PHYSICAL REVIEW A VOLUME 60, NUMBER 6

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1 PHYSICAL REVIEW A VOLUME 60, NUMBER 6 DECEMBER 1999 Relativiti many-body alulation of energy level, hyperfine ontant, eletri-dipole matrix element, and tati polarizabilitie for alkali-metal atom M. S. Safronova, W. R. Johnon, and A. Derevianko Department of Phyi, Notre Dame Univerity, Notre Dame, Indiana Reeived 23 June 1999 Removal energie and hyperfine ontant of the lowet four n, np 1/2, and np 3/2 tate in Na, K, Rb, and C are alulated; removal energie of the n 7 10 tate and hyperfine ontant of the n 7 and 8 tate in Fr are alo alulated. The alulation are baed on the relativiti ingle-double SD approximation in whih ingle and double exitation of Dira-Hartree-Fok wave funtion are inluded to all order in perturbation theory. Uing SD wave funtion, aurate value of removal energie, eletri-dipole matrix element, and tati polarizabilitie are obtained; however, SD wave funtion give poor value of the magneti-dipole hyperfine ontant for heavy atom. To obtain aurate value of the hyperfine ontant for heavy atom, we inlude triple exitation partially in the wave funtion. The preent alulation provide the bai for reevaluating parity nononerving amplitude in C and Fr. S X PACS number: Ar, Jf, Fn, Dk I. INTRODUCTION Energy level, tranition matrix element, hyperfine ontant, and tati polarizabilitie for low-lying 1/2, p 1/2, and p 3/2 tate in alkali-metal atom are tudied ytematially uing the relativiti ingle-double SD method in whih ingle and double exitation of the Dira-Hartree-Fok DHF wave funtion are inluded to all order of perturbation theory. The SD method wa applied previouly to tudy propertie of Li and Be 1, Li, Na, and C 2, C 3, and Na-like ion with Z ranging from 11 to In the latter tudy, the theoretial removal energie for Na-like ion, when orreted for the Lamb hift, agreed with experiment at the 1 20 m 1 level of auray for all tate onidered, while theoretial hyperfine ontant and dipole matrix element typially agreed with preie meaurement to better than 0.3%. Energie of alkali-metal atom have been alulated to high preiion in Ref. 5 uing the relativiti oupled-luter CC method; however, there have been no ytemati CC tudie of hyperfine ontant or tranition amplitude for alkali-metal atom. All-order method are needed for uh tudie ine orrelation orretion are large in alkali and low-order many-body perturbation theory MBPT doe not give aurate reult for atomi propertie. In K, Rb, and C, third-order MBPT give ground-tate energie in poorer agreement with experiment than eond-order MBPT, a illutrated in Table I where zeroth-order DHF energie are tabulated together with eond- and third-order MBPT orretion. The differene (k), k 0,2,3, between experimental energie and aumulated MBPT value hown in Table I oillate above and below the experimental value and how no ign of onvergene. In the SD approximation, an important ubet of MBPT diagram i iterated to all order in perturbation theory, leading to energie in exellent agreement with experiment. During the lat few year, the lifetime of the lowet p 1/2 and p 3/2 level have been meaured to high preiion for all alkali-metal atom 6 12, yielding experimental dipole matrix element aurate to 0.1% 0.25%. In the preent work, eletri-dipole matrix element for n p-n tranition in alkali from Na to Fr are evaluated for n N,N 1 and n N,...,N 3, where N i the prinipal quantum number of the ground tate. Matrix element and energie from the preent SD alulation were ued in Ref. 13 to tudy polarizabilitie of alkali-metal atom. In thi paper, we diu our alulation of tati polarizabilitie in detail and how that the ab initio SD reult are in exellent agreement with the value reommended in 13. We alo alulate Stark-indued alar and vetor tranition polarizabilitie for N-(N 1) tranition. The auray of our alulation i diued and reommended value of alar and vetor polarizabilitie are provided for C and Fr. Thee value are needed for the interpretation of experiment on parity nononervation in atom 14. A ytemati tudy of hyperfine ontant for, p 1/2, and p 3/2 level i alo preented. The auray of SD alulation of the hyperfine ontant for alkali-metal atom dereae rapidly from 0.3% for Na to 7% for C. To obtain more aurate value for heavy alkali, it wa found neeary to inlude triple exitation to the wave funtion partially. The derivation of an approximate ingle-double partial triple SDpT wave funtion i given in the following etion. In ummary, we tudy low-lying and p level in alkalimetal atom in the relativiti SD and SDpT approximation and find exellent agreement with other high-preiion alulation and with available experimental data. The SDpT TABLE I. Zeroth- or DHF, eond- and third-order MBPT removal energie E (k) in m 1 and energy differene (k) E expt E (k). K(4) Rb (5) C (6) k E (k) (k) E (k) (k) E (k) (k) E expt /99/60 6 / /$15.00 PRA The Amerian Phyial Soiety

2 PRA 60 RELATIVISTIC MANY-BODY CALCULATIONS OF wave funtion from the preent alulation will be ued later to evaluate parity nononerving PNC amplitude in eium and franium. II. TRIPLE EXCITATIONS The all-order ingle-double method wa deribed previouly in Ref Briefly, we repreent the wave funtion v of a one valene eletron atom a v v SD with a m ma SD nrb g bnr mnrab, v m E v mv SD nrab g abnr mnrvab, a b m n mnab 4 5 v SD 1 ma ma a m a a 1 2 mnab a m a n a b a a mnab m v mv a m a v mna mnva a m a n a a a v v, where v i the lowet-order atomi tate funtion, whih i taken to be the frozen-ore DHF wave funtion of a tate v. In thi equation, a i and a i are reation and annihilation operator, repetively, for tate i. Indie at the beginning of the alphabet, a, b,..., refer to oupied ore tate, thoe in the middle of the alphabet m, n,..., refer to exited tate, and v refer to valene orbital. Subtituting the wave funtion 1 