PHYSICAL REVIEW C 75, 064316 (2007)
Shape phase ansi ion in odd nuclei in a mul i- jmodel: The UB(6) ⊗UF(12) case
C. E. Alonso,1J. M. A ias,1and A. Vi u i1,2
1Depa amen o de F´
ısica A ´
omica, Molecula y Nuclea , Facul ad de F´
ısica, Uni e sidad de Se illa, Apa ado 1065, E-41080 Se illa, Spain
2Dipa imen o di Fisica Galileo Galilei and INFN, Via Ma zolo 8, I-35131 Pado a, I aly
(Recei ed 2 Ap il 2007; published 22 June 2007)
The phase ansi ion in odd nuclei when he unde lying e en-e en co e nuclei expe ience a ansi ion om
sphe ical o de o med γ-uns able shapes is in es iga ed. The odd pa icle is assumed o be mo ing in he h ee
single-pa icle o bi als j=1/2,3/2, and 5/2 . A he c i ical poin in he phase ansi ion, an analy ic solu ion
o he co esponding Boh Hamil onian, called E(5/12), is wo ked ou . Ene gy spec a and elec omagne ic
ansi ions and momen s a e p esen ed. The same p oblem is also a acked in he amewo k o he in e ac ing
boson- e mion model (IBFM). Two di e en Hamil onians a e used. The i s one is cons uc ed ad hoc so as
o mimic he si ua ion in he E(5/12) model. The second one leads o he occu ence o he OB(6) ⊗UF(12)
symme y when he boson pa app oaches he O(6) condi ion. The en i e ansi ion line is s udied wi h his
Hamil onian and, in pa icula , he c i ical poin . Bo h IBFM calcula ions a he c i ical poin a e consis en wi h
he E(5/12) esul s.
DOI: 10.1103/PhysRe C.75.064316 PACS numbe (s): 21.60.Fw, 21.60.E
I. INTRODUCTION
A omic nuclei ha e been classi ied o yea s in e ms
o collec i e de o ma ions ha a e desc ibed by in oducing
wo collec i e a iables [1]. The de o ma ion pa ame e β
measu es he axial de ia ion om sphe ici y, while he angle
a iable γcon ols he depa u e om axial symme y. Th ee
analy ical solu ions o he collec i e Boh Hamil onian ha e
been known: he ib a ional, o a ional, and de o med γ-
uns able limi s [1–3]. These si ua ions co espond, espec-
i ely, o he sphe ical, axially de o med, and de o med γ-
independen g ound s a e shapes. O cou se, ew nuclei p esen
p ope ies ha i exac ly one o hese limi ing si ua ions, and
mos o hem display a ansi ional beha io . The occu ence o
shape ansi ional beha io cha ac e izes bo h e en-e en and
odd nuclei. As he nucleon numbe changes, he s uc u e o he
sys em changes om one cha ac e o ano he , allowing one o
alk abou a g ound s a e phase ansi ion (nowadays known as
a quan um phase ansi ion). The p ope ies o he nucleus a
he icini y o he c i ical poin a y apidly, and, consequen ly,
he desc ip ion o c i ical nuclei has been hough o be e y
di icul . Recen ly, Iachello p oposed in a se ies o pape s a new
class o symme ies ha a e o mula ed in e ms o he Boh
Hamil onian and ha can be applied o c i ical poin si ua ions
[4–6]. In pa icula , a he c i ical poin om sphe ical o
γ-uns able shapes, called E(5) [4], a he c i ical poin om
sphe ical o axially de o med shapes, called X(5) [5], and
a he c i ical poin om axially de o med shapes o iaxial
shapes, called Y(5) [6]. Since he in oduc ion o hese limi s,
many heo e ical [7–12] and expe imen al [13–22] s udies
ha e been p esen ed in o de o look o nuclei ha exhibi
he p ope ies o c i icali y and o classi y he co esponding
phase ansi ions. Many s udies ha e ex ended hese o iginal
models o mo e complex si ua ions [23–27].
Because o he u he complexi y embodied in he de-
sc ip ion o he coupling o he odd pa icle o he e en co e,
he s udy o phase ansi ions in odd nuclei has only ecen ly
been a acked. The i s case conside ed by Iachello [7,28]was
he case o a j=3/2 e mion coupled o a boson co e ha
unde goes a ansi ion om sphe ical o γ-uns able si ua ion.
A he c i ical poin , an elegan analy ic solu ion, called E(5/4),
was ob ained s a ing om he Boh Hamil onian. The possible
occu ence o his symme y in 135Ba has been ecen ly
sugges ed [29]. Howe e , he es ic ion o he e mion space o
a single-jo bi makes di icul he iden i ica ion o a pa icula
nucleus as c i ical. A ecen wo k [30] conside ed he iche
case o a collec i e co e coupled o a pa icle mo ing in he
j=1/2,3/2, and 5/2 single-pa icle o bi s, wi h he boson
pa unde going a ansi ion om sphe ici y o de o med
γ-ins abili y. Fo he c i ical poin in his si ua ion, a new
analy ic solu ion, called E(5/12), has been p esen ed. Since
his si ua ion is likely o occu in se e al mass egions, E(5/12)
p o ides a be e chance o inding an example o c i icali y in
odd-e en nuclei. In his pape we ex end he p e ious s udy on
E(5/12) so as o p o ide he g ounds o expe imen al s udies
looking o c i ical beha io in odd-e en nuclei. Spec um,
elec ic quad upole, and magne ic dipole ansi ions along
wi h quad upole and magne ic momen s a e wo ked ou in
he model.
I is wo h men ioning ha all hese s udies ha e been
pe o med in he amewo k o he collec i e model. Howe e ,
a pa allel ea men can be done wi hin he in e ac ing boson
model (IBM) [31]. Al hough his model is o mula ed om
he beginning in second quan ized o m, i is possible o
ob ain i s geome ic pic u e by using cohe en s a es and
shape a iables [32–34]. Th ee dynamical IBM symme ies a e
known: U(5), SU(3), and O(6) co esponding o geome ical
sphe ical, axially de o med, and γ-uns able de o med shapes,
espec i ely [35]. T ansi ional classes among hese limi s ha e
been ho oughly s udied since he in oduc ion o he model.
