Core excitation effects in the breakup of the one-neutron halo nucleus 11Be on a proton target

PHYSICAL REVIEW C 85, 054613 (2012) Co e exci a ion e ec s in he b eakup o he one-neu on halo nucleus 11Be on a p o on a ge A. M. Mo o1,*and R. C espo2,3,† 1Depa amen o de F´ ısica A ´ omica, Molecula y Nuclea , Uni e sidad de Se illa, Apa ado 1065, E-41080 Se illa, Spain 2Cen o de F´ ısica Nuclea , Uni e sidade de Lisboa, A enida do P o esso Gama Pin o 2, P-1649-003 Lisboa, Po ugal 3Depa amen o de F´ ısica, Ins i u o Supe io T´ ecnico, Uni e sidade T´ ecnica de Lisboa, A enida do P o esso Ca aco Sil a, Taguspa k, P-2780-990, Oei as, Po ugal (Recei ed 6 Ma ch 2012; published 9 May 2012) We in es iga e he phenomenon o co e exci a ion in he inelas ic and b eakup o wo-body halo nuclei, composed by a alence nucleon and a co e. To e alua e he impo ance o his e ec we p opose a simple eac ion model based on an ex ension o he s anda d dis o ed wa e Bo n app oxima ion (DWBA) me hod. The model akes in o accoun co e-exci ed admix u es in he s a es o he composi e p ojec ile, as well as he possibili y o dynamic co e exci a ion due o he in e ac ion o he co e wi h he a ge . As an applica ion o he model, we p esen calcula ions o he b eakup o 11Be on a p o on a ge a an inciden ene gy o 63.7 MeV/nucleon, compa ing i wi h he a ailable da a o his eac ion. We ind ha he da a a e well ep oduced by he model and ha he e ec o dynamic co e exci a ion is essen ial o explain he obse ed c oss sec ion. DOI: 10.1103/PhysRe C.85.054613 PACS numbe (s): 24.50.+g, 25.60.Gc, 27.20.+n I. INTRODUCTION B eakup and inelas ic sca e ing a e now s anda d ech- niques o s udy he s uc u e o halo nuclei. In pa icula , he p o on a ge is e y appealing o spec oscopic s udies because he exci a ion occu s mainly due o he nuclea in e - ac ion, he eby exci ing se e al mul ipola i ies and e ealing he esonan s uc u e o he p ojec ile con inuum [1,2]. To ex ac eliable in o ma ion om hese da a i is impo an o iden i y he ele an exci a ion mechanisms. In he case o a p ojec ile composed by a alence pa icle weakly bound o a co e, i is commonly assumed ha he main exci a ion mechanism is due o he exci a ion o he alence nucleon ou side he co e. This single-pa icle pic u e is behind many ew-body eac ion o malisms used in he analysis o eac ions induced by halo and o he weakly bound nuclei, such as he con inuum-disc e ized coupled-channels (CDCC) me hod [3], he adiaba ic app oxima ion [4,5], he Al , G assbe ge , Sandhas (AGS) o mula ion o he Faddee equa ions [6,7], and a a ie y o semiclassical app oaches [8–13]. Despi e he ela i e success o hese me hods, hey ypically igno e he ac ha he p ojec ile s a es will con ain, in gene al, signi ican admix u es o se e al co e componen s and, he e o e, he exci a ion o he p ojec ile may be caused also by ansi ions be ween hese co e s a es. These e ec s a e no included in he s anda d o mula ions o he me hods discussed abo e, al hough some e o s ha e been made in ecen yea s owa d his di ec ion. A ecen example is he ex ended e sion o he CDCC me hod (named XCDCC) ecen ly p oposed by he au ho s o Re . [14]. The e ec o co e exci a ion has also been s udied [15], using an ex ension o he adiaba ic model o Re . [16]. The o malism was applied o he elas ic sca e ing o 8Bona *[email p o ec ed] †[email p o ec ed] ca bon a ge , and some con ibu ion was ound a la ge angles due o he exci a ion o he 7Be co e. I is he pu pose o his wo k o e alua e he in luence o hese co e exci a ion e ec s in he inelas ic and b eakup o halo nuclei using a simple model, bu e aining ne e heless he key physics. The model p esen ed he e is an ex ension o he co e exci a ion model p esen ed in a p e ious wo k [17]. Unde he assump ion ha he DWBA app oxima ion is alid, and ha co e- ecoil e ec s can be neglec ed, his model gi es a simple ela ion be ween he exci a ion o he p ojec ile and ha o he co e on he same a ge . The model was applied o he b eakup o 11Be on a p o on a ge a 63.7 MeV/nucleon inciden ene gy, and compa ed wi h he da a om Re . [1]. Due o he expe imen al ene gy esolu ion, he expe imen al angula dis ibu ions o he exclusi e b eakup we e ex ac ed o wo in e als o neu on-10Be co e ene gy: (i) E el =0– 2.5 MeV and (ii) E el =2.5–5.0 MeV. In he i s in e al one expec s an impo an con ibu ion coming om he low-lying na ow 5/2+ esonance a a ela i e ene gy o 1.28 MeV [18]. This esonance has a dominan 10Be(0+)⊗ν1d5/2pa en age and a small componen 10Be(2+)⊗ν2s1/2. The c oss sec ion o he second in e al con ains, p esumably, con ibu ions coming om se e al esonances, namely, Ex=2.64 MeV (3/2−), 3.40 MeV (3/2−,3/2+), 3.89 MeV (5/2−), and 3.95 (3/2−)[18]. In a p e ious wo k, we analyzed hese da a using a simple e sion o he co e exci a ion model [17], and showed ha he main con ibu ion o he second in e al comes om he esonance a 3.40 MeV, o which we assumed a 3/2+assignmen , ollowing he sugges ion o he au ho s o Re . [19]. These calcula ions p o ided a easonable ag eemen wi h he da a o Re . [1] and, mos impo an ly, e idenced he impo ance o he co e exci a ion mechanism in he b eakup o halo nuclei. The calcula ions pe o med by he au ho s o Re . [17] assumed a simple single-pa icle con igu a ion o he alence neu on in he 11Be nucleus ela i e o he 10Be co e. I is he pu pose o his wo k o ex end his model o he mo e 054613-1 0556-2813/2012/85(5)/054613(8) ©2012 Ame ican Physical Socie y A. M. MORO AND R. CRESPO PHYSICAL REVIEW C 85, 054613 (2012) ealis ic si ua ion in which he p ojec ile s a es consis o a supe posi ion o di e en alence con igu a ions and co e s a es. The pape is s uc u ed as ollows. In Sec. II we in oduce he Hamil onian used in his wo k. In Sec. III, we desc ibe he co e exci a ion eac ion model used in he sca e ing calcula ions. In Sec. IV we de ail he s uc u e model used o desc ibe he 11Be s a es. In Sec. Vwe p esen he calcula ions o he 11Be +p eac ion. Finally, Sec. VI is le o summa y and conclusions. II. THE FEW-BODY HAMILTONIAN The h ee-body Hamil onian o he sys em has he o m H=TR+Hp oj +Vc +V ,(1) whe e TR ep esen s he kine ic ene gy ope a o o he p ojec ile- a ge ela i e mo ion, Vc and V a e he co e- a ge and he alence- a ge in e ac ions, and Hp oj is he in e nal Hamil onian o he p ojec ile, gi en by Hp oj =T +V c( ,  ξ)+hco e( ξ),(2) whe e  is he ela i e coo dina e be ween he alence and he co e, T he co e- alence kine ic ene gy ope a o , and hco e( ξ) he in insic Hamil onian o he co e, whose ene gies and eigens a es a e labeled by he angula momen um (I) and i s p ojec ion (Mc), ha is, hco e|IMc=εI|IMc.(3) Addi ional quan um numbe s, equi ed o ully speci y he co e s a es, a e omi ed o simplici y in he no a ion. The co e- alence and co e- a ge in e ac ions may depend upon he in e nal co e deg ees o eedom  ξ. A gi en p ojec ile s a e wi h o al angula momen um and p ojec ion JM will consis on a linea supe posi ion o se e al alence-co e con igu a ions JM( ,  ξ)= α=I,,j ψJ α( )⊗I( ξ)JM,(4) whe e ψJ α( ) a e wa e unc ions desc ibing he mo ion o he alence neu on ela i e o a gi en co e s a e. The se o quan um numbe s s,, and j e e o he in insic spin o he alence pa icle, i s o bi al angula momen um ela i e o he co e, and hei sum, espec i ely. The wa e unc ions ψJ α( ) will depend upon he adop ed s uc u e model. Fo ou case s udy his will be desc ibed in Sec. IV. III. THE REACTION APPROACH: CORE EXCITATION REACTION MODEL In his sec ion we epo in de ail he co e-exci ed eac ion o malism (C-ex) ha shall be used o desc ibe he exci a ion o a p ojec ile nucleus (p), om i s g ound s a e wi h o al angula momen um J o a inal s a e Jdue o i s in e ac ion wi h a a ge ( ). The p ojec ile nucleus is assumed o be well desc ibed by a alence pa icle ( ) o bi ing a ound a co e o nucleons (c). The a ge is assumed o be ine . In ou Rc R ξ c R FIG. 1. (Colo online) Rele an coo dina es o he sca e ing o a wo-body sys em, composed o a co e (c) and a alence pa icle ( ), sca e ed o a a ge nucleus ( ). wo king example, his will co espond o he exci a ion o 11Be (10Be +n) on a p o on a ge o inal s a es in he con inuum, bu he o malism could be equally applied o he ansi ion be ween bound s a es. Fo simplici y, he e we igno e he spin o he p o on a ge . The di e en ial c oss sec ion o he exci a ion om an ini ial s a e |i JM o a inal s a e | JMo he p ojec ile is gi en in gene al by dσp d =1 ˆ J2 μp (2π¯h2)2 K K M,M TJM,JM p 2(5) wi h ˆ J=√2J+1, μp is