Ex eme quad upole de o ma ion and clus e iza ion
J. Da ai1,a, J. Cseh2, G. Adamian3, and N. An onenko3
1Depa men o Expe imen al Physics, Uni e si y o Deb ecen, Deb ecen, P . 105, Hunga y-4010
2Ins i u e o Nuclea Resea ch, Hunga ian Academy o Sciences, Deb ecen, P . 51, Hunga y-4001
3Bogoliubo Labo a o y, Join Ins i u e o Nuclea Resea ch, 141980 Dubna, Russia
Abs ac . We discuss a simple symme y-adap ed me hod o he de e mina ion o he shape isome s, and o
he s udy o hei possible agmen a ion. In o he wo ds he connec ion be ween he quad upole (collec i e) and
dipole (clus e ) deg ees o eedom is conside ed in e ms o an easily applicable, ye mic oscopic me hod. The
ene ge ics is aken in o accoun by he double- olding me hod. Special a en ion is ocused on hose cases in
which he heo e ical p edic ions ha e a di ec compa ison wi h expe imen al obse a ion.
1 In oduc ion
The beha io o he ini e nuclea ma e unde ex eme
condi ions aises many exci ing ques ions. One o hem
conce ns he case o he la ge quad upole de o ma ion. Su-
pe de o med (SD) s a es ha e been obse ed in se e al nu-
clei, while he appea ance o hype de o ma ion (HD) is a
ho opic o a discussion. Especially in e es ing is hei
exis ence in he N=Znuclei, in which he ole o pai -
ing, qua e ing, isospin, can be s udied. The expe imen al
obse a ion o he SD s a es in hese nuclei s a ed app ox-
ima ely a decade.
The in es iga ion o clus e iza ion o a speci ic nuclea
s a e is impo an om wo aspec s. Fi s i con ibu es o
ou unde s anding o he nuclea s uc u e. Second i gi es
aluable in o ma ion on he nuclea eac ions, which can
popula e he s a e in ques ion. In pa icula , a bina y clus-
e con igu a ion wi h he clus e s in hei g ound in insic
s a e is uniquely ela ed o a eac ion channel, like a ge
and p ojec ile in hei g ound s a e. Ac ually, one o he
possible de ini ion o a bina y clus e con igu a ion is p o-
ided by his kind o eac ion channel. (Since i is di ec ly
ela ed o i s obse a ion, p obably his is he bes way o
de ine a clus e con igu a ion.)
In his con ibu ion we p esen a simple me hod, ha
we ha e applied ecen ly o he de e mina ion o he shape
isome s o some ligh nuclei, and o in es iga ing hei
possible bina y clus e iza ions as well. In bo h s eps sym-
me y-conside a ions played an essen ial ole.
The nex sec ion desc ibes b ie ly he me hodology, whi-
le in Sec. 3 we p esen some esul s. We ocus on hose
cases in which he heo e ical p edic ions o esul s can be
compa ed wi h he expe imen al obse a ion.
2 Me hods
2.1 P elimina ies
The basic connec ion be ween he quad upole shape and
clus e iza ion was ound in he i ies, ia he b idge o
ae-mail: [email p o ec ed]
he shell model. This ela ion is based on he S U(3) sym-
me y. Ellio has shown [1], how he quad upole de o ma-
ion and collec i e o a ion can be de i ed om he sphe -
ical shell model: he s a es belonging o a collec i e band
a e de e mined by hei speci ic S U(3) symme y. Wilde -
mu h and Kanellopoulos [2] es ablished he ela ion be-
ween he shell and clus e models. They ha e shown ha
he Hamil onians o he wo models can be ew i en in o
each o he exac ly in he ha monic oscilla o app oxima-
ion. This ela ion esul s in a close connec ion be ween he
co esponding eigen ec o s, oo: he wa e unc ion o one
model is a linea combina ion o hose o he o he , which
belong o he same ene gy. La e on his ela ion was in-
e p e ed by Bayman and Boh [3] in e ms o he S U(3)
symme y. As a consequence, he clus e bands can be con-
side ed om he shell model iewpoin , like a se o s a es
ha ing special S U(3) symme y, jus like he collec i e o-
a ional bands.