E en g ound s a e phase ansi ions we e s udied soon a e
[33,36–38], bu hese s udies ha e been e i alized ecen ly
wi h he abo e discussed wo ks on quan um phase ansi ions
0556-2813/2007/75(6)/064316(15) 064316-1 ©2007 The Ame ican Physical Socie y
C. E. ALONSO, J. M. ARIAS, AND A. VITTURI PHYSICAL REVIEW C 75, 064316 (2007)
wi hin he Boh Hamil onian amewo k. Nume ical s udies
o e en-e en nuclei on he c i ical poin om U(5) o O(6)
and om U(5) o SU(3) ha e been p esen ed, and he analysis
o he phase ansi ion cha ac e is ics has been he objec
o many pape s [39–61]. A ecen e iew on he opic can
be ound in Re . [62]. Fo he case o odd-e en nuclei, he
simples IBM ex ension is called he in e ac ing boson- e mion
model (IBFM) [63]. The IBFM has been used o s udy he
phase ansi ion along he U(5)-O(6) line o he case o
a e mion in a single j=3/2 o bi al [64,65], which is
he si ua ion s udied in he E(5/4) model. The choice o
he j=3/2 o bi al allows one o eco e a he γ-so
ex eme he OB(6) ⊗SUF(4) boson- e mion symme y. This
is because in his case he ull boson- e mion quad upole-
quad upole e m in he Hamil onian can be exp essed in
e ms o he Casimi ope a o s o he gene al UBF(5), OBF(5),
SpinBF(6), and SpinBF(5) algeb as. The case o he j=3/2
is no , howe e , unique. I is known in he li e a u e [63,66]
ha also in he mul i-jcase wi h j=1/2,3/2,5/2, wi h a
p ope choice o he boson- e mion Hamil onian, one can
ob ain a ansi ional Hamil onian exp essed in e ms o Casimi
ope a o s associa ed wi h he OBF(6) algeb a. This has allowed
one o s udy wi hin he IBFM a simila si ua ion o ha
s udied in he E(5/12) model. The IBFM s udy has been
shown o be use ul o check he E(5/4) [64] and E(5/12) [30]
cases. In his pape , we p esen o j=1/2,3/2,5/2 he
equi alen E(5/12) esul s o selec ed IBFM Hamil onians.
The con on a ion o E(5/12) and IBFM a he co esponding
c i ical poin will p o ide use ul guidelines o expe imen al
s udies.
The s uc u e o he pape is as ollows. In Sec. II, he
E(5/12) model is p esen ed in de ail and he wa e unc ions
a e used o calcula e di e en spec oscopic p ope ies o
expe imen al in e es . In Secs. III and IV, wo IBFM model
Hamil onians p oduced o s udy he c i ical poin wi hin he
IBM amewo k a e p esen ed. We will he e o e conside
wi hin he IBFM he case o he coupling o an odd pa icle
mo ing in he j=1/2,3/2, and 5/2 o bi als o a boson
co e unde going a ansi ion om U(5) o O(6) si ua ions.
We ollow along he ansi ion he e olu ion o he spec um
and s udy in de ail ene gies and ansi ions in co espon-
dence o he c i ical poin . We will show ha he esul s
o IBFM and E(5/12) models gi e compa able beha io s
p oducing obus esul s indica ing c i icali y in odd-e en
nuclei.
II. THE BOHR HAMILTONIAN AT THE
CRITICAL POINT AND THE E(5/12) MODEL
In his sec ion, we s udy he coupling o an e en co e
o a pa icle mo ing in he 1/2, 3/2, and 5/2 o bi als. We
show ha i can also display analy ic solu ions wi hin he
Boh collec i e model, in analogy o he case o he j=3/2
pa icle, desc ibed by Iachello wi hin he E(5/4) c i ical poin
symme y. We will name his case as E(5/12). Bu i s , we
would like o p esen b ie ly he E(5) case o e en-e en nuclei,
since i will be an impo an e e ence o ou s udy. The E(5)
c i ical poin si ua ion co esponds o he solu ion o he Boh
Hamil onian
H=−
¯h2
2B1
β4
∂
∂β β4∂
∂β +1
β2sin 3γ
∂
∂γ sin 3γ∂
∂γ
−1
4β4
κ
Q2
κ
sin2γ−2
3πκ+U(β,γ),(2.1)
in which he po en ial ene gy Uis γindependen and modeled
as an in ini e squa e well in β
U(β)=0,β<β
w,
U(β)=∞,β⩾βw.
(2.2)
This si ua ion is supposed o mimic he c i ical poin om
sphe ical (minimum o he ene gy su ace a β=0) o
de o med γuns able (minimum o he ene gy su ace a β= 0
and γindependen ). In his si ua ion, analy ical solu ions
a e ound. Ene gy le els a e di ec ly ela ed o ze os o
app op ia e Bessel unc ions [4,7]. In Fig. 1, he spec um
and a ew selec ed E2 ansi ion p obabili ies a e displayed.
Fo no a ion and mo e de ails, we e e o Re s. [4,7]. We
mus men ion ha in his si ua ion, because o he symme y
o he p oblem, s a es a e classi ied by he o ally symme ic
i educible ep esen a ions o he O(5) and O(3) g oups, (τ,0)
and L, espec i ely. In addi ion, an ex a label ξis in oduced
o ma k di e en solu ions o he adial (β) equa ion. Thus,
s a es a e labeled by |ξ,τ,L. The solu ions o Eq. (2.1)a e
ψ(β,γ,θi)= ξ,τ(β)τµLM(γ,θi),(2.3)
whe e he  unc ions we e de e mined in Re s. [3,67]. These
unc ions a e eigens a es o any quad upole Hamil onian which
displays O(5) ⊃O(3) symme y, i.e., o U(β,γ)=U(β).
The (β) unc ions depend on he selec ion o he βpo en ial.
