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
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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
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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. M. acknowledges
inancial suppo by he Jun a de Andalucı
´a.
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