The 1958 S U(3) connec ion was based on he symme-
y o he sphe ical shell model. In ligh o he ac ha
oday e y la gely de o med s a es a e known expe imen-
ally, i is an impo an ques ion, wha happens o his sym-
me y wi h inc easing de o ma ion. The supe de o med s a-
es e.g. ep esen a si ua ion which is close o he axially
symme ic sphe oid wi h a ios o main axis o 2:1:1, he
hype de o med s a es co espond o 3:1:1.
In [4] i was shown ha he symme y algeb a o he
aniso opic ha monic oscilla o is S U(3), when i s equen-
cies a e commensu a e, i.e. exp essed as a ios o in ege
numbe s. As a special case i includes he sphe ical oscilla-
o , as well as he supe de o med o hype de o med shapes.
Mo e de ails, conce ning he axially symme ic case wi h
2:1, 3:1, and 3:2 a ios a e discussed in [5], and o he i-
axially de o med oscilla o in [6]. (Fo p e ious wo ks on
his p oblem we e e o he ci a ions in hese pape s.)
The connec ion be ween he aniso opic ha monic os-
cilla o and clus e iza ion ha e been discussed in [7–9].
2.2 Shape isome s
In wha ollows we pay a en ion o he p oblem o mo e
ealis ic in e ac ions. In hese conside a ions he Nilsson
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model plays an impo an ole, which gi es single nucleon
o bi als o mo e ealis ic in e ac ions, including spin-o bi ,
as well as l2 e ms. Soon a e he expe imen al disco e y
o he supe de o med s a es, Sugawa a-Tanabe e al. ha e
ealized ha he L−Scoupling eco e s o he supe de-
o med shape, i s in a simple model [10], hen in mo e
ealis ic calcula ions [11]. In pa icula hey ound ha he
L−Scoupled sphe ical wa e unc ion componen s became
mo e ha 85% o he o al. The eason o his is he domi-
nan ole o he quad upole in e ac ion due o he la ge de-
o ma ion. This phenomenon p o ides an explana ion o
he appea ance o he pa i y double le els.
We in es iga e he su i al (o appea ance) o he S U(3)
symme y sys ema ically as a unc ion o he quad upole
de o ma ion (including iaxial de o ma ion). We do so in
e ms o he Nilsson-model combined wi h he concep o
he quasi-dynamical (o effec i e) U(3) symme y [12]. I
is a gene aliza ion o he concep o he eal U(3) symme-
y, known o be app oxima ely alid in ligh nuclei [1].
The quasi-dynamical symme y is mo e gene al in he ol-
lowing sense. The Hamil onian b eaks he symme y in
such a way ha he U(3) quan um numbe s a e no alid
o i s eigen ec o s ei he (con a y o he case o he eal
U(3) dynamical symme y). In o he wo ds nei he he op-
e a o is symme ic (i.e. i is no a U(3) scala ), no i s
eigen ec o s a e (i.e. hey do no ans o m acco ding o a
single i educible ep esen a ion) [13]. Ye , he symme y
is p esen is some sense.
An asymp o ic Nilsson-s a e se es as an in insic s a e
o he quasi-dynamical S U(3) ep esen a ion. Thus he e -
ec i e quan um numbe s a e de e mined by he Nilsson-
s a es in he egime o la ge de o ma ion [14]. When he
de o ma ion is no la ge enough, hen we can expand he
Nilsson-s a es in he asymp o ic basis, and calcula e he e -
ec i e quan um numbe s based on his expansion [15].
The S U(3) quan um numbe s uniquely de e mine he
quad upole shape o he nucleus [16], hus we ob ain he
shape isome s om hem. In pa icula , a sel -consis ency
calcula ion is pe o med wi h espec o he quad upole
shape o nucleus, which is an al e na i e o he mo e usual
ene gy-minimum calcula ions. I is based on he applica-
ion o he quasi-dynamical U(3) quan um numbe s [17],
and in hose cases when a de ailed compa ison can be made
wi h he adi ional ene gy-minimum calcula ions, he e-
sul s a e e y simila [17–19].
