Polarizabilities and hyperfine structure constants of the low-lying levels of barium

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1 Eur. Phys. J. D 5, ) THE EUROPEAN PHYSICAL JOURNAL D c EDP Sciences Società Itlin di Fisic Springer-Verlg 1999 Polrizbilities nd hyperfine structure constnts of the low-lying levels of brium M.G. Kozlov nd S.G. Porsev Petersburg Nucler Physics Institute, Gtchin, Leningrd district, , Russi Received: 22 June 1998 / Revised: 2 September 1998 / Accepted: 15 September 1998 Abstrct. The results of b initio clcultion of energies, hyperfine structure constnts nd sttic polrizbilities for severl low-lying levels of brium re reported. The effective Hmiltonin for the vlence electrons H eff hs been constructed in the frme of CI+MBPT method nd solutions of mny electron eqution H eff Φ n = E nφ n were found. Using the wve functions obtined the hyperfine structure constnts nd sttic polrizbilities were clculted. PACS Ar Ab initio clcultions Dk Electric nd mgnetic moments, polrizbility Fn Fine nd hyperfine structure 1 Introduction In this pper we report results of n b initio clcultion of sttic polrizbilities nd hyperfine structure hfs) constnts for severl low-lying levels of brium. In such clcultions the ccurcy of tomic wve function is tested twice, t lrge nd short distnces. Indeed, t short distnces tomic wve function cn be checked by the comprison of the clculted hfs constnts with the experimentl ones. The ltter re usully known to very good ccurcy, providing good test of the qulity of the wve function ner the nucleus. Oscilltor strengths nd polrizbilities re, in contrst, determined by the wve function behvior t lrge distnces. Usul experimentl ccurcy for the oscilltor strengths nd sclr polrizbilities is on the level of few percent. This is close or even less thn the ccurcy of precise tomic clcultions see, for exmple, clcultions for Cs [1]). On the other hnd, tensor polrizbilities cn be mesured with the ccurcy of 1% or better see, for instnce, [2,3]). Thus, it is possible to test tomic wve function t lrge distnces t 1% level. Note tht 1% ccurcy is crucil for clcultions of prity non-conservtion effects in toms, becuse it llows to test predictions of the Stndrd model t smll momentum trnsfer [1, 4]. So fr such precision hs been chieved only for one-electron toms Cs nd Fr [5 7]. In this work we consider much more complicted B tom. We consider brium s two electron tom with xenon-like core. Vlence-vlence correltions re tken into ccount by configurtion interction CI) method, while core-vlence nd core-core correltions re treted within the mny body perturbtion theory MBPT). The e-mil: porsev@thd.pnpi.spb.ru ltter is used to construct n effective Hmiltonin for the CI problem in the vlence spce. The detils of the method cn be found in the ppers [8,9]. Appliction of this method to clcultions of hfs constnts hs been discussed in [10]. Here we pply the sme technique for clcultions of tomic polrizbilities. In prticulr, we clculted sclr nd tensor polrizbilities for five even-prity sttes 1 S 0 6s 2 ), 3 D J ), nd 1 D 2 ) nd four oddprity sttes 3 P o J ) nd1 P o 1 ). 2 Generl formlism Sttic polrizbility of the sublevel, J, M in DC electric field E = Eẑ is defined s: E,J,M = 1 2 α,j,me 2 = 1 2 3M 2 ) JJ +1) α 0,,J + α 2,,J E 2, J2J 1) 1) where E,J,M is n energy shift nd α 0, α 2 define sclr nd tensor polrizbilities correspondingly. Being second order property, α,j,m cn be expressed s sum over unperturbed intermedite sttes: α,j,m = 2 n, J, M D z n, J n,m 2 E E n, 2) where D is dipole moment opertor, E n is n unperturbed energy of level n, nd the sum runs over ll sttes of opposite prity. The formlism of the reduced mtrix elements llows to write explicit expressions for the sclr

