We have calculated atomic helium spontaneous decay rates and absorption oscillator strengths for the spin-forbidden transitions from 3′,3P 1,2 and 3′,1P 1 to all lower 1S 0 and 3S 1 states. In particular we found A 10=44.33(4)′, s -1 for the E1 transition 3′,3P 1-1′,1S 0 and 0.1147(1) s -1 for the M2 transition 3′,3P 2-1′,1S 0. © 2011 American Physical Society

Drake, Gordon W. F.

Morton, D. C.

Scholarship at UWindsor

University of WindsorScholarship at UWindsorPhysics Publications Department of Physics2011Spin-forbidden radiative decay rates from the 3′,3P1,2 and 3′,1P 1 states of heliumD. C. MortonGordon W. F. DrakeUniversity of WindsorFollow this and additional works at: http://scholar.uwindsor.ca/physicspubPart of the Physics CommonsThis Article is brought to you for free and open access by the Department of Physics at Scholarship at UWindsor. It has been accepted for inclusion inPhysics Publications by an authorized administrator of Scholarship at UWindsor. For more information, please contact scholarship@uwindsor.ca.Recommended CitationMorton, D. C. and Drake, Gordon W. F.. (2011). Spin-forbidden radiative decay rates from the 3′,3P 1,2 and 3′,1P 1 states of helium.Physical Review A - Atomic, Molecular, and Optical Physics, 83 (4), 42503-1-42503-3.http://scholar.uwindsor.ca/physicspub/169PHYSICAL REVIEW A 83, 042503 (2011)Spin-forbidden radiative decay rates from the 3 3 P1,2 and 3 1 P1 states of heliumDonald C. Morton1 and G. W. F. Drake21Herzberg Institute of Astrophysics, National Research Council, Victoria, British Columbia, Canada V9E 2E72Department of Physics, University of Windsor, Windsor, Ontario, Canada N9B 3P4(Received 17 February 2011; published 6 April 2011)We have calculated atomic helium spontaneous decay rates and absorption oscillator strengths for the spin-forbidden transitions from 3 3P1,2 and 3 1P1 to all lower 1S0 and 3S1 states. In particular we found A10 =44.33(4) s−1 for the E1 transition 3 3P1–1 1S0 and 0.1147(1) s−1 for the M2 transition 3 3P2–1 1S0.DOI: 10.1103/PhysRevA.83.042503 PACS number(s): 32.70.Cs, 12.20.DsI. INTRODUCTIONThe study of spin-forbidden transitions in helium and theheliumlike ions has a long history as a fundamental testingground for the interactions between radiation and matter. In thelimit of LS coupling, electric dipole (E1) transitions betweensinglet and triplet states are strictly forbidden in the lowestorder due to spin orthogonality because the leading terms in theelectric dipole transition operator are just −e(r1 + r2), whichis not spin dependent. However, relativistic spin-dependentcorrections to the wave functions and retardation correctionsto the transition operator both contribute nonvanishing termsof relative order (αZ)2, where Ze is the nuclear charge and αis the fine-structure constant.The best-studied example is the astrophysically important1s2p 3P1–1s2 1S0 transition of helium and the heliumlike ions,where the subscript is the eigenvalue of the total angular mo-mentum operator J = L + S. For this case, the intermixing ofthe 1s2p 3P1 and 1s2p 1P1 states is the dominant effect, but thecontribution from all the higher-lying 1snp 3P1 states and final1s2 1S0 state perturbations due to mixing with doubly-excitedP states of the form npn′p 3P e1 are also significant. Followingthe earlier work by Elton [1] and Drake and Dalgarno [2],Drake [3] performed a complete calculation, which includedspin-dependent perturbations to both the initial and final states.This work also demonstrated the equivalence of the lengthand velocity forms of the transition operator, provided that anextra spin-dependent term is added to the velocity form of thetransition operator, as further discussed in the present work.Also, the spin-dependent part of the magnetic quadrupole (M2)operator directly enables 1s2p 3P2–1s2 1S0 [4].A general derivation of relativistic corrections to allowedand spin-forbidden electric dipole transitions was carried outfrom