Shape phase transition in odd nuclei in a multi- j model: The UB(6) ⊗ UF(12) case

PHYSICAL REVIEW C 75, 064316 (2007) Shape phase ansi ion in odd nuclei in a mul i- jmodel: The UB(6) ⊗UF(12) case C. E. Alonso,1J. M. A ias,1and A. Vi u i1,2 1Depa amen o de F´ ısica A ´ omica, Molecula y Nuclea , Facul ad de F´ ısica, Uni e sidad de Se illa, Apa ado 1065, E-41080 Se illa, Spain 2Dipa imen o di Fisica Galileo Galilei and INFN, Via Ma zolo 8, I-35131 Pado a, I aly (Recei ed 2 Ap il 2007; published 22 June 2007) The phase ansi ion in odd nuclei when he unde lying e en-e en co e nuclei expe ience a ansi ion om sphe ical o de o med γ-uns able shapes is in es iga ed. The odd pa icle is assumed o be mo ing in he h ee single-pa icle o bi als j=1/2,3/2, and 5/2 . A he c i ical poin in he phase ansi ion, an analy ic solu ion o he co esponding Boh Hamil onian, called E(5/12), is wo ked ou . Ene gy spec a and elec omagne ic ansi ions and momen s a e p esen ed. The same p oblem is also a acked in he amewo k o he in e ac ing boson- e mion model (IBFM). Two di e en Hamil onians a e used. The i s one is cons uc ed ad hoc so as o mimic he si ua ion in he E(5/12) model. The second one leads o he occu ence o he OB(6) ⊗UF(12) symme y when he boson pa app oaches he O(6) condi ion. The en i e ansi ion line is s udied wi h his Hamil onian and, in pa icula , he c i ical poin . Bo h IBFM calcula ions a he c i ical poin a e consis en wi h he E(5/12) esul s. DOI: 10.1103/PhysRe C.75.064316 PACS numbe (s): 21.60.Fw, 21.60.E I. INTRODUCTION A omic nuclei ha e been classi ied o yea s in e ms o collec i e de o ma ions ha a e desc ibed by in oducing wo collec i e a iables [1]. The de o ma ion pa ame e β measu es he axial de ia ion om sphe ici y, while he angle a iable γcon ols he depa u e om axial symme y. Th ee analy ical solu ions o he collec i e Boh Hamil onian ha e been known: he ib a ional, o a ional, and de o med γ- uns able limi s [1–3]. These si ua ions co espond, espec- i ely, o he sphe ical, axially de o med, and de o med γ- independen g ound s a e shapes. O cou se, ew nuclei p esen p ope ies ha i exac ly one o hese limi ing si ua ions, and mos o hem display a ansi ional beha io . The occu ence o shape ansi ional beha io cha ac e izes bo h e en-e en and odd nuclei. As he nucleon numbe changes, he s uc u e o he sys em changes om one cha ac e o ano he , allowing one o alk abou a g ound s a e phase ansi ion (nowadays known as a quan um phase ansi ion). The p ope ies o he nucleus a he icini y o he c i ical poin a y apidly, and, consequen ly, he desc ip ion o c i ical nuclei has been hough o be e y di icul . Recen ly, Iachello p oposed in a se ies o pape s a new class o symme ies ha a e o mula ed in e ms o he Boh Hamil onian and ha can be applied o c i ical poin si ua ions [4–6]. In pa icula , a he c i ical poin om sphe ical o γ-uns able shapes, called E(5) [4], a he c i ical poin om sphe ical o axially de o med shapes, called X(5) [5], and a he c i ical poin om axially de o med shapes o iaxial shapes, called Y(5) [6]. Since he in oduc ion o hese limi s, many heo e ical [7–12] and expe imen al [13–22] s udies ha e been p esen ed in o de o look o nuclei ha exhibi he p ope ies o c i icali y and o classi y he co esponding phase ansi ions. Many s udies ha e ex ended hese o iginal models o mo e complex si ua ions [23–27]. Because o he u he complexi y embodied in he de- sc ip ion o he coupling o he odd pa icle o he e en co e, he s udy o phase ansi ions in odd nuclei has only ecen ly been a acked. The i s case conside ed by Iachello [7,28]was he case o a j=3/2 e mion coupled o a boson co e ha unde goes a ansi ion om sphe ical o γ-uns able si ua ion. A he c i ical poin , an elegan analy ic solu ion, called E(5/4), was ob ained s a ing om he Boh Hamil onian. The possible occu ence o his symme y in 135Ba has been ecen ly sugges ed [29]. Howe e , he es ic ion o he e mion space o a single-jo bi makes di icul he iden i ica ion o a pa icula nucleus as c i ical. A ecen wo k [30] conside ed he iche case o a collec i e co e coupled o a pa icle mo ing in he j=1/2,3/2, and 5/2 single-pa icle o bi s, wi h he boson pa unde going a ansi ion om sphe ici y o de o med γ-ins abili y. Fo he c i ical