2 60 The Europen Physicl Journl D nd tensor prts of the polrizbility: 2, J D n, J n 2 α 0,,J =, 3) 32J +1) E n E n ) 1/2 5J2J 1) α 2,,J =4 62J + 3)2J +1)J+1) { } 1) J+Jn+1 J 1 J n, J D n, J n 2, 4) 1 J 2 E n E n nd reduced mtrix elements re defined s follows: n, J,M D q, J, M ) = 1) J M J 1 J n, J D, J. 5) M qm In order to use equtions 2 4) in clcultions one needs to know the complete set of eigensttes of the unperturbed Hmiltonin. It becomes prcticlly impossible when dimension of CI spce exceeds few thousnd determinnts. It is known, tht it is much more convenient to solve inhomogeneous eqution insted of the direct summtion over the intermedite sttes [11]. The generl technique of the solution of n inhomogeneous eqution ws reported in [12]. Let us consider the solution of the following eqution: E H) X,M = D q, J, M, 6) where q =0,±1ndM =M+q. Obviously, the right hnd side in 2) cn be expressed in terms of the function X,M note tht r 0 r z ): α,j,m = 2, J, M D 0 X,M. 7) If we wnt to rewrite equtions 3, 4) in terms of the function X,M, we need to decompose the ltter in terms, tht correspond to prticulr ngulr moment J i. Generlly speking, there cn be three such terms with J i = J, J ± 1: X,M = X,J 1,M + X,J,M + X,J+1,M. 8) Now, with the help of the functions X,J,M the equtions 3, 4) re reduced to: α 0,,J = 1) q J +1) ) 2 J 1 J, J, M D q X M,J,M, qm J =J,J± 1 9) ) 1/2 α 2,,J =4 1) q+1 5J2J 1) 62J + 3)2J +1)J+1) { } 2 ) 2 J 1 J J 1 J 1) J+J 1 J 2 M qm J =J,J± 1, J, M D q X,J,M. 10) Note, tht these equtions re vlid only if 3j-symbols on the right hnd side do not turn to zero. Thus, it is usully necessry to solve 6) for q = ± 1, rther thn for q = 0. Indeed, for q =0the3j-symbol tht correspond, for instnce, to J = J =1ndM =M= 0, turns to zero. If we know the solution of the eqution 6) nd its decomposition 8), then expressions 9, 10) llow us to find both sclr nd tensor polrizbilities of the stte, J. Moreover, the sme functions X,J,M cn be lso used to find other second order tomic properties, such s mplitudes of Strk-induced E1 trnsitions or prity nonconserving E1 trnsitions between the sttes of the sme nominl prity see, for exmple [13]). 3 Clcultion detils In this section we give brief description of the clcultion procedure putting specil emphsis to the solution of the eqution 6). Detils of the construction of the effective Hmiltonin with the help of the MBPT digrmmtic technique cn be found in [8, 9]. Effective opertors for hfs nd dipole mplitudes include rndom phse pproximtion RPA) corrections nd severl smller MBPT corrections s described in [10]. 3.1 Orbitl bsis set nd CI spce This clcultion is done in the V N pproximtion, tht mens tht core orbitls re obtined from the Dirc- Hrtree-Fock DHF) equtions for neutrl tom we use DHF computer code [14]). The bsis set for the vlence electrons includes 6s, 6pnd 5d DHF orbitls nd 7s 16s, 7p 16p, 6d 13d, 4f 12f, nd5g 7gvirtul orbitls. The ltter were formed in two steps. On first step we construct orbitls with the help of recurrent procedure, which is similr to tht suggested in [15] nd described in [9,13]. After tht we digonlize the V N DHF opertor to obtin the finl set of orbitls. For the described orbitl bsis set the complete CI is mde for both even-prity nd odd-prity levels. Mny electron wve functions re the liner combintions of the Slter determinnts with given J z. Tht mens tht no symmetriztion with respect to ngulr momentum J is mde. 3.2 Effective opertors Within the CI+MBPT method the eqution 6) is pproximted by the eqution in the vlence spce E H eff ) X,M = D eff,q, J, M, 11) with the effective opertors, which re found by mens of the MBPT. The effective Hmiltonin for the two vlence electrons is formed within the second order MBPT [8]. Only excittions of the 26 uppermost core electrons from

