Clus e s and he quasi-dynamical symme y
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Clus e s and he quasi-dynamical symme y
J Cseh1, P O Hess2, J Da ai3, A Algo a1,4and H Yepez-Ma inez5
1Ins i u e o Nuclea Resea ch, Hung. Acad. Sci., Deb ecen, POB 51, Hunga y-4001
2Ins i u o de Ciencias Nuclea es, UNAM, A.P. 70-543, Mexico Ci y, 04510 D.F. Mexico
3Ins . o Expe imen al Phys., Uni . o Deb ecen, Deb ecen, POB 105, Hunga y-4010
4IFIC (CSIC-Uni . o Valencia), Valencia, Spain
5Uni . Au . Ciudad de Mexico, P olong. San Isid o 151, Mexico Ci y, 09790 D.F. Mexico
E-mail: 1cseh@a omki.hu
Abs ac . The possible ole o he quasi-dynamical symme y in nuclea clus e iza ion is
discussed. Two pa icula examples a e conside ed: i) he phases and phase- ansi ions o
some algeb aic clus e models, and ii) he clus e iza ion in hea y nuclei. The in e ela ion
o exo ic (supe de o med, hype de o med) nuclea shapes and clus e -con igu a ions a e also
in es iga ed bo h o ligh , and o hea y nuclei, based on he dynamical and quasi-dynamical
SU(3) symme ies, espec i ely.
1. In oduc ion
Clus e iza ion is a ich phenomenon, which shows up among di e en ci cums ances in a omic
nuclei. In he p esen con ibu ion we also conside a ious aspec s o i , he e o e we s a he e
by ecalling (one o ) i s basic de ini ion(s). I seems mos na u al o app oach he p oblem om
he side o he expe imen al obse a ion, hus (simila ly o o he simple s a es o a omic nuclei)
we call a s a e clus e s a e, i i s wa e unc ion la gely o e laps wi h ha o an obse a ion
channel. [1]. Se e al o he de ini ions a e possible, and a e in use in he li e a u e, including
mo e heo e ical ones. Thei in e ela ion is an in e es ing opic, bu i ob iously goes much
beyond he limi s o his sho pape .
Ano he impo an ea u e o clus e iza ion is ha i is a he complex, he e o e i s exac
heo e ical desc ip ion is e y complica ed, o in many cases impossible. The comple ely
mic oscopic me hods inco po a e all he ele an aspec s o he p oblem, bu hei applicabili y
is limi ed o simple sys ems, e.g. wi h small nucleon numbe s. Some phenomenological models
a e widely applied, bu he p ice we ha e o pay o i is he applica ion o s ongly simpli ying
assump ions, which a e no always well-unde s ood. The semimic oscopic app oaches, including
mos o he discussion o he p esen con ibu ion, y o inco po a e he mos impo an
consequences o he mic oscopic s uc u e, i.e. he Pauli-p inciple in a well-con olled way,
when s ill conside able simpli ica ion o he many-body-p oblem is done.
The s uc u e o his pape is as ollows. Fi s we e iew e y b ie ly he concep s o he
well-known dynamical symme y, and i s ex ension o quasi-dynamical symme y. Then we
ecall some applica ions o he dynamical symme y in nuclea s uc u e s udies in gene al,
and in clus e physics in pa icula . The main pa deals wi h he ole o he quasi-dynamical
symme y in clus e iza ion. Two pa icula examples a e shown: he ques ion o phases and
phase- ansi ions o clus e s a es, and he p oblem o he exo ic clus e iza ion o hea y nuclei,
9 h In e na ional Con e ence on Clus e ing Aspec s o Nuclea S uc u e and Dynamics IOP Publishing
Jou nal o Physics: Con e ence Se ies 111 (2008) 012043 doi:10.1088/1742-6596/111/1/012043
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wi h an a emp o inco po a e he consequences o he exclusion p inciple on an app oxima e
le el. Then he exci ing ques ion o he in e ela ion o he exo ic nuclea de o ma ion and
clus e iza ion is add essed bo h in ligh and in hea y nuclei, by applying selec ion ules, based
on eal and quasi-dynamical symme ies, espec i ely. Finally some conclusions a e d awn.