2.3 Clus e iza ion
Once he shape isome s ha e been ound, he nex ques ion
is how hey a e ela ed o clus e con igu a ions. In simple
cases he S U(3) connec ion could p o ide he answe [20].
I was wo king e y simple (e.g. in he g ound-s a e egion)
due o wo ac s ha : i) he S U(3) symme y was app ox-
ima ely alid, and ii) he connec ion be ween he clus e
and shell model s a es was simple, i.e. expanding he clus-
e basis s a e in e ms o shell model basis only a e y
ew (some imes a single) basis s a es ga e con ibu ion [21,
22]. The symme y-b eaking in e ac ion, howe e , mixes
he S U(3) s a es. Fu he mo e, he shell-model expansion
o a clus e s a e is usually e y complica ed, oo; i.e. many
shell-model basis s a es gi e con ibu ion [21,22].
Wha happens wi h he shape isome s? As seen abo e,
he S U(3) symme y eco e s, e en o he symme y-b ea-
king in e ac ions like spin-o bi . I is no necessa ily a eal
S U(3), mo e o en i is a quasi-dynamical S U(3), bu o
ligh nuclei many imes he wo coincide wi h each o he .
Wha can we expec om a selec ion ule, based on he
quasi-dynamical S U(3) symme y? Fi s i has a de ini e
geome ical con en in spi e o he abs ac algeb aic o -
mula ion. I shows how much he quad upole shapes o
he shell-model s a e and he clus e con igu a ion a e con-
sis en wi h each o he [18]. This consis ency is ob ained
om a clus e pic u e, in which he Pauli-p inciple is aken
in o accoun a an app oxima e le el. Namely, single shell-
model con igu a ions can be associa ed o he in e nal s uc-
u e o he clus e s [14], and o hese con igu a ions he
exclusion p inciple is alid. O cou se, when he quasi-
dynamical symme y educes o a eal one, he selec ion
ule gi es an exac esul ( o he ex en he U(3) symme y
is alid). Thus we can say ha he Pauli-p inciple is inco -
po a ed only in an app oxima e, bu well-de ined way, and
i s alidi y can be checked by making a compa ison wi h
he esul s o he ully mic oscopic desc ip ion, whe e hey
a e a ailable.
The selec ion ule is only a necessa y condi ion o he
appea ance o he allowed clus e iza ion, bu no a suffi-
cien one. Ano he simple p esc ip ion is p o ided by Ha -
ey [23] which is also based on he use o he ha monic
oscilla o basis, he e o e, i is easily applicable, oo. This
is also a necessa y condi ion. The con en o he wo e-
qui emen is no he same, in some sense, hey a e comple-
men a y o each o he . Thei ela ion is discussed mo e in
de ail in [19]. The e o e, one should apply bo h p esc ip-
ion in a combined way, and ha is how we used hem in
he cases p esen ed la e on. When a clus e con igu a ion
is o bidden, we cha ac e ize i s o biddenness in a quan i-
a i e way [24].
In sho his pa can be summa ized, by saying ha
condi ion i), i.e. he alidi y o he U(3) symme y is sa -
is ied (in an app oxima e way). I also u ns ou ha con-
di ion ii), i.e. he simple ela ion be ween he clus e and
shell model basis is also alid many imes, in spi e o he
ac ha he shape isome s a e usually highly exci ed s a es.
I is a consequence o he ac ha he highly de o med
s a es a e ealized only in a e y limi ed numbe o shell-
con igu a ions.
