Eur. Phys. J. D 49, 157–160 (2008)DOI: 10.1140/epjd/e2008-00150-y THE EUROPEAN

PHYSICAL JOURNAL D

Fine-structure energy levels, oscillator strengthsand lifetimes in Co XV�,��

G.P. Gupta1,a and A.Z. Msezane2

1 Department of Physics, S. D. (Postgraduate) College, Muzaffarnagar, 251 001, India(affiliated to Chowdhary Charan Singh University, Meerut, 250 004, India)

2 Department of Physics and Center for Theoretical Studies of Physical Systems, Clark Atlanta University,Atlanta, Georgia 30314, USA

Received 8 April 2008 / Received in final form 25 June 2008Published online 1st August 2008 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2008

Abstract. Excitation energies from ground state for 97 fine-structure levels as well as oscillator strengthsand radiative decay rates for all electric-dipole-allowed and intercombination transitions among thefine-structure levels of the terms belonging to the (1s22s22p6)3s23p, 3s3p2, 3s23d, 3p3, 3s3p3d, 3p23d,3s3d2, 3s24s, 3s24p, 3s24d, 3s24f , and 3s3p4s configurations of Co XV are calculated, using extensiveconfiguration-interaction (CI) wave functions, obtained with the CIV3 computer code of Hibbert. Theimportant relativistic effects in intermediate coupling are included through the Breit-Pauli approximationvia spin-orbit, spin-other-orbit, spin-spin, Darwin and mass correction terms. Small adjustments to thediagonal elements of the Hamiltonian matrices have been made. Our calculated excitation energies, in-cluding their ordering, are in excellent agreement with the experimental results and the experimentallycompiled energy values of the National Institute for Standards and Technology (NIST) wherever avail-able. The mixing among several fine-structure levels is found to be very strong, with most of the stronglymixed levels belonging to the (1s22s22p6)3p23d and 3s3d2 configurations. The strong mixing among sev-eral fine-structure levels makes it very difficult to identify them uniquely. Perhaps, that may be the reasonfor the lack of both experimental and theoretical results for these levels. We believe that our extensivecalculated values can guide experimentalists in identifying the fine-structure levels in their future work.From our radiative decay rates we have also calculated radiative lifetimes of some fine-structure levels. Inthis calculation we also predict new data for several fine-structure levels where no other theoretical and/orexperimental results are available.

PACS. 32.10.Fn Fine and hyperfine structure – 32.70.Cs Oscillator strengths, lifetimes, transition moments– 95.30.Ky Atomic and molecular data, spectra, and spectral parameters

1 Introduction

The study of Al-like spectra of ions of the elements ofthe iron group has received a great deal of attention bothexperimentally as well as theoretically. These ions areof particular interest in astrophysics and controlled ther-monuclear fusion. Many of these ions are also known toproduce resonance transitions in the EUV solar spectra [1]and in the spectra of many solar-type stars as well as inother astrophysical objects. These resonance transitionsare also widely detected in the spectra of laboratory plas-

� Tables 1–5 are only available in electronic form athttp://www.epj.org�� Table 5 which includes our calculated wavelengths (in A),oscillator strengths in both the length fL and velocity fV formsand the transition probabilities in the length form AL for all1527 transitions is also available with the first author.

a e-mail: g p guptal@yahoo.co.in

mas, including tokamaks [2]. The identification of reso-nances in intercombination transitions in these ions wasaccomplished by a number of measurements [2–7].

Emission lines of the ions of Cobalt have been ob-served, as listed by Dere et al. [8] or in the CHIANTIdatabase at http://wwwsolar.nrl.navy.mil/chianti.html/. Additionally, Co is an importantimpurity element in fusion plasmas, and hence atomicdata such as excitation energies, absorption oscillatorstrengths and radiative decay rates are required forthe analysis of power loss in the reactors and also fordiagnostic and modeling of the plasmas.

Most of the theoretical energy levels, oscillatorstrengths, transition probabilities and lifetimes for ions ofthe Al isoelectronic sequence, available in the literature,are limited to allowed transitions involving a few low-lyingstates of the n = 3 subshell or lack accuracy [9–18]. Re-cently, Gupta and Msezane [19] calculated the excitation

158 The European Physical Journal D

energies, oscillator strengths, transition probabilities andlifetimes for a few Al-like ions using the configuration-interaction code [20] and the Breit-Pauli Hamiltonian [21].The energy spectra of Co XV have been compiled byShirai et al. [22] and the National Institute of Standardand Technology (NIST), which are available at the websitehttp://physics.nist.gov/PhysRefData.

