Stabilization method in two-body systems with core excitations

S abiliza ion me hod in wo-body sys ems wi h co e exci a ions J.A. Lay∗, J.M. A ias∗, J. Gómez-Camacho∗,† and A.M. Mo o∗ ∗Depa amen o de FAMN, Facul ad de Física, Uni e sidad de Se illa, Apdo. 1065, E-41080 Se illa, Spain †Cen o Nacional de Acele ado es, A da. Thomas A. Edison, E-41092, Se illa, Spain Abs ac . The alidi y o he s abiliza ion me hod in co e+ alence sys ems including he possibili y o exci ing he co e is s udied. A pseudos a e me hod, based on he ans o med ha monic oscilla o basis, is ex ended o include he co e deg ees o eedom. The me hod is applied o he case o 11Be s uc u e conside ing he 0+g ound s a e and he 2+fi s exci ed s a e o he 10Be co e. The s abiliza ion me hod is defined in e ms o one pa ame e ha can be chosen ei he disc e e o con inuous. In he applica ion o 11Be, bo h cases a e analyzed. Keywo ds: Exo ic nuclei, con inuum disc e iza ion, esonances PACS: 24.10.Eq, 25.10.+s, 25.45.De INTRODUCTION Recen ly, he s udy o eac ions in ol ing loosely bound exo ic nuclei ha e been one o he mos ac i e fields in Nuclea Physics. This kind o eac ions is known o be s ongly influenced by he coupling o he unbound s a es o he weakly bound nucleus. The e o e he e has been a g ea in e es in finding a ealis ic desc ip ion o he con inuum o hese nuclei. In eac ion calcula ions, one-neu on halo nuclei a e usually desc ibed using a wo- body model, comp ising an ine co e and one alence pa icle. This Hamil onian can be sol ed using pseudos a es (PS). The PS me hod consis s in diagonalizing he Hamil o- nian in a squa e-in eg able basis. The eigens a es ob ained a e a fini e app oxima ion o he bound and unbound s a es o he nucleus. Mo eo e , he posi i e ene gy eigens a es can help o unde s and he s uc u e o he con inuum and i s influence in eac ions in- ol ing hese nuclei. Se e al bases ha e been es ed in he PS me hod. He e we use a ans o med ha monic oscilla o (THO) basis, which is o med by applying he ollow- ing ans o ma ion s( )=⎡ ⎢ ⎣ 1 1 4+1 γ √ 4 ⎤ ⎥ ⎦ 1 4 ,(1) o he solu ions o he iso opic ha monic oscilla o . This ans o ma ion was fi s in o- duced in [1]. This basis is easy o calcula e and, in mos o he cases, is able o educe he minimum numbe o unc ions needed o con e gence o he sca e ing obse ables wi h espec o o he bases [2]. Beau y in Physics: Theo y and Expe imen AIP Con . P oc. 1488, 436-440 (2012); doi: 10.1063/1.4759428 © 2012 Ame ican Ins i u e o Physics 978-0-7354-1100-5/$30.00 436 As men ioned abo e, in he desc ip ion o halo nuclei usually an ine co e is consid- e ed. Howe e , ecen expe imen al and heo e ical de elopmen s sugges ha possible exci a ions o he co e may a ec no only o he s uc u e bu also o he eac ion dy- namics. In a ecen wo k [3] we ha e gene alized he THO basis in o de o include he exci a ions o he co e in a simple pa icle- o o model. He e we s udy in mo e de ail he so-called s abiliza ion me hod ha allows o iden i y esonances in he con inuum using PS basis [4]. In his me hod, we assume he e is a pa ame e ela ed o he ex ension o he basis. The esul ing eigen alues depend on his pa ame e , so ha hey ypically dec ease in ene gy when he basis ex ension is inc eased. When an eigen alue c osses he ene gy o a esonance, his beha iou changes. The ene gy o he eigen alue is s abilized