2. Dynamical and quasi-dynamical symme y
A quan um mechanical sys em go e ned by a ime-independen Hamil onian His said o ha e an
exac dynamical symme y desc ibed by he Lie-algeb a Li he basis ec o s o Lcommu e wi h
H[2]. E.g. he h ee dimensional ha monic oscilla o p oblem has U(3) as an exac dynamical
symme y. I H=T+V, and he elemen s o Lcommu e no only wi h H, bu wi h Tand V
as well, he symme y is called geome ical one. E.g. he oscilla o has O(3) as a geome ical
symme y. When exac symme y holds hen no only he ope a o is symme ic, bu so a e i s
eigen ec o s, oo (i.e. hey ans o m acco ding o an i educible ep esen a ion) [3]. In such a
case he Hamil onian can con ain he elemen s o he symme y Lie-algeb a only ia i s in a ian
ope a o s.
I he Hamil onian is exp essed in e ms o he in a ian ope a o s o a nes ed chain o
subalgeb as ( a he han o a single algeb a as be o e), we speak abou a b oken dynamical
symme y. In such a case he Hamil onian is no symme ic (scala ) any mo e, bu i s
eigen ec o s a e symme ic [4]. E.g. in he Ellio -model (see nex sec ion) he Hamil onian is
w i en in e ms o he Casimi in a ian s o he U(3) ⊃SU(3) ⊃SO(3) algeb a-chain, he e o e
U(3) and SU(3) a e b oken dynamical symme ies, while SO(3) is an exac (geome ical) one.
In his case he o iginal degene acy co esponding o he U(3) spli s up. The eigen alue-p oblem
o a Hamil onian wi h b oken dynamical symme y s ill has an analy ical solu ion (simila ly o
ha o he exac symme y). The e a e many use ul examples o his kind o b oken dynamical
symme ies in nuclea physics. (In many pape s hese dynamical symme ies a e called exac
ones.)
When he symme y-b eaking in e ac ion is e en s onge such ha i no only spli s up
di e en basis s a es bu e en mixes hem wi h each o he , and ye he symme y su i es
( o some o he s a es) we speak abou quasi-dynamical symme y [5]. In his case nei he he
ope a o , no i s eigen ec o s a e symme ic [4]. The symme y-b eaking in e ac ion which leads
o such a si ua ion is s ill no qui e a bi a y, o cou se.
3. Dynamical symme y and clus e iza ion
The basic assump ion o he clus e models is ha he ele an deg ees o eedom o he a omic
nucleus a e classi ied in o wo ca ego ies: some o hem desc ibe he in e nal s uc u e o he
clus e s (e.g. in e ms o a shell model), while o he s accoun o hei ela i e mo ion.
The Ellio -model [6] is an algeb aic shell model wi h a U(3) dynamical symme y belonging
o he spa ial deg ees o eedom. The spin-isospin sec o is accoun ed o by Wigne ’s UST (4)
g oup [7]. The an isymme iza ion in his scheme is done exac ly.
The ela i e mo ion o wo clus e s can be desc ibed by he ib on model [8], which is
an algeb aic model o he dipole collec i e mo ion. I has an U(4) algeb aic s uc u e wi h
wo limi ing cases co esponding o he algeb a-chains: U(4) ⊃U(3) ⊃SU(3) ⊃SO(3),
U(4) ⊃O(4) ⊃SO(4) ⊃SO(3).
By combining he ib on model and he Ellio -model (o some o he algeb aic model o
he desc ip ion o he in e nal clus e s uc u e) one can cons uc an algeb aic clus e model,
and i has been done bo h on he phenomenological and on he semimic oscopical le el. The
dis inc ion is made by he ac whe he o no he Pauli- o bidden s a es a e excluded om
he model space. This complica ion a ises in spi e o he ac ha he Ellio -model has
an isymme ised wa e unc ions, because he an isymme iza ion is no ca ied ou wi h espec
o he in e change o nucleons om di e en clus e s. When he wo models a e coupled on
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he U(3) le el, he p oblem can be sol ed in a ela i ely simple way, due o he ela ion o he
uni a y and pe mu a ional g oups [9]. In his way one can ob ain a semimic oscopic algeb aic
clus e model (SACM) [10], in which he model space is mic oscopic, i.e. i is ee om he
Pauli- o bidden s a es, bu he physical ope a o s (exp essed in e ms o g oup-gene a o s) a e
ea ed phenomenologically, i.e. hey con ain pa ame e s, which a e i ed o expe imen al da a.