poin in his si ua ion, a new analy ic solu ion, called E(5/12), has been p esen ed. Since his si ua ion is likely o occu in se e al mass egions, E(5/12) p o ides a be e chance o inding an example o c i icali y in odd-e en nuclei. In his pape we ex end he p e ious s udy on E(5/12) so as o p o ide he g ounds o expe imen al s udies looking o c i ical beha io in odd-e en nuclei. Spec um, elec ic quad upole, and magne ic dipole ansi ions along wi h quad upole and magne ic momen s a e wo ked ou in he model. I is wo h men ioning ha all hese s udies ha e been pe o med in he amewo k o he collec i e model. Howe e , a pa allel ea men can be done wi hin he in e ac ing boson model (IBM) [31]. Al hough his model is o mula ed om he beginning in second quan ized o m, i is possible o ob ain i s geome ic pic u e by using cohe en s a es and shape a iables [32–34]. Th ee dynamical IBM symme ies a e known: U(5), SU(3), and O(6) co esponding o geome ical sphe ical, axially de o med, and γ-uns able de o med shapes, espec i ely [35]. T ansi ional classes among hese limi s ha e been ho oughly s udied since he in oduc ion o he model. E en g ound s a e phase ansi ions we e s udied soon a e [33,36–38], bu hese s udies ha e been e i alized ecen ly wi h he abo e discussed wo ks on quan um phase ansi ions 0556-2813/2007/75(6)/064316(15) 064316-1 ©2007 The Ame ican Physical Socie y C. E. ALONSO, J. M. ARIAS, AND A. VITTURI PHYSICAL REVIEW C 75, 064316 (2007) wi hin he Boh Hamil onian amewo k. Nume ical s udies o e en-e en nuclei on he c i ical poin om U(5) o O(6) and om U(5) o SU(3) ha e been p esen ed, and he analysis o he phase ansi ion cha ac e is ics has been he objec o many pape s [39–61]. A ecen e iew on he opic can be ound in Re . [62]. Fo he case o odd-e en nuclei, he simples IBM ex ension is called he in e ac ing boson- e mion model (IBFM) [63]. The IBFM has been used o s udy he phase ansi ion along he U(5)-O(6) line o he case o a e mion in a single j=3/2 o bi al [64,65], which is he si ua ion s udied in he E(5/4) model. The choice o he j=3/2 o bi al allows one o eco e a he γ-so ex eme he OB(6) ⊗SUF(4) boson- e mion symme y. This is because in his case he ull boson- e mion quad upole- quad upole e m in he Hamil onian can be exp essed in e ms o he Casimi ope a o s o he gene al UBF(5), OBF(5), SpinBF(6), and SpinBF(5) algeb as. The case o he j=3/2 is no , howe e , unique. I is known in he li e a u e [63,66] ha also in he mul i-jcase wi h j=1/2,3/2,5/2, wi h a p ope choice o he boson- e mion Hamil onian, one can ob ain a ansi ional Hamil onian exp essed in e ms o Casimi ope a o s associa ed wi h he OBF(6) algeb a. This has allowed one o s udy wi hin he IBFM a simila si ua ion o ha s udied in he E(5/12) model. The IBFM s udy has been shown o be use ul o check he E(5/4) [64] and E(5/12) [30] cases. In his pape , we p esen o j=1/2,3/2,5/2 he equi alen E(5/12) esul s o selec ed IBFM Hamil onians. The con on a ion o E(5/12) and IBFM a he co esponding c i ical poin will p o ide use ul guidelines o expe imen al s udies. The s uc u e o he pape is as ollows. In Sec. II, he E(5/12) model is p esen ed in de ail and he wa e unc ions a e used o calcula e di e en spec oscopic p ope ies o expe imen al in e es . In Secs. III and IV, wo IBFM model Hamil onians p oduced o s udy he c i ical poin wi hin he IBM amewo k a e p esen ed. We will he e o e conside wi hin he IBFM he case o he coupling o an odd pa icle mo ing in he j=1/2,3/2, and 5/2 o bi als o a boson co e unde going a ansi ion om U(5) o O(6) si ua ions. We ollow along he ansi ion he e olu ion o he spec um and s udy in de ail ene gies and ansi ions in co espon- dence o he c i ical poin . We will show ha he esul s o IBFM and E(5/12) models gi e compa able beha io s p oducing obus esul s indica ing c i icali y in odd-e en nuclei. II. THE BOHR HAMILTONIAN AT THE CRITICAL POINT AND THE E(5/12) MODEL In his sec ion, we s udy he coupling o an e en co e o a pa icle mo ing in he 1/2, 3/2, and 5/2 o bi als. We show ha i can also display analy ic solu ions wi hin he Boh collec i e model, in analogy o he case o he j=3/2 pa icle, desc ibed by Iachello wi hin he E(5/4) c i ical poin symme y. We will name his case as E(5/12). Bu i s , we would