o a ce ain ange o alues o he pa ame e gi ing ise o a pla eau a ound he ene gy o he esonance. Mo eo e , he pseudos a e co esponding o his s abilized ene gy is ound o ep oduce well he p ope ies o he esonance. THE THO APPROACH TO PARTICLE-ROTOR MODEL In he weakly coupling limi , when we conside a co e+ alence sys em including di e - en exci a ions o he co e, we ha e a Hamil onian o he o m: H=T +V c( , ξ )+hco e( ξ ),(2) whe e hco e( ξ ) ep esen s he in e nal Hamil onian o he co e, wi h eigens a es φ I( ξ ). V c( , ξ )is he in e ac ion be ween he pa icle and he co e and i is esponsible o he coupling be ween he alence pa icle mo ion and he exci a ions o he co e. I depends on he model assumed o he exci a ions o he co e. In a pa icle- o o model, we suppose he co e o be pe manen ly de o med. In his case he in e ac ion wi h he alence pa icle would depend on he s a e o o a ion o he de o med co e and he ela i e o ien a ion be ween pa icle and o o . I he coupling is weak we can conside he o al wa e unc ion as a combina ion o co e s a es and alence configu a ions coupled o he co esponding o al angula momen um J: Ψ ε ;JM( , ξ )=∑ α RJ ε , α ( )Ysj(ˆ )⊗ φ I( ξ )JM .(3) Each e m o his sum is called channel and i is cha ac e ized by a gi en se o quan um numbe s α ={,s,j,I}. I we now cons uc a THO unc ion o each channel in he o m: Φ α n,JM( , ξ )=RTHO n, α ( )Ysj(ˆ )⊗ φ I( ξ )JM ,(4) we can eadily diagonalize he Hamil onian in a unca ed basis o N THO unc ions so ha he final PS will be: 437 1.6 2 2.4 2.8 γ ( m1/2) 0 1 2 3 4 5 6 ε (MeV) 0246810 N J=5/2+ spec um FIGURE 1. (Colo online) Eigen alues ob ained om he diagonaliza ion o he 11Be Hamil onian in a THO basis, as a unc ion o he LST con inuum pa ame e ( γ ) in he le panel, and as a unc ion o he numbe o oscilla o s a es included in he basis (N) in he igh panel. The dashed line indica es he ene gy o he 5/2+ esonance and he do ed line, he ene gy o he 10Be(2+)+n h eshold. Ψ(N) i,JM( , ξ )= N ∑ n=1∑ α ci n, α ,JΦ α n,JM( , ξ ).(5) APPLICATION TO 11BE Pa ame e s γ and Ncon ol he adial ex ension o he THO basis. The e o e, he s abiliza ion me hod can be applied wi h bo h pa ame e s. In p e ious wo ks we ha e used N o his pu pose, whe eas γ was al eady fixed in o de o ailo he densi y o s a es o he ene gy egion o in e es . The main d awback is ha Nis a disc e e a iable. The e is no gua an ee ha he poin whe e he ene gy is s abilized coincides wi h a alue o N. Tha is he main eason o use also γ , a con inuous a iable. Howe e , he fi s de i a i e o he e olu ion o he eigen alue is ela ed o he wid h o he esonance. This implies ha , o na ow esonances, he ene gy does no a y in a egion nea he s abilized poin and he nea es alue o Nis a good app oxima ion o he s abilized ene gy. In o de o analyze he alidi y o γ and Nas s abiliza ion pa ame e s in wo-body sys ems wi h exci a ions o he co e, we apply he PS THO basis o 11Be conside ing a pa icle- o o model. He e we conside ha he co e 10Be has a quad upole de o ma ion wi h β =0.67. We include only he fi s exci ed s a e o he co e 10Be, 2+, in addi ion o i s 0+g ound s a e. Fu he de ails o he po en ial can be ound in [5]. The fi s esonance in 11Be has spin and pa i y 5/2+. In figu e 1, we show he spec um ob ained a ying bo h γ and Npa ame e s. We see ha he s abiliza ion appea s a he ene gy o he esonance (1.2 MeV abo e he 10Be(0+)+n h eshold), so ha he s abiliza ion me hod emains alid in his exci ed co e amewo k. 438 0102030 ( m) -0.2 0 0.2 0.4 R( ) N=10 γ=1.84 m1/2 N=10 γ=2.75 m1/2 N=5 γ=1.84 m1/2 |0+ x d5/2 > |2+ x s1/2 > |2+ x d3/2 > |2+ x d5/2 > FIGURE 2. (Colo online) Radial wa e unc ions o he 5/2+ esonance in 11Be o h ee di e en s abilized se o pa ame e s Nand γ . Howe e i is necessa y o check how ep esen a i e a e he co esponding eigens a es when we a y Nand i is no possible o ob ain he exac s abilized poin . In o de o do so, we show in figu e 2, wo di e en wa e unc ions o wo di e en alues o N close o his poin . We also compa e wi h a wa e unc ion whe e we a y γ o a simila Nwhe e we can ge he exac pa ame e o he s abiliza ion. We see ha he main di e ence be ween he h ee wa e unc ions is i s ex ension, so ha he in e io is always he same, hus e aining he p ope ies o he esonance. The e o e, he selec ion o he exac pa ame e o s abiliza ion is no c ucial o ob ain a pseudos a e ep esen a i e o he esonance. This fi s esonance has mainly a d5/2configu a ion o he halo neu on wi h he 10Be co e in i s 0+g ound s a e. The e o e i is a case e y simila o he wo-body case wi h an ine co e. A mo e complica ed case a e esonances whose main configu a ion is bound. This is he case o he fi s 3/2+ esonance in 11Be. The main con ibu ion o his esonance is an s1/2wa e wi h he 10Be co e in i s 2+exci ed s a e. This esonance is below he 10Be(2+)+n h eshold so ha his configu a ion is e ec i ely bound. Ne e heless, we find in figu e 3 ha he eigen alues a e also s abilized o he ene gy o his esonance (3.0 MeV abo e he 10Be(0+)+n h eshold). CONCLUSIONS We ha e seen how we can cons uc a THO basis o wo-body sys ems wi h exci ed co e in o de o apply he PS me hod. The s abiliza ion me hod is used o iden i y he esonan ene gies o a wo-body halo nucleus, including he e ec o co e exci a ions. As an example, he me hod has been applied o he 5/2+ esonance in 11Be, using a pa icle- o o model o desc ibe his nucleus. The s abiliza ion in he e olu ion o he eigenene gies p o ides he alue o he esonance, as seen in wo-body sys ems wi h an 439 1.6 2 2.4 2.8 γ ( m1/2) 0 1 2 3 4 5 6 ε (MeV) 0246810 N J=3/2+ spec um FIGURE 3. (Colo online) Eigen alues ob ained om he diagonaliza ion o he 11Be Hamil onian in a THO basis, as a unc ion o he LST con inuum pa ame e ( γ ) in he le panel, and as a unc ion o he numbe o oscilla o s a es included in he basis in he igh panel. The dashed line indica es he ene gy o he 3/2+ esonance and he do ed line, he ene gy o he 10Be(2+)+n h eshold. ine co e. We also ound ha he co esponding pseudos a es o di e en s abilized se s o pa ame e s a e ep esen a i e o his esonance. E en using he disc e e pa ame e N whe e we canno selec he he exac alue o s abiliza ion, we ob ain good esul s. Fo he 3/2+ esonance in 11Be we ound he same s abiliza ion pa e n. The e o e he s abiliza ion me hod applies also o esonances whose main configu a ion is bound wi h espec o he co esponding co e+ alence h eshold. Mo eo e , we no e ha he 3/2− esonance p edic ed by he pa icle- o o model used he e gi es simila esul s, al hough we ha e no include i in he discussion [3]. ACKNOWLEDGMENTS This wo k is p esen ed on he occasion o F anco Iachello’s 70 h bi hday. 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