Fo he in ini e squa e well p oposed, he wa e unc ions in β
ake he simple o m
ξ,τ(β)=Cξ,τβ−3/2Jτ+3/2xξ,τ
βω
β,(2.4)
0+
0+
2+
6+4+3+0+
2+
4+
4+
2+
2+
0
1
2.20
3.59
3.03
4.80
6.78
0
1
2
3
0
1
2
τ E
τ E
100 167 217
114
75 124
167
103
155
217
87
124
62
E(5)
ξ=1
ξ=2
FIG. 1. Ene gy le els (no malized o he ene gy o he i s exci ed
s a e) and B(E2) ansi ion p obabili ies o E(5).
064316-2
SHAPE PHASE TRANSITION IN ODD NUCLEI IN A . . . PHYSICAL REVIEW C 75, 064316 (2007)
whe e τ=0,1,2,... labels he SO(5) g oup, ξis an index
ha enume a es he successi e ze os o he Bessel unc ions,
xξ,τ is he ξ h ze o o a gi en τ,Cξτ is a no maliza ion,
and βωis he ange o he squa e po en ial. The ull s a es in
Eq. (2.3) can hen be deno ed by
|ξτµLM,(2.5)
whe e µis a label ha dis inguishes be ween epea ed L
o a gi en alue o τ. Once he wa e unc ions a e known,
he ansi ion p obabili ies a e ela ed o he educed ma ix
elemen o he ele an elec omagne ic ope a o ,
B(E/M,λ)=1
2Li+1ξ τ µ L ||T(λ)
C||ξiτiµiLi2,(2.6)
i s calcula ion includes an in eg al o p oduc s o (β)
unc ions on he β a iable, plus in eg als o p oduc s o
τµLM(γ,θi) unc ions in he γ a iable and he Eule
angles. De ails o hese in eg a ions can be ound in
Re . [7].
Nex we go o ou case o in e es , in which a single
pa icle ha can occupy he j=1/2,3/2, and 5/2 o bi als is
coupled o an E(5) e en-e en co e. In his si ua ion, one can
use he sepa a ion o he single-pa icle angula momen um
in o a pseudospin and pseudo-o bi al angula momen um
(0 and 2) and assume no coupling due o he pseudospin.
In his si ua ion, an E(5) co e is coupled o a wo-le el
sys em ep esen ing he odd pa icle. The lowe le el wi h
LF=0(j=1/2) has ze o ene gy, and he uppe one wi h
LF=2(j=3/2,5/2 degene a e) has some κene gy. The
pseudo-o bi al pa ans o ms as he ep esen a ions [τF,0] o
OF(5), wi h τFbeing ei he 0 o 1. One can use in his case a
model Hamil onian o he o m
H=−
¯h2
2B1
β4
∂
∂β β4∂
∂β +1
β2sin 3γ
∂
∂γ sin 3γ∂
∂γ
−1
4β4
κ
Q2
κ
sin2γ−2
3πκ+u(β)+kg(β)
×[2 ˆ
LB◦ˆ
LF]+kg(β)ˆ
L2
F,
(2.7)
wi h
u(β)=0,β<β
w,
u(β)=∞,β⩾βw,
g(β)=¯h2
2Bβ2.
(2.8)
No e ha ˆ
LBand ˆ
LFa e he i e-dimensional boson and
e mion angula momen a, and ha he do ◦indica es he
i e-dimensional scala p oduc . He e and in he ollowing we
will use he callig aphic le e ˆ
L o deno e he i e-dimensional
O(5) angula momen a and he i alic le e L o deno e he
usual h ee-dimensional O(3) angula momen a. I is wo h
no ing ha he Hamil onian (2.7) is a gene aliza ion o he
one p oposed o he E(5/4) case. Since in E(5/12) he e
a e wo OF(5) ep esen a ions in ol ed, in Eq. (2.7) helas
e m has been in oduced o include he ene gy di e ence
be ween j=1/2 and j=3/2,5/2 single-pa icle o bi als.
The coupling scheme is p esen ed in Fig. 2.In helowes
pa o he igu e he e en-e en deg ees o eedom, gi en
by he E(5) model, ha e o be coupled o he odd-pa icle
deg ees o eedom. These las ones a e sepa a ed in o a
pseudo-o bi al pa ( wo-le el sys em) plus a pseudospin pa .
Fi s , he co e E(5), whose s a es a e classi ied by (ξ,τB,L
B)
as explained abo e, is coupled o he pseudo-o bi al pa
o he odd pa icle, labeled (τF,L
F). This gi es ise o a
BFS
(0) 0
(τΒ) LB(τF) LF
⊗⊗
s
(1) 2
(2) 4,2
(0) 0
(1) 2
1/2
τΒ,τF(τ1,τ2) LBF
0,0(0,0) 0
1,0(1,0) 2
2,0(2,0) 4,2
5/2,3/2
5/2,3/2
0,1(1,0) 2
1,1(2,0) 4,2
1,1(1,1) 3,1
1,1(0,0) 0
9/2,7/2,5/2,3/2
1/2
9/2,7/2,5/2,3/2 7/2,5/2,3/2,1/2 1/2
ull BF
pseudo-o bi al BF
JFIG. 2. Coupling scheme o E(5/12) model.
Lowe pa : an e en-e en E(5) co e ( ep esen ed
by he lowes h ee s a es wi hin he g ound s a e
band, ξ=1) is coupled i s o a wo-le el
sys em ha ep esen s he pseudo-o bi al angula
momen um o he odd pa icle. The esul ing
o al o bi al angula momen um o he odd-e en
sys em is inally coupled o he spin pa o he
odd pa icle.
