Isotope shift of the 3 2S1/2−2 2S1/2 transition in lithium and the nuclear polarizability

Iso ope Shi o he 32S1=2-22S1=2T ansi ion in Li hium and he Nuclea Pola izabili y M. Puchalski, 1 A. M. Mo o, 2 and K. Pachucki 1 1 Ins i u e o Theo e ical Physics, Wa saw Uni e si y, Hoz ˙a 69, 00-681 Wa saw, Poland 2 Depa amen o de FAMN, Uni e sidad de Se illa, Apa ado 1065, 41080 Se illa, Spain (Recei ed 29 July 2006; published 25 Sep embe 2006) High p ecision calcula ion o he iso ope shi o he 32S1=2-22S1=2 ansi ion in li hium is p esen ed. The wa e unc ion and ma ix elemen s o ela i is ic ope a o s a e ob ained by using ecu sion ela ions. Apa om he ela i is ic con ibu ion, we ob ain he nuclea pola izabili y co ec ion o 11Li. The esul ing di e ence o he squa ed cha ge adii 11Li-7Li based on he measu emen s o Sa ´nchez e al. [Phys. Re . Le . 96, 033002 (2006)] is  2 ch 0:15781 m2, which signi ican ly di e s om he p e ious e alua ion. DOI: 10.1103/PhysRe Le .97.133001 PACS numbe s: 31.30.J , 21.10.F The ecen ad ances in p ecise spec oscopy o a omic sys ems make possible he de e mina ion o nuclea cha ge adii om iso ope shi measu emen s [1–4]. In spi e o he ac ha he nuclea size is 5 o de s o magni ude smalle han he a omic size, he achie ed expe imen al p ecision o ansi ion equencies allows one o de e mine nuclea cha ge adii much mo e accu a ely han om elec on sca e ing measu emen s. This, howe e , equi es he heo- e ical calcula ions o be pe o med wi h high ela i e p ecision, o example, a leas 106 o li hium iso opes. The accu acy o heo e ical p edic ions achie ed o hyd o- gen [5] leads o he de e mina ion o he p o on cha ge adius [5,6], which is a mo e accu a e han om he elec on sca e ing measu emen s. A simila expe imen al accu acy achie ed o deu e ium allowed one o de e mine e y accu a ely he deu e on adius [1]. Su p isingly, he a omic measu emen s lead o a sligh ly di e en alue om he elec on sca e ing de e mina ion, which s imu- la ed a eanalysis o elec on sca e ing da a. Wha makes he deu e on di e en om o he ypical nuclei is i s low binding ene gy o abou 2.226 MeV. Such a so nucleus is dis o ed by a su ounding elec on, which leads o a shi in a omic ansi ion equencies. E en smalle is he wo- neu on sepa a ion ene gy in 11Li [7], which indica es he possible signi icance o nuclea s uc u e e ec s on he iso ope shi . In his Le e , we p esen signi ican ly imp o ed calcu- la ions o ini e nuclea mass con ibu ions o he iso ope shi in he li hium 32S1=2-22S1=2 ansi ion. Such calcu- la ions ha e al eady been pe o med by Yan and D ake in Re s. [8–10] and we e used o de e mine nuclea cha ge adii o a ious li hium iso opes on he basis o ecen iso ope shi measu emen s [3,4]. Ou esul o he ela- i is ic ecoil co ec ion is abou 10 imes smalle han ha epo ed in Re s. [8,9]. Apa om he known non ela i - is ic, leading ela i is ic, and QED co ec ions, we include highe o de ecoil co ec ions and he nuclea pola izabil- i y e ec Epol, he las being signi ican o he 11Li nu- cleus. Finally, we use ou combined esul s o calcula e new nuclea cha ge adii o he li hium iso opes, aking he expe imen al iso ope shi esul s om Re s. [3,4]. Le us deno e =M. The expansion o an ene gy le el in he ine s uc u e cons an is E; m2E2m4E4m5E5 m6E6O7Epol E s;(1) whe e E s is he nuclea ini e size co ec ion, and m,M, a e he elec on, nucleus, and he educed