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Cha ge adii o hallium iso opes nea he =126 shell closu e
© Published by he Ame ican Physical Socie y
Published e sion
IDS Collabo a ion
IDS Collabo a ion. (2024). Cha ge adii o hallium iso opes nea he =126 shell closu e. Physical
Re iew C, 110(3), A icle 034315. h ps://doi.o g/10.1103/phys e c.110.034315
2024
PHYSICAL REVIEW C 110, 034315 (2024)
Cha ge adii o hallium iso opes nea he N=126 shell closu e
Z. Yue ,1,*A. E. Ba zakh ,†A. N. And eye ,1,2,‡I. N. Bo zo ,§J. G. Cubiss ,1A. Algo a,3M. Au ,2,4M. Balogh,5
S. Ba a,6R. A. Ba k,7C. Be ne d,2,6M. J. G. Bo ge ,8D. B ugna a ,5K. Ch ysalidis,2,4T. E. Cocolios,6H. De Wi e,6
Z. Fa ie ,2L. M. F aile ,9H. O. U. Fynbo,10 A. Go a do,5R. G zywacz,11 R. Heinke,2A. Illana ,9,12,13 P. M. Jones,7
D. S. Judson,14 A. Ko gul,15 U. Kös e ,2,16 M. Labiche,17 L. Le,2R. Liˇ
ca,2,18 M. Madu ga,11 N. Ma ginean,18 B. A. Ma sh,2
C. Mihai,18 E. Náche ,3C. Neacsu,18 C. Ni a,18 B. Olaizola ,2,8J. N. O ce,19 C. A. A. Page ,1R. D. Page ,14
J. Paka inen,2,12,13 P. Papadakis,17 A. Pe ea,8M. Pie sa-Siłkowska,20 Zs. Podolyák ,2,21 E. Reis,2,22 S. Ro he,2M. Sedlak,5
C. So y,18 S. S egemann,2O. Tengblad,8S. V. Tolokonniko ,§J. M. Udías,9P. Van Duppen,6N. Wa ,23 and W. Woj aczka6
(IDS Collabo a ion)
1School o Physics, Enginee ing and Technology, Uni e si y o Yo k, Yo k YO10 5DD, Uni ed Kingdom
2CERN, 1211 Gene a 23, Swi ze land
3Ins i u o de Fisica Co puscula , CSIC-Uni e sidad de Valencia, E-46071 Valencia, Spain
4Johannes Gu enbe g-Uni e si ä , Saa s . 21, 55099 Mainz, Ge many
5INFN, Labo a o i Nazionali di Legna o (LNL), Viale dell’Uni e si à 2, 35020 Legna o (PD), I aly
6Ins i uu oo Ke n- en S alings ysica, KU Leu en, B-3001 Leu en, Belgium
7iThemba LABS, Na ional Resea ch Founda ion, P.O. Box 722, Some se Wes 7129, Sou h A ica
8Ins i u o de Es uc u a de la Ma e ia, CSIC, 28006 Mad id, Spain
9G upo de Física Nuclea and IPARCOS, Uni e sidad Complu ense de Mad id, CEI Moncloa, E-28040 Mad id, Spain
10Depa men o Physics and As onomy, Aa hus Uni e si y, DK-8000 Aa hus C, Denma k
11Depa men o Physics and As onomy, Uni e si y o Tennessee, Knox ille, Tennessee 37966, USA
12Depa men o Physics, Uni e si y o Jy äskylä, P.O. Box 35, FI-40014 Jy äskylä, Finland
13Helsinki Ins i u e o Physics, Uni e si y o Helsinki, P.O. Box 64, FIN-00014 Helsinki, Finland
14Oli e Lodge Labo a o y, Uni e si y o Li e pool, Li e pool L69 7ZE, Uni ed Kingdom
15Ins i u e o Expe imen al Physics, Wa saw Uni e si y, Wa szawa PL 00-681, Poland
16Ins i u Laue-Lange in, F-38042 G enoble, F ance
17STFC Da esbu y Labo a o y, Da esbu y, WA4 4AD Wa ing on, Uni ed Kingdom
18Ho ia Hulubei Na ional Ins i u e o Physics and Nuclea Enginee ing (IFIN-HH), R-077125 Bucha es , Romania
19Depa men o Physics, Uni e si y o he Wes e n Cape, P/B X17 Bell ille 7535, Sou h A ica
20Ins i u e o Expe imen al Physics, Wa saw Uni e si y, Wa saw PL 02-093, Poland
21Depa men o Physics, Uni e si y o Su ey, Guild o d GU2 7XH, Uni ed Kingdom
22Uni e si ä Duisbu g-Essen, 45141 Duisbu g, Ge many
23Ins i u ü Ke nphysik, Uni e si ä zu Köln, D-50937 Köln, Ge many
(Recei ed 20 July 2024; accep ed 27 Augus 2024; published 13 Sep embe 2024)
The changes in he mean-squa ed cha ge adius o 209Tlg(N=128) and 207Tlm(N=126) ela i e o 205Tl
ha e been measu ed o he i s ime using he in-sou ce lase esonance-ioniza ion spec oscopy echnique
wi h he Lase Ion Sou ce and T ap (LIST) a ISOLDE (CERN). The applica ion o he LIST supp esses he
dominan backg ound om isoba ic ancium iso opes and allows access o hallium nuclides wi h A⩾207.