like o p esen b ie ly he E(5) case o e en-e en nuclei, since i will be an impo an e e ence o ou s udy. The E(5) c i ical poin si ua ion co esponds o he solu ion o he Boh Hamil onian H=− ¯h2 2B1 β4 ∂ ∂β β4∂ ∂β +1 β2sin 3γ ∂ ∂γ sin 3γ∂ ∂γ −1 4β4 κ Q2 κ sin2γ−2 3πκ+U(β,γ),(2.1) in which he po en ial ene gy Uis γindependen and modeled as an in ini e squa e well in β U(β)=0,β<β w, U(β)=∞,β⩾βw. (2.2) This si ua ion is supposed o mimic he c i ical poin om sphe ical (minimum o he ene gy su ace a β=0) o de o med γuns able (minimum o he ene gy su ace a β= 0 and γindependen ). In his si ua ion, analy ical solu ions a e ound. Ene gy le els a e di ec ly ela ed o ze os o app op ia e Bessel unc ions [4,7]. In Fig. 1, he spec um and a ew selec ed E2 ansi ion p obabili ies a e displayed. Fo no a ion and mo e de ails, we e e o Re s. [4,7]. We mus men ion ha in his si ua ion, because o he symme y o he p oblem, s a es a e classi ied by he o ally symme ic i educible ep esen a ions o he O(5) and O(3) g oups, (τ,0) and L, espec i ely. In addi ion, an ex a label ξis in oduced o ma k di e en solu ions o he adial (β) equa ion. Thus, s a es a e labeled by |ξ,τ,L. The solu ions o Eq. (2.1)a e ψ(β,γ,θi)= ξ,τ(β)τµLM(γ,θi),(2.3) whe e he  unc ions we e de e mined in Re s. [3,67]. These unc ions a e eigens a es o any quad upole Hamil onian which displays O(5) ⊃O(3) symme y, i.e., o U(β,γ)=U(β). The (β) unc ions depend on he selec ion o he βpo en ial. Fo he in ini e squa e well p oposed, he wa e unc ions in β ake he simple o m ξ,τ(β)=Cξ,τβ−3/2Jτ+3/2xξ,τ βω β,(2.4) 0+ 0+ 2+ 6+4+3+0+ 2+ 4+ 4+ 2+ 2+ 0 1 2.20 3.59 3.03 4.80 6.78 0 1 2 3 0 1 2 τ E τ E 100 167 217 114 75 124 167 103 155 217 87 124 62 E(5) ξ=1 ξ=2 FIG. 1. Ene gy le els (no malized o he ene gy o he i s exci ed s a e) and B(E2) ansi ion p obabili ies o E(5). 064316-2 SHAPE PHASE TRANSITION IN ODD NUCLEI IN A . . . PHYSICAL REVIEW C 75, 064316 (2007) whe e τ=0,1,2,... labels he SO(5) g oup, ξis an index ha enume a es he successi e ze os o he Bessel unc ions, xξ,τ is he ξ h ze o o a gi en τ,Cξτ is a no maliza ion, and βωis he ange o he squa e po en ial. The ull s a es in Eq. (2.3) can hen be deno ed by |ξτµLM,(2.5) whe e µis a label ha dis inguishes be ween epea ed L o a gi en alue o τ. Once he wa e unc ions a e known, he ansi ion p obabili ies a e ela ed o he educed ma ix elemen o he ele an elec omagne ic ope a o , B(E/M,λ)=1 2Li+1ξ τ µ L ||T(λ) C||ξiτiµiLi2,(2.6) i s calcula ion includes an in eg al o p oduc s o (β) unc ions on he β a iable, plus in eg als o p oduc s o τµLM(γ,θi) unc ions in he γ a iable and he Eule angles. De ails o hese in eg a ions can be ound in Re . [7]. Nex we go o ou case o in e es , in which a single pa icle ha can occupy he j=1/2,3/2, and 5/2 o bi als is coupled o an E(5) e en-e en co e. In his si ua ion, one can use he sepa a ion o he single-pa icle angula momen um in o a pseudospin and pseudo-o bi al angula momen um (0 and 2) and assume no coupling due o he pseudospin. In his si ua ion, an E(5) co e is coupled o a wo-le el sys em ep esen ing he odd pa icle. The lowe le el wi h LF=0(j=1/2) has ze o ene gy, and he uppe one wi h LF=2(j=3/2,5/2 degene a e) has some κene gy. The pseudo-o bi al pa ans o ms as he ep esen a ions [τF,0] o OF(5), wi h τFbeing ei he 0 o 1. One can use in his case a model Hamil onian o he o m H=− ¯h2 2B1 β4 ∂ ∂β β4∂ ∂β +1 β2sin 3γ ∂ ∂γ sin 3γ∂ ∂γ −1 4β4 κ Q2 κ sin2γ−2 3πκ+u(β)+kg(β) ×[2 ˆ LB◦ˆ LF]+kg(β)ˆ L2 F, (2.7) wi h u(β)=0,β<β w, u(β)=∞,β⩾βw, g(β)=¯h2 2Bβ2. (2.8) No e ha ˆ LBand ˆ LFa e he i e-dimensional boson and e mion angula momen a, and ha he do ◦indica es he i e-dimensional scala p oduc . He e and in he ollowing we will use he callig aphic le e ˆ L o deno e he i e-dimensional O(5) angula momen a and he i alic le e L o deno e he usual h ee-dimensional O(3) angula momen a. I is wo h no ing ha he Hamil onian (2.7) is a gene aliza ion o he one p oposed o he E(5/4) case. Since in E(5/12) he e a e wo OF(5) ep esen a ions in ol ed, in Eq. (2.7) helas e m has been in oduced o include he ene gy di e ence be ween j=1/2 and j=3/2,5/2 single-pa icle o bi als. The coupling scheme is p esen ed in Fig. 2.In helowes pa o he igu e he e en-e en deg ees o eedom, gi en by he