064316-3
C. E. ALONSO, J. M. ARIAS, AND A. VITTURI PHYSICAL REVIEW C 75, 064316 (2007)
boson- e mion pseudo-o bi al coupling p esen ed in he uppe
le panel. The s a es a e labeled τBcha ac e izing he OB(5)
g oup and τFcha ac e izing he OF(5) g oup. These wo
ep esen a ions a e coupled o a common OBF(5) g oup whose
i educible ep esen a ions a e labeled (τ1,τ
2). I he coupling
is wi h τF=0,L
F=0, all he ob ained s a es a e as in he E(5)
e en-e en case. When he coupling is wi h τF=1, he allowed
coupled ep esen a ions a e (τ1,τ
2)=(τB+1,0),(τB,1), and
(τB−1,0). Once he OBF(5) ep esen a ions a e known, he
usual educ ion o angula momen um is pe o med. In he
uppe le panel o Fig. 2, such a pseudo-o bi al coupling is
ep esen ed o he ξ=1 s a es in he E(5) pa , he le
band comes om he coupling o he E(5) s a es o he
(τF=0,L
F=0) s a e and hen, as men ioned abo e, a e as in
he E(5) model. The o he h ee bands come om he coupling
o he E(5) pa o he (τF=1,L
F=2) s a e. This coupling o
he τB=0 p oduces jus one s a e (τ1=1,τ
2=0) a he same
ene gy as he exci a ion ene gy o he τF=1 wi h espec o
he τF=0 s a e o he odd pa icle. This s a e is labeled
τB,τ
F(τ1,τ
2)LBF =0,1(1,0)2 (LBF is he boson- e mion
pseudo-o bi al angula momen um). The coupling o he boson
τB=1 s a es o he single-pa icle pseudo-o bi al τF=1 s a e
gi e h ee possibili ies (τ1,τ
2)=(2,0),(1,1),and (0,0); he
co esponding educ ion ules gi e LBF =4,2,L
BF =3,1,
and LBF =0, espec i ely. This is ep esen ed schema ically
in Fig. 2(uppe le panel). Finally, hese s a es a e coupled
o he pseudospin pa and gi e he spec um o he ull
boson- e mion sys em, p esen ed in he uppe igh pa o
Fig. 2. Each line ep esen s degene a e s a es wi h o al angula
momen um gi en by J, p oduced by he coupling o he
boson- e mion pseudo-o bi al angula momen um LBF o he
pseudospin s.
Wi h his choice o he coupling, he o al wa e unc ion
can be ac o ized in he o m
=F(β)[(γ,θi,η
o b)⊗χ(ηspin)]JBF ,(2.9)
whe e ηo b and ηspin ep esen he pa icle coo dina es asso-
cia ed wi h he pseudo-o bi al and pseudospin spaces. The
wa e unc ion is cha ac e ized by good quan um numbe s
(τB,τ
F,τ
1,τ
2,L
BF,M
L) and is gi en by
τB,τF,(τ1,τ2),LBF ,ML(γ,θi,η
o b)
=
LB,MLB,LF,mLF(τB,0) (τF,0) |(τ1,τ
2)
LBLF|LBF 
×LBLF|LBF
MLBmLF|MLφτB,LB,MLB(γ,θi)
×XτF,LF,mLF(ηo b),(2.10)
whe e he symbols a e he summa ion a e isoscala ac o s
o he O(5)-O(3) and he O(3)-O(2) educ ions. The unc ion
X(ηo b) desc ibes he i e-dimensional pseudo-o bi al pa ,
while χ(ηspin) is he pseudospin wa e unc ion, which does no
con ibu e o he ene gy. The equa ion o F(β) hen acqui es
he amilia o m
−¯h2
2B
1
β4
∂
∂β β4∂
∂β +¯h2
2B

β2+u(β)F(β)=EF(β),
(2.11)
wi h
=τB(τB+3) +k[τ1(τ1+3) +τ2(τ2+1)
−τB(τB+3) −τF(τF+3)] +kτF(τF+3).(2.12)
The solu ions o his equa ion, as in he case o he E(5/4)
model, a e gi en in e ms o Bessel unc ions o o de ν=
√+9/4, in he o m
Fξ,{τB,τF,(τ1,τ2),LBF }(β)
=cξ,{τB,τF,(τ1,τ2),LBF }β−3/2Jν(xξ,{τB,τF,(τ1,τ2),LBF }β/βw),
(2.13)
whe e cξ,{τB,τF,(τ1,τ2),LBF }is a no maliza ion cons an , and
xξ,{τB,τF,(τ1,τ2),LBF } he ξ h ze o o Jν(z). The co esponding
eigen alues a e gi en as
Eξ,{τB,τF,(τ1,τ2),LBF }=¯h2
2Bxξ,{τB,τF,(τ1,τ2),LBF }
βw2
.(2.14)
The co esponding spec um is gi en in Fig. 3 o he
choice o he pa ame e s k=−1/4 and k=5/2. The
s a es a e labeled acco ding o he quan um numbe s
ξ,τB,τ
F,(τ1,τ
2),L
BF. One should emembe ha each s a e
ac ually ep esen s a mul iple o degene a e le els, whose
angula momen a can be ex ac ed om Table I.
The pa ame e s kand kcon ol he o de o he bands. In
Fig. 4we p esen he e olu ion o he bandheads as a unc ion
o he kand kpa ame e s.
Once he wa e unc ions a e known, one can calcula e
any o he spec oscopic obse able o in e es . In gene al, he
elec omagne ic ope a o o he odd sys em will ha e bo h
con ibu ions om he collec i e co e and om he single
e mion. We will ha e o calcula e educed ma ix elemen s
 ||T(λ||i,(2.15)
whe e a e gi en by Eq. (2.9). Since in his o mula ion
he o al angula momen um o he e mion is no a good
quan um numbe , we p e e o pe o m a simple change o
coupling ans o ma ion gi en by
|(LB,L
F),L
BF,s;J
=
j
(−1)LB+LF+1/2+J(2LBF +1)(2j+1)
×


LBLFLBF
1/2Jj


|LB,(LF,s),j;J.(2.16)
Wi h his i is easy o use s anda d angula momen um coupling
o calcula e bo h collec i e, T(λ)
C, and single pa icle, T(λ)
F, pa s
o any elec omagne ic ope a o :
064316-4
SHAPE PHASE TRANSITION IN ODD NUCLEI IN A . . . PHYSICAL REVIEW C 75, 064316 (2007)
TABLE I. Reduc ion (b anching ules) om he OBF(5) i educible ep esen a ions
(τ1,τ
2) ( i s column) o he OBF(3) i educible ep esen a ions LBF (second column). They
can be ound, o ins ance, in Re . [63] (p. 73, Eqs. 3.85 and 3.86). In he las column, he
angula momen um con en JBF in each OBF(5) ep esen a ion is p esen ed. I comes om
he coupling o LBF o he e mion pseudospin s=1/2. The supe sc ip on a gi en LBF (o
J) means he numbe o imes ha LBF (o J) is ob ained.
