Phase transitions in the interacting boson fermion model: The γ -unstable case

RAPID COMMUNICATIONS PHYSICAL REVIEW C 72, 061302(R) (2005) Phase ansi ions in he in e ac ing boson e mion model: The γ-uns able case C. E. Alonso,1J. M. A ias,1L. Fo una o,2and A. Vi u i2 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 5 Sep embe 2005; published 27 Decembe 2005) The phase ansi ion a ound he c i ical poin in he e olu ion om sphe ical o de o med γ-uns able shapes is in es iga ed in odd nuclei wi hin he in e ac ing boson e mion model. We conside he pa icula case o an odd j=3/2 pa icle coupled o an e en-e en boson co e ha unde goes a ansi ion om sphe ical U(5) o γ-uns able O(6) si ua ion. The pa icula choice o he j=3/2 o bi al p ese es in he odd case he condi ion o γ-ins abili y o he sys em. As a consequence, ene gy spec um and elec omagne ic ansi ions, in co espondence o he c i ical poin , display beha io s quali a i ely simila o hose o he e en co e. The esul s a e also in quali a i e ag eemen wi h he ecen ly p oposed E(5/4) model, al hough ew di e ences a e p esen , due o he di e en na u e o he wo schemes. DOI: 10.1103/PhysRe C.72.061302 PACS numbe (s): 21.60.Fw, 21.60.E The s udy o phase ansi ions has ecen ly ecei ed pa icula a en ion in nuclea s uc u e. The concep o c i ical poin symme y has been i s p oposed in a numbe o cases by Iachello [1–3]. These symme ies apply when a quan al sys em unde goes ansi ions be ween adi ional dynamical symme ies, as o example hose cha ac e izing si ua ions desc ibed in e ms o ha monic ib a ions o igid o a ions. Al hough hese symme ies ha e been ob ained wi hin he o malism based on he Boh Hamil onian [4], hei concep has also been used in connec ion wi h he in e ac ing boson model (IBM) [5]. One o hese c i ical poin symme ies is associa ed wi h he ansi ion be ween sphe ical and γ-uns able shapes. Wi hin he IBM his can be ob ained, o example, om he Hamil onian HB=xˆ nd−1−x N ˆ QB·ˆ QB,(1) which p oduces, a ying he pa ame e x om 1 o 0, a ansi ion be ween he wo ex eme si ua ions cha ac e is ic o U(5) and O(6) symme ies. The co esponding second-o de shape phase ansi ion has been in es iga ed wi hin he Boh collec i e model in Re . [6]. The ope a o s appea ing in he Hamil onian abo e a e gi en by ˆ nd= µ d† µdµ,(2) ˆ QB=(s†×˜ d+d†×˜ s)(2),(3) and Nis he o al numbe o bosons. Fo any alue o x his Hamil onian main ains he ypical degene acies o he O(5) symme y. Consis en ly wi h his, wi hin he IBM cohe en s a e o malism [7–9], 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 ace displays a sphe ical minimum in β=0 o xla ge han he c i ical alue xc=4N−8 5N−8, while ha ing a de o med minimum o alues o x smalle han he c i ical alue. A he c i ical poin , he ene gy su ace acqui es a β4beha io [10,11], which is app oxima ed by an in ini e squa e well in he E(5) c i ical poin symme y [1] wi hin he amewo k o he collec i e Boh Hamil onian. Recen ly Iachello has discussed a supe symme ical ex en- sion o his concep , in oducing he so-called E(5/4) model, whe e he boson pa has a γ-independen squa e well po en ial and he boson- e mion coupling is aken as a spin (5) scala in e ac ion [12]. This solu ion desc ibes he spec al p ope ies o odd-e en nuclei a he ansi ion be ween sphe ical and γ-uns able shapes. In his Rapid Communica ion we wan o discuss he e olu ion o odd nuclei and he co esponding beha io a he c i ical poin , wi hin he amewo k o he in e ac ing boson e mion model (IBFM) [13], in which a