In our calculation here we used the Slater-type orbitalsin program CIV3 [20] to construct the large configuration-interaction (CI) wavefunctions. The important relativisticeffects in intermediate coupling are incorporated by meansof the Breit-Pauli Hamiltonian [21]. These extensive CIwavefunctions in the intermediate coupling scheme arethen used to calculate the excitation energies, oscillatorstrengths, and transition probabilities for electric-dipole-allowed and inter-combination transitions among the(1s22s22p6)3s23p(2Po), 3s3p2(2S, 2P, 2D, 4P), 3s23d(2D),3p3(4So, 2Po, 2Do), 3s3p(3Po)3d(2Po, 2Do, 2Fo, 4Po,4Do, 4Fo), 3s3p(1Po)3d(2Po, 2Do, 2Fo), 3p2(1S)3d(2D),3p2(1D)3d(2S, 2P, 2D, 2F, 2G), 3p2(3P)3d(2P, 2D, 2F, 4P,4D, 4F), 3s3d2(2S, 2P, 2D, 2F, 2G, 4P, 4F), 3s24s(2S),3s24p(2Po), 3s24d(2D), 3s24f(2Fo), 3s3p(3Po)4s(2Po,4Po), and 3s3p(1Po)4s(2Po) states of Co XV, mainly tocomplete the void in the existing data and provide manyadditional new and accurate data for various optically al-lowed and intercombination transitions. From our tran-sition probabilities we have also calculated the radiativelifetimes of some fine-structure levels. We have investi-gated the effects of electron correlations on our calculateddata, particularly on the intercombination transitions, byincluding orbitals with up to the n = 5 quantum num-ber. We considered up to three electron excitations fromthe valence electrons of the basic configurations and in-cluded a large number of configurations (1164) to ensureconvergence. These configurations represent all major in-ternal, semi-internal and all-external electron correlationeffects [23]. In order to determine configuration mixing,the important relativistic terms of the order (αZ)2 areincluded in the final Hamiltonian, with α being the fine-structure constant. The wave function obtained by diag-onalizing the Hamiltonian matrix, is an eigen function ofthe total angular momentum (J) and parity (π) of theatomic system.

2 Choice of wavefunctions

The 44 LS states belonging to the (1s22s22p6)3s23p,3s3p2, 3s23d, 3p3, 3s3p3d, 3p23d, 3s3d2, 3s24s, 3s24p, 3s24d,3s24f , and 3s3p4s configurations of Co XV give riseto 98 fine-structure levels corresponding to various Jvalues – see Table 2. The atomic state wave functionsare represented by the J-dependent CI expansions of theform [21]

Ψi(JMJ) =K∑

j=1

bijφj(αjLjSjJMJ), (1)

where each of the K single-configuration functions φj isconstructed from one-electron functions and αj defines

the coupling of the orbital Lj and the spin Sj angularmomenta to give the total angular momentum J . Themixing coefficients bij are the eigenvector components ofthe Hamiltonian matrix 〈φi |H |φj〉 with the basis φj [21].The Hamiltonian is represented by the non-relativisticelectrostatic interactions plus the Breit-Pauli terms suchas one-body mass correction, Darwin term, and spin-orbit, spin-other-orbit, and spin-spin operators. The com-plete description of the individual terms of the Breit-PauliHamiltonian can be found in the book of Froese Fischeret al. [24], Chapter 7 and references therein. The detailsof the implementation of the Breit-Pauli Hamiltonian inProgram CIV3 are given in Glass and Hibbert [21]. The in-clusion of the mass correction, Darwin term, and spin-spincontact shifts the energy of a configuration as a whole. Thespin-orbit, spin-other-orbit and spin-spin terms also con-tribute to the fine-structure splitting. The radial parts ofthe one-electron functions are expressed in analytic formsas a sum of Slater-type orbitals

Pnl =K∑

i=1

Ci

[(2ξi)2pi+1/(2pi)!

]1/2rpi exp(−ξir), (2)

where n and l, respectively, are the principal and orbitalquantum numbers. The parameters Ci, ξi and pi in equa-tion (2) are respectively the expansion coefficients, expo-nents, and powers of r of the radial functions. In the opti-mization process the parameters, namely the coefficients(Ci) and the exponents (ξi) are determined variationallyas described by Hibbert [20] while pi, being integral pow-ers, are generally kept fixed. The wave functions given byequation (1) are then used to calculate the excitation ener-gies of the fine-structure levels, length and velocity formsof oscillator strengths, and transition probabilities amongthe fine-structure levels.