mass, espec- i ely. In he ollowing, we ob ain hese expansion coe - icien s. The non ela i is ic Hamil onian o he li hium a om in a omic uni s is HH1HM;(2) H1X ap2 a 2Z aX a>b 1 ab ;(3) HMX a p2 a 2MX a>b ~ pa~ pb M:(4) We sol e he Sch o ¨dinge equa ion o H1and ea HM pe u ba i ely. Following Re . [11], he Sch o ¨dinge wa e unc ion is ep esen ed in he Hylle aas basis se  n1 23 n2 31 n3 12 n4 1 n5 2 n6 3ew1 1w2 2w3 3;(5) wi h all combina ions o ni, such ha P6 i1ni and 3;...;12. De ails can be ound in Re . [12]. The ma ix elemen s o he non ela i is ic Hamil onian can all be exp essed by a Hylle aas in eg al n1;n 2;n 3;n 4;n 5;n 6Zd3 1 4Zd3 2 4Zd3 3 4ew1 1w2 2w3 3 n11 23 n21 31 n31 12 n41 1 n51 2 n61 3;(6) wi h in ege alues o ni. We calcula e analy ically o alues o ni0;1in sex uple p ecision and use ecu sion PRL 97, 133001 (2006) PHYSICAL REVIEW LETTERS week ending 29 SEPTEMBER 2006 0031-9007=06=97(13)=133001(4) 133001-1 ©2006 The Ame ican Physical Socie y ela ions o ni>1[13]. Resul s o non ela i is ic ene - gies ob ained wi h he la ges numbe o basis unc ions, namely, 9576, a e so a he lowes ones in he li e a u e E22S1=27:478 060 323 890;(7) E32S1=27:354 098 421 380;(8) and ini e mass co ec ions a e p esen ed in Table I. The high quali y o he non ela i is ic wa e unc ion allows one o ob ain accu a e esul s o ma ix elemen s, o example, X a 3 a3S X a 3 a2S 0:106 108 0:(9) Rela i is ic co ec ions a e ob ained om he B ei - Pauli Hamil onian, which includes he e ini e mass co ec- ions [8] E4X a~ p4 a 8Z 23 am M Z 2pi aij a  i a j a 3 aX b pj bM X a>b3 ab1 2pi aij ab  i ab j ab 3 ab pj bM ;(10) whe e h...iMdeno es a ma ix elemen , calcula ed wi h he wa e unc ion ha includes he ini e nuclea mass. Ma ix elemen s o E4in ol e ex ended Hylle aas in eg als, whe e one nibecomes nega i e. High p ecision calcula- ions o Hylle aas in eg als wi h in e se powe s o aand ab a e qui e di icul and a e one o he p incipal achie e- men s o his wo k. We use he in eg a ion by pa s iden- i ies o exp ess 1;0;0; 0;0;0in e ms o one- dimensional in eg als which a e pe o med wi h 48 digi accu acy, and 1;n 2;n 3;n4;n 5;n 6a e ob ained by e- cu sion ela ions [14]. Simila ly, 0;0;0;1;n 5;n 6a e exp essed as a one-dimensional in eg al which is pe - o med nume ically, and n1;n 2;n 3;1;n 5;n 6a e ob- ained by ecu sion ela ions [12]. Resul s o ela i is ic ecoil co ec ions a e p esen ed in Table I. We no e a s ong cancella ion be ween educed mass, mass pola iza ion, and he di ec elec on-nucleus B ei in e ac ion. Ou esul o 0.038 MHz is mo e han 10 imes smalle han he p e ious one in Re s. [8,9], bu i is in ag eemen wi h he ecen ecalcula ion by Yan and D ake [15], which gi es 0.024(19) MHz. Leading QED ecoil co ec ions E5ha e been s udied in Re . [16] and calcula ed o li hium in Re s. [9,10]. They can be ep esen ed (in a omic uni s) as: E519 30 2lnlnk0M4Z 3X a 3 aM 164 15 14 3lnX a>b 3 abM 7 6X a>b 1 3 abM m M2 3ln62 98 3lnk0X a 3 a 7Z2 6X a 1 3 aOm M2;(11) whe e lnk0is he Be he loga i hm and 1= 3has implici sub ac ion o ln", wi h "being he cu o and  he Eule cons an . 