single e mion is coupled o he e en-e en bosonic co e. The sys em will hen be desc ibed by he Hamil onian H=HB+HF+VBF ,(4) whe e he e m VBF couples he bosonic and e mionic pa s. In ou case we will assume he boson Hamil onian o be o he o m gi en in Eq. (1). Fo he e mion and boson- e mion pa s we will ake he pa icula choice o a pa icle mo ing in a single j-shell j=3/2 and a coupling e m o he o m VBF =−21−x N ˆ QB·ˆ qF,(5) whe e ˆ QB[ aken o he o m gi en in Eq. (3)] and ˆ qF= (a† 3/2ט a3/2)(2) a e he boson and e mion quad upole ope a- o s, espec i ely. This choice o he e mion space and he boson- e mion in e ac ion is such ha one eco e s, in he ex eme cases, he Bose-Fe mi symme y [13] associa ed wi h he spinBF(6) g oup ( o x=0) and he ib a ional UB(5) ⊗SUF(4) case ( o x=1). In analogy wi h he o e all O(5) s uc u e in he e en case, his selec ion gua an ees he p ese a ion o he degene acies associa ed wi h he spinBF (5) symme y o any alue o x. This can be mo e clea ly seen wi hin he in insic ame o malism [7–9]. In o de o ge ene gy eigen alues o he case o a j=3/2 pa icle coupled o he g ound s a e |B(β,γ)boson condensa e one has o diagonalize he Hamil onian in a space o dimension ou wi h basis ec o s: |B(β,γ)⊗|j,m. The pu e boson pa o he case conside ed he e is only a unc ion o βand gi es a 0556-2813/2005/72(6)/061302(4)/$23.00 061302-1 ©2005 The Ame ican Physical Socie y RAPID COMMUNICATIONS ALONSO, ARIAS, FORTUNATO, AND VITTURI PHYSICAL REVIEW C 72, 061302(R) (2005) 0 0.2 0.4 0.6 0.8 1 1-x 0 1 2 3 4 5 6 E / E21 + 0 0.2 0.4 0.6 0.8 1 1-x 0 1 2 3 4 5 6 E / E1s exci ed N=7 N=7 1/2,5/2,7/2 3/2,5/2,7/2 3/2 (3/2,1/2) (5/2,1/2) (1/2,1/2) (3/2,1/2) (7/2,1/2) 3/2 (1/2,1/2) (5/2,1/2) 0+ 2+ 2+ ,4 + 0+ ,3 + ,4 + ,6 + 0+ 2+ (0) (1) (2) (3) (1) (0) 15 15 15 15 13 13 13 2σ1 c i ical poin poin c i ical (1/2,1/2) (3/2,1/2) (5/2,1/2) (4) (0) 9/2,11/2 (2) FIG. 1. Ene gy le els (no malized o he ene gy o he i s exci ed s a e) o he e en and odd sys ems a e displayed as a unc ion o he pa ame e (1 −x) o he boson (1) and boson- e mion (4) Hamil onians. A numbe N=7 o bosons has been assumed in bo h cases, while he odd pa icle has been aken in he j=3/2 o bi al. In he le panel (e en case) we indica e o each le el he τquan um numbe (in pa en hesis), spin and pa i y. In he igh panel (odd case) we quo e he (τ1,τ 2) quan um numbe s (in pa en hesis) and spin. In he ex eme x=0 case we also indica e he σ1quan um numbe . The posi ion o he e en c i ical poin is ma ked. global diagonal con ibu ion. The pu e e mionic pa is jus a cons an . Thus, he only emaining pa , VBF , has o be diagonalized. I s eigen alues, doubly degene a e, u n ou o be γ-independen , E±(β,γ)=E±(β)=±2(1 −x)β 1+β2.(6) In o he wo ds, he addi ion o he odd pa icle does no des oy he γ-ins abili y o he sys em, gi ing ise o ene gy su aces o he di e en odd in insic s a es ha a e s ill γ-independen [14]. The pa icula beha io o he j=3/2 o bi al was i s pu in e idence by Bayman and Sil e be g [15] wi hin he collec i e model. The esul ing ene gy spec a in he odd sys em a e shown in he igh panel o Fig. 1 as a unc ion o he con ol pa ame e 1−x. The o al numbe o bosons, N, has been assumed o be equal o 7. Fo a be e compa ison, we also show in he le panel o he igu e he co esponding e olu ion o he spec um in he e en co e. I should be emembe ed ha he use o a ini e numbe o bosons does no lead o he same esul s as he co esponding ones ob ained in he equi alen si ua ion wi hin he Boh Hamil onian, which is only eached