The absorption oscillator strengths in the length andvelocity forms are calculated using the relations [25]

fL = [2ΔE/(3gi)]|〈ψj |r|ψi〉|2 (3)

fV = [2/(3ΔEgi)]|〈ψj |∇|ψi〉|2 (4)

where ΔE = Ej −Ei and gi is the statistical weight of thelower state ψi. The radiative lifetime of an excited stateis obtained from our radiative transition probabilities Aji

asτj = 1/ΣiAji (5)

where the sum over i is over all accessible final states.

3 Results and discussion

In our calculation we used the 15 orthogonal one-electronorbitals up to n = 5. The 1s, 2s, 2p, 3s and 3p radialfunctions are chosen as the Hartree-Fock (HF) func-tions of the ground state (1s22s22p6)3s23p of Co XVgiven by Clementi and Roetti [26]. The 3d, 4s, 4p, 4d,and 4f functions are chosen as flexible spectroscopictype and are optimized on the excited states 3s23d(2D),

G.P. Gupta and A.Z. Msezane: Fine-structure energy levels, oscillator strengths and lifetimes in Co XV 159

3s24s(2S), 3s24p(2Po), 3s24d(2D), and 3s24f(2Fo), re-spectively. To improve the degree of correlation, we alsoincluded the 5s, 5p, 5d, 5f , and 5g orbitals in our calcu-lations. These are chosen as correlation type and are op-timized on 3s3p2(2S), 3s3p2(2P), 3s24d(2D), 3s24f(2Fo),and 3s23p(2Po), respectively. In order to represent all theenergy levels by a single set of orthogonal functions, it isnecessary to use the correlation functions in addition tothe spectroscopic functions. In all cases we chose k = n− lso that the coefficients Ci are uniquely specified by the or-thogonality condition on Pnl [20]. The parameters of theoptimized radial functions are displayed in Table 1.

We used several test calculations to investigate theconvergence of the CI expansions for different LS symme-tries. In our largest calculation, we considered up to threeelectron excitations from the basic configurations namely(1s22s22p6)3s23p, (1s22s22p6)3s3p2, (1s22s22p6)3p3, and(1s22s22p6)3s3p3d. This produced several hundred con-figurations in each LS symmetry. In this calculation weretained all configurations within the n = 5 complex thathad weight higher than 0.001 for the states that mixedstrongly and 0.002 for the rest of the states. Dropping theother configurations changed the LS energies by only lessthan 1.0%. The inclusion of higher orbitals with n = 6and 7 had no significant effect on the calculated data.

In Table 2 our calculated excitation energies of the fine-structure levels relative to the ground level for Co XV arecompared with the experimental values compiled by Shiraiet al. [22] and the compiled values of the National Instituteof Standard and Technology (NIST), which are availableat the website http://physics.nist.gov/PhysRefData.The differences between the two data sets for energies(Ref. [22] and NIST) are better than 0.01% in most cases(wherever available). The mixing among some of the rel-ativistic levels is very strong. In general, our ab initio cal-culation (a) is in good agreement with the experimentalvalues [22] and the NIST data. However, we note the sig-nificant gaps in both the experimental and the NIST datasets. In order to keep our calculated energies as close aspossible to the experimental as well as the NIST levels,we have made small J-dependent adjustments to the di-agonal elements of the Hamiltonian matrices. These ad-justments improve the accuracy of the mixing coefficientsbij , which depend in part on the accuracy of the eigen-values. This is a fine-tuning technique [27] that has beenfurther justified in a paper on Na III [28] and most recentlyon Cl I [29]. These adjustments also affect the composi-tion of the eigenvectors slightly. In a way, they correct theab initio approach for the neglected core-valence correla-tion, which has been shown to contribute significantly inneutral magnesium (see, for instance: Jonsson et al. [30]).Our adjusted theoretical energies, “Present (b)”, are alsolisted in Table 2 and are in excellent agreement with theexperimental [22] as well as the NIST levels. We have ar-ranged our adjusted energy levels in ascending order. Thelast column in this table represents the leading percent-age composition of the various levels (corresponding toPresent (b)). The first number of each entry in this columnrepresents the leading percentage of the level correspond-

ing to the level number under the first column, followedby a set of numbers of the form M(N). These mean thatthe next leading percentage is M% of the level number Nin the first column and so on. As can be seen from thelast column, the mixing among several fine-structure lev-els is very strong, with most of the strongly mixed levelsbelonging to the 3p23d and 3s3d2 configurations.

