Forbidden transitions in B II, C III, 0 V, Ne VII and Mg IX

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1 Mon. Not. R. Astron. Soc. 279, ) Forbidden transitions in B II, C III, 0 V, Ne VII and Mg IX J. Fleming,t K. L. Bell,t A. Hibbert,t N. Vaeck 2 * and M. R. Godefroid 2 t ldepartment of Applied Mathematics and Theoretical Physics, The Queen's University of Belfast, Belfast B17 INN 2Laboratoire de Chimie Physique Mo/eculaire, Universite Libre de Bruxelles, CP , 50, avenue F. D. Roosevelt, B-I050 Bruxelles, Belgium Accepted 1995 November 14. Received 1995 November 13; in original form 1995 July 17 ABSTRACT Intercombination line transition rates between n = 2 levels of B II, 0 v, Ne VII and Mg IX have been calculated using the configuration interaction code CIV3. The recommended A-values for the 2s2p3~-2s21S0 intercombination lines are 9.73 S-1 for B 11,2280 S-1 for 0 V, S-1 for Ne VII and S-1 for Mg IX. We estimate that the accuracy of these values is better than 2 per cent. Calculations of the M2, E2 and M1 transitions have also been carried out for these ions and for C III. Key words: atomic data. 1 INTRODUCTION Interest in accurate determinations of intercombination lines of highly abundant low-z atoms, in the first few ionization stages, increased with the launch of the International Ultraviolet Explorer IUE) and again more recently with the launch of the Hubble Space Telescope HST). Such developments in space technology enable these lines to be observed, since they fall in the ultraviolet spectrum. These transitions are important in the determination of the abundances of the ions, in, for example, the interstellar medium, and the accuracy of these abundances depends on the accuracy of the data used to calculate them. It is the aim of this paper to provide intercombination line and other forbidden line transition probabilities accurate to within a few per cent for some low-z Be-like ions. Specifically, we investigate in this paper some transitions which are forbidden in LS coupling intercombination line, M2, E2 and M1) in the Be-like ions B II, C III, 0 v, Ne VII and Mg IX. These transitions have been previously reported for N IV Fleming et al. 1995b) as has the intercombination line of C III Fleming, Hibbert & Stafford 1994). Several previous studies have been performed for the intercombination line of these ions. Recently, Ynnerman & Froese Fischer 1995) have used the Multi-Configuration Dirac-Fock MCDF) method to improve, for C III, N IV and 0 V, the earlier work of Cheng, Kim & Desclaux 1979) who also used the MCDF method, but with very limited correlation, to study the ions C III up to Mg IX. Glass 1979, 1982) carried out a restricted configuration-interaction calculation for these ions, whilst Laughlin, Constantinides & Victor 1978) studied the ions B II up to Ne VII within a model potential approximation. 2 METHOD 2.1 Configuration interaction wave functions In our calculations we use configuration interaction CI) wave functions of the form M 'f/lsj) = L aicj>iixilis;l) 1) i=l determined using the code CIV3 Hibbert 1975; Glass & Hibbert 1978). The {IXi} define the angular momentum coupling scheme of the configuration state functions CSFs) {cj>j. The {aj are the eigenvector components of the Hamiltonian matrix whose typical element is Hij = <cj>iihicj>j)' The Hamiltonian consists of the usual nonrelativistic operators plus the relativistic operators associated with the Breit-Pauli approximation: spin-orbit, spin-other-orbit, spin-spin and the Darwin and mass-correction terms. The CSFs of all states are constructed from a common set of one-electron orbitals whose radial functions Pnlr) are expressed analytically as a sum of normalized) Slater-type orbitals: *Senior Research Assistant of the Belgian National Fund for Scientific Research FNRS). tsenior Research Associate of the Belgian National Fund for Scientific Research FNRS). The orthonormality of the orbitals is preserved by requiring 1996 RAS