The ene ge ic p e e ence ep esen s a complemen a y
iewpoin o he selec ion o clus e iza ion. We inco po-
a e i in wo diffe en ways: i) by applying simple binding-
ene gy a gumen s [25], and ii) by pe o ming double- ol-
ding calcula ions, acco ding o he dinuclea sys em model
[26,27]. In his model he cha ge (mass) asymme y ηZ=
(Z1−Z2)/(Z1+Z2) (η=(A1−A2)/(A1+A2)), whe e Z1(A1)
and Z2(A2) a e he cha ge (mass) numbe s o he hea y and
ligh nuclei o he dinuclea sys em, is he ele an collec-
i e a iable desc ibing he pa i ion o nucleons among he
nuclei o ming he dinuclea sys em. The wa e unc ion
in ηZis hough o be a supe posi ion o he mononucleus
con igu a ion wi h |ηZ|=1 and diffe en clus e - ype con-
igu a ions. The ela i e con ibu ion o each clus e com-
ponen o he o al wa e unc ion is uled by he po en ial
U(ηZ) a |ηZ|<1 [27,28]:
U(ηZ)=V(R=Rm, ηZ)+B1(ηZ)+B2(ηZ)B.(1)
In he case o Z=Nnuclei, ηZ=η. The in e nuclea dis-
ance o Rm=R +0.5 m co esponds o he minimum
16001-p.2
NSRT12
o he nucleus-nucleus po en ial V. He e R is he ouch-
ing dis ance be ween he clus e s, which depends on hei
ela i e o ien a ion. The nega i e quan i ies B1and B2a e
he binding ene gies o he clus e s and Bis he binding
ene gy o he pa en nucleus. The expe imen al g ound-
s a e masses and quad upole de o ma ion pa ame e s [29,
30] a e used in he p esen calcula ions. Since he alues
o Ua e gi en wi h espec o B, U(|ηZ|=1) =0. The
double- olding me hod o he calcula ion o Vas well as
he pa ame e s used a e p esen ed in [28,31]. The mu ual
o ien a ion o nuclei in he dinuclea sys em co esponds
app oxima ely o he esul o he mic oscopic conside a-
ion.
3 Resul s
3.1 28Si
The e has been much discussion ecen ly on he supe de-
o med s a e in 28Si, bo h om he heo e ical and om
he expe imen al sides. An AMD s udy by Taniguchi e al.
[32] delinea es an SD band in 28Si wi h a momen o ine ia
o ≈6~2/MeV. I s s uc u e is domina ed by 24Mg+αand
12C+16O clus e ing. Ano he ecen calcula ion in e ms o
he mac oscopic-mic oscopic model [33] also p edic s an
SD band wi h s ong 12C+16O clus e ing.
F om he expe imen al side, he e a e a se o s a es in
28Si iden i ied in he 12C(20Ne,α)28Si eac ion which ha e
been a ibu ed o he supe de o med band by Kubono e
al. [34]. This sequence has no, howe e , he smoo h cha -
ac e is ics expec ed o such a band.
Jenkins e al. ha e e iewed he a ailable expe imen al
da a, and ex ended hem wi h new γ- ansi ions [35]. As a
esul , hey p opose a new candida e o he SD band. In
pa icula he 6+s a e was iden i ied by B enneisen e al.
[36] a 12.86 MeV, which is popula ed in (α, γ) eac ion,
bu no in (p, γ). In a ecen Gammasphe e measu emen
he 12C(20Ne,α)28Si eac ion was s udied [35]. Double and
iple γ-coincidence we e measu ed, and nea ly all s a es
below 10 MeV ha e been loca ed, as well as essen ially
all known γ-decaying high-spin (J>4) s a es. This wo k
con i ms he loca ion and he decay b anching o he can-
dida e s a e by B ennens ein e al. As a esul , he new
SD candida e band has s a es wi h 2+, 4+, and 6+spin-
pa i ies. Thei γ-decay is s ongly e a ded o he obla e
g ound s a e band, and enhanced o he p ola e band.
Inspi ed by his exci ing si ua ion wi h open ques ions
on he SD s a e, we ha e ca ied ou an independen heo-
e ical analysis o highly-de o med s uc u es in 28Si [37].
In pa icula we pe o med a Nilsson-calcula ion, combined
wi h quasi-dynamical U(3) conside a ions, and de e mined
he allowed bina y clus e iza ions o he shape isome s. As
a esul we ha e ob ained 8 shape isome s om he g ound
s a e up o he alpha-chain s a es. The low-ene gy sec ion,
con aining he g ound (GS), p ola e (P ) and supe de o med-
p ola e (SD) s a es is shown in Table 1.
Fi e ( om ou eigh ) shape isome s we e seen p e i-
ously in ene gy-calcula ions o he Nilsson model [38].
