Supe and hype de o med s a es, and eac ions o popula e hem
J. Cseh1,a, J. Da ai2, A. Algo a3, N. An onenko4, and G. Adamian4
1Ins i u e o Nuclea Resea ch, Hunga ian Academy o Sciences, Deb ecen, P . 51, Hunga y-4001
2Ins i u e o Expe imen al Physics, Uni e si y o Deb ecen, Deb ecen, P . 105, Hunga y-4010
3IFIC, CSIC-Uni e sidad de Valencia, A. C. 22085, E 46071, Valencia, Spain
4Bogoliubo Labo a o y, Join Ins i u e o Nuclea Resea ch, 141980 Dubna, Russia
Abs ac . We s udy he possible bina y clus e con gu a ions o he supe de o med and hype de o med s a es
o some N=Znuclei. These in es iga ions con ibu e o he s uc u al unde s anding o he shape isome s, and
indica e some eac ion channels o popula e hem.
1 In oduc ion
The in es iga ion o clus e iza ion o a speci c nuclea s a e
is impo an om wo aspec s. Fi s i con ibu es o ou un-
de s anding o he nuclea s uc u e. Second i gi es alu-
able in o ma ion on he nuclea eac ions, which can pop-
ula e he s a e in ques ion. In pa icula , a bina y clus e
con gu a ion wi h he clus e s in hei g ound in isic 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 possi-
ble de ni ion o a bina y clus e con gu 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 ne a
clus e con gu a ion.) The e o e, by s udying bina y clus-
e con gu a ions o a nuclea s a e we ge a s aigh o -
wa d indica ion on some possible eac ion channels which
can each i .
The in es iga ion o he ex emely de o med s a es, i.e
supe de o med (SD), o hype de o med (HD) s a es in nu-
clei is a long-s anding opic. This kind o shape isome s o
he N=Znuclei, howe e , a e known only ecen ly. The
expe imen al obse a ion o he SD s a es in hese nuclei
s a ed a decade ago o so, while he ques ion o he hype -
de o ma ion is mainly a opic o heo e ical p edic ions.
In his con ibu ion we in es iga e he SD and HD s a es
o N=Znuclei om a heo e ical iewpoin , and discuss
some aspec s o hei expe imen al obse a ion as well.
2 Me hods o in es iga ion
Bo h in ou de e mina ion o he shape isome s, and in he
in es iga ion o hei possible clus e iza ions symme y-
conside a ions play an impo an ole. In pa icula , he quasi-
dynamical (o e ec i e) U(3) symme y is used [1]. I is a
gene aliza ion o he concep o he eal U(3) symme y,
known o be app oxima elly alid in ligh nuclei [2]. The
ae-mail: cseh@a omki.hu
quasi-dynamical symme y is mo e gene al in he ollow-
ing 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 ope a o
is symme ic (i.e. i is no a U(3) scala ), no i s eigen ec-
o s (i.e. hey do no ans o m acco ding o a single i e-
ducible ep esen a ion) [3]. 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 SU(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 [4]. 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 [5].
The SU(3) quan um numbe s uniquely de e mine he
quad upole shape o he nucleus [6], 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. I is based on he applica ion o he quasi-
dynamical U(3) quan um numbe s [7], and in hose cases
when a de ailed compa ison can be made wi h he mo e a-
di ional ene gy-minimum calcula ions, he esul s a e e y
simila [8,9,7].
Once he shape isome s ha e been ound, he nex ques-
ion is how hey a e ela ed o clus e con gu a ions. To
nd hei connec ion we use he Ha ey p esc ip ion [10]
and he U(3) selec ion ule [9]. They can inco po a e he
e ec s o he exclusion p inciple, only in an app oxima e
way, o cou se. Bu i is a well-de ned way, and i s a-
lidi 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. In geome ical e ms he U(3) selec ion ule
exp esses he simila i y o he quad upole-de o ma ion o
he clus e con gu a ion and he shell-model (o collec i e
model) s a e.
Ene ge ic p e e ence ep esen s a complemen a y iew-
poin o he selec ion o clus e iza ion. We inco po a e
EPJ Web o Con e ences 17, 16001 (2011)
DOI: 10.1051/epjcon /20111716001
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i in wo di e en ways: i) by applying simple binding-
ene gy a gumen s [11], and ii) by pe o ming double- olding
calcula ions, acco ding o he dinuclea sys em model [12,
13].
3 Case s udies
3.1 36A
The SD band o he 36A nucleus was epo ed in [14]. Fol-
lowing he expe imen al obse a ion a conside able heo-
e ical e o has been concen a ed on his band. In [15]
e.g. he possible bina y clus e iza ions o his s a e was
s udied sy ema ically. Simila s udies ha e been done also
o he g ound, and he hype de o med band. The la e one
had been p edic ed om alpha-clus e model calcula ions
[16]. One o he in e es ing conclusions o his wo k [15],
was ha he HD s a e o he 36A nucleus could be popu-
la ed in he 24Mg+12C and 20Ne+16O eac ions.
A ecen analysis o he 24Mg+12C elas ic sca e ing
[17] 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
phase-shi s udy, as well as Regge-pole and ene gy-dependen
esonance calcula ions. The exis ence o e esonances
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 [17],
as shown in he uppe pa o Fig. 1. I s momen o ine ia
is in a e y good ag eemen wi h ha o he HD shape p e-
dic ed om alpha-clus e model [16]. The simila 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 ne he p e e ed clus e -con gu a ions o he HD
shape sugges ha he ecen ly obse ed band in Fig. 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 su-
pe de o med bands a e indica ed in Fig. 1.
