Three-body model for the analysis of quasifree scattering reactions in inverse kinematics

PHYSICAL REVIEW C 92, 044605 (2015) Th ee-body model o he analysis o quasi ee sca e ing eac ions in in e se kinema ics A. M. Mo o* Depa 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 (Recei ed 29 July 2015; published 12 Oc obe 2015) A new me hod o calcula e c oss sec ions o (p,pn)and(p,2p) eac ions measu ed unde in e se kinema ics condi ions is p oposed. The me hod uses he p io o m o he sca e ing ansi ion ampli ude and eplaces he exac h ee-body wa e unc ion appea ing in his exp ession wi h an expansion in e ms o p-no p-ps a es, co e ing he physically ele an exci a ion ene gies and pa ial wa es. A p ocedu e o disc e iza ion, simila o ha used in con inuum-disc e ized coupled-channels calcula ions, is applied o make his expansion ini e and nume ically ac able. The p oposed o malism is non ela i is ic, bu se e al ela i is ic kinema ical co ec ions a e applied o ex end i s applicabili y o ene gies o cu en in e es . The unde lying op ical po en ials o he en ance and exi channels a e gene a ed mic oscopically by olding an e ec i e densi y-dependen Gma ix wi h he densi y o he composi e nucleus. Nume ical calcula ions o 12C(p,2p), 12C(p,pn), and 23O(p,pn)a ∼400 MeV/nucleon a e p esen ed o illus a e he me hod. The ole o inal-s a e in e ac ions and Pauli p inciple be ween he ou going nucleons is also discussed. DOI: 10.1103/PhysRe C.92.044605 PACS numbe (s): 25.60.Gc,24.10.Eq,25.45.De,25.40.Ep I. INTRODUCTION Quasi ee sca e ing (QFS) expe imen s o he o m (p,pn) and (p,2p) [he ea e (p,pN)] ha e been used ex ensi ely as a ool o ex ac spec oscopic in o ma ion o p o on-hole and neu on-hole s a es in nuclei, such as sepa a ion ene gies, spin-pa i y assignmen s, and occupa ion p obabili ies. In hese eac ions, an ene ge ic p o on beam (E>100 MeV) collides wi h a s able a ge nucleus, emo ing one o mo e nucleons, and lea ing a esidual nucleus, ei he in i s g ound s a e o in an exci ed s a e. Recen ly, he echnique has been ex ended o he s udy o uns able nuclei, using in e se kinema ics, i.e., bomba ding a hyd ogen a ge wi h an ene ge ic adioac i e beam. This echnique is analogous o he knockou expe imen s wi h composi e a ge s used ex ensi ely in he pas yea s [1–5]. Al hough bo h kinds o expe imen s a e mean o p o ide simila in o ma ion, he e a e impo an di e ences be ween hem. In knockou eac ions, a nucleon is suddenly emo ed om he as -mo ing p ojec ile a e colliding wi h a ligh a ge nucleus, like 9Be. Owing o he s ongly abso p i e na u e o he co e- a ge in e ac ion a hese ene gies, he p ocess is highly pe iphe al and hence mainly dependen on he ail o he wa e unc ion o he emo ed nucleon o , mo e co ec ly, on he o e lap unc ion be ween he p ojec ile and esidual co e wa e unc ions. The no m o his o e lap is he spec oscopic ac o , a quan i y ha is di ec ly ela ed o he occupa ion p obabili y o a gi en single-pa icle o bi al. Because his no m depends on he ull o e lap, and no jus on i s ail, his aises he ques ion o he eliabili y o he spec oscopic in o ma ion ex ac ed om a p ocess ha is only sensi i e o a small piece o he wa e unc ion. Howe e , (p,pN) eac ions a e expec ed o be mo e sensi i e o deepe po ions o he wa e unc ion and so hey should help o educe hese ambigui ies. Consequen ly, he in o ma ion ob ained *[email p o ec ed] om hese expe imen s will be complemen a y o ha ob ained om hea y-ion knockou eac ions a in e media e ene gies and om ans e eac ions a lowe ene gies. Some leading acili ies, such as GSI (Ge many), RIKEN (Japan) and NSCL/MSU (USA), ha e plans o pe o m in e se kinema ics expe imen s wi h exo