The spin-isospin deg ees o eedom a e desc ibed by he UST (4) g oup. When, howe e , only
a single sec o o he isospin is conside ed, i s gene a o s do no play any ole in he physical
ope a o s, and in his sense we can say ha e.g. a bina y clus e sys em can be cha ac e ized
by he chain:
UC1(3) ⊗UR(4) ⊗UC2(3) ⊃UC(3) ⊗UR(3) ⊃U(3) ⊃SU(3) ⊃SO(3),(1)
whe e Cand Rs and o clus e , and ela i e mo ion, espec i ely.
The SACM p o ed o be success ul in desc ibing he de ailed spec a o some clus e sys ems
[11]. I is also wo h men ioning ha he uni ied desc ip ion o di e en clus e sys ems can also
be ca ied ou in his amewo k by he ex ensions o he dynamical symme y (1). Di e en
clus e -con igu a ions o he same nucleus can be ea ed on an equal oo ing by applying
he mul ichannel dynamical symme y [12], while simila clus e iza ions (e.g. co e-plus-alpha-
pa icle) o di e en nuclei can be desc ibed by he supe symme ic model [13]. By applying
la ge mul iple -s uc u es o he mic oscopic model space, and uni ied physical ope a o s,
hese schemes handle he p oblem wi h se ious cons ain s, and consequen ly hey ha e s ong
p edic i e powe , oo. (I is ema kable ha cu en ly ano he clus e supe symme y-scheme
has been es ablished on he phenomenological le el [14].)
4. Quasi-dynamical symme y and clus e iza ion
4.1. Phases and phase- ansi ions
Phases and phase- ansi ions a e usually in es iga ed in sys ems wi h e y la ge numbe s o
deg ees o eedom. Mo e ecen ly, howe e , much in e es has been concen a ed on he phase-
ansi ions in ini e quan um sys ems e.g. a omic nuclei [15, 16]. Algeb aic models seem o
be especially use ul in his kind o s udies, and hey we e in es iga ed ho oughly conce ning
he quad upole collec i e mo ion. He e we in es iga e he p oblem om he iewpoin o
clus e iza ion (i.e. dipole collec i i y) based on he esul s o [17].
In algeb aic models one usually conside s ini e numbe (N) o pa icles, bu i is possible
o go o he la ge Nlimi , whe e eal phase- ansi ions can ake place. Fo ini e Ni can
be in es iga ed, whe he o no some less obus changes su i e. As a con ol pa ame e one
has he ela i e weigh o he Hamil onians belonging o he di e en dynamical symme ies
(analy ically sol able limi s). The ene gy-minimum is in es iga ed as a unc ion o he con ol
pa ame e , and he deg ee o i s de i a i e showing discon inui y (in he la ge Nlimi ) de ines
he o de o he phase- ansi ion.
In [17] we concen a ed on he ela i e mo ion o some bina y clus e sys ems. The e a e wo
ele an algeb a-chains: (1) abo e, and
UC1(3) ⊗UR(4) ⊗UC2(3) ⊃UC(3) ⊗OR(4) ⊃SOC(3) ⊗SOR(3) ⊃SO(3).(2)
Thei physical con en a e he ollowing. F om he collec i e mo ion iewpoin (1) co esponds o
a so ib a o wi h sphe ical equilib ium shape, while (2) desc ibes a igid o o wi h pe manen
dipole de o ma ion. F om he mic oscopic iewpoin (1) co esponds o shell-model-like clus e s,
while (2) desc ibes localized clus e s.
We ha e in es iga ed bina y clus e sys ems (wi h ze o, one and wo open-shell clus e s)
bo h in a phenomenological and in a semimic oscopical model (in o de o s udy he in luence
o he Pauli-p inciple on he ques ion o phase- ansi ion). I u ned ou ha i s o de
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phase ansi ion akes place a a c i ical poin bo h in he phenomenological and in he
semimic oscopical model in he la ge Nlimi . Fo ini e sys ems he ansi ion is smoo hed
ou somewha , bu s ill obse able. (The la ge he model space he mo e ab up he ansi ion
is.)