he p ojec ile- a ge educed mass,  K( K) he ini ial ( inal) linea momen um, and TJM,JM p is he ansi ion ampli ude. Wi hin he DWBA app oach, he la e is gi en by TJM,JM p =χ(−)  K( R) JM( , ξ)VTχ(+)  K( R)i JM( ,  ξ),(6) whe e χ(+)  K( R) and χ(−)  K( R) a e dis o ed wa es desc ibing he p ojec ile- a ge ela i e mo ion in he ini ial and inal channels, espec i ely. In a co e + alence model o he p ojec ile, he ansi ion po en ial is VT=V ( R )+Vc ( Rc , ξ),(7) whe e V ( R ) and Vc ( Rc , ξ) a e he alence- a ge and co e- a ge po en ials. The ele an coo dina es o his p oblem a e depic ed in Fig. 1. The alence- a ge in e ac ion is assumed o be cen al, and hence V ( R )=V (R ). On he o he hand, he co e- a ge in e ac ion con ains bo h cen al and noncen al pa s, and is w i en as he mul ipola expansion Vc ( Rc , ξ)= L,M V(L) c (Rc ,ξ)Y∗ LM(ˆ R)YLM(ˆ ξ).(8) We assume ha he adial o m ac o does no depend on he in e nal coo dina es o he co e [i.e., VL(Rc ,ξ)=VL(Rc )], which is alid o he o a ional model used he e. F om Eqs. (6) o (8) he DWBA ampli ude is exp essed as a sum o wo e ms: TJM,JM p ( K, K)=TJM,JM al +TJM,JM co ex .(9) The i s e m, ha we deno e alence ampli ude o sho ness, is explici ly gi en by TJM,JM al ( K, K)=χ(−)  K( R) JM( , ξ)V (R ) +V(0) c (Rc )χ(+)  K( R)i JM( ,  ξ).(10) 054613-2 CORE EXCITATION EFFECTS IN THE BREAKUP OF THE ... PHYSICAL REVIEW C 85, 054613 (2012) This e m con ains only he cen al pa s o he agmen - a ge po en ials V and Vc and he e o e i canno induce ansi ions in ol ing exci a ions o he co e. Using he gene al o m (4) o he ini ial and inal s a es, his ampli ude can be ew i en as TJM,JM al ( K, K)= α,αχ(−)  K( R)ψJ α( )V (R ) +V(0) c (Rc )χ(+)  K( R)ψJ α( )δI,I.(11) Fo each s a e Io he co e con ained in bo h he ini ial and inal s a es o he p ojec ile, his ampli ude is simila o ha ound in con en ional CDCC calcula ions and he e o e can be e alua ed using he s anda d codes a ailable o his kind o calcula ions. The second e m o he ansi ion ampli ude is due o he noncen al pa o he co e in e ac ion, ha is, TJM,JM co ex ( K, K)= L>0,Mχ(−)  K( R) JM( , ξ)V(L) c (Rc ) ×Y∗ LM(ˆ Rc )YLM(ˆ ξ)χ(+)  K( R)i JM( ,  ξ). (12) This e m accoun s o he dynamic exci a ion o he co e du ing he collision. I is he pu pose o he p esen wo k o e alua e his e m app oxima ely, bu e aining he key physics, o p o ide a simple es ima ion o he co e exci a ion e ec s. In addi ion o he co e exci a ion mechanism, he ampli- ude (12) may also p oduce exci a ions in he co e- alence ela- i e mo ion. This is because he ansi ion po en ial is e alua ed a  Rc . This coo dina e can be exp essed as  Rc = R+γ [γ=m /(m +mc)]. The dependence on he co e- alence coo dina e ( ) p oduces a co e- ecoil e ec ha can induce exci a ions o he p ojec ile. Howe e , because in ou es case γ1, we will assume ha his e ec can be neglec ed in his ampli ude [al hough i is ully aken in o accoun in he ampli ude (11)], so we can make he app oxima ion  R≈ Rc in Eq. (12), which means ha he dis o ed wa es in Eq. (12) a e e alua ed a he  Rc coo dina e. This app oxima ion leads o wha we shall call he ea e he co e exci a ion (C-exc) eac ion model. We no e ha his model can be deduced om he XCDCC o malism [14] by calcula ing he ansi ion in he Bo n app oxima ion and neglec ing he co e- ecoil e ec s o L>0. The sca e ing ampli ude o he C-exc app oach ac o izes hen in o a sum o p oduc s o a eac ion and a s uc u e e m TJM,JM co ex = L>0,M TLM( K, K) × JM( , ξ)YLM(ˆ ξ)i JM( ,  ξ),(13) whe e we ha e in oduced he quan i ies TLM( K, K)=χ(−)  K( Rc )|V(L)(Rc )Y∗ LM(ˆ Rc )|χ(+)  K( Rc ), (14) which con ain he dependence on he eac ion pa o he ansi ion ampli ude. Using he Wigne -Ecka heo em in he o m [20], we can w i e o he s uc u e componen  JMYL,M (ˆ ξ)i JM=JM|JMLM J||YL( ξ)||i J. (15) No ing ha he ope a o appea ing in he educed ma ix elemen depends only on he co e coo dina es and ha he ini ial and inal s a es a e exp essed in he o m (4), his ma ix elemen can be w i en as (see, o ins ance, Re . [20], Appendix VI)  J||YL( ξ)||i J= α,αRJ αRJ αG(L) αJ,αJI||YL( ξ)||I, (16) whe e we in oduced he geome ic ac o G(L) αJ,αJ=δj,j(−1)L+j+J+Iˆ Jˆ IJJL II j.