ic beams. Consequen ly, i is o imely impo ance o e isi heo e ical me hods o analyze hese kinds o p ocesses. Theo e ical analyses o he (p,pN) eac ions wi h s able nuclei ha e been commonly done using he dis o ed-wa e impulse app oxima ion (DWIA) [6]. Roughly speaking, he impulse app oxima ion (IA) means ha he binding po en ial o he emo ed pa icle can be neglec ed in compa ison wi h he p ojec ile- a ge kine ic ene gy (see, e.g., Re . [7], Chap. 11). A he ene gies usually employed in QFS expe imen s (se e al hund eds o MeV pe nucleon) his app oxima ion is expec ed o be jus i ied. The DWIA me hod is usually o mula ed in e ms o he nucleon-nucleon ansi ion ma ix (Tma ix he ea e ); he IA in ol es he eplacemen o his ope a o wi h a ee Tma ix be ween he inciden p o on and he s uck nucleon. P ac ical implemen a ions o he DWIA o malism commonly in ol e u he app oxima ions, such as he subs i u ion o his Tma ix by i s on-shell alue, o i s ep esen a ion by a ze o- ange ope a o . I eliable s uc u e in o ma ion is o be ex ac ed om hese expe imen s, hese app oxima ions need o be e isi ed and es ed. Consequen ly, he alidi y o he DWIA o malism should be in es iga ed compa ing wi h mo e elabo a e eac ion heo ies [8–10]. A h ee-body eac ion amewo k which does no make use o he IA is he con inuum-disc e ized coupled-channels (CDCC) me hod [11]. This me hod has been e y success ul in he analysis o eac ions induced by weakly bound p ojec iles a low and medium ene gies. The me hod has been also applied o one-neu on emo al eac ions on p o on a ge s a in e media e ene gies (∼40–70 MeV/nucleon) [12,13]. Fo a knockou eac ion o he o m A(p,pN)C, he s anda d CDCC me hod aims a expanding he h ee-body wa e unc ion o he sys em in e ms o N−Ceigens a es. To make he 0556-2813/2015/92(4)/044605(13) 044605-1 ©2015 Ame ican Physical Socie y A. M. MORO PHYSICAL REVIEW C 92, 044605 (2015) expansion ini e, he N−Ccon inuum mus be unca ed in ene gy and angula momen um and disc e ized. Howe e , he la ge angula momen um and ene gy ans e ound in hese eac ions makes ha con e gence o he obse ables equi es a e y la ge model space. Fo ene gies o cu en in e es , o se e al hund eds o MeV/u, he me hod becomes unp ac ical. Benchma k calcula ions wi h he Faddee /AGS me hod [14] o he 11Be(p,pn) eac ion a ∼35 MeV/u showed ha , while he CDCC expansion o he b eakup s a es in e ms o n-10Be s a es con e ged e y slowly wi h he size o he model space, he al e na i e expansion in e ms o p-ns a es equi ed a much smalle model space o ep oduce he dominan pa o he 10Be inclusi e c oss sec ions. This al e na i e expansion makes use o he p io - o m ep esen a ion o he ansi ion ampli ude, in which he h ee-body wa e unc ion is app oxima ed by an expansion in a basis o p+Ns a es. A p ocedu e o con inuum disc e iza ion, simila o ha used in s anda d CDCC calcula ions, is used o he p-Ns a es. The esul an exp ession is o mally simila o he coupled-channel Bo n app oxima ion (CCBA) exp ession commonly used in ans e eac ions and, he e o e, in some p e ious applica ions [15], he me hod has been e e ed o as ans e o he con inuum me hod. This allows i s implemen a ion in s anda d coupled-channels codes a e some sui able modi ica ions. In his pape , I p esen some explo a o y calcula ions o illus a e he applica ion o his me hod o he in e p e a ion o (p,2p) and (p,pn) expe imen s in in e se kinema ics. Al hough he gene al o mula ion o he me hod has been p esen ed be o e [14,15], i is desc ibed in mo e de ail he e and some sui able p esc ip ions o i s inpu ing edien s (in e nal wa e unc ions, op ical po en ials, e c.), as well as se e al ela i is ic kinema ic co ec ions ( o make