Ano he in e es ing inding was ha in bo h models he quasi-dynamical U(3) symme y
p o ed o be alid be ween he endpoin o he eal dynamical symme y and he c i ical poin ;
i.e. h oughou he whole phase. This obse a ion, combined wi h some simila esul s in
ela ion wi h he quad upole model [18] indica es ha he exis ence o he quasi-dynamical
symme y could be conside ed as he gene al de ini ion o he phase in ini e quan um sys ems.
The si ua ion esembles o ha o he phase- ansi ions in Landau’s heo y, whe e he di e en
phases a e cha ac e ized by di e en symme ies [19].
Le us no e he e, ha he localized ( igid o o , O(4)) and he shell-like (so ib a o , U(3))
clus e phases show e y ema kable simila i ies o he ”solid” and ”liquid” phases discussed
in o he con ex a his con e ence [16, 20], despi e he di e ences in he applied heo e ical
me hods. In his espec i is an in e es ing open ques ion, whe he o no he hi d possible
limi o he SACM
UC1(3) ⊗UR(4) ⊗UC2(3) ⊃UC(3) ⊗UR(3) ⊃SOC(3) ⊗SOR(3) ⊃SO(3),(3)
which co esponds o he weak coupling be ween he ela i e mo ion and in e nal deg ees o
eedom, has any ea u es o he ”gas-like” phase [21].
Ano he in e es ing ques ion (in which he li e a u e is no unic ocal), i i is wo h speaking
abou shell-like clus e s a all, o we jus could call clus e s he localized ones, and conside
he o he s as shell model s a es. I seems o us, howe e , mo e consequen o dis inguish
be ween localized and shell-like clus e s because hese s a es wi h app oxima ely good ( eal)
U(3) symme ies may ha e e y la ge clus e spec oscopic ac o s o some clus e iza ion [22],
and negligible o o he s. Thus hei ela ion o clus e iza ion is e y s uc u ed, which may
e y well ha e di e ences in expe imen al obse a ion, oo.
4.2. Clus e iza ion in hea y nuclei
In case o ligh nuclei one can o mula e a selec ion ule based on he app oxima e ( eal)
U(3) symme y. I says ha a bina y clus e -con igu a ion (C1+C2) is allowed in a s a e
cha ac e ized by he [n1, n2, n3] quan um numbe s, i his se appea s in he di ec p oduc :
[nC1
1, nC1
2, nC1
3]⊗[nC2
1, nC2
2, nC2
3]⊗[nR
1,0,0]. O he wise i is o bidden (up o he app oxima ion
he U(3) symme y holds). I can be e y use ul in in es iga ing he simila i y be ween
quad upole de o ma ion and clus e iza ion, o o aking in o accoun he Pauli-p inciple wi hou
ca ying ou he an isymme iza ion (by checking i he clus e s a e can be ound among he
an isymme ic shell model s a es).
In medium and hea y nuclei, howe e , he U(3) symme y is no alid in i s o iginal o m,
due o he impo ance o he symme y-b eaking in e ac ions, like spin-o bi and pai ing.
Ne e heless, i was ound in [23] ha in spi e o he s ong symme y-b eaking in e ac ions
he e ec i e, o quasi-dynamical U(3) symme y, may su i e e en o hea y nuclei. In [24] a
me hod was de eloped o he de e mina ion o he e ec i e U(3) quan um numbe s, based on
he occupa ion o he asymp o ic Nilsson o bi s. The p ocedu e, which was o iginally in en ed
o he la ge p ola e de o ma ion was ex ended in [25] o he obla e shape and small de o ma ions
as well, based on he expansion o single-pa icle o bi als in e ms o asymp o ic Nilsson-s a es.
The concep o e ec i e symme y is applicable also o ligh nuclei, and when he simple
leading ep esen a ion app oxima ion is alid, he eal and e ec i e U(3) quan um numbe s
usually coincide [25]. This ci cums ance gi es a s aigh o wa d way o he ex ension o he
simple selec ion ule conside a ion. Due o he a e age na u e o hese quan um numbe s,
9 h In e na ional Con e ence on Clus e ing Aspec s o Nuclea S uc u e and Dynamics IOP Publishing
Jou nal o Physics: Con e ence Se ies 111 (2008) 012043 doi:10.1088/1742-6596/111/1/012043
4

howe e , he e ec o he selec ion ule is di e en om ha o he eal U(3) selec ion ule. I
gi es in o ma ion on he ma ching, o misma ching o he a e age nucleon dis ibu ions in he
clus e -con igu a ion and in he shell-model-s a e. The e o e, i ac s like a sel -consis ency check
o he quad upole de o ma ion and he clus e iza ion.