(17) The educed ma ix elemen I||YL( ξ)||Iappea ing in Eq. (16) depends on he s uc u e model assumed o he co e and will be speci ied la e . Collec ing esul s, he sca e ing ampli ude o he co e exci a ion eads TJM,JM co ex = L>0,MJMLM|JMTLM( K, K) × α,αRJ α|RJ αG(L) αJ,αJI||YL( ξ)||I.(18) The di e en e ms ha en e in his ansi ion ampli ude a e ela i ely s aigh o wa d o calcula e. The ampli udes TLM( K, K), de ined by Eq. (14), a e hose appea ing in s an- da d DWBA calcula ions wi h local o m ac o s, o example, in inelas ic sca e ing calcula ions. The adial unc ions RJ α( ) a e he solu ion o a coupled se o di e en ial equa ions and he echniques o sol e his p oblem a e desc ibed elsewhe e (see, e.g., he Appendix VI o Re . [21]). The es o he e ms a e jus kinema ical and geome ical ac o s. The ampli ude (18) can also be ela ed o he wo-body inelas ic ampli udes o a co e- a ge sca e ing p oblem. To make his ela ion explici , le us conside a gi en ansi ion IMc→IM c o he inelas ic exci a ion o he co e sca e ed o he same a ge . In DWBA, he ampli ude o his p ocess eads, simila ly o Eq. (6), TIMc,I M c c =χ(−)  K( Rc ) IM c( ξ)Vc ( Rc , ξ) ×χ(+)  K( Rc )i IMc( ξ).(19) Using he Wigne -Ecka heo em, his ampli ude can be also w i en as TIMc,I M c c =IM c|IMcLM T(LM) c (I→I),(20) wi h he educed ampli udes  T(LM) c (I→I)=TLM( K, K)I||YL(ˆ ξ)||I,(21) wi h TLM( K, K) gi en by Eq. (14). 054613-3 A. M. MORO AND R. CRESPO PHYSICAL REVIEW C 85, 054613 (2012) Compa ing wi h Eq. (18) and making use again o he no- ecoil app oxima ion (  Rc ≈ R) we ge he ela ion TJM,JM co ex = L>0,MJM|JMLM × α,αRJ αRJ αG(L) αJ,αJ T(LM) c (I→I).(22) This ampli ude is equi alen o ha gi en by Eq. (18),bu emphasizes in a clea way he ela ion be ween he h ee- body sca e ing ampli ude, Eq. (12), and he co esponding wo-body ampli udes, unde he app oxima ions assumed by he model. Equa ion (22) p o ides also a p ac ical way o nume ically implemen ing he co e-exci ed model since he wo-body ampli udes can be ob ained om any s anda d DWBA code. In he ex eme si ua ion in which he alence exci a ion is small compa ed wi h he co e exci a ion mechanism, he ansi ion ampli ude is gi en en i ely by Eq. (18) and he co esponding c oss sec ion is ob ained inse ing his ampli ude in o Eq. (5). Fu he mo e, i he e is a con ibu ion om a single mul ipole ansi ion L, and a single con ig- u a ion in he ini ial and inal s a es, he p ojec ile- a ge c oss sec ion becomes p opo ional o he c oss sec ion o he exci a ion o he co e mul iplied by a geome ic ac o and he adial o e lap, ha is, dσp d co ex =ˆ J2 ˆ J2 ˆ I2 ˆ I2G(L) αJ,αJRJ α|RJ α2dσc d (I→I). (23) This pa icula case was used in a p elimina y applica ion o his model o 11Be +p esonan b eakup [17]. No e, howe e , ha in gene al bo h he alence and he co e exci a ion mechanisms will con ibu e o he b eakup p ocess. In his case, he co esponding ampli udes need o be added cohe en ly and hence in e e ence e ec s will a ise. In he p esen wo k we ake in o accoun simul aneously he alence and co e exci a ion con ibu ions. IV. STRUCTURE MODEL To calcula e he ini ial and inal s a es o he p ojec ile one needs o speci y he s uc u e model o he co e. In he calcula ions p esen ed in his wo k o he 11Be +psys em, he p ojec ile is ea ed wi hin he pa icle- o o model o Boh and Mo elson [21]. Then we assume a o a ional model o he 10Be co e wi h a pe manen quad upole de o ma ion, which, o simplici y, is aken o be axially symme ic. The e o e, we can cha ac e ize he de o ma ion by a single pa ame e β2.In he body- ixed ame, he su ace adius is hen pa ame ized as R(ˆ ξ)=R0[1 +β2Y20(ˆ ξ)], wi h R0an a e age adius, o be speci ied la e . S a ing om a cen al po en ial, V(0) c ( ), he ull alence-co e in e ac ion is ob ained by de o ming his cen al in e ac ion V c( , ˆ ξ)=V(0) c [ −δ2Y20(ˆ ξ)],(24) wi h δ2=β2R0being he de o ma ion leng h. T ans o ming o he space- ixed ame o e e ence, and expanding in sphe ical ha monics, his de o med po en ial eads V c( ,  ξ)= L,M V(L) c ( )Y∗ LM(ˆ )YLM(ˆ ξ).(25) The in e nal s a es o he p ojec ile a e expanded acco ding o Eq. (4). I we w i e ψJ α( )=RJ α( )Ysjm(ˆ ), wi h Ysjm(ˆ )= [Y(ˆ )⊗χs]jm, he p ojec ile s a es become JM( ,  ξ)= α RJ α( )[Ysj (ˆ )⊗I( ξ)]JM.