3 M.G. Kozlov nd S.G. Porsev: Polrizbilities nd hfs constnts of the low-lying levels of brium 61 the shells n =4,5 were included. We used RPA for the effective dipole moment opertor see, for exmple, [16]). We hve checked tht MBPT corrections to D eff,which re not included in RPA re smll, if RPA equtions re solved in the V N 2 pproximtion, i.e. 6s electrons re excluded from the self-consistency procedure. The more detiled description of the effective opertor formlism is given elsewhere [10]. In order to evlute the right hnd side of the eqution 11), the mtrix eigenvlue eqution with the effective Hmiltonin hs been solved for five low-lying even-prity sttes 1 S 0 6s 2 ), 3 D J ), J =1,2,3, nd 1 D 2 ) nd four odd-prity sttes 3 P o J ), J = 0, 1, 2nd 1 P o 1 ). 3.3 Inhomogeneous eqution be n nth pproximtion nd R n) is the corresponding residue here we omit quntum number M): Techniclly there is no difference between equtions 6, 11), nd to simplify the nottions we will spek bout the solution of eqution 6). This is mtrix eqution with lrge, sprse nd symmetricl mtrix. For the orbitl bsis set, in which DHF opertor is digonl, the nondigonl mtrix elements correspond to the residul two-electron interction nd re smll. Tht llows to look for n itertive solution of the mtrix eqution 6). Let X n) R n) = Y E H)X n), 12) where Y is the right hnd side of 6). We now use the digonl of the mtrix H to form probe vector C n) nd new pproximte solution X n+1) C n) X n+1) : =E digh)) 1 R n), 13) N = k 1 X n) + k 2 C n) + k l+2 Ψ l, 14) where coefficients k i re found from minimum residue condition nd Ψ l re some pproximte) eigenfunctions of the effective Hmiltonin. The procedure 12 14) provides the monotonous decrese of the residue. For the ground stte = 0), this procedure converges rther rpidly even for N = 0. For excited sttes convergence cn be significntly slowed down when there re eigenvlues of the effective Hmiltonin, which re close to E. Suppose tht there re N eigenfunctions of the opposite prity Ψ l nd E l E 1, l =1,...N. We cn speed up the convergence by dding these functions to the right hnd side of 14) nd find minimum of the residue with respect to ll N + 2 coefficients k i. Note tht it does not require ny dditionl time-consuming computtions becuse the product HΨ l hs to be clculted only once. Such itertive procedure works fine, provided tht ll most importnt eigenvectors nd eigenvlues hve been found beforehnd. l=1 Tble 1. Energy levels for B in V N pproximtion. Level E vl.u.) cm 1 ) Experiment cm 1 ) 1 S b D D D D P o P o P o P o Reference [21]. b The bsolute vlue of this energy corresponds to the sum of the two first ioniztion potentils of brium, for which experiment gives u. [21]. 3.4 Decomposition of the function X,M 0 When the function X,M is found, the decomposition 8) is esily done with the help of the following opertors ˆP i : X,Ji,M = ˆP i X,M, 15) ˆP Ĵ2 ) i = N i J k J k +1))Ĵ2 J l J l +1), 16) N i =J i J i +1) J k J k +1)) 1 J i J i +1) J l J l +1)) 1, where Ĵ is the ngulr momentum opertor nd J i J k J l {J 1,J,J +1}. Let us point out tht the opertor ˆP i is not projector on subspce with J = J i, but it works s such when pplied to three component function like X,M. Indeed, ˆP i elimintes components with J k J i,j k {J 1,J,J+1} nd leves the component with J k = J i unchnged. If, J, M in the right hnd side in 6) is not n exct eigenvector of the opertor H, orx,m is not n exct solution of the eqution 6), then the ltter cn hve dmixtures with other ngulr moment. In this cse functions defined by 15) re not eigenfunctions of Ĵ2 nd re not orthogonl to ech other. This provides us with simple check whether the function X,M cn be decomposed in three components with definite ngulr moment or not. It is sufficient to check normliztion of the right hnd side nd the left hnd side of the eqution 8). So, opertors ˆP i llow not only to decompose the function X,M in terms with definite ngulr moment, but lso to check the qulity of this function. 4 Results nd discussion Results of the solution of the eigenvlue problems for evenprity nd odd-prity levels re presented in Tble 1.

