The convoluted theoreticalcross sections have been compared with the measured cross sec-tions in Figs. 3–5. The convoluted theoretical cross sections forthe intercombination transition in Fig. 3 are within experimentalerror bars for most incident electron energies except a few en-ergies around 12 eV and 15 eV where experimental results aresomewhat larger than the theory.

The excitation cross sections for the resonance 2s22p 2P!–2s2p2 2D and 2S transitions have been compared with measuredcross sections in Figs. 4–5. Our theory shows good agreementwith the measured cross sections for the 2P!–2s2p2 2D transitionin Fig. 4 at most incident electron energies. However, discrepan-cies exist at a few energies. Similar agreement between theoryand experiment exists for the 2P!–2s2p2 2S transition shown inFig. 5. The theoretical cross sections are within experimental un-certainty for all energies except two incident electron energiesaround 14 eV. The 8-state R-matrix theoretical cross sectionsfrom the work of Smith et al. (1996) have also been displayed inFigs. 3–5 by open squares. The two calculations normally agreeto about 20%, except for the 2P!–2s2p2 2S transition close tothreshold energy where 8-state calculation overestimates. Theagreement between theory and experiment provides some ad-ditional indication that our collision strengths are likely to be

lowest 29 levels included in our calculation over a temperature rangethat is suitable for astrophysical plasma modeling calculations.

The theoretical approach and codes used in the calculation ofcollision strengths have been described by Zatsarinny (2006), andhere we present a brief outline. The wave function describing thetotal e+Si ii system in the internal region is expanded in terms ofenergy-independent functions

!k ! AX

ij

aijk"iuj(r); "3#

where "i are channel functions formed from the multiconfigu-rational functions of the 31 target levels, and uj are the radialbasis functions describing the motion of the scattering electron.The operator A antisymmetrizes the wave function and expan-sion coefficients aijk are determined by diagonalizing the (N $ 1)electronHamiltonian. In our calculation, the radial functions uj areexpanded in the B-spline basis as

uj(r) !X

i

aijBi(r); "4#

and the coefficients aij (which now replace aijk in eq. [3]) are de-termined by diagonalizing the (N $ 1) electron Hamiltonian in-side the R-matrix box that contained all bound atomic orbitalsused for the description of Si ii levels. The relativistic effects in thescattering calculations have been incorporated in the Breit-PauliHamiltonian through the use of the Darwin, mass correction, andspin-orbit operators. The radius of theR-matrix boxwas chosen tobe 45.6a0, and 95 B-splines were used for the expansion of con-tinuum orbitals. These parameters were sufficient to obtain con-verged results for a wide energy range up to about 10.95 ryd. TheB-spline R-matrix calculations were carried out for partial wavesup to J ! 36. These partial waves were estimated normally togive converged cross sections for forbidden transitions. For theallowed transitions a top-up procedure based on the Coulomb-Bethe approximation (Burgess & Sheorey 1974) was employedto estimate the contributions for J larger than 36. The top-up con-tributions for the nondipole transitions have been estimated by as-suming that the collision strengths form a geometric progressionin J.

In many astrophysical applications it is convenient to use exci-tation rate coefficients or thermally averaged collision strengths asa function of electron temperature. The excitation rates are ob-

tained by averaging collision strengths over a Maxwellian distri-bution of electron energies. The excitation rate coefficient for atransition from state i to state f at electron temperature Te is givenby

Ci f !8:629 ; 10%6

giT1=2e

!i f (Te) exp%#Ei f

kTe

! "cm3 s%1; "5#

where gi is the statistical weight of the lower level i, #Eif !Ef % Ei is the excitation energy, and !i f is a dimensionless quan-tity called effective collision strength given by

!i f (Te) !Z 1

0

$i f exp%Ef

kTe

! "d

Ef

kTe

! "; "6#

where Ef is the energy of incident electron with respect to theupper level f. If the collision strength is assumed to be indepen-dent of the incident electron energy, we have !i f ! $i f . The ef-fective collision strengths are calculated by integrating collisionstrengths for fine-structure levels over a Maxwellian distributionof electron energies. The integration in equation (6) should be car-ried out using energy-dependent collision strengths from thresh-old to infinity (Burgess & Tully 1992). The collision strengths at

Fig. 2.—Collision strength for intercombination 3s 23p2Po3/2Y3s3p

2 4P3/2 tran-sition as a function of electron energy in ryd.

Fig. 1.—Collision strength for forbidden 3s23p2Po1/2Y3s

23p2Po3/2 transition

as a function of electron energy in ryd.

Fig. 3.—Collision strength for allowed 3s 23p2Po3/2Y3s3p

2 2D5/2 transition as afunction of electron energy in ryd.