(26) The adial unc ions RJ α( ) a e hen ob ained by sol ing he Sch ¨ odinge equa ion using he po en ial (25) and wi h he app op ia e bounda y condi ions. Fo bound s a es, hese adial unc ions decay exponen ially o →∞gi ing ise o squa e-in eg able unc ions. Fo con inuum s a es, he unc ions RJ α( ) a e also ob ained by sol ing a se o coupled adial equa ions, bu subjec o he bounda y condi ion ha inciden wa es occu in a gi en open channel cha ac e ized by a se o quan um numbe s α={, s, j, I}. Al hough he calcula ions p esen ed in his wo k could be pe o med wi h he sca e ing s a es hemsel es, i is nume ically ad an ageous o adop a binning p ocedu e, simila o ha used in CDCC calcula ions. Then, he con inuum spec um is di ided in o ene gy in e als. Fo each ene gy in e al, o bin, a ep esen a i e squa e-in eg able s a e is cons uc ed by a weigh ed supe posi ion o sca e ing s a es wi hin he bin in e al. The bin wa e unc ion o a gi en incoming wa e αis gi en by a squa e-in eg able unc ion, wi h a s uc u e simila o ha o he bound s a es (26) bin [k1k2]αJM( ,  ξ)= α RJ [k1k2]αα( ) ×[Ysj (ˆ )⊗I( ξ)]JM,(27) whe e [k1k2] deno es he momen um in e al de ining he bin. V. APPLICATION TO 11Be +pRESONANT BREAKUP We now apply he co e-exci ed model o he b eakup o 11Be on a p o on a ge and compa e wi h he da a o [1]. Be o e p esen ing he calcula ions wi h his model, we conside he case igno ing he e ec o de o ma ion in bo h he 11Be s uc u e and in he co e- a ge in e ac ion. A. Calcula ions wi hou de o ma ion The calcula ions wi h no-de o ma ion we e done wi h he s anda d CDCC me hod, using he FRESCO code [22]. These calcula ions a e simila o hose p esen ed in ou p e ious wo k [23]. The alence-co e in e ac ion con ains cen al +spin-o bi e ms, o Woods-Saxon shape, wi h pa ame e s adjus ed o ep oduce he g ound s a e sepa a ion ene gy, he bound exci ed s a e (1/2−), and he posi ion o he low-lying 5/2+ esonance. The co e- a ge in e ac ion co esponds o he Wa son pa ame iza ion o Re . [24]. Fo he in e ac ion be ween he alence neu on and he p o on a ge , we conside ini ially a simple Gaussian in e ac ion 054613-4 CORE EXCITATION EFFECTS IN THE BREAKUP OF THE ... PHYSICAL REVIEW C 85, 054613 (2012) 0246 E el (MeV) 0 20 40 dσ/dE el (mb/MeV) Faddee : 3S1 Gaussian Faddee : CD Bonn CDCC: 3S1 Gaussian CDCC: Modi ied Gaussian FIG. 2. (Colo online) Di e en ial c oss sec ion as a unc ion o he n-10Be ela i e ene gy calcula ed wi h he Faddee /AGS and CDCC me hods, using di e en choices o he alence neu on- p o on in e ac ion. The solid line is he Faddee calcula ion using he ealis ic CD Bonn in e ac ion. The do ed and do ed-dashed lines a e, espec i ely, he Faddee and CDCC calcula ions using a p-n Gaussian in e ac ion adjus ed o he 3S1s-wa e phase-shi s: V( )= −72.15 exp[−( /1.484)2]. The dashed line is he CDCC calcula ions using he modi ied Gaussian in e ac ion wi h he same geome y, bu wi h he educed s eng h V0=−45 MeV. All hese calcula ions igno e he 10Be de o ma ion. V( )=−72.15 exp[−( /1.484)2][3], which ep oduces he deu e on binding ene gy and low-ene gy s-wa e iple (3 S1) phase shi s. The 11Be con inuum was disc e ized in ene gy in e als using he s anda d binning p ocedu e. We included n-10Be pa ial wa es up o =2. In Fig. 2we ep esen he calcula ed ene gy spec um dσ/dE el ha eme ges by in eg a ing he b eakup c oss sec ion o e he solid angle dc.m.. To assess he eliabili y o he CDCC me hod in his case, we include also he Faddee calcula ion (quo ed om Re . [23]), pe o med wi h he same wo-body in e ac ions. The CDCC calcula ion (do -dashed line) ep oduces ai ly well he Faddee esul indica ing ha , a leas in he limi o no-de o ma ion, he CDCC is a eliable ool o analyze his eac ion. I was ound by he au ho s o Re . [23] ha he b eakup c oss sec ions a e e y sensi i e o he in e ac ion be ween he alence neu on and he p o on a ge . This is illus a ed by he solid line in Fig. 2, which ep esen s he Faddee calcula ion pe o med eplacing he simple Gaussian po en ial by he ealis ic CD Bonn in e ac ion. This p oduces a sizable educ ion o he b eakup c oss sec ion. Consequen ly, o ge e- liable esul s i is manda o y o use a ealis ic nucleon-nucleon (NN) in e ac ion. Un o una ely, exis ing implemen a ions o he CDCC me hod