into the many-body Shrödinger equation, where the Hamiltonian i taken to be the relativiti no-pair Hamiltonian with Coulomb interation 15, one obtain the oupled equation for ingle- and double-exitation oeffiient mv, ma, mnva, and mnab. The oupled equation are olved iteratively for the exitation oeffiient. We ue the reulting SD wave funtion to evaluate hyperfine ontant and eletri-dipole matrix element. It wa hown in 4 that the SD energie are not omplete in third order and that the miing third-order energy ontribution are aoiated with the omitted triple exitation. In 4, we alulated the miing third-order term eparately and added them to the final energie. It an be hown that one-body matrix element alulated uing SD wave funtion are omplete through third order. A mentioned in the Introdution, the hyperfine ontant for heavy alkali-metal atom are not determined to high preiion in the SD approximation. However, by adding the two triple-exitation term mnrvab and mnrab to the SD wave funtion, we automatially inlude the miing third-order energy and alo ubtantially improve the auray of our alulation of hyperfine ontant. The orreted wave funtion i v SD v 1 6 mnrvab a m a n a r a b a a a v mnrab 1 18 mnrab a m a n a r a a b a a v, mnrab where v SD i ingle-double wave funtion in Eq. 1. The addition of the ore term mnrab i neeary to preerve the ymmetry relation mnva nmav. Carrying out the alulation, we obtain the following equation for the energy and ingle- and double-exitation oeffiient: E v SD mnab g abmn mnvvab, SD rd g dar mnrbd rd g dbr nmrad r g mr nrba r g nr mrab, a v m n E v mnva SD rb g bar mnrvb r g bvr mnrab rb g bmr rnvba rb g bnr rmvab. In the above equation, we write out only thoe term ariing from the triple exitation. The quantitie i are inglepartile energie and E v i the orrelation orretion to the valene energy. Below, we ue the notation mn m n, g abd g abd g abd, and abd abd abd. The ontribution from the ingle- and double-exitation oeffiient, deignated by SD above, are given in Ref. 4. We require that the triple-exitation oeffiient mnrvab and mnrab be antiymmetri with repet to any non-yli permutation of the indie mnr, vab and mnr, ab, repetively, and we obtain the following equation for the triple-exitation oeffiient: a b m n r mnrab 123mnr ab 1 g g d g 1d1 2 23d3 a b v m n r E v mnrvab 123mnr vab 1 g triple, g g triple, 9 where triple group together term ontaining mnrvab or mnrab. In the above equation, the notation 123mnr deignate ymbolially that the indie 123 range over all ix permutation of the indie mnr; even permutation ontribute with a poitive ign while odd permutation ontribute with a negative ign. The relatively mall ontribution from ingle and triple exitation on the right-hand ide of Eq. 8 and 9 are omitted in the preent tudy.

3 4478 M. S. SAFRONOVA, W. R. JOHNSON, AND A. DEREVIANKO PRA 60 The dominant triple orretion arie from the triple ontribution to E v and mv given in Eq. 3 and Eq. 5, repetively. Solving the equation for mnrvab and ubtituting the reulting expreion into Eq. 3, wefind E v SD mnab g abmn ab mn g mav nvb g nva mvb g vbv mna g mvv nba g mab vnv g mna vvb g mnv vba g vba mnv. 10 Repeating thee tep for mv, we obtain from Eq. 5 v m E v mv SD nrab g abnr ab nr g nva rmb g rma nvb g mva nrb g rmv nab g nab rmv g nra mvb g nrv mab g mab nrv. 11 In our numerial tudie, we ue the approximation g abmn ab mn mnab in Eq. 10 and 11. In the preent alulation, we inlude triple in the mv and E v equation only. A diued above, only double-exitation term are onidered in the equation for the triple-exitation oeffiient. Finally, expliit triple-exitation orretion to matrix element are omitted; only indiret orretion aued by modifiation of E v and mv are inluded. The modified matrix element are evaluated a deribed in Ref. 4. In the approximation ued here, all third-order orretion to E v are automatially inluded. III. RESULTS AND DISCUSSIONS A. Removal energie and fine truture The SD equation are et up in a finite bai and olved iteratively to give the ingle- and double-exitation oeffiient ma, mv, mnab, and mnva, and the orrelation energy E v. The bai orbital ued to define the inglepartile tate are linear ombination of B-pline 16. For eah angular momentum tate, the bai et onited of 40 bai orbital ontruted from 40 B-pline of order 7. In our iterative alulation, we ued only 35 of the 40 orbital. The B-pline bai orbital were interpolated onto a 250 FIG. 1. Comparion of MBPT and SD orrelation orretion to the ground-tate energie for alkali-metal atom. point nonlinear radial grid. All orbital were ontrained to a large pherial avity; the avity radii were hoen to be 110 a.u. for Na, 100 a.u. for K and Rb, 75 a.u. for C, and 90 a.u. for Fr. Suh large avitie were needed to aommodate the highly exited tate onidered here. The DHF energie of the lowet three to four and p tate were reprodued to five or more ignifiant digit by the B-pline bai funtion. Generally, the larger value of n had lower auray, whih i unimportant owing to the dereaing ize of orrelation orretion with inreaing n. Term in the angular momentum deompoition with angular momentum l from0to6 were retained in the bai and the partial-wave ontribution were extrapolated to give the final value of the orrelation energy. The extrapolation proedure i deribed in 4. For the ae of Fr, only partial wave with l 5 were retained beaue of omputational limitation, and the extrapolation proedure wa implified, leading to omewhat lower auray. Contribution to the energy from the Breit interation with all-order orrelation orretion were obtained a expetation value of the Breit operator uing SD wave funtion, a deribed in 4. Breit orretion were found to be le than 15 m 1 in all ae. In Fig. 1, value of E v for the ground tate of the alkali, orreted for miing triple, are ompared with the eond-order energy E (2), the third-order energy E (2) E (3), and with the experimental orrelation energy Expt. Differene between E v and E (2) E (3) are from fourthand higher-order term in the MBPT expanion. It i lear from the figure that the SD proedure reolve the problem of poor onvergene of MBPT diued previouly and hown in Table I, and that the SD ground-tate orrelation energie are in good agreement with experimental value. Contribution from fourth- and higher-order orretion inreae from 8% of the total orrelation energy for Na to 24% for Fr. Differene with meaurement for the ground-tate orrelation energy range from 0.1% for Na to 2.7% for Fr. The SD approximation, therefore, aount for a dominant fration of the fourth- and higher-order orrelation energy. Correlation orretion for lowet p 1/2 tate are about 3 time maller than thoe for the ground tate. The relative