The cha ac e is ic kink in he cha ge adii a he N=126 neu on shell closu e, as well as he odd-e en e ec
simila o ha in he adjacen bismu h, lead, and me cu y iso opic chains, ha e been obse ed. The sel -consis en
heo y o ini e Fe mi sys ems based on he ene gy densi y unc ional by Fayans e al. ep oduces he beha io
o cha ge adii in hese iso opic chains nea N=126. The compa ison wi h calcula ions in he amewo k o he
ela i is ic mean ield (RMF) app oach is also p esen ed. In he case o he Fayans unc ional i is a speci ic o m
o pai ing in e ac ion wi h he dependence on he densi y g adien ha is essen ial o p o ide ag eemen wi h he
*Con ac au ho : [email p o ec ed]
†Con ac au ho : [email p o ec ed]; A ilia ed wi h an ins i u e co e ed by a coope a ion ag eemen wi h CERN a he ime o he
expe imen .
‡Con ac au ho : and ei.and eye[email p o ec ed]
§A ilia ed wi h an ins i u e co e ed by a coope a ion ag eemen wi h CERN a he ime o he expe imen .
Published by he Ame ican Physical Socie y unde he e ms o he C ea i e Commons A ibu ion 4.0 In e na ional license. Fu he dis ibu ion
o his wo k mus main ain a ibu ion o he au ho (s) and he published a icle’s i le, jou nal ci a ion, and DOI. Open access publica ion
unded by CERN.
2469-9985/2024/110(3)/034315(9) 034315-1 Published by he Ame ican Physical Socie y
Z. YUE e al. PHYSICAL REVIEW C 110, 034315 (2024)
expe imen al cha ge adii. In pa icula , he kink is ep oduced wi hou he in e sion o g9/2and i11/2neu on
single-pa icle s a es, which is a p e equisi e o co ec ly desc ibe he kink in he RMF models.
DOI: 10.1103/PhysRe C.110.034315
I. INTRODUCTION
I is well es ablished ha he slope o he change in
he mean-squa ed cha ge adii (δ 2) along iso opic chains
inc eases ab up ly when c ossing he neu on magic num-
be s ( he so-called “kink in cha ge adii” o “shell e ec in
adii” [1,2]). This e ec p o ed o ha e an uni e sal cha ac e
being obse ed nea almos all neu on magic numbe s includ-
ing N=126 [1,2].
Ano he uni e sal end in he cha ge- adii beha io is he
odd-e en s agge ing (OES): in mos cases, nuclea cha ge
adii o odd-Niso opes a e smalle han he a e age o hei
e en-Nneighbo s [1,2]. A ew excep ions a e usually ex-
plained by he in luence o he s a ic (quad upole o oc upole)
de o ma ion (see Re s. [3,4] and e e ences he ein).
In he icini y o he N=126 and Z=82 shell closu es,
well-documen ed e idence o he kink exis s o he me cu y
(Z=80) [5,6], lead (Z=82) [7], and bismu h (Z=83) [8]
iso opic chains. This egion is especially sui able o compa e
o di e en heo e ical app oaches o desc ibing he kink and
OES, since he de o ma ion e ec s which could dis o hese
δ 2 ends, a e expec ed o be small. This is in con as
wi h, e.g., he si ua ion nea N=82 whe e apid g ow h o
de o ma ion in he close icini y o he magic numbe was
obse ed (see, o example, Fig. 4 in Re . [9] and expe imen al
da a compila ion in Re . [10]).
Co espondingly, he i s aim o he p esen wo k is o
ill he Z=81 gap in he δ 2sys ema ics nea N=126 by
measu ing δ 2 o hallium iso opes (Z=81) wi h N>127.
These alues a e de i ed om he iso ope shi s (IS) o he
a omic op ical lines.
P e iously, IS and hype ine s uc u e (h s) in neu on-
de icien hallium iso opes down o 179Tl we e s udied [11–13]
using he esonance ioniza ion lase ion sou ce echnique [14].
Iso opes wi h N>127 emained inaccessible due o he o e -
whelming su ace-ionized isoba ic ancium con amina ion.
The applica ion o he esonance-ioniza ion spec oscopy in
he Lase Ion Sou ce and T ap (LIST) [15–17] in he p esen
wo k supp esses his backg ound and p o ides access o hea y
hallium iso opes.