E(5) model, ha e o be coupled o he odd-pa icle deg ees o eedom. These las ones a e sepa a ed in o a pseudo-o bi al pa ( wo-le el sys em) plus a pseudospin pa . Fi s , he co e E(5), whose s a es a e classi ied by (ξ,τB,L B) as explained abo e, is coupled o he pseudo-o bi al pa o he odd pa icle, labeled (τF,L F). This gi es ise o a BFS (0) 0 (τΒ) LB(τF) LF ⊗⊗ s (1) 2 (2) 4,2 (0) 0 (1) 2 1/2 τΒ,τF(τ1,τ2) LBF 0,0(0,0) 0 1,0(1,0) 2 2,0(2,0) 4,2 5/2,3/2 5/2,3/2 0,1(1,0) 2 1,1(2,0) 4,2 1,1(1,1) 3,1 1,1(0,0) 0 9/2,7/2,5/2,3/2 1/2 9/2,7/2,5/2,3/2 7/2,5/2,3/2,1/2 1/2 ull BF pseudo-o bi al BF JFIG. 2. Coupling scheme o E(5/12) model. Lowe pa : an e en-e en E(5) co e ( ep esen ed by he lowes h ee s a es wi hin he g ound s a e band, ξ=1) is coupled i s o a wo-le el sys em ha ep esen s he pseudo-o bi al angula momen um o he odd pa icle. The esul ing o al o bi al angula momen um o he odd-e en sys em is inally coupled o he spin pa o he odd pa icle. 064316-3 C. E. ALONSO, J. M. ARIAS, AND A. VITTURI PHYSICAL REVIEW C 75, 064316 (2007) boson- e mion pseudo-o bi al coupling p esen ed in he uppe le panel. The s a es a e labeled τBcha ac e izing he OB(5) g oup and τFcha ac e izing he OF(5) g oup. These wo ep esen a ions a e coupled o a common OBF(5) g oup whose i educible ep esen a ions a e labeled (τ1,τ 2). I he coupling is wi h τF=0,L F=0, all he ob ained s a es a e as in he E(5) e en-e en case. When he coupling is wi h τF=1, he allowed coupled ep esen a ions a e (τ1,τ 2)=(τB+1,0),(τB,1), and (τB−1,0). Once he OBF(5) ep esen a ions a e known, he usual educ ion o angula momen um is pe o med. In he uppe le panel o Fig. 2, such a pseudo-o bi al coupling is ep esen ed o he ξ=1 s a es in he E(5) pa , he le band comes om he coupling o he E(5) s a es o he (τF=0,L F=0) s a e and hen, as men ioned abo e, a e as in he E(5) model. The o he h ee bands come om he coupling o he E(5) pa o he (τF=1,L F=2) s a e. This coupling o he τB=0 p oduces jus one s a e (τ1=1,τ 2=0) a he same ene gy as he exci a ion ene gy o he τF=1 wi h espec o he τF=0 s a e o he odd pa icle. This s a e is labeled τB,τ F(τ1,τ 2)LBF =0,1(1,0)2 (LBF is he boson- e mion pseudo-o bi al angula momen um). The coupling o he boson τB=1 s a es o he single-pa icle pseudo-o bi al τF=1 s a e gi e h ee possibili ies (τ1,τ 2)=(2,0),(1,1),and (0,0); he co esponding educ ion ules gi e LBF =4,2,L BF =3,1, and LBF =0, espec i ely. This is ep esen ed schema ically in Fig. 2(uppe le panel). Finally, hese s a es a e coupled o he pseudospin pa and gi e he spec um o he ull boson- e mion sys em, p esen ed in he uppe igh pa o Fig. 2. Each line ep esen s degene a e s a es wi h o al angula momen um gi en by J, p oduced by he coupling o he boson- e mion pseudo-o bi al angula momen um LBF o he pseudospin s. Wi h his choice o he coupling, he o al wa e unc ion can be ac o ized in he o m =F(β)[(γ,θi,η o b)⊗χ(ηspin)]JBF ,(2.9) whe e ηo b and ηspin ep esen he pa icle coo dina es asso- cia ed wi h he pseudo-o bi al and pseudospin spaces. The wa e unc ion is cha ac e ized by good quan um numbe s (τB,τ F,τ 1,τ 2,L BF,M L) and is gi en by τB,τF,(τ1,τ2),LBF ,ML(γ,θi,η o b) = LB,MLB,LF,mLF(τB,0) (τF,0) |(τ1,τ 2) LBLF|LBF  ×LBLF|LBF MLBmLF|MLφτB,LB,MLB(γ,θi) ×XτF,LF,mLF(ηo b),(2.10) whe e he symbols a e he summa ion a e isoscala ac o s o he O(5)-O(3) and he O(3)-O(2) educ ions. The unc ion X(ηo b) desc ibes he i e-dimensional pseudo-o bi al pa , while χ(ηspin) is he pseudospin wa e unc ion, which does no con ibu e o he ene gy. The equa ion o F(β) hen acqui es he amilia o m −¯h2 2B 1 β4 ∂ ∂β β4∂ ∂β +¯h2 2B  β2+u(β)F(β)=EF(β), (2.11) wi h =τB(τB+3) +k[τ1(τ1+3) +τ2(τ2+1) −τB(τB+3) −τF(τF+3)] +kτF(τF+3).(2.12) The solu ions o his equa ion, as in he case o he E(5/4) model, a e gi en in e ms o Bessel unc ions o o de ν= √+9/4, in he o m Fξ,{τB,τF,(τ1,τ2),LBF }(β) =cξ,{τB,τF,(τ1,τ2),LBF }β−3/2Jν(xξ,{τB,τF,(τ1,τ2),LBF }β/βw), (2.13) whe e cξ,{τB,τF,(τ1,τ2),LBF }is a no maliza ion cons an , and xξ,{τB,τF,(τ1,τ2),LBF } he ξ h ze o o Jν(z). The co esponding eigen alues a e gi en as Eξ,{τB,τF,(τ1,τ2),LBF }=¯h2 2Bxξ,{τB,τF,(τ1,τ2),LBF } βw2 .