4 PRA 60 RELATIVISTIC MANY-BODY CALCULATIONS OF ontribution of higher-order orretion are found to be approximately the ame for n and np tate. A detailed omparion of removal energie for and p 1/2 tate with experiment i given in Table II. The experimental data ued in thi omparion are from Ref. 17 exept for Fr, where experimental energie ompiled in 18 and reult of reent meaurement 19 are ued. For Na, our theoretial unertainty from extrapolation range from 0.4 m 1 for the 3 tate to 0.04 m 1 for the 6 tate; thi unertainty inreae for heavier alkali. The agreement with experiment i exellent for Na, where the 6 energy differ from experiment by 0.14 m 1 and the 3 energy differ by 2 m 1. The orreponding differene are 5 48 m 1 in K, 7 42 m 1 in Rb, m 1 in C, and m 1 in Fr. Agreement with experiment improve ubtantially with n ine orrelation orretion dereae. Our reult for the removal energie of np 1/2 tate are in exellent agreement with experiment for all tate onidered. For np 1/2 tate, the differene with experiment are m 1 in Na, 2 4 m 1 in K, 1 7 m 1 in Rb, 9 24 m 1 in C, and m 1 in Fr. The removal energie of np tate are expeted to be in better agreement with experiment beaue of the maller ize of orrelation orretion. We make predition of 9p 1/2 and 10p 1/2 energie in Fr where there are no experimental value in the lat row of Table II. Thee predition are baed on omparion of SD energie with experimental energie for other np tate in C and Fr. We expet our predition to be aurate to about 5 m 1. Experimental energie for all tate, exept the np tate of Na, are larger than theoretial value; in other word, orrelation orretion are generally underetimated in the SD approximation. The SD energie are alo ompared with the relativiti CC alulation from Ref. 5 and with MBPT alulation from 18 in Table II. The CC alulation agree better with experiment for n tate exept for the ae of Na, where the CC energy differ from experiment by about 100 m 1 for the 3 ground tate. For the np tate, the preent alulation are in better agreement with experiment than the CC alulation, epeially for the 6p 1/2 tate of Rb and the 7p 1/2 tate of C. The fine-truture interval np 3/2 -np 1/2 are ompared with experiment and with relativiti CC alulation 5 in Table III. Predition for the fine-truture interval of the 8p and 9p tate in Fr, baed on omparion of other interval in C and Fr, are alo given in the table. The theoretial finetruture interval are een to be in uniformly exellent agreement with experiment. In ummary, the relativiti SD approximation give aurate value for n removal energie in alkali-metal atom, the agreement with experiment being better for lighter element. Removal energie for np tate and np 3/2 -np 1/2 finetruture interval are alo in exellent agreement with experiment. B. Eletri-dipole matrix element Eletri-dipole matrix element for n p 1/2 -n and n p 3/2 -n tranition are evaluated in the SD approximation TABLE II. Comparion of SD alulation of n and np 1/2 removal energie with the CC alulation from Ref. 5, many-body alulation from Ref. 18, and with experimental energie from Ref in unit of m 1. Na Theory Expt CC Na 3p 1/2 4p 1/2 5p 1/2 6p 1/2 Theory Expt CC K Theory Expt CC K 4p 1/2 5p 1/2 6p 1/2 7p 1/2 Theory Expt CC Rb Theory Expt CC Rb 5p 1/2 6p 1/2 7p 1/2 8p 1/2 Theory Expt CC C Theory Expt CC C 6p 1/2 7p 1/2 8p 1/2 9p 1/2 Theory Expt CC Fr Theory Expt CC Fr 7p 1/2 8p 1/2 9p 1/2 10p 1/2 Theory Expt a 3795 a CC a Predition baed on SD alulation.

5 4480 M. S. SAFRONOVA, W. R. JOHNSON, AND A. DEREVIANKO PRA 60 TABLE III. Comparion of SD fine-truture interval in Na, K, Rb, C, and Fr with experiment and with theoretial CC value from Ref. 5. Unit: m 1. Thi work Expt. 5 Na 3p 3/2-3p 1/ p 3/2-4p 1/ p 3/2-5p 1/ p 3/2-6p 1/ p 3/2-7p 1/ K 4p 3/2-4p 1/ p 3/2-5p 1/ p 3/2-6p 1/ p 3/2-7p 1/ Rb 5p 3/2-5p 1/ p 3/2-6p 1/ p 3/2-7p 1/ p 3/2-8p 1/ C 6p 3/2-6p 1/ p 3/2-7p 1/ p 3/2-8p 1/ p 3/2-9p 1/ Fr 7p 3/2-7p 1/ p 3/2-8p 1/ p 3/2-9p 1/ a 10p 3/2-10p 1/ a a Predition baed on SD alulation. uing the formalim laid out in Ref. 1. In brief, the onepartile matrix element Z i repreented in eond quantization a Z ij z ij a i a j, 12 where z ij i the matrix element of the dipole operator z between ingle-partile orbital. In the SD approximation, matrix element of Z are obtained by ubtituting the SD wave funtion from Eq. 1 into the matrix element w Z v. Correting for normalization, one obtain SD Z val SD w Z SD v wv Z ore 1N SD w 1N SD v, 1/2 13 where the firt term ontribute for alar operator only. The term Z SD val i the um Z SD val z wv z (a) wv z (t) wv, 14 where z wv i the DHF matrix element and the remaining 20 term are linear or quadrati funtion of the ingle- and double-exitation oeffiient ma, mv, mnab, and mnva. Expreion for the term z (i) wv and the normalization ontant N SD v are given in Ref. 1. Matrix element for n p 1/2 -n and n p 3/2 -n tranition with nn N 3 and n N,N 1, where N i the prinipal quantum number of the ground tate, are alulated