1= 3 e ms and lnk0including mass pola - iza ion co ec ions ha e been calcula ed by Yan and D ake in Re s. [9,10], and we use hei esul s he e. The comple e m5con ibu ion is p esen ed in Table I. I is sligh ly di e en om ha in Re s. [9,10], due o including he addi ional e m om educed mass scaling o 1= 3 ab in Eq. (11). Conside ing highe o de ecoil co ec ions m2=M6, hey a e known exac ly only o hyd ogenic sys ems [17– 19]. Some o hem a e known o be p opo ional o he Di ac del a unc ion, such as he adia i e o adia i e ecoil, so he ex ension o li hium is simple. Pu e ecoil co ec ions a e mo e complica ed. The e a e s a e depen- den e ms which a e known o hyd ogen [17,19], bu he ex ension o mo e han one-elec on a oms has no ye been achie ed. We neglec hem he e and associa e oughly 25% unce ain y. This co ec ion is al eady qui e small, so he app oxima e ea men is su icien . The o mula we use o E6is E6 m3Z2427 96 2 ln2 m MZ235 36 448 2722 ln2 63 2 m MZ34 ln2 7 2X a 3 a;(12) and nume ical esul s using Eq. (9) a e p esen ed in Table I, in ag eemen wi h he es ima ion om Re . [9]. Apa om ela i is ic and QED e ec s, he e is a ini e nuclea size and nuclea pola izabili y which con ibu e o he iso ope shi . Since one uses he iso ope shi measu e- men o de e mine nuclea cha ge adius, one shall es ima e he e ec o nuclea pola izabili y, which we ind he e o be signi ican o 11Li. Fo his, we assume ha he in e ac ion o he nucleus wi h he elec omagne ic ield can be de- sc ibed as ollows: Hin qA0~ d~ E~ ~ Bq 6h 2i 2A0:(13) TABLE I. Con ibu ions o he Li iso ope shi =M. Con ibu ion 32S1=2-22S1=2 11Li-7Li (MHz) m20.133 764 842 25 104.489(21) m220.123 659 8 2:968 m40.003 78(3) 0.038 m51:43 0:104 m638:100:0205 Epol 0.039(4) PRL 97, 133001 (2006) PHYSICAL REVIEW LETTERS week ending 29 SEPTEMBER 2006 133001-2 The domina ing nuclea exci a ions a e E1 ansi ions by he elec ic dipole coupling ~ d~ E[20]. The ene gy shi due o he wo-pho on exchange in he empo al gauge is Epol ie2 20Zd! 2Zd3k 23!2ik kikk !2 !2k2 jl kjkl !2 !2k2 T j1 p6 k6 mii1 p6 k6 mj0I 4 hNjdk1 ENHN!dljNi;(14) whe e 20m3hPa3 ai,pm; ~ 0, and we used plane wa e app oxima ion o he elec ons, since he cha ac e is ic pho on momen um kis much la ge han he in e se Boh adius. A e pe o ming kin eg a ion and eplacing !iw, one ob ains Epol m4X a 3 am3~pol;(15) whe e ~pol is a kind o elec ic pola izabili y o he nu- cleus, which is gi en by he ollowing double in eg al: ~pol 16 3Z1 ET dE 1 e2jhNj~ djEij2 Z1 0 dw w E E2w2 1 ?11 1?1 1 11 ?1;(16) whe e  12im=w p, and Eis he exci a ion ene gy o he nucleus wi h espec o he g ound s a e wi h h eshold alue ET. This gene al o mula ag ees in he limi Em wi h ha de i ed p e iously o he nuclea pola izabili y e ec in deu e ium [21]. ~pol in ol es a squa e o he ansi ion dipole momen . This can be ela ed o he BE1dis ibu ion which was ecen ly accu a ely mea- su ed o 11Li in Re . [20] jhj~ djEij24 3 dBE1 dE (17) in uni s e2 m2MeV, which explains he p esence o e2in he denomina o in Eq. (16). Wi h he new da a om Re . [20] (see Fig. 1) and he wo-neu on sepa a ion ene gy ET0:3765MeV [7], one ob ains ~pol 60:96:1 m31:060:11106m3(18) and a pola izabili y co ec ion o he 32S1=2-22S1=2 an- si ion equency o pol 394kHz. Pola izabili y co - ec ion o 9Li and ligh e iso opes is expec ed o be a leas 10 imes smalle and is, hus, negligible. The las signi ican con ibu ion o he iso ope shi is due o he ini e size o he nucleus E s 2 3Z4m3h 2 chiX a 3 a:(19) This nuclea olume e ec can now be ex ac ed om he iso ope shi measu emen s, o ob ain nuclea cha ge adii. We addi ionally accoun o he leading ela i is ic co ec- ion 201Z2lnm chZ o