basic QED by Drake [5], including the effects of theelectron-electron interaction and negative energy states. It wasproven there that the nonrelativistic Pauli form of the Breitinteraction remains valid for off-diagonal matrix elements inthe presence of radiation emission, and that a semiclassicalrepresentation of the radiation field remains valid up to termsof relative order α2Z2. The former point is important becausethe Breit interaction is normally taken to be correct only as afirst-order perturbation correction to the nonrelativistic energy[6]. These theoretical results were recently confirmed by Łachand Pachucki [7], and much more accurate numerical valuesfor matrix elements were obtained by them, as well as byMorton, Moffatt, and Drake [8].The purpose of the present work is to extend the resultsof Ref. [8] on the calculation of He I spin-forbidden A andf values for the electric dipole (E1) decays 3 3P1–n 1S0 and3 1P1–n 3S1, as well as the magnetic quadrupole (M2) decays3 3P2–n 1S0 and 3 1P1–n 3S1. Since Baldwin and his colleagues[9,10] are considering measuring 3 3P1–1 1S0 and 3 3P2–1 1S0,this theoretical study is timely. In emission, the spin-forbiddentransitions are dominated by ordinary allowed E1 transitionsto lower states, but in absorption, the spin-forbidden transitionscan readily be observed.II. CALCULATIONSAs outlined in the Introduction, if spin-orbit coupling isweak, spin-forbidden transitions can be calculated to order(αZ)2 relative to allowed E1 transitions from the perturbationof the nonrelativistic wave function |nPSLJ 〉 by the Breitspin-orbit (SO) and spin-other-orbit (SOO) operators. Thespin-spin operator does not contribute to S = 1 to S = 0transitions because it is a scalar formed by coupling irreducibletensors of rank 2. The perturbed wave function can be ex-panded in terms of a complete set of virtual intermediate states.The spin-changing transitions come from the virtual states withopposite spin but the same parity P and angular momentum J .As described in [8], we approximate these intermediate states,including the continuum, by the complete set of N nonrela-tivistic pseudostates derived from the variational solution ofthe energy eigenvalue equations for a basis set with N terms.Specifically, for the spin-changing transition 3 3P1–1 1S0,we computed the sums〈1 1S0 | H1 | 3 3P1〉=N∑m〈1 1S0 | H1 | m 1P1〉 〈m1P1 | B | 3 3P1〉(3 3P1) − (m 1P1)+N∑n〈1 1S0∣∣B∣∣n 3P e0〉(1 1S0) − (n 3P e0)〈n 3P e0∣∣H1∣∣3 3P1〉, (1)where H1 = z1 + z2 is the interaction operator, B is the sumof the SO and SOO operators, and all the wave functions onthe right are nonrelativistic. In the atomic units used here,lengths r and energies  are related to laboratory valuesby R = a0r and E = α2mec2, where a0, α, me, and c arethe usual atomic constants. The only states affecting 1 1S0are the doubly-excited ones beginning with 2p2 3P e0 above thefirst ionization of helium. Similarly for 2 3S1, the intermediate042503-11050-2947/2011/83(4)/042503(3) ©2011 American Physical SocietyDONALD C. MORTON AND G. W. F. DRAKE PHYSICAL REVIEW A 83, 042503 (2011)TABLE I. Matrix Elements M and f- and A-values for 3 3P1,2–1 1S0, 2 1S0, and 3 1S0. The error on each entry indicates the numericaluncertainty in the convergence as the size of the basis sets is increased and does not include any estimate for the theoretical approximations.E1 Transition 3 3P1–1 1S0 3 3P1–2 1S0 3 3P1–3 1S0∞ (a.u.) theory 0.845 643 292 8 0.087 892 961 78 3.190 905 466 × 10−3 (a.u.) actual 0.845 494 46 0.087 878 693 0 3.188 316 × 10−3λ (nm) actual 53.889 594 518.489 09 14 290.72Length calculationa1〈n 1S0|z1 + z2|3 3P1〉0 −0.008 24(1) × 10−4 −0.000 523(3) × 10−4 −0.000 043 9(2) × 10−30〈n 1S0|z1 + z2|3 3P1〉1 1.021 445(3) × 10−4 5.487 346(3) × 10−4 3.100 340(1) × 10−3Total ML 1.013 20(1) × 10−4 5.486 823(4) × 10−4 3.100 296(1) × 10−3Velocity calculationa1〈n 1S0|∂/∂z1+ ∂/∂z2|3 3P1〉0/∞ 0.021 72(1) × 10−4 0.018 80(4) × 10−4 0.021 93(9) × 10−30〈n 