uing thi method. The reulting matrix element are ubequently ued to evaluate polarizabilitie. In Fr, the eletridipole matrix element of n p-9 tranition are alo alulated to provide additional data for thi leat tudied alkalimetal atom. We performed alulation of eletri-dipole matrix element both in length and veloity gauge. The matrix element for the prinipal tranition in the two gauge differ by 0.01% for Na and by 0.7% 1% for Fr. The gauge dependene tem from the nonloality of the tarting DHF potential and from the limited number of MBPT diagram inluded in the formalim. Only length gauge reult are lited in the following table ine the length gauge i generally more reliable for orrelation alulation a diued in Ref. 21. In Table IV, we ompare the SD matrix element for the prinipal Np 1/2 -N and Np 3/2 -N tranition in Na, K, Rb, C, and Fr with the high-preiion experimental reult from 7,9,12. The differene between the preent SD alulation and experiment range from 0.1% in Na to 0.5% 0.8% in Fr. The SD reult for the prinipal tranition are in all ae in better agreement with experiment than the thirdorder MBPT value from 20 beaue of the more omplete treatment of higher-order orretion. In C, whih ha been extenively tudied during the pat 15 year, all-order reult from Ref. 21 and 22 are alo available. Comparion of our reult with thee theoretial alulation will be given below. Redued matrix element for tranition from all n p 1/2 and n p 3/2 tate to N and (N 1) tate of Na, K, and Rb are given in Table V. Thee matrix element are ued later to evaluate polarizabilitie. Exept for the prinipal tranition, no high-preiion experimental value are available for thee matrix element. It i poible to inlude effet of triple exitation indiretly by uing the valene ingle- and double-exitation o- TABLE IV. Comparion of SD alulation of redued dipole matrix element a.u. for the prinipal tranition in alkali-metal atom with experimental value. Na K Rb C Fr 3p 1/2-3 Ref. 4p 1/2-4 Ref. 5p 1/2-5 Ref. 6p 1/2-6 Ref. 7p 1/2-7 Ref. Preent Expt p 3/2-3 4p 3/2-4 5p 3/2-5 6p 3/2-6 7p 3/2-7 Preent Expt

6 PRA 60 RELATIVISTIC MANY-BODY CALCULATIONS OF TABLE V. SD value of redued dipole matrix element a.u. in Na, K, and Rb. Na K Rb 3p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 1/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ p 3/ effiient mv and mnva modified a explained previouly to inlude triple partially. Equation 14 itelf i not modified in thi proedure; thu, effet of the triple are inluded only indiretly. We ue thi proedure to obtain SDpT value for hyperfine ontant. We found that inluding triple indiretly doe not improve the agreement with experiment for matrix element of prinipal tranition, exept for Na; for tranition other than the prinipal one the auray improve lightly. To improve the auray of the matrix element further, one mut inlude triple exitation expliitly in the matrix element, i.e., alulate matrix element in Eq. 14 uing the SDT wave funtion given in Eq. 2. A a reult, the expreion for Z val will be modified: Z SDT val Z SD val triple, 15 where triple are term ontaining the triple-exitation oeffiient mnrvab and mnrab. It i poible to etimate the ontribution of ome omitted higher-order term. The dominant orrelation orretion to mot of the tranition matrix element are from the Bruekner-orbital BO term defined in 4 and diued in Ref. 3 and 20. To etimate the effet of omitted higherorder orretion to the BO term, we aled the ingleexitation oeffiient mv, a deribed in Ref. 21 : the oeffiient were multiplied by the ratio of the experimental to theoretial orrelation energie. In Table VI, aled reult for C matrix element are ompared with our SD data, with the all-order alulation of Ref. 21,22 and with experiment. The experimental data for 6p-6 tranition in thi table are from the mot reent meaurement 12. For the other tranition, with the exeption of 7p-7, the experimental data ompiled in Ref. 22 are ued. Matrix element for 7p 1/2-7 and 7p 3/2-7 tranition an be determined aurately from a reent high-preiion meaurement of the Stark hift 23. Value determined in thi way deribed more ompletely in the following etion are lited intead of experimental data for thee two tranition ine no aurate experimental value are available. We alo lit matrix element from Ref. 24, where experimental and theoretial data were ompiled to provide bet value. A we an ee from the Table VI our ab initio SD alulation provide aurate value for all of the matrix element with the exeption of np-6. For np-6 tranition, omitted higher-order orretion are very large, but an be etimated uing the aling proedure deribed above. Reult from Ref. 21 and 22 were obtained uing imilar aling proedure; however, the relative importane of aling i different in eah ae owing to the different treatment of orrelation orretion. In Ref. 21, aling gave mall 0.2% 0.4% ontribution for all tranition 3, while in Ref. 22 aling led to 5.5% and 4% hange in 7p 1/2-6 and 7p 3/2-6 matrix element and 0.1% 0.7% hange in the other. For our SD alulation, aling hange matrix element for 7p 1/2-6 and 7p 3/2-6 tranition by 6% and 4%, repetively, and reult for all other tranition by 0.5% 1.2%. Our aled matrix element in C are in exellent agreement with other aurate theoretial reult and with experimental value for tranition other than the prinipal tranition. For the prinipal tranition, the preent aled value are in poorer agreement with experiment, ine aling doe not aount for miing fourth- and higher-order random phae approximation RPA term 4 that ontribute ignifiantly in thi ae. TABLE VI. Comparion of SD redued dipole matrix element a.u. for C with other theoretial value and with experiment. Tranition SD Saled Ref. 21 Ref. 22 Expt. Ref. 24 6p 1/ p 3/ p 1/ p 3/ p 1/ p 1/ p 1/ p 3/ p 1/ a p 3/ a a Predition baed on the experimental value of the Stark hift 23.