E en be e ag eemen is ecognized wi h B ink-model e-
sul s [39–41]. These s udies ga e also eigh shape isome s,
and se en o hem a e in a one- o-one co espondence wi h
ou esul s.
Table 1. Shape isome s in he 28Si nucleus om he Nilsson
model +quasi-dynamical S U(3) calcula ion. ‘e’ s ands o effec-
i e U(3) quan um numbe s, ‘h’ indica es he s a es co espond-
ing o simple ha monic oscilla o con igu a ions. (p) and (o) mean
p ola e and obla e, espec i ely. The iaxiali y, γ, is gi en in de-
g ees. The penul ima e column indica es he axis a io, while he
inal column p esen s he momen o ine ia (in uni s o ~2/MeV).
S Mod ~ωU(3) β2γa:b:c J
e(p) -1 [13,13,9] 0.17 60 1.2:1.2:1 3.4
GS e(o) 0 [16,15,5] 0.44 55 1.6:1.5:1 4.8
3.5
3.3
h 0 [16,16,4] 0.50 60 1.7:1.7:1 3.4
P e 0 [19,9,8] 0.44 5 1.5:1:1 4.4
4.3
2.8
h 0 [20,8,8] 0.50 0 1.5:1:1 4.5
SD e 4 [27,8,5] 0.81 7 2.2:1.1:1 5.8
5.5
2.3
h 4 [28,8,4] 0.88 9 2.3:1.2:1 6.1
5.7
2.2
We ha e s udied sys ema ically he allowed bina y clus-
e con igu a ions o he shape isome s. In doing so he
clus e s we e conside ed o ha e de o ma ions, like he g o-
und s a es o he co esponding nuclei, and no cons ain
was applied o hei ela i e o ien a ion. The alpha-like
clus e iza ions p o ed o be ene ge ically a o ed.
The GS allows hose clus e iza ions o which he ligh e
clus e is ei he 4He and 8Be, while he P and SD s a es
allow se e al clus e con igu a ions (bo h he effec i e and
he ha monic oscilla o s a es can be clus e ized). Fo he
SD s a e, which was he ocus o ou wo k, we ound ha
he 4He+24Mg, and he 12C+16O clus e con igu a ions a e
he mos p obable, aking accoun o bo h he selec ion
ules and ene ge ic p e e ence. This inding is in line wi h
he ecen AMD [32] and mac oscopic-mic oscopic calcu-
la ions [33], as well as wi h he new expe imen al esul s
[35]. I s p edic ed momen -o -ine ia is nea -iden ical o
ha o he AMD calcula ion, and showed by he expe i-
men , hus i gi es suppo o he new candida e o he SD
s a e [35].
3.2 36A
The SD band o he 36A nucleus was epo ed in [42]. Fol-
lowing he expe imen al obse a ion a conside able heo-
e ical effo has been concen a ed on his band. In [43]
e.g. he possible bina y clus e iza ions o his s a e was
s udied sys ema ically. Simila s udies ha e been done also
o he g ound, and he hype de o med bands. The la e
one had been p edic ed om alpha-clus e model calcula-
ions [44]. One o he in e es ing conclusions o his wo k
[43] was ha he HD s a e o he 36A nucleus could be
popula ed in he 24Mg+12C and 20Ne+16O eac ions.
A ecen analysis o he 24Mg+12C elas ic sca e ing
[45] e ealed he ac ha he c oss sec ion can be de-
sc ibed only by supposing esonances on op o he po-
en ial sca e ing. This e y ca e ul analysis inco po a ed
16001-p.3
EPJ Web o Con e ences
Fig. 1. Exci ed s a es in 36A , obse ed as esonances in he
12C+24Mg (open ci cles) and 16O+20Ne ( ull ci cles) eac ions.