The de e mina ion o he δ 2 alues o hallium iso opes
wi h N>126 also p obes he possible ole ha low- jp o on
o bi als could play in p oducing he kink. Indeed, o he bis-
mu h iso opic chain abo e Z=82, i is he high- jh
9/2p o on
o bi al ha domina es he s uc u e o he g ound s a e, while
he low- js
1/2o bi al is ac i e in g ound s a es o hallium
iso opes.
His o ically, conside able heo e ical e o s ha e been di-
ec ed o he desc ip ion o he kink in he lead chain a ound
N=126. I was ound ha s anda d non ela i is ic Ha ee-
Fock (NRHF) calcula ions wi h Sky me o Gogny o ces we e
no able o ep oduce he kink [18–20]. Fu he nume ous
a emp s o ake in o accoun g ound-s a e co ela ions using
beyond mean- ield app oaches ( andom phase app oxima ion,
gene a o -coo dina e me hod, sel -consis en sum ule ap-
p oach) ail o ep oduce he obse ed i egula i y (see
Re s. [9,19,21,22] and e e ences he ein).
The OES also became a benchma k o es he heo e ical
models. As in he case o he kink, he i s heo e ical a emp s
ied o accoun o a educ ion o quad upole ib a ions in
odd-Nnuclei due o blocking, which can be ega ded as a kind
o beyond mean- ield app oach. Howe e , expe imen al da a
we e only quali a i ely explained wi hin he co esponding
pai ing-plus-quad upole model o sphe ical nuclei [23].
The ailu e o he beyond Ha ee-Fock a emp s o ex-
plain he kink and OES, indica es ha a co ec desc ip ion
o IS (a leas in he NRHF calcula ions) is no de e -
mined by he highe -o de co ela ions and a solu ion should
be sea ched o h ough he modi ica ion o he e ec i e
o ces hemsel es [19]. In pa icula , i was shown ha he
expe imen al kink and OES can be ep oduced in he ame-
wo k o he Sky me-Ha ee-Fock (SHF) app oach wi h a
densi y-dependen pa ame iza ion o he pai ing pa o he
in e ac ion [19,24] o wi h he modi ica ion o he spin-o bi
in e ac ion, see Re s. [20,25–29]. Howe e , i was shown ha
i emains di icul o ep oduce bo h he masses and he kink
accu a ely wi hin one SHF amewo k wi h he gene aliza ion
o he spin-o bi in e ac ion [30]. In he case o he pai ing
modi ica ion i also emains unclea whe he he modi ied
unc ionals p o ide an accu a e global desc ip ion o nuclea
masses.
A he same ime, bo h kink and OES a e well ep o-
duced in he amewo k o he ela i is ic mean ield (RMF)
app oach wi hou in oducing new unc ional membe s o ad-
jus ing pa ame e s ([18,31], see also comp ehensi e e iews
which co e s nea ly he whole nuclide cha [32,33]). How-
e e , his success is achie ed a he expense o he in e sion
o he νi11/2and νg9/2neu on single-pa icle s a es posi ion
in con adic ion wi h expe imen [5,6,32].
In con as wi h he heo e ical app oaches discussed
abo e, in he sel -consis en ini e Fe mi sys ems heo y
(TFFS) [34] based on he gene alized ene gy densi y unc-
ional (EDF) by Fayans e al. [21,35–38], nuclea adii in
he icini y o he neu on shell closu es a e di ec ly in-
cluded in he op imiza ion p o ocol. Two ecen e sions o he
Fayans- ype unc ional (DF3-a [37] and Fy( ,HFB) [38])
success ully desc ibe he shell e ec in adii and OES o
K[39,40], Ca [38,41], Cu [42,43], Sn [44], Ca-Zn [45], and
some o he iso opic chains. In con as wi h he RMF calcu-
la ion, in he TFFS-Fayans amewo k he kink is ep oduced
wi hou he in e sion o he νi11/2and νg9/2s a es. Thus, we
ha e wo main compe i i e app oaches o desc ibing he kink
and OES: RMF and non ela i is ic Fayans- unc ional heo-
ies [46]. I is impo an ha bo h app oaches desc ibe hese
peculia i ies o he δ 2beha io wi hou loss o accu acy in
he ep oduc ion o he nuclea masses in con as wi h he
majo i y o he (modi ied) NRHF app oaches [38,47]. In he
case o he Fayans unc ional i is a speci ic o m o pai ing
034315-2
CHARGE RADII OF THALLIUM ISOTOPES NEAR … PHYSICAL REVIEW C 110, 034315 (2024)
in e ac ion wi h he dependence on he densi y g adien ha is
essen ial o p o ide ag eemen wi h he expe imen al cha ge
adii [35,44], whe eas he kink and OES a e al eady p esen
in he RMF calcula ions wi hou pai ing [5,6].