7 4482 M. S. SAFRONOVA, W. R. JOHNSON, AND A. DEREVIANKO PRA 60 TABLE VII. Comparion of SD redued dipole matrix element a.u. for Fr with other theoretial value and with experiment. SD Saled Ref. 25 Ref. 18 a Ref. 18 b Ref. 20 Expt. 9 7p 1/ p 1/ p 1/ / p 3/ p 3/ p 3/ p 3/ p 1/ p 1/ p 1/ p 1/ p 3/ p 3/ p 3/ p 3/ p 1/ p 1/ p 1/ p 1/ p 3/ p 3/ p 3/ p 3/ a Inlude ontribution from non-bruekner diagram extrapolated from C reult. b Predition given in Ref. 18. For other tranition, aling of the SD reult ubtantially inreae the auray, allowing u to make reaonable predition for the orreponding tranition in Fr where no experimental reult are available and to etimate the auray of Fr polarizability alulation. In Table VII, we ompare our reult for n -np 1/2 and n -np 3/2, matrix element in franium with theoretial alulation from Ref. 18,20,25 and with experiment 9. A for other alkali-metal atom, the preent all-order reult agree better with experiment than the MBPT reult from Ref. 20. The reult from the all-order alulation of 18 are hown in olumn a of Table VII and the predition from 18 are hown in olumn b. Our SD reult for the 8-7p tranition are between the value hown in olumn a and b, while SD data for 8-8p tranition are very loe to value from b. The only tranition for whih there i a large direpany between the SD value and thoe from Ref. 18 i 7-8p 1/2. A previouly noted, there i a large ontribution to thi matrix element from triple exitation that an be etimated by aling. The aled SD matrix element for thi tranition, lited in eond olumn of Table VII, i in muh better agreement with Ref. 18 ; it differ by 5% from the reult a and by 2% from the predition b. Thi tranition i partiularly hard to alulate ine the total orrelation orretion i about 40% about twie a large a for the 7-7p 1/2 tranition, and a more aurate treatment of higher-order orretion i required. Our SD reult for the 7-8p 3/2 tranition agree with Ref. 18 to 1%. Reently, a large number of n p-n matrix element in Fr were evaluated uing a emiempirial model potential method 25. Thee emiempirial value agree with the ab initio SD alulation to better than 1% with the exeption of the 7-8p and 7-9p tranition, where ontribution from orrelation orretion are very large. The aled SD data, whih are more aurate for thee four tranition, are in good agreement with 25. In onluion, the all-order SD method give aurate data for a wide range of n p-n matrix element for all alkali-metal atom with exeption of ome tranition, uh a 7-8p in Fr, whih have mall dipole matrix element and large orrelation orretion. The auray for uh tranition i ignifiantly improved by aling ingle exitation oeffiient. To ahieve higher preiion for eletri-dipole matrix element and to improve the auray of other matrix element in C and Fr, a more omplete treatment of triple exitation i neeary. C. Stati polarizabilitie A mentioned in the Introdution, SD matrix element and energie were ued to alulate tati polarizabilitie, van der Waal oeffiient, and atom-wall interation oeffiient of alkali-metal atom in 13. We diu the alulation of the tati polarizabilitie in more detail here. The polarizabilitie of the ground tate of alkali-metal atom are given by the um of two term, v a, where v i the

8 PRA 60 RELATIVISTIC MANY-BODY CALCULATIONS OF TABLE VIII. Contribution to tati polarizabilitie a.u. of alkali-metal atom and omparion with reommended value from Ref. 13. Na K Rb C Fr v SD Reomm Expt a b b b main v tail v a Referene 28. b Weighted average of experimental data from Ref. 29,30. ontribution from valene exited tate and a i the ontribution from ore exited autoionizing tate. The ontribution of the autoionizing tate an be well approximated by, the polarizability of the ioni ore. We write a v, where v i a ounterterm ompenating for Pauli-priniple-violating exitation from the ore to the valene hell. For an alkali atom in it N ground tate, thee ontribution are given by v 1 Nzn p 1/2 2 Nzn p 2 3/2 3 n E n p 1/2 E N E n p 3/2 E, N ma v 1 3 a azm 2 E m E a, 17 azn 2 E a E N. 18 The expreion for and v above are written in the ingle-partile approximation. The dominant term i the valene ontribution v. Thi term i evaluated by umming over the firt few value of n in Eq. 16 expliitly and approximating the remainder. Thu, v v main v tail. In the term v main, we inluded n p tate with n3 7 for Na, n4 7 for K, n5 8 for Rb, n6 9 for C, and n7 10 for Fr. All matrix element were alulated uing SD wave funtion. Thee tate aount for more than 99% of v ; the mall remainder v tail wa evaluated in the DHF approximation and i expeted to be aurate to better than 15% for Na and 50% for Fr. The ore polarizability whih ontribute le than 10% of the total in all ae wa alulated uing the relativiti RPA. Value of for Na,K,Rb, and C were taken from Ref. 26 and the RPA value for Fr wa obtained in a eparate alulation 13. The reulting value of are expeted to be aurate to better than 5% baed on omparion with reommended value from Miller and Bederon 27. The muh maller valene-ore ontribution v were evaluated uing DHF wave funtion. A breakdown of ontribution to ground-tate polarizabilitie i given in Table VIII, together with a omparion with reommended value from 13 and experiment In thi table and in the paragraph below, value of the polarizabilitie are given in atomi unit (a 3 0 ). The SD reult for Na, K, Rb, and C are in exellent agreement with the value reommended in Ref. 13 whih were obtained uing highpreiion experimental matrix element for the prinipal tranition and experimental energie. In the ae of Fr, the differene i 1%; however, the auray of the reommended value i 0.75%. The differene in Fr i in part due to the lower auray of the SD dipole matrix element for the prinipal tranition ompared to the auray of thee matrix element for other alkali. Stark-indued alar and vetor polarizabilitie S and S for tranition from N to the (N 1) tate were alo alulated. The vetor polarizability S i important for the interpretation of PNC experiment 14. In addition, we evaluated differene between polarizabilitie of the (N 1) tate and the N ground tate. Formula for S and S are given in 21. Ceium i the only alkali-metal atom for whih experimental data are available for all three of thee parameter. The preent alulation provide ueful referene data for the lighter alkali-metal atom and for Fr. Contribution to S and S are lited in Table IX together with omparion with experiment and with emiempirial alulation from Ref. 24. The ore ontribution vanih for the tranition polarizabilitie but the ore-valene ontribution v and v do not. The term tail S, v, tail S, and v were evaluated in the DHF approximation, whih i uffiient ine thee term give mall fration of the total. The data in the row labeled SD S and SD S were obtained uing SD data for energie and matrix element. The SD value for the alar tranition polarizability S in C differ from the experimental value by 1.5%. A we ee from Table IX, S i very mall for Na but inreae rapidly with Z. Our value of for C i in good agreement with the latet experimental value S (43) expt (67) theory from Ref. 31. The vetor polarizability S i epeially diffiult to alulate preiely, ine np 1/2 and np 3/2 term ontribute with oppoite ign. For example, the 6p 1/2 ontribution i and the 6p 3/2 ontribution i A a reult, even mall unertaintie in the value of matrix element an lead to large error. The prinipal unertaintie in S are from 7-6p and 7p-6 matrix element. It hould be noted that it i uffiient to aurately know the ratio of (np 3/2 -n )/(np 1/2 -n ) matrix element to ignifiantly redue the error.