5. Exo ic shapes and clus e s
In o de o s udy he de o ma ion-dependence we ha e in es iga ed he appea ance o clus e -
con igu a ions in he g ound, supe de o med and hype de o med s a es o some nuclei. The i s
wo (a leas in some cases) a e known expe imen ally, he hype de o med s a es we e p edic ed
heo e ically.
The main mo i a ion o hese s udies was ha in addi ion o aking in o accoun he ene ge ic
p e e ences o di e en clus e -con igu a ions we ied o inco po a e he consequences o he
exclusion p inciple as well. This la e one is done by he applica ion o he eal o quasi-
dynamical SU(3) symme y o ligh and hea y nuclei, espec i ely, as men ioned abo e. (Please,
no e ha he SU(3) symme y is known o eco e o he supe - and hype de o med shapes
[26, 27].) In his way he e ec o he Pauli-p inciple is handled only app oxima ely, o cou se,
bu in a mic oscopic and well-con olled way, and i s esul s can be es ed by compa ing wi h
hose o he ully mic oscopic calcula ions, whe e hey a e a ailable. The o biddenness o he
clus e -con igu a ions a e cha ac e ized quan i a i ely. The de o ma ion o he clus e s (and
pa en nuclei) a e aken in o accoun , and no cons ain is applied o hei ela i e o ien a ion.
The ene ge ic p e e ence o he clus e iza ion is measu ed by he binding-ene gy di e ence
(combined wi h he no-dipole cons ain ) o [28], on he one side, and in some cases wi h he
mo e de ailed double- olding po en ial ene gy o he dinuclea sys em model [29] on he o he
side. This la e quan i y is de e mined bo h o he usual pole- o-pole con igu a ion, and o
he one, which is p e e ed by he selec ion ule.
We ha e conside ed he possible bina y con igu a ions o he 36A , 40Ca and 252C nuclei,
[30, 31], and some e na y con igu a ions [32] o he 36A and 252C .
The main conclusion o hese calcula ions can be summa ized as ollows. The p e e ence o he
exclusion p inciple and he ene gy-calcula ion do no necessa ily coincide. The e o e, we hink
ha when sea ching o he mos p obable clus e -con igu a ion(s), one has o ake in o accoun
no only he ene ge ic ci cums ances, bu he exclusion p inciple, oo. I also u ned ou ha
some imes he same clus e iza ion can be p esen bo h in he g ound and in he supe de o med,
as well as in he hype de o med s a e. The di e ence be ween hem is he spa ial a angemen
o he de o med clus e s.
6. Conclusion
In clus e s udies, jus like in many o he physical p oblems, symme y-conside a ions can help o
ind a simple solu ion o a complex p oblem. In his con ibu ion we ha e ied o illus a e how
he quasi-dynamical symme y, which is one o he mos gene al symme y concep o quan um
mechanics, can be applied. The wo phenomena we ha e conside ed, he phases and phase-
ansi ions o clus e s a es, and he exo ic clus e iza ion in hea y nuclei a e o u mos in e es .
Ob iously much wo k emains o be done un il we each hei p ope heo e ical unde s anding,
and i seems ha symme y-a gumen s can be ui ul along his line.
Acknowledgmen s
This wo k was suppo ed by he OTKA (G an No. 46791), DGAPA, CONACyT, and by
he MTA-CONACyT, as well as by he MTA-CSIC exchange p og ammes. One o us (J Cs)
acknowledges ui ul discussions wi h d N I agaki on he phases o nuclea s a es.
9 h In e na ional Con e ence on Clus e ing Aspec s o Nuclea S uc u e and Dynamics IOP Publishing
Jou nal o Physics: Con e ence Se ies 111 (2008) 012043 doi:10.1088/1742-6596/111/1/012043
5
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Jou nal o Physics: Con e ence Se ies 111 (2008) 012043 doi:10.1088/1742-6596/111/1/012043
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