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 [7]. we ha e ca ied ou Nilsson-
model+quasi-dynamical SU(3) calcula ion in o de o nd
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 .
3.2 56Ni
The s able elonga ed shapes o he 56Ni nucleus, which a e
ele an o clus e iza ion, om a simila Nilsson-model
+quasi-dynamical U(3) symme y calcula ion, is shown
in Figu e 3. In his gu e i is no he minima, a he he
ho izon al pla eaus, which co espond 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 de o ma ions ul ll he sel -
consis ency a gumen be ween he inpu and ou pu de o ma ion-
pa ame e s o some app oxima ion.)
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.
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 gu 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 [16],
whe e he con ou co esponds o he HD s a e o he Nilsson-
model.
As i is seen om he gu e, he iaxial g ound-s a e
( o which he expe imen al de o ma ion is β2=0.173)
is ollowed 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 su-
pe de o med shape. This s a e ep esen s 4 nucleon exci a-
ion, being e y much in line wi h [18,19]. Then appea s
an e en mo e de o med s a e wi h iaxial shape, and wo
p onounced 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 [19], and in ou (Nilsson-model-based)
quasi-dynamical symme y conside a ion. The supe de o med
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FUSION11
Fig. 3. Quad upole de o ma ion o he 56Ni nucleus om he
Nilsson-model wi h he e ec i e U(3) quan um numbe s and
schema ic illus a ions o he shape a he pla eaus.
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 [19].
We ha e ound ha he g ound s a e o 56Ni p e e s
asymme ic clus e con gu 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 gu a ion has a s ong s uc u al o biddenness [20].
The 24Mg+32S clus e con gu a ion on he o he hand, which
is de e mined by he en ance channel o he e na y ssion
expe imen [21] 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 gu a ion. This con gu a ion is allowed in he
iaxial s a e om all ci ed s udies. I he equa o - o-equa o
con gu a ion is no exac ly pa alell, 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 gu-
a ion o p ola e 28Si clus e s wi h a pole- o-pole con gu-
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 gu a ion.
I is an in e es ing nding ha di e en s a es can be
buil up om he same wo clus e s, like e.g. wo obla e
(g ound-s a e-like) 28Si can esul in he 0~ωp ola e de-
o med sa es, he supe de o med s a e wi h 4~ωexci a ion,
as well as he la gely-de o med iaxial s a e wi h many
pa icle-hole exci a ion. The di e ence in hese cases is
he ela i e o ien a ion o he wo de o med clus e s. This
obse a ion is a consequence o he ac ha he Pauli-
p inciple was aken in o accoun , and no simpli ying as-
sump ions we e made on he shapes and ela i e o ien a-
ions o he clus e s.
The ene ge ic p e e ence o he clus e con gu a ions
we e ob ained om binding-ene gy a gumen s [11], and
om calcula ions based on he Dinuclea Sys em Model
[12]. The la e ones we e pe o med bo h o he pole-
o-pole con gu a ions and o he ones de i ed om he
mic oscopic conside a ions. The 4He+co e con gu 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 di e en o de
om di e 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 [20]. In his wo k he esul s o he ene ge ic calcula ion
wi h h ee di e 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 [22] was also applied.
3.3 O he N=Znuclei
We ha e ca ied ou simila in es iga ions o o he N=Z
nuclei. Fo he 40Ca we ha e discussed he p oblem o clus-
e iza ion in [23]. Mo e ecen ly we pe o med also Nils-
son model +quasi-dynamical symme y calcula iuon o
he de e mina ion o he shape isome s. Some o he e-
sul s, oge he wi h hose conce ning he 60Zn nucleus is
shown in Table I.
4 Summa y and conclusions
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 nding he s able shapes and in de e -
mining hei ela ions o clus e con gu a ions symme y
conside a ions we e applied ex ensi ely.
We ha e de e mined he shape isome s om he quasi-
dynamical U(3) symme y, ob ained om Nilsson-calcula ions.
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 g-
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
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EPJ Web o Con e ences
Table 1. Supe and hype de o med s a es o some N=Znuclei.
Thei shell model exci a ion quan a, and he possible alpha-like
clus e iza ions (in g ound in insic s a e) a e gi en. The o, and p
subsc ip s o 28Si e e o obla e and p ola e shapes, espec i ely.
Nucl. SD HD
~ωC1+C2~ωC1+C2
36A 4 32S+4He, 28Sio,p+8Be, 12 28Sip+8Be,
24Mg+12C24Mg+12C,
20Ne+16O
40Ca 8 32S+8Be, 28Sio+12C, 16 28Sip+12C,
20Ne+20Ne 24Mg+16O,
20Ne+20Ne
56Ni 4 48C +8Be, 44Ti+12C, 24 32S+24Mg,
40Ca+16O, 36A +20Ne, 28Sip+28Sip
32S+24Mg, 28Sio,p+28Sio,p
60Zn 4 52Fe+8Be, 48C +12C, 20 32S+28Sio,p,
36A +24Mg, 32S+28Sio,p36A +24Mg
an app oxima e way, o cou se. Bu i is done in a well-
de ned p ocedu e, which can be checked in simple sys-
ems by compa ing wi h exac esul s. This app oxima ion
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 gu 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 gu a ions a e hose ones
which a e a ou 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 [23] a e e y simila o he ones om
ab ini io calcula ions o he case o he 40Ca nucleus [24],
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), NKTH
(Hunga ian-Spanish collabo a ion p ojec no. ES-26/2008),
and MTA-JINR collabo a ion (p ojec no 2009/001).
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