9 4484 M. S. SAFRONOVA, W. R. JOHNSON, AND A. DEREVIANKO PRA 60 main S tail S TABLE IX. Contribution to alar and vetor polarizabilitie S and S a.u. for alkali-metal atom. Na K Rb C Fr v SD S Reomm a a Dzuba et al Expt b main S tail S v SD S a Reomm a Dzuba et al Expt d a Value obtained by uing experimental value of energie and matrix element for the prinipal tranition and aled SD data for the eight other tranition lited in Table VI for C and Table VII for Fr. b Value obtained by ombining the meaurement of S 31 with the aurately meaured ratio / from Ref. 32. Value obtained by uing our reommended value of and the experimental / ratio from Ref. 32. d Referene 31. To etimate the auray of the SD value and to provide reommended value for alar and vetor tranition polarizabilitie in C and Fr we alo alulate S and S uing experimental energie and matrix element for the prinipal tranition and aled SD matrix element for the other tranition lited in Table VI and VII. Thi emiempirial alulation lead to the reommended value in Table IX with the exeption of the value of S in C, whih i lited in a eparate row. The auray of the value of S i alulated auming independent unertaintie in all matrix element, where the unertaintie are baed on omparion with experiment. The main ontribution to the error in S ome from unertaintie in the 7p 3/2-6 matrix element, whih i aurate to 2%. The ontribution of other unertaintie i inignifiant. The reulting value of S i in exellent agreement with the experimental value. To etimate the auray of the vetor tranition polarizability, we alulate S uing our reommended value of S and the high-preiion experimental ratio S / S 9.905(11) 32. The reulting value of S, whih i lited a the reommended value of Table IX, i with the error oming dominantly from the alulation of S. A we ee, thi value i onitent with our diret alulation of S Further improvement in the auray of value of alar and vetor polarizability will be poible when an aurate experimental value of the 7p 3/2-6 matrix element i obtained. Our reommended value of S and S in C are in exellent agreement with value obtained by Dzuba, Flambaum, and Suhkov 24. Unertaintie in the value of S and S in Ref. 24 are lower than the unertaintie of our reommended value owing to the fat that a 0.7% unertainty to the experimental value of 7p 3/2-6 matrix element i aigned in Ref. 24. We alo arried out alulation of S and S in Fr uing both method deribed above. The reult from the row labeled Reomm. are obtained by uing experimental value of energie and prinipal tranition matrix element together with aled SD data from Table VII. The SD reult S SD and S SD agree with our reommended value within 0.8% for S and 1.4% for S. A in the ae of C, the unertainty in the value of S i alulated by auming that the error in all the tranition are independent. The unertaintie are dominated by the unertainty of the 8p 3/2-7 matrix element, whih i 2% baed on omparion with C data. The final unertainty in S for Fr i 1%; the unertainty in S i alo 1% baed on a omparion with C. Table X give the ontribution to, the differene between the tati polarizabilitie of the (N 1) tate and the N ground tate of alkali-metal atom. The SD value SD of the alar tranition polarizability for C differ from the reent experimental reult by 0.6% and agree TABLE X. Contribution to the differene in tati polarizabilitie a.u. of (N 1) and the N ground tate of alkali-metal atom. Na K Rb C Fr main tail v SD Reomm a Expt b a Value obtained uing experimental energie and either experimental or aled SD matrix element. b Referene 23.

10 PRA 60 RELATIVISTIC MANY-BODY CALCULATIONS OF TABLE XI. Comparion of SDpT value of hyperfine ontant A MHz of n, np 1/2, and np 3/2 tate of alkali-metal atom with experiment. Experimental value are from Ref. 33, unle noted otherwie. Na DHF SDpT Expt Na 3p 1/2 4p 1/2 5p 1/2 6p 1/2 DHF SDpT Expt a Na 3p 3/2 4p 3/2 5p 3/2 6p 3/2 DHF SDpT Expt b K DHF SDpT Expt K 4p 1/2 5p 1/2 6p 1/2 7p 1/2 DHF SDpT Expt K 4p 3/2 5p 3/2 6p 3/2 7p 3/2 DHF SDpT Expt Rb DHF SDpT Expt Rb 5p 1/2 6p 1/2 7p 1/2 8p 1/2 DHF SDpT Expt Rb 5p 3/2 6p 3/2 7p 3/2 8p 3/2 DHF SDpT Expt C DHF SDpT Expt C 6p 1/2 7p 1/2 8p 1/2 9p 1/2 DHF SDpT Expt C 6p 3/2 7p 3/2 8p 3/2 9p 3/2 DHF SDpT Expt d a Referene 34. b Referene 35. Referene 36. d Referene 37. TABLE XII. Nuleon number A, nulear pin I, magnetization radii R m fm from Ref. 39, and magneti moment I in unit of N from Ref. 38 ued in the preparation of Table XI. A I R m I Na 23 3/ K 39 3/ Rb 85 5/ C 133 7/ within the error limit with the theoretial reult from Ref. 21. A noted previouly, the experimental value of an be ued to derive 7p-7 matrix element to high auray, ine depend almot entirely on the value of thee matrix element. The value of 7p 1/2-7 and 7p 3/2-7 matrix element were varied to yield experimental value of within experimental preiion. The ratio of thee matrix element, D(7p 3/2-7)/D(7p 1/2-7), i taken to be baed on the theoretial alulation. Experimental data were ued for 6-6p, 6-7p, and 7-6p matrix element and theoretial value were ued for all other. The reult are D(7p 1/2-7) (5) and D(7p 3/2-7) (7) auming unertainty only in the experimental value of. The only other ignifiant unertainty i from the 0.5% error in the value of 6p-7 matrix element whih reult in a 0.1% variation in the value of ). The final reult, aounting for the unertaintie in all matrix