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PHYSICAL REVIEW C 72, 061302(R) (2005)
Phase ansi ions in he in e ac ing boson e mion model: The γ-uns able case
C. E. Alonso,1J. M. A ias,1L. Fo una o,2and A. Vi u i2
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 5 Sep embe 2005; published 27 Decembe 2005)
The phase ansi ion a ound he c i ical poin in he e olu ion om sphe ical o de o med γ-uns able shapes is
in es iga ed in odd nuclei wi hin he in e ac ing boson e mion model. We conside he pa icula case o an odd
j=3/2 pa icle coupled o an e en-e en boson co e ha unde goes a ansi ion om sphe ical U(5) o γ-uns able
O(6) si ua ion. The pa icula choice o he j=3/2 o bi al p ese es in he odd case he condi ion o γ-ins abili y
o he sys em. As a consequence, ene gy spec um and elec omagne ic ansi ions, in co espondence o he
c i ical poin , display beha io s quali a i ely simila o hose o he e en co e. The esul s a e also in quali a i e
ag eemen wi h he ecen ly p oposed E(5/4) model, al hough ew di e ences a e p esen , due o he di e en
na u e o he wo schemes.
DOI: 10.1103/PhysRe C.72.061302 PACS numbe (s): 21.60.Fw, 21.60.E
The s udy o phase ansi ions has ecen ly ecei ed
pa icula a en ion in nuclea s uc u e. The concep o c i ical
poin symme y has been i s p oposed in a numbe o cases
by Iachello [1–3]. These symme ies apply when a quan al
sys em unde goes ansi ions be ween adi ional dynamical
symme ies, as o example hose cha ac e izing si ua ions
desc ibed in e ms o ha monic ib a ions o igid o a ions.
Al hough hese symme ies ha e been ob ained wi hin he
o malism based on he Boh Hamil onian [4], hei concep
has also been used in connec ion wi h he in e ac ing boson
model (IBM) [5].
One o hese c i ical poin symme ies is associa ed wi h he
ansi ion be ween sphe ical and γ-uns able shapes. Wi hin he
IBM his can be ob ained, o example, om he Hamil onian
HB=xˆ
nd−1−x
N
ˆ
QB·ˆ
QB,(1)
which p oduces, a ying he pa ame e x om 1 o 0, a
ansi ion be ween he wo ex eme si ua ions cha ac e is ic o
U(5) and O(6) symme ies. The co esponding second-o de
shape phase ansi ion has been in es iga ed wi hin he Boh
collec i e model in Re . [6]. The ope a o s appea ing in he
Hamil onian abo e a e gi en by
ˆ
nd=
µ
d†
µdµ,(2)
ˆ
QB=(s†×˜
d+d†×˜
s)(2),(3)
and Nis he o al numbe o bosons. Fo any alue o x his
Hamil onian main ains he ypical degene acies o he O(5)
symme y. Consis en ly wi h his, wi hin he IBM cohe en
s a e o malism [7–9], 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 ace displays a
sphe ical minimum in β=0 o xla ge han he c i ical alue
xc=4N−8
5N−8, while ha ing a de o med minimum o alues o x
smalle han he c i ical alue. A he c i ical poin , he ene gy
su ace acqui es a β4beha io [10,11], which is app oxima ed
by an in ini e squa e well in he E(5) c i ical poin symme y [1]
wi hin he amewo k o he collec i e Boh Hamil onian.
Recen ly Iachello has discussed a supe symme ical ex en-
sion o his concep , in oducing he so-called E(5/4) model,
whe e he boson pa has a γ-independen squa e well po en ial
and he boson- e mion coupling is aken as a spin (5) scala
in e ac ion [12]. This solu ion desc ibes he spec al p ope ies
o odd-e en nuclei a he ansi ion be ween sphe ical and
γ-uns able shapes. In his Rapid Communica ion we wan o
discuss he e olu ion o odd nuclei and he co esponding
beha io a he c i ical poin , wi hin he amewo k o he
in e ac ing boson e mion model (IBFM) [13], in which a
single e mion is coupled o he e en-e en bosonic co e. The
sys em will hen be desc ibed by he Hamil onian
H=HB+HF+VBF ,(4)
whe e he e m VBF couples he bosonic and e mionic pa s.