(2.14) The co esponding spec um is gi en in Fig. 3 o he choice o he pa ame e s k=−1/4 and k=5/2. The s a es a e labeled acco ding o he quan um numbe s ξ,τB,τ F,(τ1,τ 2),L BF. One should emembe ha each s a e ac ually ep esen s a mul iple o degene a e le els, whose angula momen a can be ex ac ed om Table I. The pa ame e s kand kcon ol he o de o he bands. In Fig. 4we p esen he e olu ion o he bandheads as a unc ion o he kand kpa ame e s. Once he wa e unc ions a e known, one can calcula e any o he spec oscopic obse able o in e es . In gene al, he elec omagne ic ope a o o he odd sys em will ha e bo h con ibu ions om he collec i e co e and om he single e mion. We will ha e o calcula e educed ma ix elemen s  ||T(λ||i,(2.15) whe e a e gi en by Eq. (2.9). Since in his o mula ion he o al angula momen um o he e mion is no a good quan um numbe , we p e e o pe o m a simple change o coupling ans o ma ion gi en by |(LB,L F),L BF,s;J = j (−1)LB+LF+1/2+J(2LBF +1)(2j+1) ×   LBLFLBF 1/2Jj   |LB,(LF,s),j;J.(2.16) Wi h his i is easy o use s anda d angula momen um coupling o calcula e bo h collec i e, T(λ) C, and single pa icle, T(λ) F, pa s o any elec omagne ic ope a o : 064316-4 SHAPE PHASE TRANSITION IN ODD NUCLEI IN A . . . PHYSICAL REVIEW C 75, 064316 (2007) TABLE I. Reduc ion (b anching ules) om he OBF(5) i educible ep esen a ions (τ1,τ 2) ( i s column) o he OBF(3) i educible ep esen a ions LBF (second column). They can be ound, o ins ance, in Re . [63] (p. 73, Eqs. 3.85 and 3.86). In he las column, he angula momen um con en JBF in each OBF(5) ep esen a ion is p esen ed. I comes om he coupling o LBF o he e mion pseudospin s=1/2. The supe sc ip on a gi en LBF (o J) means he numbe o imes ha LBF (o J) is ob ained. OBF(5) i ep OBF(3) i ep To al ang. mom. (τ1,τ 2)LBF J (0,0) 0 1/2 (1,0) 2 5/2,3/2 (2,0) 4,2 9/2,7/2,5/2,3/2 (3,0) 6,4,3,0 13/2,11/2,9/2,(7/2)2,5/2,1/2 (4,0) 8,6,5,4,2 17/2,15/2,13/2,(11/2)2,(9/2)2,7/2,5/2,3/2 (1,1) 3,1 7/2,5/2,3/2,1/2 (2,1) 5,4,3,2,1 11/2,(9/2)2,(7/2)2,(5/2)2,(3/2)2,1/2 (3,1) 7,6,52,4,32,2,1 15/2,(13/2)2,(11/2)3,(9/2)3,(7/2)3,(5/2)3,(3/2)2,1/2 ξ,τB,τ F,(τ1,τ 2),L BF ,1/2; J||T(λ) C||ξ,τ B,τ F,(τ 1,τ 2),L  BF ,1/2; J = LB,LF,L B,L F(τB,0) (τF,0) |(τ1,τ 2) LBLF|LBF (τ B,0) (τ F,0) |(τ 1,τ 2) L BL F|L BF  × j,j (−1)L B+J+j+λ(2LBF +1)(2L BF +1)(2J+1)(2J+1)(2j+1)(2j+1) ×LBLFLBF 1/2Jj L BL FL BF 1/2JjLBJj JL Bλξ,τB,L B||T(λ) C||ξ,τ B,L  BδτF,τ FδLF,L Fδj,j,(2.17) whe e ξ,τB,L B||T(λ) C||ξ,τ B,L  Bis he collec i e pa al- eady calcula ed o he e en-e en sys em. Fo he e mion pa ξ,τB,τ F,(τ1,τ 2),L BF ,1/2; J||T(λ) F||ξ,τ B,τ F,(τ 1,τ 2),L  BF ,1/2; J = LB,LF,L B,L F(τB,0) (τF,0) |(τ1,τ 2) LBLF|LBF (τ B,0) (τ F,0) |(τ 1,τ 2) L BL F|L BF  × j,j (−1)L B+J+j+λ(2LBF +1)(2L BF +1)(2J+1)(2J+1)(2j+1)(2j+1) ×LBLFLBF 1/2Jj L BL FL BF 1/2JjjJL B Jjλ ×j||T(λ) F||jδξ,ξδτB,τ BδLB,L B,(2.18) whe e j||T(λ) F||jis a single-pa icle ma ix elemen ha is ob ained om j||(a† j1טaj2)(λ)||j=− √2λ+1δj2,jδj1,j .(2.19) Wi h hese gene al exp essions, any mul ipole ansi ion and momen can be e alua ed. In pa icula , we a e s udying E2 and M1 ansi ions and momen s below. 064316-5 C. E. ALONSO, J. M. ARIAS, AND A. VITTURI PHYSICAL REVIEW C 75, 064316 (2007) IBFM E(5/12) ξ=1 ξ=1 ξ=2 ξ=2 (τ1 B+1,0) (τ1 B,1) (τ1 B-1,0) 0,0(0,0) 0,0(0,0) 1,0(1,0) 2,0(2,0) 1,0(1,0) 2,0(2,0) 3,0(3,0) 0,1(1,0) 0,1(1,0) 1,1(2,0) 2,1(3,0) 3,1(3,1) 2,1(2,1) 1,1(1,1) 2,1(1,0) 1,1(0,0) κ=−1/4 κ′=5/2 0,0(0,0) 1,0(1,0) 2,0(2,0) 3,0(3,0) 0,0(0,0) 1,0(1,0) 2,0(2,0) 0,1(1,0) 1,1(2,0) 2,1(3,0) 1,1(1,1) 2,1(2,1) 3,1(3,1) 1,1(0,0) 2,1(1,0) 0,1(1,0) 167 100 145 145 63 88 143 161 90 60 81 108 100 100 217 75 124 52 31 24 0.5 0.7 118 179 145 211 111 74 45 25 110 9 11 10 41 12 16 30 8 35 18 29 17 12 0.07 0.2 2 2 FIG. 3. Uppe ame: Ene gy le els in he odd sys em a he E(5/12) c i ical poin (no malized o he ene gy o he i s exci ed s a e). Each (degene a e) s a e is cha ac e ized by he ξ,τB,τ F,(τ1,τ 2) quan um numbe s. The alues k=−1/4andk= 5/2 we e used in he Hamil onian (2.7). The alues o he angula momen a o each degene a e s a e can be ex ac ed om Table I. Some selec ed alues o B(E2), co esponding o he ansi ions be ween he highes spins o ini ial and inal mul iple s, a e gi en in he igu e [ he B(E2) associa ed wi h he lowes 5/2-1/2 ansi ion is no malized o 100]. Lowe ame: Ene gy le els in he odd sys em wi hin he IBFM Hamil onian (3.6) a he c i ical poin . The numbe o bosons N=7 has been assumed. The alues o kand ka e he same as in he E(5/12) Hamil onian. Fo B(E2), see he uppe ame. A. E2 ansi ion and momen s Le us i s conside he elec ic quad upole ope a o . This can be w i en as he sum o he collec i e con ibu ion plus -2 -1 0 1 2 κ -2 -1 0 1 2 3 4 5 Ene gy (a bi a y uni s) -1 0 1 2 3 4 κ ′ κ ′=5/2 κ =−1/4 0,1 (1,0) 0,0 (0,0) 0,0 (0,0) 0,0 (0,0) 0,0 (0,0) 1,1 (1,1) 1,1 (0,0) 1,0 (1,0) 1,0 (1,0) 0,1 (1,0) 1,1 (0,0) 1,1 (1,1) FIG. 4. Beha io o he ew lowes ene gy E(5/12) bandheads as a unc ion o pa ame e s kand kin he Hamil onian. he odd-pa icle one: T(E2) =T(E2) C+T(E2) F.