OBF(5) i ep OBF(3) i ep To al ang. mom.
(τ1,τ
2)LBF J
(0,0) 0 1/2
(1,0) 2 5/2,3/2
(2,0) 4,2 9/2,7/2,5/2,3/2
(3,0) 6,4,3,0 13/2,11/2,9/2,(7/2)2,5/2,1/2
(4,0) 8,6,5,4,2 17/2,15/2,13/2,(11/2)2,(9/2)2,7/2,5/2,3/2
(1,1) 3,1 7/2,5/2,3/2,1/2
(2,1) 5,4,3,2,1 11/2,(9/2)2,(7/2)2,(5/2)2,(3/2)2,1/2
(3,1) 7,6,52,4,32,2,1 15/2,(13/2)2,(11/2)3,(9/2)3,(7/2)3,(5/2)3,(3/2)2,1/2
ξ,τB,τ
F,(τ1,τ
2),L
BF ,1/2; J||T(λ)
C||ξ,τ
B,τ
F,(τ
1,τ
2),L

BF ,1/2; J
=
LB,LF,L
B,L
F(τB,0) (τF,0) |(τ1,τ
2)
LBLF|LBF (τ
B,0) (τ
F,0) |(τ
1,τ
2)
L
BL
F|L
BF 
×
j,j
(−1)L
B+J+j+λ(2LBF +1)(2L
BF +1)(2J+1)(2J+1)(2j+1)(2j+1)
×LBLFLBF
1/2Jj
L
BL
FL
BF
1/2JjLBJj
JL
Bλξ,τB,L
B||T(λ)
C||ξ,τ
B,L

BδτF,τ
FδLF,L
Fδj,j,(2.17)
whe e ξ,τB,L
B||T(λ)
C||ξ,τ
B,L

Bis he collec i e pa al-
eady calcula ed o he e en-e en sys em. Fo he e mion
pa
ξ,τB,τ
F,(τ1,τ
2),L
BF ,1/2; J||T(λ)
F||ξ,τ
B,τ
F,(τ
1,τ
2),L

BF ,1/2; J
=
LB,LF,L
B,L
F(τB,0) (τF,0) |(τ1,τ
2)
LBLF|LBF (τ
B,0) (τ
F,0) |(τ
1,τ
2)
L
BL
F|L
BF 
×
j,j
(−1)L
B+J+j+λ(2LBF +1)(2L
BF +1)(2J+1)(2J+1)(2j+1)(2j+1)
×LBLFLBF
1/2Jj
L
BL
FL
BF
1/2JjjJL
B
Jjλ
×j||T(λ)
F||jδξ,ξδτB,τ
BδLB,L
B,(2.18)
whe e j||T(λ)
F||jis a single-pa icle ma ix elemen ha is
ob ained om
j||(a†
j1טaj2)(λ)||j=−
√2λ+1δj2,jδj1,j .(2.19)
Wi h hese gene al exp essions, any mul ipole ansi ion and
momen can be e alua ed. In pa icula , we a e s udying E2
and M1 ansi ions and momen s below.
064316-5

C. E. ALONSO, J. M. ARIAS, AND A. VITTURI PHYSICAL REVIEW C 75, 064316 (2007)
IBFM
E(5/12)
ξ=1 ξ=1
ξ=2
ξ=2
(τ1
B+1,0) (τ1
B,1) (τ1
B-1,0)
0,0(0,0)
0,0(0,0)
1,0(1,0)
2,0(2,0)
1,0(1,0)
2,0(2,0)
3,0(3,0)
0,1(1,0)
0,1(1,0)
1,1(2,0)
2,1(3,0)
3,1(3,1)
2,1(2,1)
1,1(1,1)
2,1(1,0)
1,1(0,0)
κ=−1/4
κ′=5/2
0,0(0,0)
1,0(1,0)
2,0(2,0)
3,0(3,0)
0,0(0,0)
1,0(1,0)
2,0(2,0)
0,1(1,0)
1,1(2,0)
2,1(3,0)
1,1(1,1)
2,1(2,1)
3,1(3,1)
1,1(0,0)
2,1(1,0) 0,1(1,0)
167
100 145
145
63 88
143
161
90
60
81
108
100
100 217
75 124
52
31
24
0.5
0.7
118
179
145 211
111
74
45
25
110
9
11
10
41
12
16
30
8
35
18
29
17
12
0.07
0.2
2
2
FIG. 3. Uppe ame: Ene gy le els in he odd sys em a
he E(5/12) c i ical poin (no malized o he ene gy o he i s
exci ed s a e). Each (degene a e) s a e is cha ac e ized by he
ξ,τB,τ
F,(τ1,τ
2) quan um numbe s. The alues k=−1/4andk=
5/2 we e used in he Hamil onian (2.7). The alues o he angula
momen a o each degene a e s a e can be ex ac ed om Table I.
Some selec ed alues o B(E2), co esponding o he ansi ions
be ween he highes spins o ini ial and inal mul iple s, a e gi en
in he igu e [ he B(E2) associa ed wi h he lowes 5/2-1/2 ansi ion
is no malized o 100]. Lowe ame: Ene gy le els in he odd sys em
wi hin he IBFM Hamil onian (3.6) a he c i ical poin . The numbe
o bosons N=7 has been assumed. The alues o kand ka e he
same as in he E(5/12) Hamil onian. Fo B(E2), see he uppe ame.