2 1290 J Fleming et al. rpnlr)pn'lr) dr=bnn The integer parameters {~nl} are held fixed while the remaining parameters {cjnl } and {jnl} are treated as variational parameters. Details of the optimization of these parameters are given for C III by Fleming et a ) and for the other ions by Fleming et a a). 2.2 Correlation effects The use of CI expansions equation 1) for the ionic states allows for the proper treatment of electron correlation not included in the Hartree-Fock approximation. It is convenient to categorize the electron correlation of Be-like ions into three distinct types. i) Valence correlation VC) for which all configurations have a common Is2 core. We therefore include CSFs of the form n ie{2,3,... }. In effect, the core is fixed so that only the motion of the valence electrons is correlated. In our calculations, we find that the use of optimized orbitals with n i :::; 5 gives sufficient convergence for valence correlation. ii) Core-valence correlation CV) which includes the polarization of the core by the valence electrons. The CSFs are of the form n ie{2,3,... }. For the ground IS state, the configurations which dominate this type of correlation are of the form [lsnp )IP 2s2p)IP] IS. The mean radius of the radial function Pnpr) associated with this core-valence correlation is much smaller than those of the valence p-orbitals. Because the dependence of these CSFs on np is linear, it is sufficient to use a single additional function which we have labelled 6p) optimized specifically to describe this effect. iii) Core-core correlation CC) for which both Is orbitals are replaced by correlation orbitals. Generally this type of correlation has only a small effect on either the oscillator strengths or the transition energies although the absolute energies of the individual states are rather slowly converging with respect to the addition of further correlation orbitals and their associated CSFs). We have therefore used only the existing orbitals n :::; 5 + 6p) to describe core-core correlation. 2.3 Use of experimental energies Theoretical energy differences are normally slightly different from experimental values even in an extensive calculation. It is possible to make use of experimental energy differences in two distinct ways in order to refine the ab initio calculation of transition probabilities. First, the transition probability depends on the transition energy de'rans through a multiplicative factor. Hence any errors in the calculated transition energy can be removed by using the experimental value or equivalently by multiplying the calculated transition probability by the factor de exp )21 + I trans de calc trans ' where A = 1 for dipole El and Ml) transitions and A = 2 for quadrupole E2 and M2) transitions. Secondly, the extent to which the experimental energy differences between interacting states e.g. 3p~ and IP~) are reproduced by the calculations is a measure of how accurately the wave functions, and in particular the CI coefficients {aj in equation 1), are determined. For intercombination lines, further refining factors which attempt to allow for possible inaccuracies can be introduced see Fleming et a b). In the context of the present calculations these factors take the form de calc )2 term de~:::n and de exp)2 de;lc ' where de'e<m =E ep~) - E CP~) and de!s =E ep~) - E ep~). Factor 3) makes allowances for inaccuracies in the mixing between the 3p~ and Ip~ states, while factor 4) seeks to correct for inaccuracies in the evaluation of the fine structure splitting between the 3po levels. An alternative to the use of equations 2) and 3), by which the diagonal Hamiltonian matrix elements are adjusted so that the eigenvalue differences coincide with the corresponding experimental energy separations, has been described and discussed by Brage & Hibbert 1989). It is particularly effective at improving an ab initio calculation of the transition probabilities of intercombination lines, especially if the ab initio calculation is already quite accurate for which reason we refer to this process as 'fine-tuning'). 3 RESULTS 3.1 The 1s22s2p 3p; -ls22s21so intercombination line The results for this transition are listed in Table 1 for B II, o v, Ne VII and Mg IX, along with experimental transition energies while the ab initio A-values from other theoretical calculations are listed in Table 2. Both ab initio and adjusted Section 2.3) transition rates are given in Table l. Table 1 displays for each ion the results of three calculations which differ in the amount of correlation considered Section 2.2). The VC calculation does not include the 6p; since this orbital was optimized to take account of the core it is only included in the CV and CC calculations. In our previous work on this intercombination line in C III and N IV Fleming et a ; Fleming et a b), the use of the multiplicative factors 2) and 3) was compared with the method of diagonal energy shift 'fine-tuning'). Both methods were found to produce similar results. It can be 2) 3) 4) 1996 RAS, MNRAS 279,