do no inco po a e he possibili y o using hese ealis ic NN in e ac ions, such as Pa is o CD Bonn. Ne e heless, we ha e ound ha he e ec o using a ealis ic NN in e ac ion can be well simula ed educing he dep h o he Gaussian in e ac ion o V0≈−45 MeV. The co esponding CDCC calcula ion, depic ed by he dashed line in Fig. 2, is seen o ep oduce ai ly well he Faddee esul wi h he CD Bonn po en ial. The co esponding angula dis ibu ions a e shown in Fig. 3. The uppe and bo om panels co espond o he ela i e 0 20 40 dσ/dΩc.m. (mb/s ) Faddee : CD Bonn CDCC: modi ied Gaussian DWBA: modi ied Gaussian 10 20 30 40 θc.m. (deg) 0 10 20 dσ/dΩc.m. (mb/s ) (b) E el=2.5-5 MeV (a) E el=0.0-2.5 MeV FIG. 3. (Colo online) Angula dis ibu ion o he b eakup o 11Be on a p o on a ge a 63.7 MeV/nucleon o E el =0–2.5MeV (uppe panel) and E el =2.5–5.0 MeV (bo om panel) calcula ed wi hou de o ma ion. The ci cles a e he expe imen al da a o Re . [1]. The solid line is he Faddee calcula ion wi h he ealis ic CD Bonn in e ac ion. The dashed line is he CDCC calcula ion using he modi ied Gaussian in e ac ion (Vg=−45 MeV) and he do -dashed line is he CDCC calcula ion calcula ed in i s o de . ene gy in e als E el =0–2.5 MeV and E el =2.5–5.0MeV, espec i ely. In each panel, he solid line is he Faddee calcula ion using he ealis ic CD Bonn in e ac ion, whe eas he dashed line is he CDCC calcula ion using he modi ied Gaussian po en ial. I is seen ha bo h calcula ions a e in easonable ag eemen . The e a e disc epancies below 15◦, whe e no da a exis ne e heless. Since he calcula ions including de o ma ion a e pe o med wi hin he DWBA app oxima ion, we ha e included also in his plo he CDCC calcula ion pe o med in i s o de , which is equi alen o a DWBA calcula ion. I is seen ha his calcula ion is e y close o he ull CDCC esul , jus i ying he use o he Bo n app oxima ion in his eac ion. This plo shows also e y clea ly ha he single-pa icle exci a ion mechanism is no adequa e o desc ibe hese da a, pa icula ly in he highe exci a ion ene gy in e al. F om he calcula ions p esen ed in his sec ion, we conclude ha he CDCC me hod (e en o i s o de ), using an adequa e e ec i e p-nin e ac ion, p o ides a good app oxima ion o he mo e sophis ica ed Faddee calcula ion wi h a ealis ic NN in e ac ion. In he ollowing sec ion, we in oduce he e ec o co e de o ma ion using he C-ex model desc ibed in Sec. III. 054613-5 A. M. MORO AND R. CRESPO PHYSICAL REVIEW C 85, 054613 (2012) B. Calcula ions wi h he co e exci a ion model To calcula e he ini ial and inal s a es o he p ojec ile [Eqs. (26) and (27)] we adop he alence-co e po en ial om Re . [25] (model Be12b) consis ing o a cen al Woods-Saxon po en ial, wi h adius R0=2.483 m, di useness a=0.65 m, and pa i y-dependen dep h, wi h V0=−54.239 MeV (V0= −49.672 MeV) o he e en (odd) wa es. This po en ial is de o med using a de o ma ion pa ame e β2=0.67, which co esponds o a de o ma ion leng h o δ2=β2R0=1.64 m. A spin-o bi e m, using he s anda d Woods-Saxon de i a i e o m o he adial shape, wi h he same adius and di useness as he cen al pa and a dep h o Vso =8.5MeV,isalso included. Fo he 10Be co e we conside only he g ound s a e (0+) and he i s exci ed s a e (Iπ=2+,Ex=3.368 MeV). The o bi al angula momen um is unca ed a max =3. We conside ed con inuum s a es wi h Jπ=1/2+,1/2−,3/2−, 3/2+, and 5/2+. Wi h he assumed Hamil onian and model space, he g ound s a e co esponds p edominan ly o a s1/2con igu a ion coupled o he co e in he g ound s a e (≈85%), bu wi h a signi ican admix u e o he |2+⊗νd5/2con igu a ion (≈13%). This po en ial p oduces also he low-lying na ow eso- nances 5/2+,3/2−, and 3/2+a ela i e ene gies o 1.2, 2.7, and 3.2 MeV, espec i ely. These esonances can be iden i ied wi h he s a es obse ed by Fukuda e al. in he 11Be + 12C eac ion a 70 MeV/nucleon [19]. The 5/2+ esonance co esponds p edominan ly o a d5/2con igu a ion coupled o he co e in he g ound s a e, whe eas he 3/2+ esonance has a dominan 10Be(2+)⊗νs1/2pa en age. Fo he in e ac ion be ween he alence neu on and he p o on a ge we use he modi ied Gaussian in e ac ion ob ained in he p e ious sec ion. Fo he cen al pa o he 10Be +pin e ac ion we keep he Wa son po en ial [24]. This in e ac ion is used o calcula e he dis o ed wa es appea ing in he DWBA ansi ion ampli ude. The noncen al pa o his in e ac ion, equi ed o allow o he dynamic exci a ion o he co e, is ob ained de o ming his cen al po en ial using he same de o ma ion leng h (δ2= 1.64 m). In Fig. 4we show he calcula ed b eakup c oss sec ion, as a unc ion o he n-10Be ela i e ene gy, in eg a ed o e he cen e -o -mass angula ange θc.m.