A. E2 ansi ion and momen s
Le us i s conside he elec ic quad upole ope a o . This
can be w i en as he sum o he collec i e con ibu ion plus
-2 -1 0 1 2
κ
-2
-1
0
1
2
3
4
5
Ene gy (a bi a y uni s)
-1 0 1 2 3 4
κ ′
κ ′=5/2 κ =−1/4
0,1 (1,0)
0,0 (0,0) 0,0 (0,0)
0,0 (0,0)
0,0 (0,0)
1,1 (1,1)
1,1 (0,0)
1,0 (1,0) 1,0 (1,0)
0,1 (1,0)
1,1 (0,0)
1,1 (1,1)
FIG. 4. Beha io o he ew lowes ene gy E(5/12) bandheads as
a unc ion o pa ame e s kand kin he Hamil onian.
he odd-pa icle one:
T(E2) =T(E2)
C+T(E2)
F.(2.20)
Fo he collec i e pa , usually a quad upole ope a o depend-
ing linea ly on βis used in he collec i e models
T(E2)
Cµ= (2)
Cα2,µ = (2)
CβD(2)
µ0(θi) cos γ
+1
√2D(2)
µ2(θi)+D(2)
µ−2(θi)sin γ,(2.21)
whe e (2)
Cis a scale ac o . Howe e , he impo ance o he nex
e m quad a ic in βhas been shown. Because o ha we will
include he nex o de in he quad upole coo dina es, which is
gi en by [67]
T(E2)
Cµ= (2)
Cα2µ+χ
2√7[α2×α2](2)
µ
= (2)
C
m
D(2)∗
µm ¯
Q(2)
m,(2.22)
whe e
¯
Q(2)
0=βcos γ−1
14χβ2cos (2γ),
¯
Q(2)
2=Q(2)
−2=1
√2βsin γ+1
14χβ2sin (2γ),
¯
Q(2)
1=¯
Q(2)
−1=0.
(2.23)
The 2√7 denomina o in Eq. (2.22) is in oduced he e o
con enience o ma ch he exp ession based on he in e ac ing
boson model (IBM) quad upole ope a o . The χpa ame e
con ols he weigh o he quad a ic ela i e o he linea e m.
The α2µ a iable beha es as he O(5) ⊃O(3) enso (τ,L)=
(1,2), while he second-o de e m [α×α](2)
µis associa ed
wi h he enso (2,2) [67]. This means ha hey ul ill s ic
selec ion ules in he φτµLM(γ,θi) collec i e space: τB=±1
o he o me and τB=0,±2 o he la e , wi h he usual
es ic ions in angula momen um coupling [31,67]. Thus he
i s e m leads o “allowed” ansi ions in i s o de , and he
second o nonze o quad upole momen s and o small alues
o “ o bidden” ones, in he E(5) scheme o Re . [4].
Fo he single-pa icle pa , in gene al,
T(E2)
F=
j,j
(2)
j,j(a†
jטaj)(2),(2.24)
whe e (2)
j,ja e pa ame e s. To educe he numbe o pa ame e s,
he ollowing o m dic a ed by he U(6/12) symme y has been
used:
T(E2)
F= (2)
F−4
5(a†
1/2טa3/2+h.c.)(2)
−6
5(a†
1/2טa5/2+h.c.)(2).(2.25)
064316-6
SHAPE PHASE TRANSITION IN ODD NUCLEI IN A . . . PHYSICAL REVIEW C 75, 064316 (2007)
1
1
1
1
5
5
5
106
5
3
33
3
53
3
7
9
977
167 17
151
134
33
50/117
100 100
118
31 15 24
52/35
0.50.5
3412
95
34/6/13 35/83
25/99
75
75
B(E2)
74/50
86/37
5
49/74
111/12
∆τ = ±1
B
E(5/12)
FIG. 5. B(E2) s eng hs induced by he linea e m o he
collec i e E2 ope a o o he E(5/12) model. Spins gi en a e wice
he ac ual alue. T ansi ions obey he selec ion ule τB=±1. The
ansi ion B(E2; 5/21→1/21) is no malized o 100. To ge absolu e
alues, all he s eng hs ha e o be mul iplied by (0.07453)( (2)
C)2β2
w.
Fo he odd sys em, con ibu ions om bo h collec i e and
single-pa icle ope a o s a e o be calcula ed. In he uppe
panel o Fig. 3,someB(E2) alues a e p esen ed o he
collec i e con ibu ion and ansi ions be ween he la ges spin
in each degene a e mul iple . In Figs. 5and 6,B(E2) alues
o τB=±1 and τB=0,±2 a e p esen ed. In he i s
case, only he linea e m is e ec i e; while in he second case,
only he quad a ic con ibu ion plays a ole.
1
1
1
1
5
5
55
5
3
33
3
53
3
7
9
977
56 12 14
122
110
100
61/15
23/52
73
E(5/12) B(E2)
∆τB = 0, ±2
0
23 56
50
118
45
26
3
116 90
254/23/177/20
21/257/39/158
355/236
0/160/45
60/91
449/300
310
310
FIG. 6. B(E2) s eng hs induced by he quad a ic e m o he
collec i e E2 ope a o o he E(5/12) model. Spins gi en a e wice
he ac ual alue. T ansi ions obey he selec ion ule τB=0,±2.
The ansi ion B(E2; 5/21→3/21) is no malized o 100. To
ge absolu e alues, all he s eng hs ha e o be mul iplied by
(0.0007855)χ2( (2)
C)2β4
w.
1
1
1
1
1
5
55
5
0.7
33
3
5
3
7
9
977
53
1.73
1.58
-0.37
-0.26
0.86 0.60
00
1.11
1.01
-0.24
-0.17
0.29
0.25
-0.41
E(5/12)
collec i e Q
3
FIG. 7. Quad upole momen s induced by he collec i e quad a ic
e m in T(E2) o he E(5/12) model. Spins gi en a e wice he ac ual
alue. The quad upole momen o he s a e 5/21is no malized o 1. To
ge absolu e alues, all he quad upole momen s ha e o be mul iplied
by (0.12229)χ (2)
Cβ2
w.