he model applicable o highe ene gies), a e discussed. The ole o Pauli p inciple and inal-s a e in e ac ions o he ou going nucleons is also discussed. The calcula ions he e p esen ed a e o ela i ely high ene gies (∼400 MeV/u), bu , because he me hod does no ely on he IA, i migh be applicable also o lowe ene gies, o which he DWIA may no be adequa e. The pape is s uc u ed as ollows. In Sec. II he heo e ical o mula ion o he me hod is p esen ed. This sec ion discusses also he choice o he NN in e ac ion, which is he mos esponsible o he (p,pN) p ocess, he cons uc ion o app o- p ia e op ical po en ials including in-medium e ec s, and some ela i is ic co ec ions applied o he model. In Sec. III he me hod is applied o he 12C(p,pN) and 23O(p,pn) eac ions a E∼400 MeV/nucleon. In Sec. IV he connec ion wi h he DWIA me hod is discussed. Finally, Sec. Vsumma izes he main esul s o his wo k. II. THEORETICAL MODEL A. The ansi ion ampli ude Le us conside a eac ion o he o m A+p→C(α)+p+N, (1) in which an inciden composi e nucleus A=C+Ncollides wi h a p o on a ge , losing a nucleon (p o on o neu on) and FIG. 1. (Colo online) Diag am o a (p,pN) eac ion in in e se kinema ics, modeled as a bina y p ocess. gi ing ise o a esidual co e nucleus (C) in some de ini e s a e αand wo ou going nucleons (p+no p+p). The p ocess is schema ically depic ed in Fig. 1. Using he p io o m o he ansi ion ampli ude, he exac ansi ion ampli ude o his eac ion can be w i en as Ti (α)=(−) α, |VpN +VpC|φA(ξA)ei KpA  R,(2) whe e φA(ξA) ep esen s he g ound-s a e wa e unc ion o he nucleus A, wi h ξAdeno ing i s in e nal coo dina es. The plane wa e ei KpA  Rdesc ibes he ela i e mo ion o he p+ Asys em. A his s age, he po en ial VpC is a many-body ope a o con aining he in e ac ion o he a ge p o on wi h all he co e nucleons. The inal unc ion (−) α, is he exac sca e ing wa e unc ion subjec o he bounda y condi ions consis ing o a plane wa e (o Coulomb wa e) in he channel (co esponding o some de ini e s a e o he ela i e mo ion o he h ee ou going agmen s), wi h he co e in s a e αand ingoing sphe ical wa es in all o he open channels. This wa e unc ion is a solu ion o he many-body Sch ¨ odinge equa ion, [E−−K −KR−VpN −V† pC −V† NC](−) α, ( , R,ξC)=0, (3) whe e E−=E−i and ξCdeno es he co e in e nal coo di- na es. No e ha he inal h ee-body s a e has been exp essed in e ms o he Jacobi coo dina es { , R}(see Fig. 1). Equa ion (2) gi es he exac ansi ion ampli ude o he many-body sca e ing p oblem. I can be o mally educed o an e ec i e h ee-body p oblem, using he app oxima ion, (−) α, ( , R,ξC)≃3b(−) ( , R)φα C(ξC),(4) whe e φα C(ξC) is he co e wa e unc ion in he s a e αand 3b(−) ( , R) is a h ee-body wa e unc ion ob ained as a solu ion o he e ec i e Sch ¨ odinge equa ion, [E−−K −KR−VpN −U† pC −U† NC]3b(−) ( , R)=0, (5) whe e UpC and UNC a e now e ec i e nucleon-nucleus in e ac ions which, in p ac ice, will be eplaced wi h op ical model po en ials a he app op ia e ene gy pe nucleon. 044605-2 THREE-BODY MODEL FOR THE ANALYSIS OF . . . PHYSICAL REVIEW C 92, 044605 (2015) In his h ee-body model, he ansi ion ampli ude is T3b i (α)=3b(−) φα C(ξC)VpN +UpCφA(ξA)ei KpA  R.(6) I he po en ial UpC is aken o be independen o he in e nal coo dina es o C(ξC), as i is usually assumed, one can pe o m he in eg al o e hese in e nal coo dina es o gi e dξCφα C(ξC)φA(ξA)=Sα,,j ϕα CA( ),(7) whe e Sα,,j φα CA( ) is an o e lap wa e unc ion, wi h φα CA( ) a uni no malized wa e unc ion depending on he ela i e coo dina e o he emo ed pa icle wi h espec o he co e and Sα,,j he spec oscopic ac o . Using Eq. (7)inEq.(6) one inds T3b i (α)=Sα,,j 3b(−) VpN +UpCϕα CAei KpA  R.