element and in the experimental value of, are D(7p 1/2-7) (15) and D(7p 3/2-7) (20). We give a reommended value for in Fr obtained in the ame way a reommended value for S. The unertainty in thi value ome almot entirely from the unertainty in the 8p-8 matrix element, whih i taken to be 0.3%. D. Hyperfine ontant The reult of our alulation of magneti-dipole hyperfine ontant A MHz for n, np 1/2, and np 3/2 tate in Na, K, Rb, and C are given in Table XI together with experimental value from The nulear magneti moment ued in the alulation are weighted average of value taken from the tabulation by Raghavan 38 ; they are lited in Table XII. The alulation inlude orretion for the finite ize of the nulear magneti moment ditribution, whih i modeled a a uniformly magnetized ball. The magnetization radii R m are obtained uing nulear parameter given in Ref. 39 and are alo lited in Table XII. The row labeled DHF in Table XI give reult alulated in the lowet-order DHF approximation. The all-order reult, inluding triple ontribution a deribed in Se. II, are lited in the row labeled SDpT. A tated in the Introdution, the SD method give poor reult for the ground-tate hyperfine ontant in alkali, exept for Na. In fat, the SD reult for the 6 hyperfine ontant in C, without orretion for triple, overetimate the experimental value by 7%, whih i wore than the orreponding third-order MBPT reult. A an be een, the SDpT value are generally in exellent agreement with experiment for n and np 1/2 tate. For the ground tate of C, the agreement with experiment improve to 1% uing SDpT wave funtion. The differene between

11 4486 M. S. SAFRONOVA, W. R. JOHNSON, AND A. DEREVIANKO PRA 60 TABLE XIII. Comparion of SDpT value of Fr hyperfine ontant A MHz with experiment and other theory. g I 0.888, R m 6.71 fm. 7 7p 1/2 7p 3/2 8 8p 1/2 8p 3/2 DHF SDpT Expt a b a Ref a Referene 41. b Referene 42. Value obtained by realing experimental value for 210 Fr MHz from Ref. 19 uing (210) 4.40 N and (211) 4.00 N. The unertainty inlude experimental unertainty of 210 Fr value MHz only. SDpT reult and experiment are greater than 1% for np 3/2 tate of Rb and C. Further improvement of the auray of the hyperfine ontant will require a more omplete treatment of triple. In the alulation deribed above, orretion due to the finite ize FS of the nulear magneti moment ditribution in Na, K, and Rb are very mall and are inluded in zerothorder only. However, FS orretion to hyperfine ontant are ignifiant for C and Fr and are, therefore, inluded to all order. The relative ize of the FS ontribution to the orrelation orretion in n tate in thee ae i found to be the ame a in the lowet-order DHF alulation. Breit orretion to the hyperfine ontant are alulated in eond order following the method outlined in 40. Thee orretion are negligible for Na and K, but grow rapidly from 0.1% for 5 tate of Rb to 0.5% for the 7 tate of Fr. The SDpT value of hyperfine ontant A for the 7, 7p 1/2,7p 3/2,8, 8p 1/2, and 8p 3/2 tate in 211 Fr are given in Table XIII, where omparion are made with experimental 19,41,42 and other theoretial data 43. It hould be noted that FS orretion ontribute 2.5% to the 7 hyperfine ontant. The value of the 7 and 7p 1/2 hyperfine ontant for 211 Fr differ from experimental value by 1.4% and 1.8%, repetively; however, the auray of the magneti moment 4.00(8) N 41 i 2%. It hould be noted that the SDpT reult for the 6 tate of C underetimate the experimental hyperfine ontant by 1% but the SDpT reult for the 7 tate of Fr overetimate the experiment value by 1.4%. The relative ontribution of orrelation for the Fr 7 hyperfine ontant i about the ame a for the C 6 hyperfine ontant. Poible reaon for the anomalou differene with experiment are unertaintie in the Fr magneti moment or magneti moment ditribution; a more preie value of the magneti moment i required to draw onluion about the auray of the orrelation orretion. The value of A for the 7p 3/2 tate in Fr differ from experiment by 3.2%; however, it i lower than the experimental value, unlike value for 7 and 7p 1/2 tate. The main oure of theoretial unertainty for the 7p 3/2 hyperfine ontant i the orrelation orretion, a it i for the C 6p 3/2 hyperfine ontant. Our reult are in good agreement with the theoretial alulation of 43, where the Fr hyperfine ontant were alulated uing MBPT. It hould be noted that our reult inlude a Breit orretion and a more omplete treatment of the orrelation and, thu, are expeted to provide more aurate reult for Fr hyperfine ontant. The dependene of the FS orretion on the value of magnetization radiu wa invetigated in lowet order. Value of A(7) for Fr obtained with magnetization radii R m 6.5 fm and R m 7.0 fm but with the ame harge radiu C nu 6.71 fm differ by 0.2%. The 7 hyperfine ontant alulated with C nu R m 6.5 fm and C nu R m 7.0 fm differ by 0.5% of the total value. E. Conluion We have preented a ytemati tudy of propertie of alkali-metal atom uing relativiti ingle-double wave funtion. Thee wave funtion give aurate value of removal energie, fine-truture interval, eletri-dipole matrix element, and polarizabilitie for alkali-metal atom from Na to Fr. The SD wave funtion, however, lead to hyperfine ontant for heavier alkali-metal atom that differ ubtantially from preie meaurement. To obtain aurate value for hyperfine ontant, it wa neeary to inlude triple partially in the wave funtion. Thi