This con adic ion encou ages a mo e de ailed compa ison
be ween hese heo ies o he kink and OES. Such an analysis
is he second aim o he p esen wo k whe eby we used he
ad anced RMF app oaches summa ized in Re . [33] wi h he
PC-L3R [48], PC-X [49], DD-MEX [49], and DD-PCX [50]
in e ac ions. The calcula ions wi h (modi ied) DF3-a Fayans-
ype unc ional we e made speci ically o he p esen wo k.
II. EXPERIMENTAL DETAILS
The da a p esen ed in his wo k o igina e om he same
expe imen as desc ibed in Re . [51], and he e o e only he
mos ele an de ails will be gi en he e.
Radioac i e hallium iso opes we e p oduced in spalla-
ion eac ions by a 1.4-GeV p o on beam (in ensi y up o
2 µA) om he CERN p o on synch o on boos e bomba d-
inga50g/cm2UCx a ge . The eac ion p oduc s di used
h ough he a ge ma e ial (T≈2000–2200 ◦C), and e used
in o he ho ca i y o he a ge -ion sou ce de ice as neu al
a oms. A he masses o in e es (A=207–209), s ong iso-
ba ic con amina ion om su ace-ionized ancium is p esen ,
which p e en ed ex ensions o ea lie expe imen s wi h hal-
lium [11–13] o hea ie masses.
To o e come his p oblem, he LIST de ice was
used [15–17]. In ou expe imen , he ancium ions we e
supp essed by a ac o o ≈104. A he same ime, he lase -
ionized hallium iso opes we e supp essed by a ac o o
≈20, ela i e o no mal lase ion sou ce ope a ion wi hou he
LIST [14]. The imp o ed signal- o-backg ound a io due o
he supp ession o he su ace-ionized ancium (see de ails in
Re . [51]) was he key condi ion which allowed us o pe o m
ou measu emen s.
Inside he LIST, hallium a oms we e esonan ly ionized
using he wo-s ep hallium ioniza ion scheme [11–13,51]. Fo
IS measu emen he i s s ep 277-nm 6p2P1/2→6d2D3/2
a omic ansi ion was scanned by a equency-doubled unable
dye lase (LIOP-TEC LiopS a ) beam wi h a linewid h o
≈3 GHz. The subsequen ioniza ion s ep was p o ided by a
equency doubled Nd : YVO4lase (Lume a Blaze, unda-
men al equency o 532 nm).
A e ioniza ion, he hallium ions we e ex ac ed and
accele a ed by a 50-kV elec os a ic po en ial, and mass-
sepa a ed by he ISOLDE gene al pu pose sepa a o . The
ions we e hen deli e ed o ei he a Fa aday cup (FC) o he
ISOLDE Decay S a ion (IDS) [52] o ion coun ing. The ion
cu en o he s able iso opes 203,205Tl om a dedica ed o en,
as well as abundan ly p oduced 207Tlg, we e di ec ly measu ed
by he FC. 205Tl was used as he e e ence iso ope o he IS
measu emen s.
III. DATA ANALYSIS AND RESULTS
A. Fi ing o he hype ine-s uc u e spec a
Examples o he h s spec a a e shown in Fig. 1.This
igu e is modi ied om Fig. 1 o Re . [51] by adding he
FIG. 1. Examples o he h s spec a o s udied hallium iso opes.
The solid ed lines a e Voig p o ile i s o he da a. Ve ical do ed
lines ma k cen e s o g a i y o he co esponding h s. The nuclea
spin and pho oion de ec ion me hod (FC o IDS), as well as γ- ay
ene gies in he case o decay-based de ec ion a IDS, a e displayed
o each iso ope. The ze o poin on he equency scale co esponds
o a wa e numbe o 36117.92 cm−1.
208Tl h s. The da a analysis p ocedu e was he same as in
ou p e ious s udies o he neu on-de icien hallium iso-
opes [12,13]. The posi ions o he hype ine componen s in
he h s spec a a e de e mined by he s anda d ela ion [12]
wi h i e pa ame e s: nuclea spin (I), iso ope shi ela i e o
he s able 205Tl (δνA,205
277nm), magne ic h s cons an s (a1and a2)
o he i s and he second le el o he ioniza ion scheme, and
he elec ic quad upole h s cons an b2 o he second le el.
No e ha b1≡0, since he i s le el in ou ioniza ion scheme
has elec onic angula momen um J=1/2.
Voig p o iles we e i ed o he expe imen al h s spec-
a using a ixed a2/a1=−0.002013(19) a io aken om
he alue o he s able iso opes 205Tl [53], and I alues
om [54]. Fo 208Tl, he cons an a1 aken om Re . [55], was
ixed.