In ou case we will assume he boson Hamil onian o be o
he o m gi en in Eq. (1). Fo he e mion and boson- e mion
pa s we will ake he pa icula choice o a pa icle mo ing in
a single j-shell j=3/2 and a coupling e m o he o m
VBF =−21−x
N
ˆ
QB·ˆ
qF,(5)
whe e ˆ
QB[ aken o he o m gi en in Eq. (3)] and ˆ
qF=
(a†
3/2ט
a3/2)(2) a e he boson and e mion quad upole ope a-
o s, espec i ely.
This choice o he e mion space and he boson- e mion
in e ac ion is such ha one eco e s, in he ex eme cases, he
Bose-Fe mi symme y [13] associa ed wi h he spinBF(6) g oup
( o x=0) and he ib a ional UB(5) ⊗SUF(4) case ( o
x=1). In analogy wi h he o e all O(5) s uc u e in he
e en case, his selec ion gua an ees he p ese a ion o he
degene acies associa ed wi h he spinBF (5) symme y o any
alue o x. This can be mo e clea ly seen wi hin he in insic
ame o malism [7–9]. In o de o ge ene gy eigen alues
o he case o a j=3/2 pa icle coupled o he g ound
s a e |B(β,γ)boson condensa e one has o diagonalize
he Hamil onian in a space o dimension ou wi h basis
ec o s: |B(β,γ)⊗|j,m. The pu e boson pa o he
case conside ed he e is only a unc ion o βand gi es a
0556-2813/2005/72(6)/061302(4)/$23.00 061302-1 ©2005 The Ame ican Physical Socie y
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0 0.2 0.4 0.6 0.8 1
1-x
0
1
2
3
4
5
6
E / E21
+
0 0.2 0.4 0.6 0.8 1
1-x
0
1
2
3
4
5
6
E / E1s exci ed
N=7 N=7
1/2,5/2,7/2
3/2,5/2,7/2
3/2
(3/2,1/2)
(5/2,1/2)
(1/2,1/2)
(3/2,1/2)
(7/2,1/2)
3/2
(1/2,1/2)
(5/2,1/2)
0+
2+
2+
,4 +
0+
,3 +
,4 +
,6 +
0+
2+
(0)
(1)
(2)
(3)
(1) (0)
15
15
15
15
13
13
13
2σ1
c i ical
poin poin
c i ical
(1/2,1/2)
(3/2,1/2)
(5/2,1/2)
(4)
(0)
9/2,11/2
(2)
FIG. 1. Ene gy le els (no malized o he
ene gy o he i s exci ed s a e) o he e en
and odd sys ems a e displayed as a unc ion
o he pa ame e (1 −x) o he boson (1)
and boson- e mion (4) Hamil onians. A numbe
N=7 o bosons has been assumed in bo h cases,
while he odd pa icle has been aken in he
j=3/2 o bi al. In he le panel (e en case) we
indica e o each le el he τquan um numbe
(in pa en hesis), spin and pa i y. In he igh
panel (odd case) we quo e he (τ1,τ
2) quan um
numbe s (in pa en hesis) and spin. In he ex eme
x=0 case we also indica e he σ1quan um
numbe . The posi ion o he e en c i ical poin is
ma ked.
global diagonal con ibu ion. The pu e e mionic pa is jus
a cons an . Thus, he only emaining pa , VBF , has o be
diagonalized. I s eigen alues, doubly degene a e, u n ou o
be γ-independen ,
E±(β,γ)=E±(β)=±2(1 −x)β
1+β2.(6)
In o he wo ds, he addi ion o he odd pa icle does no des oy
he γ-ins abili y o he sys em, gi ing ise o ene gy su aces
o he di e en odd in insic s a es ha a e s ill γ-independen
[14]. The pa icula beha io o he j=3/2 o bi al was i s
pu in e idence by Bayman and Sil e be g [15] wi hin he
collec i e model.
