Super and hyperdeformed states, and reactions to populate them

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 © Owned by he au ho s, published by EDP Sciences, 2011 This is an Open Access a icle dis ibu ed unde he e ms o he C ea i e Commons A ibu ion-Noncomme cial License 3.0, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any noncomme cial medium, p o ided he o iginal wo k is p ope ly ci ed. A icle a ailable a h p://www.epj-con e ences.o g o h p://dx.doi.o g/10.1051/epjcon /20111716001 EPJ Web o Con e ences 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 16001-p.2 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 16001-p.3 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). Re e ences 1. P. Roch o d, D. J. Rowe, Phys. Le . B 210 (1988) 5; D. J. Rowe, P. Roch o d, J. Repka, J. Ma h. Phys. 29 (1988) 572. 2. J. P. Ellio , P oc. Roy. Soc. A 245(1958) 128 and 562. 3. J. Cseh, P oc. IV In . 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