1S0|∂/∂z1+ ∂/∂z2|3 3P1〉1/∞ 1.140 782 9(1) × 10−4 5.501 376 4(3) × 10−4 3.272445(1) × 10−3Correction C/∞ −0.149 283 75(1) × 10−4 −0.033 312 00(5) × 10−4 −0.194 178 64(3) × 10−3Total MV /∞ 1.013 22(1) × 10−4 5.486 86(4) × 10−4 3.100 20(9) × 10−3E1 f01 5.787 6(2) × 10−9 1.764025(3) × 10−8 2.044 697(1) × 10−8E1 A10 (s−1) 44.326(1) 1.459 495(2) 2.229 700(1) × 10−3M2 Transition 3 3P2–1 1S0 3 3P2–2 1S0 3 3P2–3 1S0∞ (a.u.) theory 0.845 643 292 8 0.087 892 961 78 3.190 905 466 × 10−3 (a.u.) actual 0.845 494 36 0.087 878 592 9 3.188 216 × 10−3λ (nm) actual 53.889 600 518.480 07 14 291.17〈n 1S0 | z1 − z2 | 3 3P2〉 0.417 847 2(2) 2.142 383(1) 12.013 381 85(1)M2 f02 2.495035(3) × 10−11 7.364 387(7) × 10−13 1.108 032 35(1) × 10−15M2 A20 (s−1)b 0.114 654 7(1) 3.655 827(4) × 10−5 7.249 719 30(6) × 10−11aThe superscripts 1 and 0 outside the angle brackets denote the perturbation order of the adjacent wave functions.bJacobs et al. [13] had calculated 0.115 s−1 and Kundu et al. [14] 0.120 63(1) s−1 for 3 3P2–1 1S0.states begin with 2p3p 1P e1 , while for 1s2p 1P1, they are theodd states 1s2p 3P1 and higher. For these calculations wehave accurate nonrelativistic energies and wave functions forinfinite nuclear mass with successively larger basis sets up to1000 terms to assess convergence. As a test of this perturbationprocedure in Ref. [8], we obtained A10 = 177.578 s−1 forTABLE II. Matrix Elements M and f - and A-values for 3 1P1–2 3S1 and 3 3S1. The error on each entry indicates the numerical uncertaintyin the convergence as the size of the basis sets is increased and does not include any estimate for the theoretical approximations.E1 Transition 3 1P1–2 3S1 3 1P1–3 3S1∞ (a.u.) theory 0.120 083 016 14 1.354 270 538 × 10−2 (a.u.) actual 0.120 074 88 1.354 404 0 × 10−2λ (nm) actual 379.457 81 3364.088 7Length calculationa1〈n 3S1|z1 + z2|3 1P1〉0 –0.001 649 3(5) × 10−4 0.000 130 64(4) × 10−30〈n 3S1|z1 + z2|3 1P1〉1 1.213 415 8(1) × 10−4 2.898 049 27(4) × 10−3Total ML 1.211 766 5(5) × 10−4 2.898 179 91(6) × 10−3Velocity calculationa1〈n 3S1|∂/∂z1+ ∂/∂z2|3 1P1〉0/∞ 0.037 517(5) × 10−4 –0.024 429(4) × 10−30〈n 3S1|∂/∂z1+ ∂/∂z2|3 1P1〉1/∞ 1.368 098 3(1) × 10−4 2.803 014(4) × 10−3Correction C/∞ –0.193 852 94(1) × 10−4 0.119 599 95(1) × 10−3Total MV /∞ 1.211 762(5) × 10−4 2.898 185(6) × 10−3E1 f01 3.918 384(4) × 10−10 2.527 805 2(1) × 10−8E1 A10 (s−1) 0.181 543 6(2) 0.148 958 52(1)M2 Transition 3 1P1–2 3S1 3 1P1–3 3S1∞ (a.u.) theory 0.120 083 016 14 1.354 270 538 × 10−2 (a.u.) actual 0.120 074 88 1.354 404 0 × 10−2λ (nm) actual 379.457 81 3364.088 7〈n 3S1 | z1 − z2 | 3 1P1〉 0.443 407 2(1) 11.258 337 6(1)M2 f01 2.681 693(1) × 10−14 2.479 839 86(4) × 10−14M2 A10 (s−1) 1.242 462(1) × 10−5 1.461 320 16(2) × 10−7aThe superscripts 1 and 0 outside the angle brackets denote the perturbation order of the adjacent wave functions.042503-2SPIN-FORBIDDEN RADIATIVE DECAY RATES FROM . . . PHYSICAL REVIEW A 83, 042503 (2011)2 3P1–1 1S0 compared with 177.5771 s−1 calculated by Łachand Pachucki [7] and 177 ± 8 s−1 measured by Dall et al. [11].We also found A11 = 1.548 935 s−1 for 2 1P1–2 3S1, very closeto 1.548 926 s−1 in [7].This calculation uses the dipole length operator z1 + z2.Within our (αZ)2 approximation, Drake [3] has shown that asimilar one with the dipole velocity operator ∂/∂z1+ ∂/∂z2 inEq. (1), plus the correctionC =√2α24〈1 1S0| Zr31z1 − Zr32z2 + 2r312(z1 − z2)|3 3P1〉, (2)should be equivalent to the length result multiplied by theenergy difference. Note that the correction depends on onlythe initial and final nonrelativistic states and also applies to3 1P1–2 3S1.The M2 calculation also uses these states. The analysis ofDrake [4] with the energy differenceε and the reduced matrixelement in atomic units gives the statistical weight times thetransition rate in s−1 asg2A20 = α8c6a05 | 〈1 1S0 || z1 − z2 || 3 3P2〉 |2, (3)and similarly with the same factors for g1A11 for the M2 partof 1P1–3S1. For 2 3P2–1 1S0, we found A20 = 0.327 