The SD band obse ed in 36A and he g ound-s a e band a e also
included. The shapes a e om Nilsson-calcula ions.
phase-shi s udy, as well as Regge-pole and ene gy-depen-
den esonance calcula ions. The exis ence o i e eso-
nances ha e been p o ed, which ha e angula momen a 2,
4, 6, 7, 8. These s a es oge he wi h he esonances om
he 20Ne+16O eac ions seem o es ablish a o a ional band
[45], as shown in he uppe pa o igu e 1. I s momen
o ine ia is in a e y good ag eemen wi h ha o he HD
shape p edic ed om alpha-clus e model [44]. The simi-
la i y o he (p edic ed and obse ed) momen s o ine ia,
and he ac ha he esonances we e seen in exac ly hose
eac ions, which de ine he p e e ed clus e -con igu a ions
o he HD shape sugges ha he ecen ly obse ed band in
igu e 1 is a good candida e o he hype de o med shape
isome o he 36A nucleus. Fo compa ison also he g ound
and supe de o med bands a e indica ed in igu e 1. The
ene ge ical p e e ences o clus e -con igu a ions ha e been
discussed in de ail in [46].
Since a candida e o he HD s a e showed up, he ex-
ci ing ques ion a ises i such a shape can be seen in shell-
model calcula ion as well. In [17] we ha e ca ied ou Nils-
son-model+quasi-dynamical S U(3) calcula ion in o de o
ind he answe . This calcula ion gi es a HD s a e which
has exac ly he same symme y as ha om he alpha-
clus e model. The momen o ine ia om hese model
ag ees well wi h he one indica ed by he expe imen .
I was an in e es ing inding ha diffe en s a es can be
buil up om he same wo clus e s, like e.g. 12C+24Mg,
as illus a ed in igu e 2. The diffe ence in hese cases is
he ela i e o ien a ion o he wo de o med clus e s. This
esul is a consequence o he ac ha he Pauli-p inciple
was aken in o accoun , and no simpli ying assump ions
we e made on he shapes and ela i e o ien a ions o he
clus e s.
The combina ion o hese indings seems o sugges
ha he la gely de o med band in igu e 1 is a good can-
dida e o he hype de o med s a e, hus 36A may be a nu-
cleus o ha e expe imen al e idence o he exis ence o
g ound, supe de o med, as well as hype de o med bands.
The ques ion o he e na y clus e iza ion o hese shape
isome s ha e been add essed in [47]. In pa icula , hose
possible e na y clus e -con igu a ions ha e been de e min-
Fig. 2. The shape isome s o he 36A nucleus om Nilsson-
model calcula ions, and hei amalgama ion om wo clus e s.
The cen al pa shows he shell-model esul s o he g ound
(a he bo om), supe de o med, iaxial, and hype de o med (a
he op) s a es. The le column co esponds o he 24Mg+12C
clus e iza ion. The igh side illus a es he 32S+4He, 20Ne+16O,
28Si(p ola e)+8Be, 28Si(p ola e)+8Be, 20Ne+16O con igu a ions
( om he bo om), espec i ely. A he uppe mos line also he
alpha-clus e iza ion o he HD s a e is shown om he wo k [44],
whe e he con ou co esponds o he HD s a e o he Nilsson-
model.
ed, which con ain a leas one double-magic clus e s. They
a e p e e ed om he ene ge ic iewpoin .
3.3 40Ca
The si ua ion o he shape isome s o he 40Ca nucleus is
somewha simila o ha o he 36A . In pa icula he SD
band has been obse ed expe imen ally [48], and due o
his ac se e al heo e ical s udies ha e been ca ied ou .
In [49] we ha e in es iga ed he possible bina y clus e i-
za ions in he g ound, supe de o med and a hypo he ical
hype de o med band. The la e one was ob ained om
simple ha monic oscilla o shell model con igu a ions wi h
quad upole de o ma ion, which is conside ed o be app o-
p ia e o he HD s a e. The U(3) selec ion ule has been
applied, and he ene ge ic calcula ions we e ca ied ou .
Bo h o he SD and o he HD s a es he 28Si+12C
clus e iza ion u ned ou o be an impo an one. Fo he
SD shape his inding is consis en wi h he conclusion
o he ully mic oscopic AMD calcula ions [50]. Wo k is
in p og ess [51] in o de o comple e his analysis wi h
he sys ema ic de e mina ion o he shape isome s om
Nilsson-model+qusidynamical U(3)-symme y calcula ions.