(2.20) Fo he collec i e pa , usually a quad upole ope a o depend- ing linea ly on βis used in he collec i e models T(E2) Cµ= (2) Cα2,µ = (2) CβD(2) µ0(θi) cos γ +1 √2D(2) µ2(θi)+D(2) µ−2(θi)sin γ,(2.21) whe e (2) Cis a scale ac o . Howe e , he impo ance o he nex e m quad a ic in βhas been shown. Because o ha we will include he nex o de in he quad upole coo dina es, which is gi en by [67] T(E2) Cµ= (2) Cα2µ+χ 2√7[α2×α2](2) µ = (2) C m D(2)∗ µm ¯ Q(2) m,(2.22) whe e ¯ Q(2) 0=βcos γ−1 14χβ2cos (2γ), ¯ Q(2) 2=Q(2) −2=1 √2βsin γ+1 14χβ2sin (2γ), ¯ Q(2) 1=¯ Q(2) −1=0. (2.23) The 2√7 denomina o in Eq. (2.22) is in oduced he e o con enience o ma ch he exp ession based on he in e ac ing boson model (IBM) quad upole ope a o . The χpa ame e con ols he weigh o he quad a ic ela i e o he linea e m. The α2µ a iable beha es as he O(5) ⊃O(3) enso (τ,L)= (1,2), while he second-o de e m [α×α](2) µis associa ed wi h he enso (2,2) [67]. This means ha hey ul ill s ic selec ion ules in he φτµLM(γ,θi) collec i e space: τB=±1 o he o me and τB=0,±2 o he la e , wi h he usual es ic ions in angula momen um coupling [31,67]. Thus he i s e m leads o “allowed” ansi ions in i s o de , and he second o nonze o quad upole momen s and o small alues o “ o bidden” ones, in he E(5) scheme o Re . [4]. Fo he single-pa icle pa , in gene al, T(E2) F= j,j (2) j,j(a† jטaj)(2),(2.24) whe e (2) j,ja e pa ame e s. To educe he numbe o pa ame e s, he ollowing o m dic a ed by he U(6/12) symme y has been used: T(E2) F= (2) F−4 5(a† 1/2טa3/2+h.c.)(2) −6 5(a† 1/2טa5/2+h.c.)(2).(2.25) 064316-6 SHAPE PHASE TRANSITION IN ODD NUCLEI IN A . . . PHYSICAL REVIEW C 75, 064316 (2007) 1 1 1 1 5 5 5 106 5 3 33 3 53 3 7 9 977 167 17 151 134 33 50/117 100 100 118 31 15 24 52/35 0.50.5 3412 95 34/6/13 35/83 25/99 75 75 B(E2) 74/50 86/37 5 49/74 111/12 ∆τ = ±1 B E(5/12) FIG. 5. B(E2) s eng hs induced by he linea e m o he collec i e E2 ope a o o he E(5/12) model. Spins gi en a e wice he ac ual alue. T ansi ions obey he selec ion ule τB=±1. The ansi ion B(E2; 5/21→1/21) is no malized o 100. To ge absolu e alues, all he s eng hs ha e o be mul iplied by (0.07453)( (2) C)2β2 w. Fo he odd sys em, con ibu ions om bo h collec i e and single-pa icle ope a o s a e o be calcula ed. In he uppe panel o Fig. 3,someB(E2) alues a e p esen ed o he collec i e con ibu ion and ansi ions be ween he la ges spin in each degene a e mul iple . In Figs. 5and 6,B(E2) alues o τB=±1 and τB=0,±2 a e p esen ed. In he i s case, only he linea e m is e ec i e; while in he second case, only he quad a ic con ibu ion plays a ole. 1 1 1 1 5 5 55 5 3 33 3 53 3 7 9 977 56 12 14 122 110 100 61/15 23/52 73 E(5/12) B(E2) ∆τB = 0, ±2 0 23 56 50 118 45 26 3 116 90 254/23/177/20 21/257/39/158 355/236 0/160/45 60/91 449/300 310 310 FIG. 6. B(E2) s eng hs induced by he quad a ic e m o he collec i e E2 ope a o o he E(5/12) model. Spins gi en a e wice he ac ual alue. T ansi ions obey he selec ion ule τB=0,±2. The ansi ion B(E2; 5/21→3/21) is no malized o 100. To ge absolu e alues, all he s eng hs ha e o be mul iplied by (0.0007855)χ2( (2) C)2β4 w. 1 1 1 1 1 5 55 5 0.7 33 3 5 3 7 9 977 53 1.73 1.58 -0.37 -0.26 0.86 0.60 00 1.11 1.01 -0.24 -0.17 0.29 0.25 -0.41 E(5/12) collec i e Q 3 FIG. 7. Quad upole momen s induced by he collec i e quad a ic e m in T(E2) o he E(5/12) model. Spins gi en a e wice he ac ual alue. The quad upole momen o he s a e 5/21is no malized o 1. To ge absolu e alues, all he quad upole momen s ha e o be mul iplied by (0.12229)χ (2) Cβ2 w. Collec i e quad upole momen s induced by he quad a ic e m in he E2 ope a o can be calcula ed as eQ(ξ,τB,τ F,(τ1,τ 2),L BF,1/2; J) =16π 5J(2J−1) (J+1)(2J+1)(2J+3) ×ξ,τB,τ F,(τ1,τ 2),L BF,1/2; J||T(E2) C+T(E2) F|| ×ξ,τB,τ F,(τ1,τ 2),L BF,1/2; J.