Collec i e quad upole momen s induced by he quad a ic
e m in he E2 ope a o can be calcula ed as
eQ(ξ,τB,τ
F,(τ1,τ
2),L
BF,1/2; J)
=16π
5J(2J−1)
(J+1)(2J+1)(2J+3)
×ξ,τB,τ
F,(τ1,τ
2),L
BF,1/2; J||T(E2)
C+T(E2)
F||
×ξ,τB,τ
F,(τ1,τ
2),L
BF,1/2; J.(2.26)
In Fig. 7, he collec i e quad upole momen s induced by
he quad a ic e m in T(E2)
Ca e p esen ed no malized o he
quad upole momen o he 5/21s a e.
The e mionic quad upole ope a o (2.25) also con ibu es
o some ansi ions. Howe e , he s uc u e (2.25) p o ides
a selec ion ule τF=±1, and consequen ly he e a e no
con ibu ions o he e mion pa o he quad upole momen s.
Fo he same eason, he e a e no ansi ions whe e in e e ence
be ween he collec i e and he e mionic pa s exis s ( o he
collec i e pa , τF=0). In addi ion, he e mionic E2
ope a o p o ides a selec ion ule τB=0. In Fig. 8,a ew
e mionic E2 ansi ions induced by he ope a o (2.25)a e
p esen ed.
B. M1 ansi ion and momen s
The magne ic dipole ope a o can be w i en as he sum o
he collec i e con ibu ion plus he odd-pa icle one:
T(M1) =T(M1)
C+T(M1)
F.(2.27)
The collec i e con ibu ion is
T(M1)
C= (1)
C1ˆ
LB
√10 + (1)
C2α2׈
LB
√10(1)
+.... (2.28)
064316-7
C. E. ALONSO, J. M. ARIAS, AND A. VITTURI PHYSICAL REVIEW C 75, 064316 (2007)
1
1
1
1
5
5
5
5
E(5/12)
3
e mionic B(E2)
3
3
0.1/0.9
∆ τF = ± 1
3
97
7
1
1
0.8/0.2
0.3/0.7
0.2/0.8
0.9/01
0.7/0.3
0.4/0.6
0.6/0.4
FIG. 8. B(E2) s eng hs induced by he e mionic ope a o (2.25)
in T(E2) o he E(5/12) model. Spins gi en a e wice he ac ual alue.
The s eng h om he s a e [ξ=1,τ
B=0,τ
F=1,(τ1=1,τ
2=
0)LBF =2,J =5/2] o he g ound s a e is 1. To ge absolu e alues,
all s eng hs ha e o be mul iplied by ( (2)
F)2.
The i s e m is p opo ional o he collec i e angula mo-
men um ˆ
LBand is diagonal o he e en-e en co e bu no o
he odd sys em. The ansi ion s eng hs and dipole momen s
a ising om his e m a e p esen ed in Figs. 9and 10.
The e mionic e m is p opo ional o he e mion angula
momen um j. Fo he case we a e s udying, he app op ia e
ope a o is
T(M1)
F=− (1)
F1
2(a†
1/2טa1/2)(1) +√5(a†
3/2טa3/2)(1)
+35
2(a†
5/2טa5/2)(1).(2.29)
The e mionic magne ic dipole ope a o (2.29) also con ibu es
o some ansi ions wi h he selec ion ule τB=0. Since
1
1
1
1
5
5
55
5
3
33
3
53
3
7
9
97 7
111 0 100 0
0
100
5
28
100
E(5/12) B(M1)
∆τB = 0, ±1
0
25
27 21
875
321/29/400
1000/500
788/88
313/9/429
53
100*
7*
96*/11*
FIG. 9. B(M1) ansi ion s eng hs induced by he collec i e M1
ope a o (2.28) o he E(5/12) model. Spins gi en a e wice he ac ual
alue. Bo h τB=0andτB=±1 ansi ions a e included. The
τB=0 ansi ions a e no malized o he B(M1; 5/21→3/21)=
100. In o de o ge absolu e alues B(M1) wi h τB=0ha e o
be mul iplied by (0.04)( (1)
C1)2.TheτB=±1 a e ma ked wi h an ∗
and a e no malized o he B(M1; 5/22g→5/21)=100 (5/22gis
he second 5/2 wi hin he g ound s a e band). To ge absolu e alues,
B(M1) wi h τB=±1 ha e o be mul iplied by (0.02445)( (1)
C2)2.
his selec ion ule is iden ical o his pa and he collec i e
pa o he dipole ope a o , he e will be in e e ence be ween
bo h e ms. Table II p esen s combined s eng hs o selec ed
ansi ions in which in e e ence be ween collec i e and single-
pa icle e ms a e impo an .
Figu es 3–10 and Table II gi e de ailed spec oscopic
in o ma ion o he E(5/12) model. In he nex wo sec ions,
calcula ions a he c i ical poin in he ansi ion om sphe ical
TABLE II. T(M1) educed ma ix elemen s o B(M1) and magne ic momen s
in ol ing in e e ence e ec s be ween collec i e and single-pa icle con ibu ions o
some selec ed E(5/12) s a es.
Ini ial |iFinal | Reduced ma ix elemen
|ξ,τB,τ
F(τ1,τ
2)LBF,J|ξ,τB,τ
F(τ1,τ
2)LBF,J ||T(M1)||i
1,1,0 (1,0)2,5/2 1,1,0 (1,0)2,3/2 23
5 (1)
F−0.4899 (1)
C1
1,1,0 (1,0)2,5/2 1,1,0 (1,0)2,5/2 21
10 (1)
F+1.833 (1)
C1
1,1,0 (1,0)2,3/2 1,1,0 (1,0)2,3/2 −3
5 (1)
F+1.4697 (1)
C1
1,2,0 (2,0)4,9/2 1,2,0 (2,0)4,7/2 2√10
3 (1)
F−0.6667 (1)
C1
1,2,0 (2,0)2,5/2 1,2,0 (2,0)2,3/2 23
5 (1)
F−0.4899 (1)
C1
1,2,0 (2,0)4,9/2 1,2,0 (2,0)4,9/2 55
18 (1)
F+4.4222 (1)
C1
1,2,0 (2,0)4,7/2 1,2,0 (2,0)4,7/2 −√14
3 (1)
F+3.9441 (1)
C1
1,2,0 (2,0)2,5/2 1,2,0 (2,0)2,5/2 21
10 (1)
F+1.8330 (1)
C1
1,2,0 (2,0)2,3/2 1,2,0 (2,0)2,3/2 −3
5 (1)
F+1.4697 (1)
C1
064316-8
SHAPE PHASE TRANSITION IN ODD NUCLEI IN A . . . PHYSICAL REVIEW C 75, 064316 (2007)
1
1
1
1
1
5
55
5
0.9
33
3
5
3
7
9
977
53
2
1.94
1
0.9
10.9
00
1
0.97
0.5
0.45
0.75
0.71
0.25
E(5/12)
collec i e µ
3
3
0.17
0
FIG. 10. Magne ic momen s induced by he leading collec i e
e m in T(M1) o he E(5/12) model. Spins gi en a e wice he ac ual
alue. The magne ic momen o he s a e 5/21is no malized o 1. To
ge absolu e alues, all he magne ic momen s ha e o be mul iplied
by (1.29442) (1)
C1.