Fo 207Tlmwi h I=11/2, he possible quad upole spli ing
o he uppe le el 6d2D3/2o he scanned ansi ion was
034315-3
Z. YUE e al. PHYSICAL REVIEW C 110, 034315 (2024)
TABLE I. Iso ope shi s o he 277-nm ansi ion (δνA,205)and
changes in mean-squa ed cha ge adius (δ 2A,205) o hallium iso-
opes wi h A>200. The s a is ical expe imen al unce ain ies a e
gi en in pa en heses. The alues in cu ly b acke s a e he sys ema ic
e o s which s em om he unce ain y o he Fand kMS ac o s.
Whe e a ailable, li e a u e alues a e shown in i alic in a second line
o each iso ope.
AI
πδνA,205(MHz) δ 2A,205( m2)
203 1/2+−1026(104) −0.107(11){3}
−1038.5(1) [53]−0.10840(3){220} [56]
207 1/2+1023(175) 0.107(19){3}
1030(240) [12]0.1100(2){22} [56]
207m 11/2−1030(290) 0.108(31){3}
208 5+2118(190) 0.223(21){6}
0.192(13){4} [56]
0.201(14)a
209 1/2+3130(220) 0.330(23){8}
aWeigh ed mean o δ 2208,205 be ween ou esul and ha om
Re . [56].
aken in o accoun using ad anced a omic calcula ions o he
elec ic ield g adien p oduced by he elec ons a he si e o
he nucleus [51].
B. Changes in mean-squa ed cha ge adii
The δ 2 alues we e deduced using he s anda d
pa ame iza ion o he iso ope shi δνA,A=νA−νA[57]:
δνA,A=kMS1
MA−1
MA+Fδ 2A,A.(1)
He e, kMS and Fa e he mass- and ield-shi cons an s, e-
spec i ely, MAand MAa e he a omic masses o iso opes wi h
mass numbe Aand A, and δ 2A,A= 2A− 2Ais he
di e ence be ween hei nuclea mean-squa ed cha ge adii.
In his wo k, we ha e used F=9.32(23) GHz/ m2,kMS =
−575(71) GHz u, which we e ecen ly calcula ed in Re . [56]
accoun ing o ela i is ic and high-o de elec onic co ela-
ion e ec s.
In Table I, he IS o he 277-nm ansi ion and δ 2 o
he s udied hallium iso opes a e shown, along wi h alues
om li e a u e. Ou esul s o 203,207Tlgag ee wi h he li -
e a u e da a. Fo 208Tl ou alue di e s om he li e a u e by
1.2σand he weigh ed mean o he li e a u e da a and ha
om ou measu emen s is used below. Figu e 2shows he
δ 2 alues o hallium nuclei nea N=126. As seen in
Table I, he adii o 207Tlg(I=1/2) and 207Tlm(I=11/2)
a e he same wi hin unce ain ies, and would be indis in-
guishable in Fig. 2, he e o e only he δ 2 o 207Tlgis
shown.
The beha io o cha ge adii in Hg, Tl, Pb, and Bi iso opic
chains nea N=126 is s ikingly simila as seen in Fig. 3.
(See also he discussion on he kink indica o in Sec. V.)
In con as o he kink a N=82 (see Fig. 4 in Re . [9],
o example), he kink a N=126 has no clea Zdepen-
dence, a leas wi hin he limi s o he cu en expe imen al
unce ain ies. Thus, he e is no no able di e ence be ween
FIG. 2. The δ 2 alues o hallium iso opes nea N=126. The
ed iangles a e esul s om he p esen wo k, and he black squa es
a e li e a u e alues aken om [56]. The alues o e en-Nnuclei
a e connec ed by he dashed line o guide he eyes. The unce ain ies
include bo h s a is ical and sys ema ic ones.
he cases whe e alence p o ons occupy low- jo bi als, such
as in hallium (s1/2), and high- jones as in bismu h (h9/2).
The independence o he kink on he o bi al occupied by he
alence p o on is suppo ed by he negligible isome shi o
207Tlm(I=11/2,πh11/2) ela i e o 207Tlg(I=1/2,πs1/2)
(see Table I).
FIG. 3. The δ 2 alues o he iso opic chains wi h Zclose o
Z=82 in he icini y o N=126. Only e en-Niso opes a e shown
o mo e dis inc kink p esen a ion.
034315-4

CHARGE RADII OF THALLIUM ISOTOPES NEAR … PHYSICAL REVIEW C 110, 034315 (2024)
FIG. 4. Compa ison o he TFFS-Fayans calcula ions o δ 2 alues wi h he expe imen o he 80Hg,81Tl,82Pb, and 83Bi iso opic chains
nea N=126. Expe imen al da a a e aken om (a) Re s. [3,5,6]; (b) Re . [56] and p esen wo k; (c) Re . [7]; (d) Re s. [8,58].