(2.26) In Fig. 7, he collec i e quad upole momen s induced by he quad a ic e m in T(E2) Ca e p esen ed no malized o he quad upole momen o he 5/21s a e. The e mionic quad upole ope a o (2.25) also con ibu es o some ansi ions. Howe e , he s uc u e (2.25) p o ides a selec ion ule τF=±1, and consequen ly he e a e no con ibu ions o he e mion pa o he quad upole momen s. Fo he same eason, he e a e no ansi ions whe e in e e ence be ween he collec i e and he e mionic pa s exis s ( o he collec i e pa , τF=0). In addi ion, he e mionic E2 ope a o p o ides a selec ion ule τB=0. In Fig. 8,a ew e mionic E2 ansi ions induced by he ope a o (2.25)a e p esen ed. B. M1 ansi ion and momen s The magne ic dipole ope a o can be w i en as he sum o he collec i e con ibu ion plus he odd-pa icle one: T(M1) =T(M1) C+T(M1) F.(2.27) The collec i e con ibu ion is T(M1) C= (1) C1ˆ LB √10 + (1) C2α2׈ LB √10(1) +.... (2.28) 064316-7 C. E. ALONSO, J. M. ARIAS, AND A. VITTURI PHYSICAL REVIEW C 75, 064316 (2007) 1 1 1 1 5 5 5 5 E(5/12) 3 e mionic B(E2) 3 3 0.1/0.9 ∆ τF = ± 1 3 97 7 1 1 0.8/0.2 0.3/0.7 0.2/0.8 0.9/01 0.7/0.3 0.4/0.6 0.6/0.4 FIG. 8. B(E2) s eng hs induced by he e mionic ope a o (2.25) in T(E2) o he E(5/12) model. Spins gi en a e wice he ac ual alue. The s eng h om he s a e [ξ=1,τ B=0,τ F=1,(τ1=1,τ 2= 0)LBF =2,J =5/2] o he g ound s a e is 1. To ge absolu e alues, all s eng hs ha e o be mul iplied by ( (2) F)2. The i s e m is p opo ional o he collec i e angula mo- men um ˆ LBand is diagonal o he e en-e en co e bu no o he odd sys em. The ansi ion s eng hs and dipole momen s a ising om his e m a e p esen ed in Figs. 9and 10. The e mionic e m is p opo ional o he e mion angula momen um j. Fo he case we a e s udying, he app op ia e ope a o is T(M1) F=− (1) F1 2(a† 1/2טa1/2)(1) +√5(a† 3/2טa3/2)(1) +35 2(a† 5/2טa5/2)(1).(2.29) The e mionic magne ic dipole ope a o (2.29) also con ibu es o some ansi ions wi h he selec ion ule τB=0. Since 1 1 1 1 5 5 55 5 3 33 3 53 3 7 9 97 7 111 0 100 0 0 100 5 28 100 E(5/12) B(M1) ∆τB = 0, ±1 0 25 27 21 875 321/29/400 1000/500 788/88 313/9/429 53 100* 7* 96*/11* FIG. 9. B(M1) ansi ion s eng hs induced by he collec i e M1 ope a o (2.28) o he E(5/12) model. Spins gi en a e wice he ac ual alue. Bo h τB=0andτB=±1 ansi ions a e included. The τB=0 ansi ions a e no malized o he B(M1; 5/21→3/21)= 100. In o de o ge absolu e alues B(M1) wi h τB=0ha e o be mul iplied by (0.04)( (1) C1)2.TheτB=±1 a e ma ked wi h an ∗ and a e no malized o he B(M1; 5/22g→5/21)=100 (5/22gis he second 5/2 wi hin he g ound s a e band). To ge absolu e alues, B(M1) wi h τB=±1 ha e o be mul iplied by (0.02445)( (1) C2)2. his selec ion ule is iden ical o his pa and he collec i e pa o he dipole ope a o , he e will be in e e ence be ween bo h e ms. Table II p esen s combined s eng hs o selec ed ansi ions in which in e e ence be ween collec i e and single- pa icle e ms a e impo an . Figu es 3–10 and Table II gi e de ailed spec oscopic in o ma ion o he E(5/12) model. In he nex wo sec ions, calcula ions a he c i ical poin in he ansi ion om sphe ical TABLE II. T(M1) educed ma ix elemen s o B(M1) and magne ic momen s in ol ing in e e ence e ec s be ween collec i e and single-pa icle con ibu ions o some selec ed E(5/12) s a es. Ini ial |iFinal | Reduced ma ix elemen |ξ,τB,τ F(τ1,τ 2)LBF,J|ξ,τB,τ F(τ1,τ 2)LBF,J ||T(M1)||i 1,1,0 (1,0)2,5/2 1,1,0 (1,0)2,3/2 23 5 (1) F−0.4899 (1) C1 1,1,0 (1,0)2,5/2 1,1,0 (1,0)2,5/2 21 10 (1) F+1.833 (1) C1 1,1,0 (1,0)2,3/2 1,1,0 (1,0)2,3/2 −3 5 (1) F+1.4697 (1) C1 1,2,0 (2,0)4,9/2 1,2,0 (2,0)4,7/2 2√10 3 (1) F−0.6667 (1) C1 1,2,0 (2,0)2,5/2 1,2,0 (2,0)2,3/2 23 5 (1) F−0.4899 (1) C1 1,2,0 (2,0)4,9/2 1,2,0 (2,0)4,9/2 55 18 (1) F+4.4222 (1) C1 1,2,0 (2,0)4,7/2 1,2,0 (2,0)4,7/2 −√14 3 (1) F+3.9441 (1) C1 1,2,0 (2,0)2,5/2 1,2,0 (2,0)2,5/2 21 10 (1) F+1.8330 (1) C1 1,2,0 (2,0)2,3/2 1,2,0 (2,0)2,3/2 −3 5 (1) F+1.4697 (1) C1 064316-8 SHAPE PHASE TRANSITION IN ODD NUCLEI IN A . . . PHYSICAL REVIEW C 75, 064316 (2007) 1 1 1 1 1 5 55 5 0.9 33 3 5 3 7 9 977 53 2 1.94 1 0.9 10.9 00 1 0.97 0.5 0.45 0.75 0.71 0.25 E(5/12) collec