The esul ing ene gy spec a in he odd sys em a e shown in
he igh panel o Fig. 1 as a unc ion o he con ol pa ame e
1−x. The o al numbe o bosons, N, has been assumed
o be equal o 7. Fo a be e compa ison, we also show
in he le panel o he igu e he co esponding e olu ion
o he spec um in he e en co e. I should be emembe ed
ha he use o a ini e numbe o bosons does no lead
o he same esul s as he co esponding ones ob ained in
he equi alen si ua ion wi hin he Boh Hamil onian, which
is only eached in he limi o in ini e numbe o bosons.
No e also ha he geome ical limi ob ained om he boson
Hamil onian (1) o la ge alues o Nco esponds o he
case o he collec i e po en ial beha ing as β4, and no
p ecisely o he E(5) case, ha co esponds o he in ini e
squa e well. The le el e olu ion in he odd case shows a
beha io quali a i ely simila o ha o he e en case. The
g oup s uc u e o spinBF(5) wi h espec o O(5) simply leads
o a iche pa e n o he e mion case and sligh ly di e en
a ios o he ene gy le els. Fo example in he limi ing x
=0 case, co esponding o spinBF(6) and O(6) symme y
g oups o he odd and e en nuclei, espec i ely, we ob ain o
he “g ound bands,” wi h maximum σ alues (σ1=N+1/2
and σ=N o odd and e en nuclei, espec i ely), he a ios
E(τ1=5/2,τ
2=1/2)/E(τ1=3/2,τ
2=1/2) =2.4 and
E(τ1=7/2,τ
2=1/2)/E(τ1=3/2,τ
2=1/2) =4.2 wi h
espec o he alues E(τ=2)/E(τ=1) =2.5 and
E(τ=3)/E(τ=1) =4.5 o he e en case.
As in he e en case, he le els ha in he ex eme x=0 limi
e en ually co espond o he “g ound band” (σ1=N+1/2)
show a a he smoo h (o e en la ) beha io , aside om he
changes a ound he c i ical poin . Ins ead, he le els ha will
end up wi h o he alues o σ1(as he le els co esponding
o smalle alues o σin he e en case) show a mo e iolen
a ia ion in he whole ange. A ypical case is he second
(τ1=1/2,τ
2=1/2)j=3/2 s a e ha s a s a one-phonon
ene gy in he ib a ional limi o end up as he hi d j=
3/2 s a e in he opposi e limi . The posi ion o his 3/2 s a e
is he key elemen o cha ac e ize he pa icula si ua ion and
i s posi ion along he ansi ional pa h. The posi ion o his
s a e plays he same ole as he key posi ion o he i s exci ed
0+s a e in e en nuclei. The smoo he beha io o he g ound
band le els wi h espec o he o he bands con i ms he ac ha
o es ablish a de ini e c i ical si ua ion i is no a all su icien
o ely jus on he sequence o ene gy le els in he “g ound”
band, and ha some o he claims o he occu ence o de ini e
ansi ional symme ies ha e o be aken wi h se ious cau ion.
C ucial and selec i e in o ma ion on nuclea spec a comes
om he ansi ion p obabili ies, in pa icula om he elec ic
quad upole ones. These a e ob ained in e ms o he ma ix
elemen s o he elec ic quad upole ope a o . In ou IBFM
case his ope a o is gi en by
ˆ
QBF =ˆ
QB+ˆ
qF=(s†×˜
d+d†×˜
s)(2) +(a†
jט
aj)(2).
(7)
No e ha , consis en ly wi h he boson quad upole ope a o
used in he Hamil onian, we ha e no included any (d†×˜
d)(2)
e m in he ansi ion ope a o .