IV. THEORETICAL CALCULATIONS
The expe imen al da a we e compa ed wi h calcula ions in
he amewo k o he ini e Fe mi sys em heo y, based on he
gene alized EDF me hod implemen ed wi h he DF3-a unc-
ional by Fayans e al. [21,36,59]. Compa ed o he Sky me
o Gogny EDF i has a mo e complex (a ac ional-linea )
dependence on he pa icle densi ies.
The anomalous pa o he Fayans unc ionals (see de i-
ni ions in [21]) in ol es he a ac i e e m accoun ing o
pai ing in he nuclea ex e io and wo epulsi e su ace e ms
depending on he pa icle densi ies and hei g adien s. The
densi y-g adien dependen e m is c ucial o explaining he
kink and OES e ec s in cha ge adii [21,35,36,44]. I is his
e m ha made i possible o ensu e in [21] ha he 40Ca and
48Ca adii a e app oxima ely equal al eady a he mean- ield
le el. A compa ison wi h he esul s ob ained in [60] con i ms
he ac ha his g adien e m e ec i ely akes in o accoun
he phonon coupling e ec on he cha ge-densi y dis ibu ion.
The e ec i e densi y-dependen con ac pai ing wi h a
cu o a EFe mi used in ou calcula ions, is non-uni e sal,
he e o e, a numbe o (A,Z-dependen ) pai ing pa ame iza-
ions exis s e lec ing a complexi y o in e play be ween
he a o emen ioned e ms. Fo he p esen kink and OES
calcula ions, he unc ional pa ame e s in he pai ing sec o
we e eadjus ed using he a ailable neu on sepa a ion en-
e gies (Sn) and cha ge adii in me cu y, hallium, lead, and
bismu h iso opes. Fo a be e simul aneous desc ip ion o he
Snand kink da a, he ollowing pa ame e se was ound: ξ=
−1.05,hξ=0.78, ξ
∇=1.25 (designa ion o pa ame e s was
aken om [21]). This adjus men main ains a good desc ip-
ion o binding ene gies and cha ge adii o sphe ical nuclei in
he calcium, nickel and lead egions, o which he pa ame e s
o he DF3 and DF3-a unc ionals we e i ed. In gene al, he
global pe o mance o he unc ional is compa able wi h he
DF3-a pai ing unc ional se [21,36].
V. DISCUSSION
As seen in Fig. 4, he TFFS calcula ions wi h he modi ied
DF3-a unc ional sa is ac o ily desc ibe he δ 2beha -
io in hallium iso opes (mean de ia ion is 0.03 m2). The
heo e ical esul s o adjacen 80Hg,82Pb, and 83Bi chains
demons a e he simila ag eemen wi h expe imen (see
Fig. 4). I is impo an ha he change o he pai ing pa ame-
e s does no wo sen he desc ip ion o he neu on sepa a ion
ene gy as shown, e.g., o Pb and Tl chains in Fig. 5.
034315-5
Z. YUE e al. PHYSICAL REVIEW C 110, 034315 (2024)
FIG. 5. Compa ison o he TFFS-Fayans calcula ions o he neu-
on sepa a ion ene gy Snwi h he expe imen al da a o he 81Tl and
82Pb iso opic chains. Expe imen al da a we e aken om Re . [61].
In o de o quan i a i ely compa e he magni ude o he
kink and OES in di e en heo e ical app oaches, he kink (ξ)
and s agge ing (γN) indica o s we e in oduced:
ξ=δ 2128,126
δ 2126,124 ,(2)
γN=2δ 2N−1,N
δ 2N−1,N+1.(3)
Supe sc ip indices poin o he neu on numbe s. These in-
dica o s a e independen o he unce ain ies in he a omic F
ac o (usually 5–10% in he lead egion). The choice o he
e en-Niso ope nea es o he neu on magic numbe s o he
kink indica o , helps o a oid mixing o he shell e ec wi h
o he e ec s which migh con ibu e o he obse ed δ 2
alue.
In Fig. 6 he expe imen al kink indica o s o me cu y,
hallium, lead and bismu h iso opic chains a e compa ed wi h
heo e ical esul s ob ained wi hin he RMF (PC-L3R, PC-
X, DD-MEX, and DD-PCX unc ionals; see Re . [33]) and
Fayans- ype (wi h imp o ed DF3-a unc ional) app oaches.
The TFFS (DF3-a) calcula ions wi h modi ied pai ing, excel-
len ly ep oduce he expe imen al da a, whe eas he PC-L3R,
PC-X, and DD-PCX unc ionals ma kedly o e es ima e he
shell e ec in adii and he DD-MEX unc ional p edic s a
no iceable Zdependence o he kink indica o which is no
obse ed expe imen ally. Howe e , one should keep in mind
FIG. 6. Kink indica o o he iso opic chains nea Z=82. Ex-
pe imen al da a a e aken om Re s. [7] (Pb iso opes); [3,5,6](Hg
iso opes); [8,58] (Bi iso opes); [56] and p esen wo k (Tl iso opes).