in he limi o in ini e numbe o bosons. No e also ha he geome ical limi ob ained om he boson Hamil onian (1) o la ge alues o Nco esponds o he case o he collec i e po en ial beha ing as β4, and no p ecisely o he E(5) case, ha co esponds o he in ini e squa e well. The le el e olu ion in he odd case shows a beha io quali a i ely simila o ha o he e en case. The g oup s uc u e o spinBF(5) wi h espec o O(5) simply leads o a iche pa e n o he e mion case and sligh ly di e en a ios o he ene gy le els. Fo example in he limi ing x =0 case, co esponding o spinBF(6) and O(6) symme y g oups o he odd and e en nuclei, espec i ely, we ob ain o he “g ound bands,” wi h maximum σ alues (σ1=N+1/2 and σ=N o odd and e en nuclei, espec i ely), he a ios E(τ1=5/2,τ 2=1/2)/E(τ1=3/2,τ 2=1/2) =2.4 and E(τ1=7/2,τ 2=1/2)/E(τ1=3/2,τ 2=1/2) =4.2 wi h espec o he alues E(τ=2)/E(τ=1) =2.5 and E(τ=3)/E(τ=1) =4.5 o he e en case. As in he e en case, he le els ha in he ex eme x=0 limi e en ually co espond o he “g ound band” (σ1=N+1/2) show a a he smoo h (o e en la ) beha io , aside om he changes a ound he c i ical poin . Ins ead, he le els ha will end up wi h o he alues o σ1(as he le els co esponding o smalle alues o σin he e en case) show a mo e iolen a ia ion in he whole ange. A ypical case is he second (τ1=1/2,τ 2=1/2)j=3/2 s a e ha s a s a one-phonon ene gy in he ib a ional limi o end up as he hi d j= 3/2 s a e in he opposi e limi . The posi ion o his 3/2 s a e is he key elemen o cha ac e ize he pa icula si ua ion and i s posi ion along he ansi ional pa h. The posi ion o his s a e plays he same ole as he key posi ion o he i s exci ed 0+s a e in e en nuclei. The smoo he beha io o he g ound band le els wi h espec o he o he bands con i ms he ac ha o es ablish a de ini e c i ical si ua ion i is no a all su icien o ely jus on he sequence o ene gy le els in he “g ound” band, and ha some o he claims o he occu ence o de ini e ansi ional symme ies ha e o be aken wi h se ious cau ion. C ucial and selec i e in o ma ion on nuclea spec a comes om he ansi ion p obabili ies, in pa icula om he elec ic quad upole ones. These a e ob ained in e ms o he ma ix elemen s o he elec ic quad upole ope a o . In ou IBFM case his ope a o is gi en by ˆ QBF =ˆ QB+ˆ qF=(s†×˜ d+d†×˜ s)(2) +(a† jט aj)(2). (7) No e ha , consis en ly wi h he boson quad upole ope a o used in he Hamil onian, we ha e no included any (d†×˜ d)(2) e m in he ansi ion ope a o . Values o he indi idual ansi ion p obabili ies, s a e by s a e, o he odd nucleus a he c i ical poin si ua ion a e shown in Fig. 2. In i , we ha e plo ed he lowes six mul iple s, which ha e been a bi a ily spli in o de o show he possible 061302-2 RAPID COMMUNICATIONS PHASE TRANSITIONS IN THE INTERACTING BOSON . . . PHYSICAL REVIEW C 72, 061302(R) (2005) 3/2 1(1/2,1/2) 1/2 (3/2,1/2) 5/2 1 7/2 3/2 2.29 (5/2,1/2) 5/2 7/2 9/2 11/2 3/2 21.40 (1/2,1/2) 1/2 5/22.64 (3/2,1/2) 7/2 3/22.91 (1/2,1/2) (ττ 1, 2) 0 1  1 2 3 ξ 100 19.2 13.1 12.7 0.13 8.49 2.29 0.61 5.7 4.4 141.1 30.2 110.9 69.1 72.0 34.8 21.6 84.6 16.1 75.6 49.4 81.6 13.1 8.9 1.02 0.76 0.25 22. 7 0.95 0.20 0.37 0.14 0.04 1.0 0.49 0.12 0.27 1.37 0.54 5.6 5.4 6.7 0.15 3.64 10.9 0.24 0.83 0.62 0.21 13.7 14.8 11.1 3.7 0.51 j FIG. 2. Ene gy le els and quad upole ansi ion a es B(E2↓) o he odd sys em a he co e c i ical poin . Fo illus a ion pu poses he a ious mul iple s, labeled by he (τ1,τ 2) quan um numbe s in pa en hesis, ha e been a bi a ily spli acco ding o hei jquan um numbe . The label on he ex eme le is he ene gy in ela i e uni s, while he label a he ex eme igh co esponds o he label ξ,usedinRe .