A compa ison wi h he esul s o an expe imen al wo k on
he elas ic sca e ing o 28Si+12C is also being done.
3.4 56Ni
The s able elonga ed shapes o he 56Ni nucleus, ob ained
om a Nilsson-model +quasi-dynamical U(3) symme y
calcula ion, is shown in igu e 3. In his igu e i is no
he minima, a he he ho izon al pla eaus, which co e-
spond o he s able shapes. (They a e insensi i e o he
small changes o he inpu pa ame e . Fu he mo e, hese
16001-p.4
NSRT12
Fig. 3. Quad upole de o ma ion o he 56Ni nucleus om he
Nilsson-model wi h he effec i e U(3) quan um numbe s and
schema ic illus a ions o he shape a he pla eaus.
de o ma ions ul ill he sel -consis ency a gumen be ween
he inpu and ou pu de o ma ion-pa ame e s o some ap-
p oxima ion.)
As seen om he igu e, he iaxial g ound-s a e ( o
which he expe imen al de o ma ion is β2=0.173) is ol-
lowed by a p ola e-like de o med s a e o 0~ωexci a ion.
The nex egion o s abili y co esponds o he supe de-
o med shape. This s a e ep esen s 4 nucleon exci a ion,
being e y much in line wi h [52,53]. Then appea s an
e en mo e de o med s a e wi h iaxial shape, and wo p o-
nounced hype de o med shapes close o each o he .
I is ema kable ha a supe de o med, a iaxial and
a hype de o med s a e appea bo h in he alpha-clus e -
model calcula ion [53], and in ou (Nilsson-model-based)
quasi-dynamical symme y conside a ion. The supe de o -
med s a es seem o co espond o each o he exac ly, bo h
o hem being a 4~ωexci a ion. Then we obse e a la gely
de o med iaxial s a e wi h 12~ω, which is no comple ely
iden ical, bu simila o ha o he alpha-clus e model (wi h
16~ω). This la e s a e is conside ed o be a candida e
o he 28Si+28Si molecula esonances, in which he wo
obla e 28Si a e hough o ha e an equa o - o-equa o posi-
ion. The alpha-clus e -model gi es also a hype de o med
s a e, and ou calcula ion ha e wo candida es o ha . Based
on hei possible clus e -s uc u e he lowe -lying one seems
o be e y simila o ha o he wo k [53].
We ha e ound ha he g ound s a e o 56Ni p e e s
asymme ic clus e con igu a ions, om among he alpha-
like clus e iza ion only 4He+52Fe is allowed. The de o med,
supe de o med and la gely de o med iaxial s a es ma ch
wi h se e al clus e iza ions. S uc u e conside a ions sug-
ges ha he co ela ed 28Si+28Si and 40Ca+16O esonances
co espond o he supe de o med s a e o 56Ni, bu no o
he hype de o med one. In he la e case he 40Ca+16O
con igu a ion has a s ong s uc u al o biddenness [28].
The 24Mg+32S clus e con igu a ion on he o he hand, which
is de e mined by he en ance channel o he e na y ission
expe imen [54] ma ches bo h wi h he SD and HD s a es,
and wi h he la gely de o med iaxial s a e in be ween.
The iaxial s a e is o special in e es , because i is
hough o be ela ed o he molecula esonances o wo
g ound-s a e-like (obla e) 28Si clus e s in hei equa o - o-
equa o con igu a ion. This con igu a ion is allowed in he
iaxial s a e om all ci ed s udies. I he equa o - o-equa o
con igu a ion is no exac ly pa allel, hen o he alpha-like
bina y clus e iza ions, like e.g. 24Mg+32S, a e also possi-
ble.
The hype de o med s a e bo h om he alpha-clus e
and om ou Nilsson-calcula ion p e e s a bina y con igu-
a ion o p ola e 28Si clus e s wi h a pole- o-pole con igu-
a ion. The s a e om ou quasi-dynamical conside a ions
allows 24Mg+32S as well (again close o he posi ion in
which he longes majo axes o bo h nuclei a e pa allel
wi h he molecula axis). The HD s a e om alpha-clus e
s udies does no con ain his con igu a ion.