3 Table 1. The 2s2p3P~-2s2IS0 intercombination line. BII Forbidden transitions 1291 Method Energy differences cm -I ) A-values S-I) Ll.Etran6!lE'erm!lEI' ab initio AdH Adj~ fine-tuned Adj,. VCn= CV n=5+6p CC n=5+6p Experiment d OV Ne VII MgIX VCn= CV n=5+6p CC n=5+6p Experiment' VCn= CV n=5+6p CC n=5+6p Experiment d VCn= CV n=5+6p CC n=5+6p Experiment! ascaled by equation 2). bscaled by equations 2) and 3). cprevious column scaled by equation 4). dbashkin & Stoner 1978). 'Moore 1980). 'Martin & Zalubas 1980). Table 2. Comparison of theoretical intercombination line A-values S-I). B II C III N IV 0 V Ne VII Mg IX Laughlin et al. 1978) Cheng et al. 1979) Glass 1982) Glass 1979) Ynnerman & Froese Fischer 1995) ab initio fine-tuned a Fleming et al. 1994) 104 Fleming et al. 1995b) 580 This work 'Scaled by equations 2) and 3). 96 RAS, MNRAS 279,

4 Fleming et al. Thble 3. M2, E2 and Ml transition energies au) and A-values S-1). BIT em t::..e A t::..e A OV Ne VII t::..e A t::..e A t::..e MgIX A Sp;_lSO M2: ) ) sp;_spg E2: )" ) ) ) sp;_sp~ Ml: ) ) ) ) E2: ) ) 3P~_3Pg Ml: ) ) ) ) "Powers of 10 in parentheses ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) seen from Table 1 that the same pattern is obtained for the ions considered here comparing the CC A-values in the 'Adj2' and 'fine-tuned' columns). The magnitude of the difference between the A-values from the two correction methods increases with Z, but the proportional difference decreases to much less than 1 per cent of the A-value. The difference between the two processes arises because the use of equations 2) and 3) omits corrections to the less significant) even parity energy separations between ISO, 3po) whereas the fine-tuning process incorporates these additional corrections. We note that the effect of fine-tuning is to change the best ab initio calculation by only 1-2 per cent. For each ion, the value underlined represents our best estimate of the A-value. We would draw attention to a degree of uncertainty in the experimental energy values for Ne VII. There are discrepancies between the data of Bashkin & Stoner 1978) and those of Bockasten, Hallin & Hughes 1963) quoted by Miihlethaler & Nussbaumer 1976). In Table 1, we list the ISo_3p~ transition energy of Bashkin & Stoner 1978). It is clear from Table 1 that whereas our calculated transition energy agrees almost exactly with Bashkin & Stoner 1978) for Ne VII, we are consistently below the experimental value for the other ions. Our calculations were undertaken in the same manner for each of the ions, and so we would expect a consistent though small) deviation from experiment. Hence, in obtaining our 'fine-tuned' result for Ne VII, we used diagonal matrix element shifts interpolated from those of the other ions. In Table 2 we compare the A-value obtained with 'finetuning' in our current set of calculations this work, plus Fleming et al. 1994, and Fleming et al. 1995b) with those using other methods or with a less extensive calculation also using CIV3. Laughlin et al. 1978) adopted a model potential approach which incorporates a polarization potential to account for the dominant contribution to core-valence correlation. Their results are consistently higher than ours, but only by a few per cent. Cheng et al. 1979) used the MCDF method, but with very limited correlation: the CSFs were restricted to those Thble 4. Ufetimes of the 3Fz and 3p~ levels. BII sp2 Level )c CIII ) OV ) NeVIl ) MgIX ) "A-values S-1). bufetimes s). cpowers of 10 in parentheses sp~ Level ~1) ) ) ) ) which can be constructed within the n = 2 complex of orbitals. Their results are significantly below ours, although the difference decreases somewhat with Z. This is to be expected since electron correlation effects are less pronounced as Z increases, so that the limited nature of their CI becomes less significant. Recently, Ynnerman & Froese Fischer 1995) have reported a much more extensive ab initio MCDF calculation, with orbitals up to n =9 for C III), n =8 for N IV) and n=6 for 0 v). All possible double excitations from the principal correlations are included, as well as some selected triple and quadruple excitations. For these three ions, their results are within 4 per cent of ours and the agreement improves to 1-2 per cent if we compare our results with the effect of adjusting their values using equations 2) and 3). Finally, we consider the earlier CIV3 calculations of Glass 1979, 1982). A much smaller set of orbitals and therefore CSFs) was used. The results are again consistently smaller than ours RAS. MNRAS