[12]. B(E2↓)’s ha e been no malized o he alue 100 o he ansi ions (wi h equal s eng hs) be ween he s a es o he i s mul iple and he g ound s a e. E2 ansi ions be ween he di e en s a es. In he igu e he le els ha e di e en leng hs depending on he ξ alue, which labels di e en amilies and i is shown along wi h he spin o he s a e (we con o m he e o he no a ion in oduced in Re . [12]). Al hough he gene al pa e n emains he same, he de ailed ene gy sequence depends on he numbe o in e ac ing bosons, N. Owing o he p ope ies o he spinBF(5) g oup, he τ1=0,±1,τ 2=0 selec ion ule s ill hold. I can be obse ed ha E2 ansi ions a e s onge be ween s a es wi h he same ξ alue and τ1=1. T ansi ions be ween s a es pe aining o amilies wi h di e en ξa e one o wo o de s o magni ude smalle . T ansi ions be ween s a es wi h he same ξand τ1 alues, bu di e en spins, co esponding o he same mul iple s a e also one o wo o de s o magni ude smalle han he ones be ween di e en mul iple s in he same band. The allowed quad upole ansi ions a e ske ched in Fig. 3 o he odd sys em o he spinBF(6) case and o he c i ical poin si ua ion, and o he e en case a he c i ical poin . In addi ion o he di e ence in he le el sequence, in he second case ansi ions a e allowed be ween di e en bands (al hough weake han inband ansi ions, as al eady men ioned), in a simila way as in e band ansi ions a e allowed a he c i ical poin o he e en case (as o example be ween he second 0+, wi h τ=0, and he i s 2+, wi h τ=1, as shown in he igu e). The compa ison o ou c i ical poin solu ion wi h Iachello’s one e idences a simila o e all o ganiza ion o he spec um, al hough wi h sizable di e ences in he ela i e posi ion o he di e en ξbands. I mus be ecalled ha , in p inciple, we should compa e he E(5/4) solu ion wi h he la ge Nlimi o he in e ac ing boson model, while we ha e concen a ed, along his pape , on he N=7 case. The i s τ1=5/2 mul iple in he E(5/4) lies a a ound 2.20, close o he alue 2.29 ob ained in ou model. On he o he hand, he wo i s mul iple s o he ξ=2 amily o exci ed s a es o he E(5/4) solu ion lie a a ound 3.33 and 5.02, compa ed wi h he co esponding alues 1.40 and 2.64 o ou model. This in e sion o he posi ion o 1/2 σ1=15/2 3/2 5/2 τττ 1. . . σ1= 13/2 1/2 3/2 . . . SpinBF (6) 1/2 3/2 5/2 1. . .3/2 . . . 1/2 . . . C i ical poin - Odd case 0 1 2 . . .0 1 . . . C i ical poin - E en case 1/2 FIG. 3. Schema ic illus a ion o allowed quad upole ansi ions o he odd sys em a he spinBF (6) (le panel) and c i ical poin (middle panel) cases, and o he e en case a he c i ical poin ( igh panel). Solid lines indica e s onge in-band ansi ions wi h espec o he in e band ansi ions shown as dashed lines. Ho izon al lines co espond o mul iple s including se e al j alues. 061302-3 RAPID COMMUNICATIONS ALONSO, ARIAS, FORTUNATO, AND VITTURI PHYSICAL REVIEW C 72, 061302(R) (2005) 0 0.2 0.4 0.6 0.8 1 1-x 0 1 2 E (MeV) 0 0.2 0.4 0.6 0.8 1 1-x 0 1 2 E (MeV) 3/2 1/2,5/2,7/2 3/2 3/2 1/2, 5/2, 7/2 5/2 7/2 1/2 9/2 3/2 5/2 3/2,5/2, 7/2,9/2, 11/2 FIG. 4. Ene gy le els o he odd sys em a e displayed as a unc ion o he pa ame e (1 −x) o he boson- e mion Hamil onian (4). A numbe N=7 o bosons has been assumed, while he odd pa icle has been aken in he j= 3/2 o bi al (le panel) and j=5/2 o bi al ( igh panel). The spin quan um numbe is indica ed o each s a e. he ξ=1,τ 1=5/2 and ξ=2,τ 2=1/2 mul iple s is he i s e iden