In our CIV3 calculation we identify the levels bytheir dominant eigenvector [31]. The mixing amongthe levels 3p2(1S)3d(2D1.5) & 3p2(3P)3d(2D1.5) and3p2(1D)3d(2P1.5) & 3s3d2(2P1.5) is so strong that the lev-els 3p2(1S)3d(2D1.5) and 3p2(1D)3d(2P1.5) are designatedby the eigenvectors of the second largest magnitude. Ascan be seen from Table 2, beginning from the level num-ber 41 to the end, the experimental data are availableonly toward the end of the table, viz. from level 92 to 98,while the NIST levels are also very limited in this range.It is further noted that there are significant gaps in theexperimental and the NIST data.

In Table 3 we have tabulated our calculated wave-lengths, oscillator strengths in both the length fL

and velocity fV forms and the transition probabilitiesin the length form AL for only those transitions forwhich the AL values from the NIST data are avail-able (http://physics.nist.gov/PhysRefData) to com-pare with our calculations. In calculating these parameterswe used our adjusted theoretical energy splittings, corre-sponding to the Present (b) in Table 2. The keys of thelower and upper levels involved in a transition are givenin Table 2. As can be seen, there is excellent agreementbetween our calculated AL results and the correspondingNIST values except for the transition (2–8) where the dif-ference between the two results is about 28%.

In Table 4 we have presented our calculated life-times (in seconds) for several fine-structure levels. Theseare compared with the experimental results of Trabertet al. [32] and the theoretical calculations of Huang [12]available only for the level 3s3p2(4P). For the 3s3p2(4P1/2)level, our calculated lifetime is in excellent agreement withthe experimental value [32] compared to the theoretical re-sult of Huang [12]. However, for the other two levels ourresults are in close agreement with the theoretical [12] aswell as the experimental [32] lifetimes.

In Table 5 we have tabulated our calculated wave-lengths (in A), oscillator strengths in both the length fL

and velocity fV forms and transition probabilities in thelength form AL for 1527 transitions. These parametersare calculated using our adjusted energy splittings. Wenote that in the present calculation, for all strong dipole-allowed transitions there is good agreement between thelength and velocity values of the oscillator strengths. Also,for some levels where strong mixing between singlets andtriplets exists, intercombination lines are comparable toallowed transitions, so there is reasonably good agreementbetween the fL and fV values for these transitions. How-ever, the magnitudes of the oscillator strengths for manyintercombination transitions are smaller by several ordersof magnitude than those for allowed transitions and thereis a significant difference between the fL and fV values

160 The European Physical Journal D

for several of these transitions. For intercombination linesto have length and velocity in agreement, additional rel-ativistic operators would need to be added to the veloc-ity form. This is not included in CIV3 and that is whyoften intercombination lines do not have these forms inagreement. The length form is then generally taken as thecorrect form in that case.

4 Conclusions

In conclusion, we have calculated energy splittings of98 fine-structure levels as well as oscillator strengthsand transition probabilities for transitions among thefine-structure levels of the terms belonging to the(1s22s22p6)3s23p, 3s3p2, 3s23d, 3p3, 3s3p3d, 3p23d,3s3d2, 3s24s, 3s24p, 3s24d, 3s24f , and 3s3p4s configura-tions of Co XV. In this calculation, we have used an ex-tensive set of CI wave functions and included correlationeffects in the excitation up to the 5g orbital. Our calcu-lated excitation energies, covering the ninety eight fine-structure levels, are in excellent agreement with the avail-able experimental results and the NIST levels. Also for3s3p2(4P1/2) level, our calculated lifetime is in excellentagreement with the experimental value [32] compared tothe theoretical result of Huang [12]. Finally, we believethat the present results are the most extensive and defini-tive to date and should be useful in many astrophysicalapplications and fusion plasma applications.

This research is supported by the Division of Chemical Sci-ences, Office of Basic Energy Sciences, Office of Energy Re-search, United States Department of Energy (AZM). GPGwishes to thank Professor Alan Hibbert for useful discussionon his fine-tuning technique.