4 62 The Europen Physicl Journl D Tble 2. Mgnetic dipole A) nd electric qudrupole B) hyperfine structure constnts of low-lying levels for 137 B. The electric qudrupole moment is tken to be b. 3 D 1 3 D 2 3 D 3 1 D 2 3 P o 1 3 P o 2 1 P o 1 A MHz) Dirc-Fock CI CI + MBPT Experiment b 1151 c 109 d Theory MCDF) [20] B MHz) Dirc-Fock CI CI + MBPT Experiment b 41.6 c 51 d Theory MCDF) [20] [22]; b [23]; c [24]; d [25]. These results re close to our erlier clcultions [9] nd to the results of the recent pper [17], where the sme method ws used. The typicl ccurcy for the energy intervls is better thn 100 cm 1 with the only exception of 1 P1 o level, where the error is lmost 200 cm 1.Note, tht typicl error of the coupled-cluster clcultion of the sme levels of B is cm 1 [18]. In [17] it ws shown, tht greement with the experiment cn be further improved if higher order MBPT corrections to the effective Hmiltonin re included semiempiriclly. In prticulr, in the second order of MBPT there is lrge screening of exchnge interction between 6s nd 6p electrons, which ffects the splitting between sttes 3 PJ o nd 1 P1 o.higher order corrections to this screening should be essentil [19]. When the eigenvlue problem is solved, we cn clculte the hfs constnts A nd B Tb. 2) nd the polrizbilities of the corresponding sttes Tb. 3). On the CI stge our results for the constnt A re close to those of the pper [20], where the multi-configurtionl Dirc-Fock MCDF) method ws used. It is seen tht these results strongly underestimte mgnetic hfs for ll levels. On verge, MBPT corrections contribute 30% 50% nd even more for smll constnts. The finl ccurcy of the theory for the constnt A ppers to be bout 3% 5% except 1 D 2 -stte), which is somewht less thn the ccurcy of the similr clcultions for Tl [10]. It is probbly the consequence of the smll numericl vlues of hfs constnts for B. Indeed, the lrgest hfs constnt of the level 3 P1 o is clculted with the highest ccurcy. For other levels there re cnceltions between contributions from 6s-electron nd either 5d- or6p-electron. We hve not found experimentl dt for the level 3 P2 o. MBPT corrections for this level re of the typicl size, cnceltions of different contributions re moderte, so we estimte the ccurcy of the theory for this level to be bout 3% or better. Similr sitution tkes plce for the hfs constnt B, but here our results re significntly different from those Tble 3. Sclr nd tensor polrizbilities.u.) of low-lying levels for B. Experiment Theory conf. level α 0 α 2 α 0 α 2 6s 2 1 S ) D 1 X + 164) b 531) b D 2 X + 44) b 691) b D 3 X 1214) b D 2 X 10944) b 862) b P o P o ) c P o P o ) d 43.14) e Reference [26]; b reference [27], sclr polrizbilities in this pper were mesured reltive to tht of the level 3 D 3, which is denoted by X; c reference [28]; d reference [29]; e reference [2]. of [20] lredy on the CI stge. The role of the MBPT corrections is even higher here. These corrections lmost double the nswer for the levels 1 D 2 nd 1 P1 o. So, it is not surprise, tht the ccurcy of the theory for these levels is not very high. Agin, for the level 3 P2 o the constnt B is unknown. In this cse MBPT corrections contribute more thn 60%, so we cn not gurntee the ccurcy higher thn 15%. In order to find polrizbilities we substitute eigenfunctions in eqution 11) nd solve corresponding inhomogeneous eqution. After tht equtions 9, 10) give us α 0 nd α 2. Results of these clcultions re listed