The ene ge ic p e e ence o he clus e con igu a ions
we e ob ained om binding-ene gy a gumen s [25], and
om calcula ions based on he Dinuclea Sys em Model
[26]. The la e ones we e pe o med bo h o he pole-
o-pole con igu a ions and o he ones de i ed om he
mic oscopic conside a ions. The 4He+co e con igu a ion
u ned ou o be he mos p e e ed one, ollowed by he
8Be+co e one. Then a g oup o he 12C, 28Si, and 16O clus-
e s come, wi h close-lying alues, bu in diffe en o de
om diffe en calcula ions. The 24Mg and 20Ne u ned ou
o be he leas -p e e ed alpha-like clus e s.
Mo e de ailed in es iga ions on his nucleus is p esen ed
in [28]. In his wo k he esul s o he ene ge ic calcula ion
wi h h ee diffe en me hods a e compa ed: in addi ion o
he binding ene gy, and double olding, men ioned abo e,
he e he ex ended liquid d op model [55] was also applied.
3.5 O he N=Znuclei
A compa ison o he possible (p edic ed) alpha-like bina y
clus e con igu a ions in he SD and HD s a es o hese nu-
clei, as well as o he he 60Zn nucleus is p esen ed in [56].
Tha wo k includes also u he discussion on he candida e
eac ions o popula e hese shape isome s.
4 Summa y
In his pape we ha e conside ed he elonga ed shape iso-
me s o some N=Znuclei and hei possible bina y clus-
e iza ions. Bo h in inding he s able shapes and in de e -
mining hei ela ions o clus e con igu a ions symme y
conside a ions we e applied ex ensi ely. I could be done
due o he ac s ha i) he U(3) symme y u ned ou o e-
co e o hese s a es e en o ealis ic (Nilsson- ype) in e -
ac ions, and ii) he numbe o shell-model con igu a ions
which build up he clus e s a e educe conside ably.
We ha e de e mined he shape isome s om he quasi-
dynamical U(3) symme y, ob ained om Nilsson-calcula i-
ons.
In sea ching o he possible bina y clus e iza ions o
he shape isome s we ha e aken in o accoun bo h na -
u al laws which go e n he building up o a nucleus om
smalle cons i uen s. The exclusion p inciple was aken in o
accoun by applying a selec ion ule (in combina ion wi h
Ha ey’s p esc ip ion), based on he mic oscopic con ig-
u a ion associa ed o he quasi-dynamical U(3) symme-
y. In his way he Pauli-p inciple is inco po a ed only in
an app oxima e way, o cou se. Bu i is done in a well-
de ined p ocedu e, which can be checked in simple sys-
ems by compa ing wi h exac esul s. This app oxima ion
16001-p.5

EPJ Web o Con e ences
can be illus a ed in simple geome ical e ms, in spi e o
i s abs ac algeb aic con en : i measu es, how simila he
quad upole de o ma ions a e in he clus e con igu a ion
and in he shell-model (o collec i e model) s a e.
The clus e s we e conside ed o ha e a de o ma ion,
like he co esponding ee nuclei (sphe ical, p ola e, obla e
o iaxial), and no cons ain s we e applied o hei ela-
i e o ien a ion.
The me hods we applied he e seem o be applicable in
hea ie nuclei, oo. Symme y conside a ions can be help-
ul in s udying bo h he shape isome s o nuclei, and hei
clus e iza ions. As o his la e p oblem is conce ned we
hink ha he p e e ed clus e con igu a ions a e hose ones
which a e a o ed by he ene ge ics, and which a e Pauli-
allowed. F om he iewpoin o he applica ion o hea ie
sys ems we conside i as a p omising sign, ha he esul s
o he p esen me hod [49] a e e y simila o he ones om
ab ini io calcula ions o he case o he 40Ca nucleus [50],
whe e he ully mic oscopic ea men was also applied.
Acknowledgmen
This wo k was suppo ed by he OTKA (G an No K72357),
T´
AMOP (4.2.1./B-09/1/KONV-2010-0007/IK/IT), and MTA-
JINR collabo a ion (p ojec no 2009/001).
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