ELECTRON SCATTERING FROM Si ii 537No. 2, 2008

1➜0

2➜1

2➜0

Larger by factors of 2-4 compared to previous calculations

O: 3PJ➜3PJ’

C+: 2PJ➜2PJ’

634 S. S. Tayal: Electron impact excitation collision strength for transitions in C II

0

0.2

0.4

0.6

0.8

1

1.2

4 6 8 10 12 14 16 18 20 22 24

Cro

ss s

ectio

n (1

0-16

cm

2)

Electron energy (eV)

Fig. 3. Excitation cross sections for the 2s22p 2P!–2s2p2 4P transition asa function of electron energy. Solid curve, present results; open squares,8-state R-matrix results (Smith et al. 1996); open inverted triangles, ex-perimental results of Smith et al. (1996); horizontal bar, measured re-sults of Lafyatis & Kohl (1987).

0

0.5

1

1.5

2

2.5

8 10 12 14 16 18 20 22 24 26

Cro

ss s

ectio

n (1

0-16

cm

2)

Electron energy (eV)

Fig. 4. Excitation cross sections for the 2s22p 2P!–2s2p2 2D transition asa function of electron energy. Solid curve, present results; open squares,8-state R-matrix results Smith et al. (1996); horizontal bars, measuredresults of Lafyatis & Kohl (1987); open inverted triangles, experimentalresults of (Smith et al. 1996).

R-matrix calculations of Blum & Pradhan (1992) and Wilsonet al. (2005) in Fig. 6. The present results have been shownby solid curve (42-level) and short-dashed curve (35-level)and those of Blum & Pradhan (1992) and Wilson et al. (2005)are displayed by dotted and long-dashed curves respectivelyin the temperature region from log T = 2.5 to 6.0 K. Blum& Pradhan (1992) reported e!ective collision strengths in thetemperature range from 1000 K to 40 000 K and Wilson et al.(2005) presented e!ective collision strengths for temperaturesin the range log T = 3.00–5.5 K. The results from the previouscalculations of Hayes & Nussbaumer (1984) and Lennon et al.(1985) are also shown. Hayes & Nussbaumer (1984) reportedtwo sets of e!ective collision strengths with main resonancesat calculated position and with main resonances shifted toexperimental position. We have displayed both results in Fig. 6by short-dash-dotted with calculated position and by longdash-dotted curve with resonances shifted to the experimentalposition. The 42-level and 35-level results are almost insepara-ble, indicating convergence of collision strengths. The various

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

10 12 14 16 18 20 22 24

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ss s

ectio

n (1

0-16

cm

2)

Electron energy (eV)

Fig. 5. Excitation cross sections for the 2s22p2P!–2s2p2 2S transition asa function of electron energy. Solid curve, present results; open squares,8-state R-matrix results (Smith et al. 1996); open inverted triangles, ex-perimental results (Smith et al. 1996).

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

2.5 3 3.5 4 4.5 5 5.5 6

Effe

ctiv

e C

ollis

ion

Str

engt

h

log T (K)

Fig. 6. E!ective collision strengths for the 2s22p 2P!1/2–2s22p 2P!3/2transition as a function of electron temperature. Solid curve, present42-level results; short-dashed curve, present 35-level results; long-dashed curve, 16-state R-matrix results (Wilson et al. 2005); dottedcurve, 10-state R-matrix results (Blum & Pradhan 1992); long-dash dot-ted curve, calculation of Hayes & Nussbaumer (1984) [experimentalresonance positions]; short-dash dotted curve, calculation of Hayes &Nussbaumer (1984) [calculated resonance positions]; pluses, calcula-tion of Lennon et al. (1985).

calculations except that of Hayes & Nussbaumer (1984) arenormally within 10–15% of the present results. Our resultsare lower than the other calculations. The e!ective collisionstrengths for the intercombination 2s22p 2P!1/2–2s2p2 4P3/2 and2s2p2 4P1/2–2s2p2 2D5/2 transitions have been displayed inFigs. 7 and 8 respectively where our results (42-level: solidcurve; 35-level: short dashed curve) have been compared with10-state and 16-state results together with the results fromHayes & Nussbaumer (1984) and Lennon et al. (1985). Ourresults show some qualitative and quantitative di!erenceswith previous calculations. The di!erences may have beencaused by the combined di!erences in resonance structuresand in background collision strengths which in turn may havebeen caused by the inaccuracies in target wave functions inprevious calculations. Our results are about 13% smaller thanthe calculation of Wilson et al. (2005) in the lower temperatureregion, but the two calculations show excellent agreement at

Si+: 2PJ➜2PJ’

higher energies are particularly important for the allowed transi-tions. The energy dependence of collision strengths for allowedtransitions can be properly accounted for by using an extrapola-tion technique. In the asymptotic region, the collision strengthsfollow a high-energy limiting behavior for the dipole-allowedtransitions

!i f (E ) !E!1 d ln (E ); "7#

where the parameter d is proportional to the oscillator strength.The collision strengths vary smoothly in the high-energy regionand exhibit an increasing trend for dipole-allowed transitions.The collision strength increases more rapidly for the strongerdipole-allowed transitions than the weaker transitions.