031 5 s−1compared with 0.327 032 6 s−1 in Ref. [7] and 0.324 ±0.016 s−1 measured by Hodgman et al. [9].III. RESULTS AND DISCUSSIONTables I and II summarize our results. We calculated theoscillator strengths and transition rates using the theoreticalenergy ∞ for infinite nuclear mass without singlet-tripletmixing. The actual energies and wavelengths are from thecombination of laboratory measurements and theoretical cal-culations presented by Morton, Wu, and Drake [12]. Use ofthese energies would give f and A values slightly closer to thetrue ones.The subscripts on the E1 matrix elements M label thelength (L) and velocity (V ) forms from Eq. (1), and C isthe correction of Eq. (2) with division by ∞ to matchthe length forms. In all cases, the dominant contributionfor the spin-forbidden E1 transitions comes from the perturbed3 3P1 or 3 1P1 state because of the small (3 3P1) − (3 1P1)term in the denominator of Eq. (1). In fact, this one termdominates the sum over intermediate states. This dominanceof the m = 3 term in the summation becomes even morepronounced with increasing Z because the singlet-tripletenergy difference increases only in proportion to Z.The comparison of the L and V forms of the transitionoperator is particularly interesting. Even though the individualcontributions are very different, they sum to the same finalmatrix element to within 3 parts in 105 or better, and they agreeto within the estimated convergence accuracy. This provides avery useful check on the accuracy of the results, analogous tothe corresponding comparison of L and V for ordinary allowedtransitions.Even though both the spin-forbidden E1 and M2 decayrates are of the same nominal order α8c/a0, in every case theM2 decay rate is very much smaller.The final uncertainties must also include the finite nuclearmass and other higher-order relativistic corrections. Thesecould be as large as 0.1%, so realistic estimates are A10 =44.33(4) s−1 and A20 = 0.1147(1) s−1 for E1 and M2,respectively, of 3 3P1,2–1 1S0, and likewise for all the otherresults in the tables. Since these final uncertainties are likelyto be much less than the corresponding experimental ones inthe planned experiments [9,10], the comparison will providean accurate benchmark to test experimental procedures todetermine atomic lifetimes in this regime of strongly forbiddentransitions.ACKNOWLEDGMENTSResearch support by the Natural Sciences and EngineeringResearch Council of Canada and by SHARCNET are grate-fully acknowledged.[1] R. C. Elton, Astrophys. J. 148, 573 (1967).[2] G. W. F. Drake and A. Dalgarno, Astrophys. J. 157, 459 (1969).[3] G. W. F. Drake, J. Phys. B 9, L169 (1976).[4] G. W. F. Drake, Astrophys. J. 158, 1199 (1969).[5] G. W. F. Drake, Phys. Rev. A 5, 1979 (1972).[6] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- andTwo-Electron Atoms (Springer-Verlag, Berlin, 1957), p. 173.[7] G. Łach and K. Pachucki, Phys. Rev. A 64, 042510 (2001).[8] D. C. Morton, P. Moffat, and G. W. F. Drake, Can. J. Phys. 89,129 (2011).[9] S. S. Hodgman, R. G. Dall, K. G. H. Baldwin, and A. G. Truscott,Phys. Rev. A 80, 044501 (2009).[10] K. G. H. Baldwin, S. S. Hodgman, R. G. Dall, L. J. Byron,S. J. Buckman, and A. G. Truscott, in Proceedings of the 19thICOLS Conference, Hokkaido, Japan, 2009, edited by H. Katori,H. Yoneda, K. Nakagawa and F. Shimizu (World Scientific,Singapore, 2010), p. 95.[11] R. G. Dall, K. G. H. Baldwin, L. J. Byron, and A. G. Truscott,Phys. Rev. Lett. 100, 023001 (2008).[12] D. C. Morton, Q. Wu, and G. W. F. Drake, Can. J. Phys. 84, 83(2006).[13] V. L. Jacobs, J. Phys. B 5, 213 (1972).[14] B. Kundu, P. K. Mukherjee, and H. P. Roy, Phys. Scr. 39, 722(1989).042503-3

Spin-forbidden radiative decay rates from the 3′,3P 1,2 and 3′,1P 1 states of helium

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Spin-forbidden radiative decay rates from the 3′,3P 1,2 and 3′,1P 1 states of helium

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