he squa e o he wa e unc ion a he o igin. Resul s a e summa ized in Table II. All esul s o cha ge adii di e ences a e signi i- can ly imp o ed compa ed o p e ious de e mina ions; o example, ou esul o he di e ence 11Li-7Li, 2 ch  0:15781 m2, can be compa ed o  2 ch 0:374112 m2 which is ob ained using esul s p esen ed in Re . [4]. The 0123 E (MeV) 0 1 2 dB/dE (e2 m2/MeV) Nakamu a i FIG. 1 (colo online). Elec ic dipole line s eng h by Nakamu a e al. [20] adap ed o he new alue o ET om Re . [7]. TABLE II. Summa y o iso ope shi de e mina ion o Li cha ge adii, ch7Li2:39030 m [22], m7Li7:016 003 425 645u [23]; he i s unce ain y o  he comes om unknown highe o de e ms, he second unce ain y is due o he a omic mass. Mass (u)[7,24]exp (MHz) [3,4] he (MHz)  2 ch ( m2) ch ( m) 6Li 6.015 122 794(16) 11 453:9832011 452:822200.738(13) 2.540(30) 8Li 8.022 487 36(10) 8635.782(44) 8634.990(1)(1) 0:503282.282(32) 9Li 9.026 789 5(21) 15 333.272(39) 15 331.797(3)(13) 0:938262.185(33) 11Li 11.043 715 7(54) 25 101.226(125) 25 101.473(9)(21) 0.157(81) 2.423(34) PRL 97, 133001 (2006) PHYSICAL REVIEW LETTERS week ending 29 SEPTEMBER 2006 133001-3 esul o he di e ence be ween 11Li and 9Li, 2 ch  1:098 m2, does no ag ee wi h ha ob ained on he basis o BE1da a, namely,  24 3 1 Ze2Z1 ET dE dBE1 dE 0:74 m2:(20)  2he e is he squa e o a dis ance o 9Li co e om he mass cen e . F om his, one may conclude ha he 9Li co e is signi ican ly pe u bed by he p esence o he wo a- lence neu ons in 11Li. A simila conclusion has al eady been d awn in Re . [4]. The di e ence in squa ed cha ge adii be ween 6Li and 7Li can also be ob ained om he iso ope shi measu e- men o 23S-23PJ ansi ions in Liby Riis e al. [25]. Taking he weigh ed a e age alue and heo e ical iso ope shi om Re . [26], one ob ains  2 ch 0:70260 m2,in mode a e ag eemen wi h he esul ob ained he e,  2 ch  0:74213 m2. The unce ain y o nume ical calcula ions o li hium- like a oms is negligible in compa ison o he es ima ion o unknown highe o de e ms. The pu e ecoil co ec ion o o de 6m2=M gi es he la ges heo e ical unce ain y o a ew kilohe z o li hium iso ope shi s and will become mo e signi ican o hea ie li hiumlike ions. A p esen , howe e , he domina ing sou ce o unce ain ies comes om measu emen o ansi ion equencies and om he cha ge adius o he e e ence nucleus; see Table II. Rega ding he di ec cha ge adius de e mina ion o he e e ence nucleus 7Li, calcula ions o ansi ion e- quencies a e a mo e di icul and ha e been pe o med wi h su icien p ecision only o hyd ogenic sys ems. The e o e, he spec oscopic de e mina ion o he abso- lu e cha ge adius can possibly be achie ed by measu e- men o wo-pho on ansi ions in he hyd ogenlike li hium ion o 2S-2P ansi ions in he muonic li hium. In summa y, we ha e ob ained high p ecision ini e nuclea mass co ec ions o li hium iso ope shi s and nuclea pola izabili y ( o 11Li). The esul ing nuclea cha ge adii a e signi ican ly imp o ed o e he o me de- e mina ions. Ou app oach is based on an analy ical cal- cula ion o ma ix elemen s wi h Hylle aas wa e unc ions, which can easily be applied o li hiumlike ions and, we hink, can be ex ended o 4-elec on sys ems as well. We a e g a e ul o T. Nakamu a o sending us he BE1 da a. K. P. acknowledges in e es ing discussions wi h W. No ¨ e sha ¨use and K. Rusek. A. M. 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