di e ence be ween he wo models. Ano he impo an di e ence is seen in he ela i e B(E2) alues: al hough in-band ansi ions display compa able alues, signi ican disc epancies a e obse ed in in e band ansi ion (no ably he ansi ions be ween he ξ=2,τ 1=1/2 s a e and he wo lowes mul iple s o he g ound s a e band). P ese ing he degene acies associa ed wi h he spinBF(5) symme y is no an exclusi e p ope y o he j=3/2 o bi als. O he cases a e known in he li e a u e, o example he case o he odd pa icle mo ing in he j=1/2, 3/2, 5/2 o bi als, unde he condi ion o special alues o he e mion quad upole ma ix elemen s. Also in his case one eco e s, o he x=0 limi ing case, he Bose-Fe mi symme y beha io . Bu his does no happen in mo e gene al cases wi h o he combina ions o o bi als o e mion quad upole ma ix elemen s [16]. To gi e an idea, we p esen in Fig. 4 he e olu ion o he spec um as a unc ion o he con ol pa ame e 1 −xin he cases o a single j=3/2 and j=5/2. The igu e shows a spec um o j=5/2 ( igh panel) ha is quali a i ely simila o he one displayed in he le panel o j=3/2, bu is mo e complex due o he emo al o all degene acies. To summa ize, we ha e conside ed, wi hin he in e ac ing boson e mion model, he coupling o an odd j=3/2 pa icle o a boson co e ha unde goes a ansi ion om sphe ical U(5) o γ-uns able O(6) cha ac e . The pa icula choice o he Hamil onian and o he j=3/2 o bi al p ese es in he odd case he condi ion o γ-ins abili y o he sys em, and i is e lec ed in he p ese a ion o he degene acies associa ed wi h he spinBF(5) symme y. As a consequence, he ene gy spec um and he elec omagne ic ansi ions o he odd nucleus wi h a c i ical co e display beha io s quali a i ely simila o hose cha ac e izing he phase ansi ion in he e en co e. We ha e compa ed ou esul s wi h he ecen ly p oposed E(5/4) app oach, based on he Boh Hamil o- nian. Bo h app oaches display simila quali a i e pic u es, al hough we e idence a numbe o quan i a i e di e ences, ha can be aced back o he di e en na u e o he wo schemes. We acknowledge enligh ening discussions wi h J. Ba ea and B. F. Bayman. This wo k was suppo ed by he I alian-Spanish ag eemen INFN-CICYT and by he Spanish DGICYT unde p ojec numbe FIS2005-01105. [1] F. Iachello, Phys. Re . Le . 85, 3580 (2000). [2] F. Iachello, Phys. Re . Le . 87, 052502 (2001). [3] F. Iachello, Phys. Re . Le . 91, 132502 (2003). [4] A. Boh and B. Mo elson, Nuclea S uc u e: Vol II, Nuclea De o ma ions (Benjamin, Reading, MA, 1975). [5] F. Iachello and A. A ima, The In e ac ing Boson-Model (Camb idge Uni e si y P ess, Camb idge, 1987). [6] P. S. Tu ne and D. J. Rowe, Nucl. Phys. A756, 333 (2005). [7] J. N. Ginocchio and M. W. Ki son, Nucl. Phys. A350, 31 (1980). [8] A. E. L. Diepe ink, O. Schol en, and F. Iachello, Phys. Re . Le . 44, 1747 (1980). [9] A. Boh and B. Mo elson, Phys. Sc . 22, 468 (1980). [10] J. M. A ias, C. E. Alonso, A. Vi u i, J. E. Ga c´ ıa-Ramos, J. Dukelsky, and A. F ank, Phys. Re . C 68, 041302(R) (2003). [11] J. E. Ga cia-Ramos, J. Dukelsky, and J. M. A ias, Phys. Re . C 72, 037301 (2005). [12] F. Iachello, Phys. Re . Le . 95, 052503 (2005). [13] F. Iachello and P. Van Isacke , The In e ac ing Boson Fe mion Model (Camb idge Uni e si y P ess, Camb idge, 1991). [14] C. E. Alonso, J. M. A ias, F. Iachello, and A. Vi u i, Nucl. Phys. A539, 59 (1992). [15] B. F. Bayman and L. Sil e be g, Nucl. Phys. 16, 625 (1960). [16] J. Jolie, S. Heinze, P. Van Isacke , and R. F. Cas en, Phys. Re . C70, 011305(R) (2004). 061302-4