Appendix

Table 5 which includes our calculated wavelengths (in A),oscillator strengths in both the length fL and velocity fV

forms and the transition probabilities in the length formAL for all 1527 transitions in Co XV is available in theonline version.

References

1. W.E. Behring, L. Cohen, U. Feldman, Astrophys. J. 175,493 (1972)

2. E. Hinnov, B. Denne, A. Ramsey, B. Stratton, J.Timberlake, J. Opt. Soc. Am. B 7, 2002 (1990)

3. T. Shirai, K. Mori, J. Sugar, W.L. Wiese, Y. Nakai, K.Ozawa, At. Data Nucl. Data Tab. 37, 235 (1987)

4. A. Redfors, U. Litzen, J. Opt. Soc. Am. B 6, 1447 (1989)5. V.E. Levashov, A.N. Ryabtsev, S.S. Churilov, Opt.

Spectrosc. (USSR) 69, 20 (1990)6. S.S. Churilov, V.E. Levashov, Phys. Scr. 48, 425 (1993)7. E. Trabert, C. Wagner, P.H. Heckmann, G. Moller, T.

Brage, Phys. Scr. 48, 593 (1993)8. K.P. Dere, E. Landi, P.R. Young, G.D. Zanna, Astrophys.

J. S134, 331 (2001)9. C. Froese Fischer, Can. J. Phys. 54, 740 (1976); C. Froese

Fischer, Can. J. Phys. 56, 983 (1978)10. A. Farrag, E. Luc-Koenig, J. Sinzelle, At. Data Nucl. Data

Tab. 27, 539 (1982)11. B.C. Fawcett, At. Data Nucl. Data Tab. 28, 557 (1983)12. K-N Huang, At. Data Nucl. Data Tab. 34, 1 (1986)13. C. Froese Fischer, B. Liu, At. Data Nucl. Data Tab. 34,

261 (1986)14. M. Hjorth-Jensen, K. Aashamar, Phys. Scr. 42, 309 (1990)15. U.I. Safronova, C. Namba, J.R. Albritton, W.R. Johnson,

M.S. Safronova, Phys. Rev. A 65, 022507 (2002)16. K.M. Aggarwal, F.P. Keenan, A.Z. Msezane, At. Data

Nucl. Data Tab. 85, 453 (2003)17. K.G. Dyall, I.P. Grant, C.T. Johnson, F.A. Parpia, E.P.

Plummer, Comput. Phys. Commun. 55, 424 (1989)18. C. Froese Fischer, G. Tachiev, A. Irimia, At. Data Nucl.

Data Tab. 92, 607 (2006)19. G.P. Gupta, A.Z. Msezane, Phys. Scr. 70, 235 (2004);

G.P. Gupta, A.Z. Msezane, At. Data Nucl. Data Tab. 89,1 (2005); G.P. Gupta, A.Z. Msezane, Phys. Scr. 76, 225(2007)

20. A. Hibbert, Comput. Phys. Commun. 9, 141 (1975)21. R. Glass, A. Hibbert, Comput. Phys. Commun. 16, 19

(1978)22. T. Shirai et al., J. Chem. Phys. Ref. Data 21, 23 (1992)23. I. Oksuz, O. Sinanoglu, Phys. Rev. 181, 42 (1969)24. C. Froese Fischer, T. Brage, P. Jonsson, Computational

Atomic Structure (Institute of Physics Publishing, Bristol,1997), Chap. 7

25. A. Hibbert, Atomic Structure Theory in Progress inAtomic Spectroscopy, edited by W. Hanle, H. Kleinpoppen(Plenum, New York, 1979), pp. 1–69

26. E. Clementi, C. Roetti, At. Data Nucl. Data Tab. 14, 177(1974)

27. A. Hibbert, Phys. Scr. T 65, 104 (1996)28. D. McPeake, A. Hibbert, J. Phys. B: At. Mol. Opt. Phys.

33, 2809 (2000)29. P. Oliver, A. Hibbert, J. Phys. B: At. Mol. Opt. Phys. 40,

2847 (2007)30. P. Jonsson, C. Froese Fischer, M.R. Godefroid, J. Phys. B:

At. Mol. Opt. Phys. 32, 1233 (1999)31. G.P. Gupta, K.M. Aggarwal, A.Z. Msezane, Phys. Rev. A

70, 036501 (2004)32. E. Trabert, P.H. Heckmann, R. Hutton, I. Martinson, J.

Opt. Soc. Am. B 5, 2173 (1988)