⩽60◦. Figu e 4(a) shows he con ibu ion o he dominan Jπpa ial wa es o he calcula ed ene gy spec um, as well as he o al sum o all included wa es. I is seen ha he b eakup c oss sec ion is domina ed by he esonan 3/2+and 5/2+con ibu ions and, a low exci a ion ene gies, by he 3/2−non esonan con inuum. The 1/2−and 1/2+wa es (no shown in his igu e) gi e also some con ibu ion a small exci a ion ene gies. I is also obse ed ha he 3/2− esonance (loca ed a E el =2.7MeV in his model) has a negligible e ec on he c oss sec ion. In Fig. 4(b), we show sepa a ely he alence and co e exci a ion con ibu ions o he angle-in eg a ed b eakup c oss sec ion. The low-ene gy spec um is mos ly due o he alence exci a ion, and hence he dynamic co e exci a ion is negligible a hese exci a ion ene gies. As he exci a ion ene gy inc eases he e ec o co e exci a ion becomes mo e and mo e impo an . 012345 E el (MeV) 0 20 40 60 dσ/dE el (mb/MeV) (b) alence co e alence+ co e 0 20 40 60 dσ/dE el (mb/MeV) (a) 3/2- 5/2+ 3/2+ ull θc.m.=(0o- 60o) FIG. 4. (Colo online) Di e en ial ene gy c oss sec ion o p(11Be,p)10Bena 63.7 MeV/nucleon in eg a ed o e he cen e - o -mass angula ange θc.m.⩽60◦, calcula ed wi h he C-ex model p oposed in his wo k. The uppe panel shows he indi idual con ibu ion o he main pa ial wa es and he sum o all included pa ial wa es. The bo om panel shows he sepa a e con ibu ion coming om he alence exci a ion and dynamic co e exci a ion ampli udes. See ex o de ails. In pa icula , co e dynamic e ec s a e clea ly dominan in he egion o he 3/2+ esonance and a e also impo an in he egion o he 5/2+ esonance. We hus expec signi ican changes in he co esponding angula dis ibu ions. In Fig. 5we compa e ou esul s o he b eakup c oss sec ion angula dis ibu ion dσ/dc.m.wi h he expe imen al da a o Re . [1] con aining con ibu ions wi hin ela i e neu on–10Be ene gy ange 0–2.5 MeV [Fig. 5(a)] and 2.5–5 MeV [Fig. 5(b)]. Again, we show he sepa a e con- ibu ions coming om he alence exci a ion (do -dashed line) and dynamic-co e exci a ion (dashed line), as well as hei cohe en sum (solid line). In he lowe ene gy in e al [Fig. 5(a)] he c oss sec ion is domina ed by he single-pa icle b eakup mechanism. The calcula ed angula dis ibu ion ep oduces easonably well he shape o he da a, al hough some o e es ima ion o he absolu e magni ude is obse ed. On he o he hand, in he highe ene gy in e al [Fig. 5(b)], bo h he single-pa icle and dynamic co e exci a- ion mechanisms a e impo an . The sum o bo h con ibu ions accoun s easonably well o he da a, excep o he i s da a poin , which is o e es ima ed. The main con ibu ion in his exci a ion ene gy egion comes om he popula ion o he 3/2+ esonance, as can be expec ed om Fig. 4. These esul s 054613-6 CORE EXCITATION EFFECTS IN THE BREAKUP OF THE ... PHYSICAL REVIEW C 85, 054613 (2012) 0 20 40 60 dσ/dΩc.m. (mb/s ) ull ( alence+ co e) co e alence 10 20 30 40 θc.m. (deg) 0 10 20 dσ/dΩc.m. (mb/s ) (b) E el=2.5-5 MeV (a) E el=0.0-2.5 MeV FIG. 5. (Colo online) Angula dis ibu ion o he b eakup o 11Be on a p o on a ge a 63.7 MeV/nucleon o E el =0–2.5MeV (uppe panel) and E el =2.5–5.0 MeV (bo om panel). The ci cles a e he expe imen al da a o Re . [1]. The dashed-do ed and dashed cu es ep esen he alence and co e exci a ion con ibu ions, espec i ely, whe eas he solid line co esponds o hei cohe en sum. e idence he impo ance o he co e exci a ion mechanism and con i m ou ea lie indings using a mo e simpli ied model [17]. Finally, we no e ha his eac ion has been s udied also by he au ho s o Re . [26] using an ex ended e sion o he CDCC me hod, which inco po a es equi alen co e exci a ion e ec s o hose discussed he e. Howe e , con a y o ou esul s, he e ec o co e exci a ion was ound o be e y small in ha wo k. This esul is unexpec ed gi en he la ge de o ma ion o he 10Be