5 M.G. Kozlov nd S.G. Porsev: Polrizbilities nd hfs constnts of the low-lying levels of brium 63 in Tble 3. Sclr polrizbilities for 1 S 0 -nd 3 P o 0 -levels re known from the experiment. For the D-sttes sclr polrizbilities were mesured reltive to tht of the level 3 D 3. In Tble 3 the ltter is designted by X. If we ssume tht X is equl to the corresponding theoreticl vlue, ll experimentl dt for α 0 gree with our clcultions within the experimentl uncertinty. It is seen tht typiclly α 2 is bout n order of mgnitude smller thn α 0. This is due to the strong cncelltions between three terms of the sum 10). For this reson the theoreticl ccurcy for α 2 is significntly lower thn for α 0. On the contrry, experimentl dt for α 2 re much more precise. There re two min sources of errors in the clcultions of polrizbilities. First one is the sme s for the hfs clcultions, nd connected with the inccurcy in the wve functions nd effective opertors note tht RPA corrections to the dipole opertor re much smller thn for hfs opertors). Second source of errors is the inccurcy in eigenvlues, tht is specific to the clcultions of polrizbilities. Normlly, the errors ssocited with the ltter re smll, but they cn become importnt nd even dominnt) when there re close levels of opposite prity. In prticulr, tht pplies to levels 1 D 2 nd 3 P 1, nd, to some extent, to other levels of configurtions nd. Indeed, the energy intervl between levels 1 D 2 nd 3 P 1 is underestimted by 7% see Tb. 1). This intervl enters equtions 1, 2) nd potentilly cn cuse lrge error. Fortuntely, corresponding numertors re smll, becuse singlet-triplet mplitudes re suppressed. On the other hnd, this error is reltively enhnced for the tensor polrizbility, becuse of the strong cncelltions of different contributions, mentioned bove. In conclusion let us sum up the results. We hve found pproximte solutions of mny electron eqution H eff Φ n = E n Φ n for low-lying levels of brium. Then, we tested the qulity of the wve functions obtined t lrge nd short distnces. To do this, we clculted hfs constnts nd polrizbilitilies for the levels in question. Clcultions of polrizbilities of the excited sttes were hmpered due to the proximity of the levels of opposite prity. Further improvement of theoreticl ccurcy for these polrizbilities requires better greement between theoreticl nd experimentl spectrum of the tom. Tht cn be done, if higher-order MBPT corrections to the effective Hmiltonin re included semiempiriclly [17, 19]. This work ws supported by Russin Foundtion for Bsic Reserch, Grnt No One of us SP) is grteful to St. Petersburg government for the finncil support. References 1. V.A. Dzub, V.V. Flmbum, O.P. Sushkov, Phys. Rev. A 56, R ). 2. A. Kreutztrger, G. von Oppen, Z. Phys. 265, ). 3. A. Fukumi et l., Z. Phys. D42, ). 4. W. Mrcino, J.L. Rosner, Phys. Rev. Lett. 65, ). 5. V.A. Dzub, V.V. Flmbum, O.P. Sushkov, Phys. Lett. A 141, ). 6. S.A. Blundell, W.R. Johnson, J. Spirstein, Phys. Rev. A 43, ). 7. V.A. Dzub, V.V. Flmbum, O.P. Sushkov, Phys. Rev. A 51, ). 8. V.A. Dzub, V.V. Flmbum, M.G. Kozlov, JETP Lett. 63, ); Phys. Rev. A 54, ). 9. M.G. Kozlov, S.G. Porsev, Zh. Eksp. Theor. Fiz. 111, ) [JETP 84, )]. 10. V.A. Dzub, M.G. Kozlov, S.G. Porsev, V.V. Flmbum, Zh. Eksp. Theor. Fiz. 114, ) [JETP 114 in press)]. 11. R.M. Sternheimer, Phys. Rev. 80, ); Phys. Rev. 84, ); Phys. Rev. 86, ). 12. A. Dlgrno, J.T. Lewis, Proc. Roy. Soc. 233, ). 13. M.G. Kozlov, S.G. Porsev, V.V. Flmbum, J. Phys. B 29, ). 14. V.F. Brttsev, G.B. Deynek, I.I. Tupizin, Izv. Akd. Nuk SSSR 41, ). 15. P. Bogdnovich, G. Žukusks, Sov. Phys. Coll. 23, ). 16. A. M. Mårtensson-Pendrill, Methods in Computtionl Chemistry, V. 5: Atomic nd Moleculr Properties, edited by S. Wilson Plenum Press, New-York, 1992). 17. V.A. Dzub, W.R. Johnson, Phys. Rev. A 57, ). 18. E. Eliv, U. Kldor, Phys. Rev. A 53, ). 19. M.G. Kozlov, S.G. Porsev, Opt. Spektrosk. [Opt. Spectrosc] in press, 1999). 20. T. Olsson, A. Rosen, B. Fricke, G. Torbohm, Phys. Script 37, ). 21. C.E. Moore, Atomic Energy Levels, Vol. 3 Ntl. Bur. Stnd. US, Circ. No. 467, Wshington, 1958). 22. M. Gustvsson, G. Olson, A. Rosen, Z. Phys. A 290, ). 23. S.G. Schmelling, Phys. Rev. A 9, ). 24. G. zu Putliz, Ann. Phys. 11, ). 25. H.-J. Kluge, H.Z. Suter, Z. Phys. 270, ). 26. H.L. Schwrtz, T.M. Miller, B. Bederson, Phys. Rev. A 10, ). 27. B. Schuh, C. Neureiter, H. Jäger, L. Windholz, Z. Phys. D 37, ). 28. A. Kreutztrger, G. von Oppen, W. Wefel, Phys. Lett. A 49, ). 29. J. Li, W.A. vn Wijngrden, Phys. Rev. A 51, ).