5 3.2 Other transitions As in the paper on N IV Fleming et al. 1995b) where we included calculations of other forbidden transitions, we also calculate transition probabilities for the M2 ep~-iso), E2 e~_3p~ and 3p~_3pn and M1 ep~_3p~ and 3p~_3p~) transitions in B II, C III, 0 V, Ne VII and Mg IX. Table 3 gives the results of calculations which incorporate all three correlation effects: VC, CV, CC, followed by finetuning. The orbital and CSF sets were the same as those used for the intercombination line calculation. The effect of fine-tuning was again small. In Table 4 we present the total A-value from the lifetime of the 3~ and 3p~ levels. The former level decays through very weak transitions to the ground state M2) or to the other I-levels of the configuration E2 and M1). In all the ions considered here, the effect of the E2 transition on the lifetime is negligible. For low Z, the M2 transition is dominant, but as Z increases the M1 transition gradually becomes more important. The only significant contribution to the decay of the 3p~ level is the intercombination transition to the ground state. The 3p~ level decays to the ISO ground state only through the admixture of hyperfine interactions, which we have not undertaken in this work. 4 CONCLUSION The results presented here, taken together with the work of Fleming et al. 1994) and of Fleming et al. 1995a,b), form an extensive study of various transitions between low-lying levels of Be-like ions. It is difficult to give a precise measure of the accuracy of these CIV3 calculations. Judging by the comparison given in Fleming et al. 1995b) between results obtained from CIV3 and independent Multi-Configuration Hartree-Fock calculations, as well as by the comparison between the ab initio and fine-tuned values of the present paper, we estimate that our final A-values are accurate to better than 2 per cent. ACKNOWLEDGMENTS Forbidden transitions 1293 JF wishes to thank the Department of Education for Northern Ireland for a post-graduate studentship award. MG and NY would like to thank the Belgian National Fund for Scientific Research for a research grant FRFC Convention ) and the Communaute Fran~aise of Belgium for their financial support provided by the research convention ARC-93/ The authors would like to thank the British Council-FNRS-CGRI Academic Research Collaboration Programme for support for this investigation. REFERENCES Bashkin S., Stoner J. 0.,1978, Atomic Energy-level and Grotrian Diagrams Hydrogen I-Phosphorous XV, Vol. 1). North-Holland Publishing Company, Amsterdam Bockasten K, Hallin R, Hughes T. P., 1963, Proc. Phys. Soc., 81, 522 Brage T., Hibbert A, 1989, J. Phys. B, 22, 713 Cheng K T., Kim Y.-K, Desclaux J. P., 1979, At. Data Nucl. Data Tables, 25, 111 Fleming J., Hibbert A, Stafford R P., 1994, Phys. Scr., 49, 316 Fleming J., Vaeck N., Hibbert A, Bell K L., Godefroid M. R, 1995a, Phys. Scr., in press Fleming J., Brage T., Bell K L., Vaeck N., Hibbert A, Godefroid M. R, Froese Fischer c., 1995b, ApJ, 455, 758 Glass R, 1979, J. Phys. B, 12,697 Glass R, 1982, Solar Phys., 78, 29 Glass R, Hibbert A, 1978, Comput. Phys. Commun., 16, 19 Hibbert A, 1975, Comput. Phys. Commun., 9,141 Laughlin c., Constantinides E. R, Victor G. A, 1978, J. Phys. B, 11,2253 Martin W. C., Zalubas R, 1980, J. Phys. Chem. Ref. Data, 9,1 Moore C. E., 1980, Selected Tables of Atomic Spectra, NSRDS NBS3, Section 9. US Government Printing Office, Washington, D.C. Miihlethaler H. P., Nussbaumer H., 1976, A&A, 48, 109 Ynnerman A, Froese Fischer c., 1995, Phys. Rev. A, 51, RAS, MNRAS 271),

ABSTRACT Electron collision excitation strengths of inelastic transitions among the 52 Ðne-structure levels of the

ABSTRACT Electron collision excitation strengths of inelastic transitions among the 52 Ðne-structure levels of the THE ASTROPHYSICAL JOURNAL, 530:1091È1104, 2000 February 20 ( 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A. ELECTRON COLLISION EXCITATION OF FINE-STRUCTURE LEVELS IN S