o de o med γ-uns able shapes wi hin he IBFM amewo k a e
p esen ed. Two di e en IBFM Hamil onians o desc ibing
he c i ical poin a e used in o de o see whe he he signs o
c i icali y p oduced by E(5/12) a e obus .
Be o e ending his sec ion, we would like o no e ha
ou E(5/12) model yields an in ini e numbe o le els, i
co esponds o sol ing he Boh equa ion o a γ-independen
in ini e squa e well in βpo en ial. This will no be he case
o he IBFM calcula ions, which a e associa ed wi h a ini e
numbe o bosons and consequen ly cha ac e ized by a ini e
numbe o le els. In his la e case, in addi ion, he spec um
i sel will change acco ding o he chosen numbe o bosons,
in pa icula , i s highe pa . The compa ison be ween he wo
app oaches will he e o e necessa ily be signi ican only o he
lowe pa o he spec um as in he p e ious E(5/4) model o
odd-e en nuclei, in he E(5), X(5) models o e en-e en nuclei
conce ning c i ical poin s, o , in gene al, when compa ing he
collec i e model wi h he IBM.
III. AN INTERACTING BOSON-FERMION MODEL
HAMILTONIAN THAT MIMICS E(5/12) AT
THE CRITICAL POINT
Fo he odd-e en sys em, he IBFM Hamil onian is w i en
as
H=HB+HF+VBF,(3.1)
whe e HBis he bosonic pa , HF he e mionic one, and he
e m VBF couples he boson and e mion deg ees o eedom.
Fo he bosonic pa , we will use a pa ame ized Hamil onian
ha p oduces a ansi ion be ween sphe ical and γ-uns able
shapes o he o m
HB=xˆnd−1−x
Nˆ
QB·ˆ
QB,(3.2)
whe e he ope a o s in he Hamil onian a e gi en by
ˆnd=
µ
d†
µdµ,(3.3)
ˆ
QB=(s†×˜
d+d†×˜s)(2),(3.4)
and Nis he o al numbe o bosons. This Hamil onian can be
ecas in o he o m
HB=xC1(UB(5)) −1−x
2NC2(OB(6)) −C2(OB(5)),
(3.5)
whe e C1(UB(5)) is he linea Casimi ope a o o he UB(5)
g oup, while C2(OB(6)) and C2(OB(5)) a e he quad a ic
Casimi ope a o s o he OB(6) and OB(5) g oups. We ecall
ha his Hamil onian p oduces, when a ying he pa ame e
x om 1 o 0, a phase ansi ion be ween he wo ex eme
si ua ions cha ac e is ic o U(5) and O(6) symme ies, a
xc=4N−8
5N−8. No e ha o any alue o x his Hamil onian
main ains he degene acies ypical o he O(5) symme y.
Consis en wi h his, wi hin he IBM cohe en s a e o malism
[32–34], his Hamil onian always p oduces an ene gy su ace
which is independen o he γdeg ee o eedom. In he β
a iable, he ene gy su aces display a sphe ical minimum in
β=0 o xla ge han he c i ical alue, while ha ing a
de o med minimum o alues o xsmalle han he c i ical
alues [4,41,58]. I is wo h men ioning ha when conside ing
he combined boson- e mion sys em, he ene gy su ace has
con ibu ions om bo h bosons and e mions. The con ibu ion
om he e mion is o he o de 1/N compa ed wi h he
one om he bosons. Thus, usually i is accep ed ha he
ene gy su ace o he combined sys em is go e ned by he
bosons wi h sligh con ibu ions om he e mionic deg ees o
eedom. Howe e , close o he c i ical poin , he boson ene gy
su ace is a he la , and he con ibu ion om he odd e mion
and i s coupling o he boson co e will play a undamen al ole
in de e mining he minimum in he po en ial ene gy su ace.
This is an impo an opic ha emains o be in es iga ed in
de ail. Fo he pu poses o he p esen pape , i.e., checking
he analy ical esul s ob ained o he E(5/12) model, we will
accep ha he c i ical poin o he combined sys em is jus he
c i ical poin o he boson pa and he e mion is coupled o i
wi hou changing he equilib ium con igu a ion.
Depending on he selec ion o HF+VBF in Eq. (3.1),
di e en model Hamil onians o s udy he c i ical poin a e
p oduced. In his sec ion HF+VBF is selec ed so as o ha e
a IBFM Hamil onian as close as possible o he E(5/12) one.
Thus, ou p oposal is
HF+VBF =k
2N[C2(OBF(5)) −C2(OB(5)) −C2(OF(5))]
+k
2NC2(OF(5)),(3.6)
whe e C2(OF(5)),C
2(OBF(5)) a e he quad a ic Casimi op-
e a o s o he OF(5) and OBF(5) algeb as, espec i ely. As
men ioned abo e his Hamil onian is designed o mimic as
much as possible he co esponding Hamil onian in E(5/12),
since he e ms 2 ˆ
LB◦ˆ
LFand ˆ
L2
Fin Eq. (2.7) can be w i en in
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