wa done uing the SDpT wave funtion deribed in Se. II and lead to aurate value of hyperfine ontant. Energie and tranition matrix element in Na determined here agree with thoe from the earlier SD alulation of Ref. 4 ; imilarly, the preent energie and matrix element in C are in loe agreement with the SD alulation of Ref. 21. The SD alulation for K, Rb, and Fr preented here are ompletely new. The theoretial SD ground-tate removal energie differ from experiment by amount ranging from 2 m 1 in Na to 114 m 1 in Fr, and the SD removal energie for np tate agree with experimental value to better than 30 m 1 for all tate onidered. The theoretial SD matrix element for prinipal tranition agree with reent high-preiion experiment to 0.1% 0.5%, with the exeption of the 7-7p 3/2 tranition in Fr where the differene i 0.8%. The agreement with experiment i better for lighter ytem beaue of the maller ize of the orrelation orretion. A large number of matrix element, whih were ued to alulate polarizabilitie, are tabulated for all alkali-metal atom; thee matrix element hould provide ueful referene data. The SD approximation give exellent reult for tati polarizabilitie and for Stark-indued tranition polarizabilitie. Supplementing our theoretial alulation with experimental energie and experimental matrix element for the two prinipal tranition allowed u to predit value of the Stark polarizabil-

12 PRA 60 RELATIVISTIC MANY-BODY CALCULATIONS OF itie S and S for C and Fr to high auray. The predited value for S and S in C are in exellent agreement with experimental value. Hyperfine ontant, alulated uing SDpT wave funtion, are in exellent agreement with experiment for n and np 1/2 tate of alkali-metal atom from Na to C. The differene between theoretial SDpT groundtate hyperfine ontant and experiment range from 0.3% in Na to 1.4% in Fr. The ontribution of Breit and FS orretion to the ground tate hyperfine ontant in Fr are found to be ignifiant. A more preie experimental value for the Fr nulear magneti moment i neeary to evaluate the auray of the orrelation orretion to Fr hyperfine ontant. The method developed in thi work will be ued in the future to evaluate PNC amplitude in C and Fr. ACKNOWLEDGMENT Thi work wa upported in part by National Siene Foundation Grant No. PHY and PHY S.A. Blundell, W.R. Johnon, Z.W. Liu, and J. Sapirtein, Phy. Rev. A 40, Z.W. Liu, Ph.D. thei, Notre Dame Univerity, S.A. Blundell, W.R. Johnon, and J. Sapirtein, Phy. Rev. A 43, M.S. Safronova, A. Derevianko, and W.R. Johnon, Phy. Rev. A 58, E. Eliav, U. Kaldor, and Y. Ihikawa, Phy. Rev. A 50, K.M. Jone, P.S. Julienne, P.D. Lett, W.D. Phillip, E. Tieinga, and C.J. William, Europhy. Lett. 35, U. Volz and H. Shmoranzer, Phy. Sr. T65, W. Wang P.L. Gould, and W.C. Stwalley, J. Chem. Phy. 106, J.E. Simarian, L.A. Orozo, G.D. Sproue, and W.Z. Zhao, Phy. Rev. A 57, R.J. Rafa, C.E. Tanner, A.E. Livington, K.W. Kukla, H.G. Berry, and C.A. Kurtz, Phy. Rev. A 50, R L. Young, W.T. Hill III, S.J. Sibener, S.D. Prie, C.E. Tanner, C.E. Wieman, and S.R. Leone, Phy. Rev. A 50, R.J. Rafa, C.E. Tanner, A.E. Livington, and H.G. Berry, Phy. Rev. A to be publihed. 13 A. Derevianko, W.R. Johnon, M.S. Safronova, and J.F. Babb, Phy. Rev. Lett. 82, C.S. Wood, S.C. Bennett, D. Cho, B.P. Materon, J.L. Robert, C.E. Tanner, and C.E. Wieman, Siene 275, G.E. Brown and D.G. Ravenhall, Pro. R. So. London, Ser. A 208, W.R. Johnon, S.A. Blundell, and J. Sapirtein, Phy. Rev. A 37, C.E. Moore, Atomi Energy Level, Natl. Bur. Stand. Ref. Data Ser., Natl. Bur. Stand. U.S. Cir. No. 35 U.S. GPO, Wahington, DC, 1971, Vol. I III. 18 V.A. Dzuba, V.V. Flambaum, and O.P. Suhkov, Phy. Rev. A 51, J.E. Simarian, W. Shi, L.A. Orozo, G.D. Sproue, and W.Z. Zhao, Opt. Lett. 21, ; J.E. Simarian, W.Z. Zhao, L.A. Orozo, and G.D. Sproue, Phy. Rev. A 59, W.R. Johnon, Z.W. Liu, and J. Sapirtein, At. Data Nul. Data Table 64, ; J. Sapirtein, Rev. Mod. Phy. 70, S.A. Blundell, J. Sapirtein, and W.R. Johnon, Phy. Rev. D 45, V.A. Dzuba, V.V. Flambaum, and O.P. Suhkov, Phy. Lett. A 141, S.C. Bennett, J.L. Robert, and C.E. Wieman, Phy. Rev. A 59, R V.A. Dzuba, V.V. Flambaum, and O. P. Suhkov, Phy. Rev. A 56, R M. Marineu, D. Vrineanu, and H.R. Sadeghpour, Phy. Rev. A 58, R W.R. Johnon, D. Kolb, and K.-N. Huang, At. Data Nul. Data Table 28, T.M. Miller and B. Bederon, Adv. At. Mol. Phy. 13, C.R. Ektröm, J. Shmiedmayer, M.S. Chapman, T.D. Hammond, and D.E. Prithard, Phy. Rev. A 51, R.W. Molof, H.L. Shwartz, T.M. Miller, B. Bederon, Phy. Rev. A 10, W.D. Hall and J.C. Zorn, Phy. Rev. A 10, S.C. Bennett and C.E. Wieman, Phy. Rev. Lett. 82, D. Cho, C.S. Wood, S.C. Bennett, J.L. Robert, and C.E. Wieman, Phy. Rev. A 55, W. Happer, in Atomi Phyi 4, edited by G. zu Putlitz, E.W. Weber, and A. Winnaker Plenum Pre, New York, 1974, pp W.A. Wijngaarden and J. Li, Z. Phy. D 32, W. Yei, A. Sieradzan, and M.D. Havey, Phy. Rev. A 48, R.J. Rafa and C.E. Tanner, Phy. Rev. A 56, C.E. Tanner and C. Wieman, Phy. Rev. A 38, P. Raghavan, At. Data Nul. Data Table 42, W.R. Johnon and G. Soff, At. Data Nul. Data Table 33, I.M. Savukov, A. Derevianko, H.G. Berry, and W.R. Johnon, Phy. Rev. Lett. 83, C. Ektröm, L. Roberton, A. Roén, and the ISOLDE Collaboration, Phy. Sr. 34, J.S. Groman, L.A. Orozo, M.R. Pearon, J.E. Simarian, G.D. Sproue, and W.Z. Zhao, Phy. Rev. Lett. 83, V.A. Dzuba, V.V. Flambaum, and O.P. Suhkov, J. Phy. B 17,