4. RESULTS AND DISCUSSION

Accurate description of target wave functions has been ob-tained, as indicated by an excellent agreement of calculatedexcitation energies with measured and other reliable calcula-tions. We included excited fine-structure levels arising fromthe 17 LS terms 3s23p2Po, 3s3p2 4P, 2D, 2S, 2P, 3s24s2S,3s23d 2D, 3s24p2Po, 3s25s2S, 3s24d 2D, 3s24 f 2Fo, 3s25p2Po,3s3p3d 2Do, 3s26s2S, 3s25d 2D, 3s25f 2Fo, and 3s26p2Po. InTable 2 we present collision strengths from the 3s23p2Po

1/2 and2Po

3/2 levels to the 31 levels included in our calculation at inci-dent electron energies 2.0, 4.0, 6.0, 8.0, and 10.0 ryd. Theseenergies are above the highest excitation threshold at 1.035 rydwhere collision strengths exhibit smooth behavior. The collisionstrength for the various transitions shows the expected behavior athigher electron energies. It may be noted that some very weakdipole-allowed transitions do not show an increasing trend withthe increasing incident electron energies. For example, the dipole-allowed 3s23p 2Po

1/2Y3s3p2 2D3/2, 3s23p 2Po

3/2Y3s3p2 2D3/2,

and 3s23p 2Po3/2Y 3s3p

2 2D5/2 transitions have very small oscil-lator strengths, and they do not show an increasing trend at higherenergies. The collision strength for magnetic quadruple forbiddentransitions remains almost constant, while the collision strengthdecreases as the incident electron energy is increased for inter-combination transitions.

The resonant collision strengths in the threshold energy regionbelow the highest excitation threshold at 1.035 ryd for the for-

bidden 3s23p 2Po1/2Y3s

23p 2Po3/2, intercombination 3s23p 2Po

3/2Y3s3p2 4P3=2, and allowed 3s23p 2Po

3/2Y3s3p2 2D5/2 transitions

have been displayed as a function of electron energy in Figures 1Y3. It is clear from these figures that the resonance structures arecomplex andmaymake significant enhancements in the collisionstrengths. The nonresonant background collision strength for theallowed transitions is generally larger than the forbidden tran-sitions. However, a decreasing trend in the background collisionstrength away from resonances with increasing energy can benoted for the weak resonance 3s23p 2Po

3/2Y3s3p2 2D5/2 transition

in Figure 3. The resonance enhancement in collision strengths forthe forbidden transitions is normally larger than for allowed transi-tions. The relativistic effects appear to be small for these transitions.We chose a fine energy mesh for collision strength calculation inthe thresholds energy region to delineate the resonance structures.The collision problem in the external region was solved using anenergy mesh of 0.00025 ryd in the closed-channels energy re-gions up to 1.035 ryd. Resonance structures are quite dense in theenergy region up to the 3s3p2 2P3/2 threshold around 0.775 ryd.Our calculations properly include the important short-range cor-relation effect to obtain correct resonance positions in the low-energy region.

We have calculated thermally averaged collision strengths usingequation (6) at 14 electron temperatures in the range log (Te) $3:4Y5:4 K. In Table 3 we present effective collision strengths forall transitions between the lowest 29 fine-structure levels. Thekeys of lower and upper levels of a transition are given in Table 1.The experimental wavelengths of transitions are also listedin Table 3. The results for transitions to the levels 30 and 31(3s25f 2Fo

5/2;7/2) are not given because the coupling from higherexciting levels above level 31 may significantly influence thecollision strengths for these transitions and our results for thesetransitions may be in error.

The effective collision strength for the allowed transitions in-creaseswith increasing temperature. The effective collision strengthfor the intercombination transitions decreases rapidly with in-creasing temperature in the high-temperature regime. These tran-sitions can only occur through electron exchange. The strength ofthe spin-orbit interaction appears to be small, particularly for low-lying excited levels. Our effective collision strengths for the for-bidden 3s23p 2Po

1/2Y2Po

3/2 transition have been compared withthe eight-stateR-matrix calculations of Dufton&Kingston (1991)

Fig. 4.—Effective collision strength for forbidden 3s23p2Po1/2Y3s

23p2Po3/2

transition as a function of electron temperature. Solid curve: Our 31 level calcu-lation; dashed curve: eight-state calculation of Dufton & Kingston (1991).

Fig. 5.—Effective collision strength for 3s 23p2Po1/2Y3s3p

2 4P3/2 transitionsas a function of electron temperature. Solid curve: Our 31 level calculation; dashedcurve: eight-state calculation of Dufton & Kingston (1991).

ELECTRON SCATTERING FROM Si ii 539