nucleus. VI. CONCLUSION In conclusion, we s udied he p oblem o co e exci a ion in he b eakup sca e ing o halo nuclei. To accoun o his e ec in a quan i a i e way, we de eloped a co e exci a ion eac ion model, based on he DWBA app oxima ion, which akes in o accoun he e ec o co e de o ma ion in he s uc u e o he halo nucleus, as well as he possibili y o dynamic co e exci a ion du ing he collision. We showed ha , igno ing co e- ecoil e ec s, he con ibu ion o he sca e ing ampli ude a ising om he co e exci a ion can be w i en in e ms o a supe posi ion o wo-body ampli udes co esponding o he inelas ic sca e ing o he co e sca e ed by he same a ge . As an illus a ion o he model, we pe o med calcula ions o he b eakup o 11Be on p o ons a an inciden ene gy o 63.7 MeV/nucleon. The ini ial and inal s a es a e ea ed wi hin he pa icle- o o model and hence hey a e conside ed as a supe posi ion o se e al alence con igu a ions coupled o he 10Be co e in ei he he g ound s a e (0+)o he i s exci ed s a e (2+). The noncen al pa o he 10Be + p o on in e ac ion, which is esponsible o he dynamic co e exci a ion, is ob ained de o ming he 10Be +p o on po en ial. We ind ha he co e exci a ion mechanism gi es an impo an con ibu ion and i s inclusion pe mi s a sui able desc ip ion o he da a om Re . [1]. We also showed ha he impo ance o dynamic co e exci a ion becomes mo e impo an a inc easing exci a ion ene gies and is in ac essen ial o accoun o he ene gy-in eg a ed angula dis ibu ion a E el =2.5–5.0MeV. F om he calcula ions p esen ed in his wo k, we may con- clude ha hese co e exci a ion e ec s will be also impo an in o he eac ions induced by weakly bound p ojec iles wi h de o med cons i uen s. The me hod p oposed he e can p o ide a use ul and simple es ima e o hese e ec s in hose si ua ions in which he assump ions o he model (i.e., he alidi y o he Bo n app oxima ion and he possibili y o neglec ing co e- ecoil) a e jus i ied. ACKNOWLEDGMENTS This wo k has been pa ially suppo ed by he FCT G an No. PTDC/FIS/103902/2008, by he Spanish Minis e io de Ciencia e Inno aci´ on unde P ojec No. FPA2009-07653, and by he Spanish Consolide -Ingenio 2010 P og amme CPAN (CSD2007-00042). [1] A. Sh i as a a e al. ,Phys. Le . B 596, 54 (2004). [2] Y. Sa ou e al.,Phys. Le . B 660, 320 (2008). [3] N. Aus e n, Y. Ise i, M. Kamimu a, M. Kawai, G. Rawi sche , and M. Yahi o, Phys. Rep. 154, 125 (1987). [4] P. Bane jee and R. Shyam, Phys. Re . C 61, 047301 (2000). [5] J. A. Tos e in, S. Rugmai, and R. C. Johnson, Phys.Re .C57, 3225 (1998). [6] L. D. Faddee , Zh. Eksp. Theo . Fiz. 39, 1459 (1960) [So . Phys. JETP 12, 1014 (1961)]. [7] E. O. Al , P. G assbe ge , and W. Sandhas, Nucl. Phys. B 2, 167 (1967). [8] S. Typel and G. Bau , Phys.Re .C50, 2104 (1994). [9] H. Esbensen and G. F. Be sch, Nucl. Phys. A 600, 37 (1996). [10] T. Kido, K. Yabana, and Y. Suzuki, Phys. Re . C 50, R1276 (1994). [11] S. Typel and G. Bau , Phys. Re . C 64, 024601 (2001). [12] P. Capel, G. Golds ein, and D. Baye, Phys. Re . C 70, 064605 (2004). [13] A. Ga cia-Camacho, A. Bonacco so, and D. M. B ink, Nucl. Phys. A 776, 118 (2006). [14] N. C. Summe s, F. M. Nunes, and I. J. Thompson, Phys.Re .C 74, 014606 (2006). [15] K. Ho ii, M. Takashina, T. Fu umo o, Y. Saku agi, and H. Toki, Phys.Re .C81, 061602 (2010). 054613-7 A. M. MORO AND R. CRESPO PHYSICAL REVIEW C 85, 054613 (2012) [16] R. C. Johnson, J. S. Al-Khalili, and J. A. Tos e in, Phys. Re . Le . 79, 2771 (1997). [17] R. C espo, A. Del u a, and A. M. Mo o, Phys. Re . C 83, 044622 (2011). [18] J. H. Kelley e al.,Nucl. Phys. A 880, 88 (2012). [19] N. Fukuda e al.,Phys. Re . C 70, 054606 (2004). [20] D. B ink and G. Sa chle , Angula Momen um, 2nd ed., (Cla endon P ess, Ox o d, 1968). [21] A. Boh and B. Mo elson, Nuclea S uc u e, edi ed by W. A. Benjamin, (New Yo k, 1969). [22] I. Thompson, Compu . Phys. Rep. 7, 167 (1988). [23] E. C a o, R. C espo, A. M. Mo o, and A. Del u a, Phys. Re . C 81, 031601 (2010). [24] B. A. Wa son, P. P. Singh, and R. E. Segel, Phys. Re . 182, 977 (1969). [25] F. M. Nunes, J. A. Ch is ley, I. J. Thompson, R. C. Johnson, and V. D. E os, Nucl. Phys. A 609,43 (1996). [26] N. C. Summe s and F. M. Nunes, Phys. Re . C 76, 014611 (2007). 054613-8