Extreme quadrupole deformation and clusterization

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 EPJ Web o Con e ences DOI: 10.1051/ C Owned by he au ho s, published by EDP Sciences, 2012 , epjcon 20123/ 38 816001 16001 (2012) This is an Open Access a icle dis ibu ed unde he e ms o he C ea i e Commons A ibu ion License 2 0 , which .pe mi s un es ic ed use, dis ibu i and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. on, A icle a ailable a h p://www.epj-con e ences.o g o h p://dx.doi.o g/10.1051/epjcon /20123816001 EPJ Web o Con e ences 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. 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