i e µ 3 3 0.17 0 FIG. 10. Magne ic momen s induced by he leading collec i e e m in T(M1) o he E(5/12) model. Spins gi en a e wice he ac ual alue. The magne ic momen o he s a e 5/21is no malized o 1. To ge absolu e alues, all he magne ic momen s ha e o be mul iplied by (1.29442) (1) C1. o de o med γ-uns able shapes wi hin he IBFM amewo k a e p esen ed. Two di e en IBFM Hamil onians o desc ibing he c i ical poin a e used in o de o see whe he he signs o c i icali y p oduced by E(5/12) a e obus . Be o e ending his sec ion, we would like o no e ha ou E(5/12) model yields an in ini e numbe o le els, i co esponds o sol ing he Boh equa ion o a γ-independen in ini e squa e well in βpo en ial. This will no be he case o he IBFM calcula ions, which a e associa ed wi h a ini e numbe o bosons and consequen ly cha ac e ized by a ini e numbe o le els. In his la e case, in addi ion, he spec um i sel will change acco ding o he chosen numbe o bosons, in pa icula , i s highe pa . The compa ison be ween he wo app oaches will he e o e necessa ily be signi ican only o he lowe pa o he spec um as in he p e ious E(5/4) model o odd-e en nuclei, in he E(5), X(5) models o e en-e en nuclei conce ning c i ical poin s, o , in gene al, when compa ing he collec i e model wi h he IBM. III. AN INTERACTING BOSON-FERMION MODEL HAMILTONIAN THAT MIMICS E(5/12) AT THE CRITICAL POINT Fo he odd-e en sys em, he IBFM Hamil onian is w i en as H=HB+HF+VBF,(3.1) whe e HBis he bosonic pa , HF he e mionic one, and he e m VBF couples he boson and e mion deg ees o eedom. Fo he bosonic pa , we will use a pa ame ized Hamil onian ha p oduces a ansi ion be ween sphe ical and γ-uns able shapes o he o m HB=xˆnd−1−x Nˆ QB·ˆ QB,(3.2) whe e he ope a o s in he Hamil onian a e gi en by ˆnd= µ d† µdµ,(3.3) ˆ QB=(s†×˜ d+d†×˜s)(2),(3.4) and Nis he o al numbe o bosons. This Hamil onian can be ecas in o he o m HB=xC1(UB(5)) −1−x 2NC2(OB(6)) −C2(OB(5)), (3.5) whe e C1(UB(5)) is he linea Casimi ope a o o he UB(5) g oup, while C2(OB(6)) and C2(OB(5)) a e he quad a ic Casimi ope a o s o he OB(6) and OB(5) g oups. We ecall ha his Hamil onian p oduces, when a ying he pa ame e x om 1 o 0, a phase ansi ion be ween he wo ex eme si ua ions cha ac e is ic o U(5) and O(6) symme ies, a xc=4N−8 5N−8. No e ha o any alue o x his Hamil onian main ains he degene acies ypical o he O(5) symme y. Consis en wi h his, wi hin he IBM cohe en s a e o malism [32–34], his Hamil onian always p oduces an ene gy su ace which is independen o he γdeg ee o eedom. In he β a iable, he ene gy su aces display a sphe ical minimum in β=0 o xla ge han he c i ical alue, while ha ing a de o med minimum o alues o xsmalle han he c i ical alues [4,41,58]. I is wo h men ioning ha when conside ing he combined boson- e mion sys em, he ene gy su ace has con ibu ions om bo h bosons and e mions. The con ibu ion om he e mion is o he o de 1/N compa ed wi h he one om he bosons. Thus, usually i is accep ed ha he ene gy su ace o he combined sys em is go e ned by he bosons wi h sligh con ibu ions om he e mionic deg ees o eedom. Howe e , close o he c i ical poin , he boson ene gy su ace is a he la , and he con ibu ion om he odd e mion and i s coupling o he boson co e will play a undamen al ole in de e mining he minimum in he po en ial ene gy su ace. This is an impo an opic ha emains o be in es iga ed in de ail. Fo he pu poses o he p esen pape , i.e., checking he analy ical esul s ob ained o he E(5/12) model, we will accep ha he c i ical poin o he combined sys em is jus he c i ical poin o he boson pa and he e mion is coupled o i wi hou changing he equilib ium con igu a ion. Depending on he selec ion o HF+VBF in Eq. (3.1), di e en model Hamil onians o s udy he c i ical poin a e p oduced. In his sec ion HF+VBF is selec ed so as o ha e a IBFM Hamil onian as close as possible o he E(5/12) one. Thus, ou p oposal is HF+VBF =k 2N[C2(OBF(5)) −C2(OB(5)) −C2(OF(5))] +k 2NC2(OF(5)),(3.6) whe e C2(OF(5)),C 2(OBF(5)) a e he quad a ic Casimi op- e a o s o he OF(5) and OBF(5) algeb as, espec i ely. As men ioned abo e his Hamil onian is designed o mimic as much as possible he co esponding Hamil onian in E(5/12), since he e ms 2 ˆ LB◦ˆ LFand ˆ L2 Fin Eq. (2.7) can be w i en in 064316-9