Theo e ical alues (RMF wi h di e en unc ionals) a e de i ed om
he calcula ion in Re . [33].
ha accoun ing o he cen e -o -mass co ec ions gi e sizable
impac s on he cha ge adius and can imp o e desc ip ion o
kink in he RMF model [62].
I should be no ed ha he single-pa icle spec um o
neu ons in 208Pb ex ac ed om he exci a ion spec a o he
neighbo ing odd iso opes 207Pb and 209Pb, is well ep oduced
in he TFFS-Fayans calcula ions (see Fig. 1 in Re . [37]),
whe eas he RMF app oach ails o ep oduce he o de o
he neu on single-pa icle s a es nea N=126. In pa icula ,
in he RMF calcula ions he νi11/2s a e is lowe in ene gy
han he νg9/2s a e in con adic ion wi h expe imen [5,6,32].
In con as wi h he RMF calcula ion, in he TFFS-Fayans
app oach he kink is ep oduced wi hou he in e sion o he
νi11/2and νg9/2s a es.
The con adic ion o he RMF app oach wi h expe i-
men (in e ed posi ion o νg9/2and νi11/2) is elimina ed o
odd-Nnuclei by aking in o accoun he coupling o single-
pa icle mo ion wi h phonons (pa icle- ib a ion coupling)
(see [63,64]). A he same ime, in he RMF amewo k he
g ound s a es o he e en-Niso opes a N>126 a e de e -
mined by he illing o he lowe lying νi11/2shell [5]in
con adic ion o he nume ous shell-model calcula ions whe e
he νg9/2o bi al is ixed below he νi11/2one by 779 keV
(see, o example, Re s. [65–67] and e e ences he ein). I
should be no ed, ha hese shell-model calcula ions qui e
success ully desc ibe spec a and ansi ion p obabili ies o
he majo i y o he e en-Nnuclei in he icini y o he shell
closu es a N=126 and Z=82. E iden ly, he TFFS cal-
cula ions wi h he Fayans unc ionals a e ee om hese
p oblems.
In Fig. 7 he expe imen al s agge ing indica o alues o
di e en iso opic chains a e shown. The heo e ical alues a e
gi en only o he lead and hallium iso opes no o o e bu den
he igu e. Resul s o he o he chains a e simila o hose
o lead and hallium iso opes. As in he case o he kink,
one can see he s iking simila i y in he s agge ing indica o s
034315-6
CHARGE RADII OF THALLIUM ISOTOPES NEAR … PHYSICAL REVIEW C 110, 034315 (2024)
FIG. 7. S agge ing indica o o he iso opic chains nea Z=82
and N=126. Expe imen al da a a e aken om Re s. [7](Pbiso-
opes); [3,5,6] (Hg iso opes); [8,58] (Bi iso opes); [56] and p esen
wo k (Tl iso opes).
o Z=80,81,82,83. In pa icula , o each chain he e is a
clea dependence o γNon he shell occupied by he alence
neu on (ν 5/2,νp1/2,νg9/2). The sel -consis en TFFS calcu-
la ions ep oduce s agge ing indica o and i s neu on-numbe
dependence wi h he excep ion o he alues a N>126 when
he νg9/2shell is illing. As i was shown in Re . [41], he
luc ua ing con ibu ion due o he quasipa icle-phonon in e -
ac ion can be impo an o N>Nmagic.
In he RMF calcula ions wo di e en p ocedu es labeled
as “LES” and “EGS” we e used o he blocking in odd-
Anuclei [5,6]. In he LES p ocedu e, he lowes -ene gy
con igu a ion was used, which is simila o he TFFS-
Fayans calcula ions o OES [21,38]. In he EGS p ocedu e,
he con igu a ion wi h he spin and pa i y o he blocked
s a e co esponding o hose o he expe imen al g ound
s a e is employed, al hough i is no necessa ily he lowes
in ene gy.
I was ound ha in he RMF app oach he OES is
desc ibed easonably well when he EGS p ocedu e is ap-
plied. Wi h he LES p ocedu e, he expe imen al OES is
signi ican ly unde es ima ed [5,6]. A he same ime, in he
TFFS-Fayans calcula ions he e a e no addi ional condi ions
o he choice o he blocked s a e in he odd-Nnuclei due o
he good ep oduc ion o he neu on single-pa icle spec um
in con as wi h he RMF heo y.