Values o he indi idual ansi ion p obabili ies, s a e by
s a e, o he odd nucleus a he c i ical poin si ua ion a e
shown in Fig. 2. In i , we ha e plo ed he lowes six mul iple s,
which ha e been a bi a ily spli in o de o show he possible
061302-2
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3/2 1(1/2,1/2)
1/2
(3/2,1/2) 5/2 1
7/2
3/2
2.29 (5/2,1/2) 5/2
7/2
9/2
11/2
3/2 21.40 (1/2,1/2)
1/2
5/22.64 (3/2,1/2) 7/2
3/22.91 (1/2,1/2)
(ττ
1, 2)
0
1

1
2
3
ξ
100
19.2
13.1
12.7
0.13
8.49
2.29
0.61
5.7
4.4
141.1
30.2
110.9
69.1
72.0
34.8
21.6
84.6
16.1
75.6
49.4
81.6
13.1
8.9
1.02
0.76
0.25
22. 7
0.95
0.20
0.37
0.14
0.04
1.0
0.49
0.12
0.27
1.37
0.54
5.6
5.4
6.7
0.15
3.64
10.9
0.24
0.83
0.62
0.21
13.7
14.8
11.1
3.7
0.51
j
FIG. 2. Ene gy le els and quad upole ansi ion a es B(E2↓) o he odd sys em a he co e c i ical poin . Fo illus a ion pu poses he
a ious mul iple s, labeled by he (τ1,τ
2) quan um numbe s in pa en hesis, ha e been a bi a ily spli acco ding o hei jquan um numbe .
The label on he ex eme le is he ene gy in ela i e uni s, while he label a he ex eme igh co esponds o he label ξ,usedinRe .[12].
B(E2↓)’s ha e been no malized o he alue 100 o he ansi ions (wi h equal s eng hs) be ween he s a es o he i s mul iple and he
g ound s a e.
E2 ansi ions be ween he di e en s a es. In he igu e he
le els ha e di e en leng hs depending on he ξ alue, which
labels di e en amilies and i is shown along wi h he spin
o he s a e (we con o m he e o he no a ion in oduced in
Re . [12]). Al hough he gene al pa e n emains he same, he
de ailed ene gy sequence depends on he numbe o in e ac ing
bosons, N. Owing o he p ope ies o he spinBF(5) g oup,
he τ1=0,±1,τ
2=0 selec ion ule s ill hold. I can be
obse ed ha E2 ansi ions a e s onge be ween s a es wi h
he same ξ alue and τ1=1. T ansi ions be ween s a es
pe aining o amilies wi h di e en ξa e one o wo o de s o
magni ude smalle . T ansi ions be ween s a es wi h he same
ξand τ1 alues, bu di e en spins, co esponding o he same
mul iple s a e also one o wo o de s o magni ude smalle
han he ones be ween di e en mul iple s in he same band.
The allowed quad upole ansi ions a e ske ched in Fig. 3
o he odd sys em o he spinBF(6) case and o he c i ical
poin si ua ion, and o he e en case a he c i ical poin . In
addi ion o he di e ence in he le el sequence, in he second
case ansi ions a e allowed be ween di e en bands (al hough
weake han inband ansi ions, as al eady men ioned), in a
simila way as in e band ansi ions a e allowed a he c i ical
poin o he e en case (as o example be ween he second
0+, wi h τ=0, and he i s 2+, wi h τ=1, as shown in he
igu e).
The compa ison o ou c i ical poin solu ion wi h Iachello’s
one e idences a simila o e all o ganiza ion o he spec um,
al hough wi h sizable di e ences in he ela i e posi ion o
he di e en ξbands. I mus be ecalled ha , in p inciple, we
should compa e he E(5/4) solu ion wi h he la ge Nlimi o he
in e ac ing boson model, while we ha e concen a ed, along
his pape , on he N=7 case. The i s τ1=5/2 mul iple in
he E(5/4) lies a a ound 2.20, close o he alue 2.29 ob ained
in ou model. On he o he hand, he wo i s mul iple s o
he ξ=2 amily o exci ed s a es o he E(5/4) solu ion lie a
a ound 3.33 and 5.02, compa ed wi h he co esponding alues
1.40 and 2.64 o ou model. This in e sion o he posi ion o
1/2
σ1=15/2
3/2
5/2
τττ
1.
.
.
σ1= 13/2
1/2
3/2
.
.
.
SpinBF (6)
1/2
3/2
5/2
1.
.
.3/2
.
.
.
1/2
.