VI. CONCLUSION
The δ 2 alues o 207Tlm(Iπ=11/2−) and 209Tlg(Iπ=
1/2+) ha e been measu ed o he i s ime, using lase
esonance-ioniza ion spec oscopy in he LIST ion sou ce a
ISOLDE (CERN). The cha ac e is ic kink in he g ound-s a e
hallium cha ge adii a he N=126 shell closu e has been
obse ed. I was ound ha he kink indica o o hallium
coincides wi hin he limi o unce ain ies wi h ha o he
neighbo ing me cu y, lead, and bismu h iso opic chains. Thus,
he e is no no able di e ence be ween he cases whe e alence
p o ons occupy low- jo bi als, such as in hallium (s1/2), and
high- jones as in bismu h (h9/2). The independence o he kink
on he o bi al occupied by he alence p o on is suppo ed
by he negligible isome shi o 207Tlm(I=11/2,πh11/2)
ela i e o 207Tlg(I=1/2,πs1/2) (see Table I).
Measu emen o he IS o 209Tl also enabled us o com-
ple e he sys ema ics in he OES in he conside ed egion. As
in he case o he kink, he OES o hallium iso opes nea
N=126 coincides in he limi s o unce ain ies wi h ha o
he neighbo ing iso opic chains.
The de ailed compa ison o he comple e se o he ex-
pe imen al δ 2da a nea N=126,Z=82 and he heo y
was made. Calcula ions using he DF3-a unc ional by Fayans
e al. [21,36,37] wi h adjus men o pa ame e s in he pai ing
sec o demons a e a good ag eemen wi h he expe imen al
da a o he Sn, kink and OES alues in he me cu y, hallium,
lead, and bismu h chains wi h he excep ion o he OES a N>
126. The la e de iciency can be connec ed wi h neglec ing
he phonon coupling, which has an impac in his egion [41].
The ecen ad anced RMF unc ionals (PC-L3R, PC-X,
DD-MEX, and DD-PCX; see Re . [33]) o e es ima e he kink-
indica o alue and show i s Zdependence which is absen
in expe imen al da a. The e y p esence o he kink in he
RMF heo y is de e mined by he in e sion o he posi ion
o he νg9/2and νi11/2neu on shells. In he TFFS-Fayans
calcula ions he o de o he single-pa icle neu on s a es nea
N=126 co esponds well wi h he expe imen al da a and
he e is no in e sion o he νg9/2and νi11/2s a es. Fu he
heo e ical and expe imen al e o s a e equi ed in o de o
elucida e he in luence o he neu on single-pa icle spec um
on he cha ge adius e olu ion. I should be no ed ha in
con as wi h ou TFFS-Fayans calcula ions, he RMF unc-
ionals we e no adjus ed o p o ide he kink desc ip ion in
his speci ic mass egion.
The comple e se o da a on cha ge adii in he close
icini y o magic Z=82 and N=126 helps o u he elu-
cida e he pai ing pa o he Fayans unc ional which would
enable a be e desc ip ion o he kink and OES. No e, ha
o he nuclea obse ables e eal mo e weak dependence on
he g adien pai ing s eng h [21]. The sea ch o a uni e sal
pa ame iza ion in he pai ing sec o which would keep he
global pe o mance o he Fayans unc ional along wi h a
good simul aneous desc ip ion o he iso opic beha io o
bo h geome ic (δ 2) and ene ge ic (Sn) di e en ial obse -
ables h oughou he nuclide cha is he aim o u u e wo k.
ACKNOWLEDGMENTS
We would like o acknowledge he suppo o he ISOLDE
collabo a ion and echnical eams. The esea ch leading o
hese esul s has ecei ed unding om he Eu opean Union’s
Ho izon Eu ope Resea ch and Inno a ion P og amme unde
G an Ag eemen No. 101057511. The wo k was suppo ed
by he STFC G an s No. ST/V001035/1, No. ST/V001108/1,
No. ST/V001027/1, and No. ST/P004598/1, by he Roma-
nian Nucleu P ojec No. PN 23 21 01 02 and IFA G an
No. CERN/ISOLDE, by he Resea ch Founda ion Flan-
de s (FWO, Belgium), by BOF KU Leu en (G an No.
C14/22/104), by he Spanish unding agency MICIN/AEI
10.13039/501100011033 (FEDER, EU) ia P ojec s No.
034315-7
Z. YUE e al. PHYSICAL REVIEW C 110, 034315 (2024)
RTI2018-098868-B-I00 and No. PID2021-126998OB-I00,
by Ge man BMBF unde Con ac No. 05P21PKCI1 and
Ve bundp ojek 05P2021, and by he Polish Na ional Sci-
ence Cen e unde G an No. 2020/39/B/ST2/02346, by he
Polish Minis y o Educa ion and Science unde Con ac
No. 2021/WK/07. M.A. acknowledges unding om he
Eu opean’s Union Ho izon 2020 Resea ch and Inno a ion
P og am unde G an Ag eemen No. 861198 p ojec ‘LISA’
(Lase Ioniza ion and Spec oscopy o Ac inides) Ma ie
Skłodowská-Cu ie Inno a i e T aining Ne wo k (ITN).
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