.
.
C i ical poin - Odd case
0
1
2
.
.
.0
1
.
.
.
C i ical poin - E en case
1/2
FIG. 3. Schema ic illus a ion o allowed quad upole ansi ions
o he odd sys em a he spinBF (6) (le panel) and c i ical poin
(middle panel) cases, and o he e en case a he c i ical poin ( igh
panel). Solid lines indica e s onge in-band ansi ions wi h espec
o he in e band ansi ions shown as dashed lines. Ho izon al lines
co espond o mul iple s including se e al j alues.
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0 0.2 0.4 0.6 0.8 1
1-x
0
1
2
E (MeV)
0 0.2 0.4 0.6 0.8 1
1-x
0
1
2
E (MeV)
3/2
1/2,5/2,7/2
3/2
3/2 1/2,
5/2,
7/2
5/2
7/2
1/2
9/2
3/2
5/2
3/2,5/2,
7/2,9/2,
11/2
FIG. 4. Ene gy le els o he odd sys em
a e displayed as a unc ion o he pa ame e
(1 −x) o he boson- e mion Hamil onian (4).
A numbe N=7 o bosons has been assumed,
while he odd pa icle has been aken in he j=
3/2 o bi al (le panel) and j=5/2 o bi al ( igh
panel). The spin quan um numbe is indica ed o
each s a e.
he ξ=1,τ
1=5/2 and ξ=2,τ
2=1/2 mul iple s is he i s
e iden di e ence be ween he wo models. Ano he impo an
di e ence is seen in he ela i e B(E2) alues: al hough
in-band ansi ions display compa able alues, signi ican
disc epancies a e obse ed in in e band ansi ion (no ably
he ansi ions be ween he ξ=2,τ
1=1/2 s a e and he wo
lowes mul iple s o he g ound s a e band).
P ese ing he degene acies associa ed wi h he spinBF(5)
symme y is no an exclusi e p ope y o he j=3/2 o bi als.
O he cases a e known in he li e a u e, o example he case o
he odd pa icle mo ing in he j=1/2, 3/2, 5/2 o bi als, unde
he condi ion o special alues o he e mion quad upole
ma ix elemen s. Also in his case one eco e s, o he x=0
limi ing case, he Bose-Fe mi symme y beha io . Bu his
does no happen in mo e gene al cases wi h o he combina ions
o o bi als o e mion quad upole ma ix elemen s [16]. To gi e
an idea, we p esen in Fig. 4 he e olu ion o he spec um as
a unc ion o he con ol pa ame e 1 −xin he cases o a
single j=3/2 and j=5/2. The igu e shows a spec um o
j=5/2 ( igh panel) ha is quali a i ely simila o he one
displayed in he le panel o j=3/2, bu is mo e complex
due o he emo al o all degene acies.
To summa ize, we ha e conside ed, wi hin he in e ac ing
boson e mion model, he coupling o an odd j=3/2 pa icle
o a boson co e ha unde goes a ansi ion om sphe ical
U(5) o γ-uns able O(6) cha ac e . The pa icula choice o
he Hamil onian and o he j=3/2 o bi al p ese es in he
odd case he condi ion o γ-ins abili y o he sys em, and i
is e lec ed in he p ese a ion o he degene acies associa ed
wi h he spinBF(5) symme y. As a consequence, he ene gy
spec um and he elec omagne ic ansi ions o he odd
nucleus wi h a c i ical co e display beha io s quali a i ely
simila o hose cha ac e izing he phase ansi ion in he
e en co e. We ha e compa ed ou esul s wi h he ecen ly
p oposed E(5/4) app oach, based on he Boh Hamil o-
nian. Bo h app oaches display simila quali a i e pic u es,
al hough we e idence a numbe o quan i a i e di e ences,
ha can be aced back o he di e en na u e o he wo
schemes.
We acknowledge enligh ening discussions wi h J. Ba ea and
B. F. Bayman. This wo k was suppo ed by he I alian-Spanish
ag eemen INFN-CICYT and by he Spanish DGICYT unde
p ojec numbe FIS2005-01105.
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