(8) As done in ans e calcula ions, i is con enien o in oduce an auxilia y po en ial in he incoming channel, UpA( R), so he p e ious equa ion ans o ms o T3b i (α)=Sα,,j 3b(−) VpN +UpC −UpAϕα CAχ(+) pA ,(9) whe e χ(+) pA is he dis o ed wa e gene a ed by he po en ial UpA( R). No e ha , when 3b is he exac solu ion o Eq. (5), he ampli ude (9) is s ic ly independen o he choice o UpA. In p ac ical calcula ions, in which 3b mus be app oxima ed somehow, he esul will, howe e , depend on his po en ial. The usual choice is o use o UpA an op ical po en ial desc ibing he elas ic sca e ing o he p+Asys em. Wi h his choice, one expec s ha he di e ence UpC −UpA ( he so-called emnan e m) will con ibu e li le o he in eg al and hence he ma ix elemen will be mos ly de e mined by he VpN in e ac ion. To educe Eq. (5) o a ac able o m, 3b(−)is expanded in e ms o p+Neigens a es, i.e., 3b(−) ( , R)= jπdkφjπ(k, )χj,π( K,  R),(10) whe e  kis he ela i e wa e numbe o he p+Npai ,  Kis ha o he ela i e mo ion be ween he esidual nucleus C and he p+Npai , and χj,π( K,  R) is a unc ion desc ibing he ela i e mo ion o he p+Nsys em wi h espec o he esidual nucleus, when he o me is in a gi en inal s a e {k,jπ}. (No e ha , in ac ual calcula ions, his expansion will be done in e ms o s a es o o al angula momen um J.To simpli y he discussion, he no a ion used he e is somewha schema ic.) The s a es φjπ(k, ) a e he eigens a es o he p+NHamil onian using he po en ial VpN ( ). In he (p,pn) case, he expansion (10) will con ain also a e m o he deu e on g ound s a e. This e m is omi ed o simplici y o he no a ion. In he (p,2p) case, he p+pwa e unc ion mus be an isymme ized o accoun o he indis inguishabili y o he ou going p o ons. In p ac ice, his es ic s he allowed alues o jπand in oduces a ac o o 2 in he calcula ed c oss sec ions wi h espec o he unsymme ized calcula ion.1 Ene gy conse a ion in he inal channel (non ela i is ic by now) implies ha E=epN +Ec.m.=2k2 2μpN +2K2 2μpN,C ,(11) whe e Eis he o al ene gy o he sys em, epN he ela i e ene gy o he p+Npai (epN =−2.22 MeV o he deu e on g ound s a e), and Ec.m. he kine ic ene gy associa ed wi h he ela i e mo ion o he co e wi h espec o he cen e o mass (c.m.) o he p+Npai . The unc ions χj,π( K,  R) could, in p inciple, be ob ained by inse ing he expansion (10) in o Eq. (5). Howe e , because hese unc ions depend upon he con inuous pa ame e K, his would gi e ise o an in ini e numbe o equa ions. In p ac ice, one may use a disc e iza ion p ocedu e simila o ha used in he CDCC me hod [11], in which he inal p+Ns a es a e g ouped (binned) in ene gy o momen um in e als as 3b(−) ≈CDCC = n,j,π φjπ n(kn, )χn,j,π ( Kn, R),(12) whe e kna e some a e age alues o he disc e ized p-N ene gies and φjπ n(kn, ) he bin wa e unc ions. No e ha he subsc ip in CDCC e ains he in o ma ion on he inal s a e, co esponding o some de ini e alues o n,j, and π. De ails on he cons uc ion o hese bins can be ound elsewhe e [11,16]. The angula di e en ial c oss sec ion o a gi en inal disc e ized bin ={n,j,π}and a gi en co e s a e αis dσn,j,π (α) dc=1 (2sp+1)(2JA+1) ×μiμ (2π2)2 Kn Ki σT3b i, (α) 2,(13) wi h μi(μj) he educed mass o he ini ial ( inal) mass pa i- ion and T3b i, (α) he ansi ion ampli ude ob ained by eplacing he CDCC expansion (12) in he ansi ion ampli ude (9). The sum in σincludes he spin p ojec ions o he ou going pN pai and o he esidual nucleus C. The angle speci ied by cis he sca e ing angle o he co e in he c.m. ame. The double di e en ial c oss sec ion, wi h espec o he sca e ing angle o he co e and he in e nal ene gy o he p+Nsys em, can be ob ained a he disc e ized ene gies epN =en pN as d2σj,π(α) depN dcepN =en pN ≃1 n dσn,j,π (α) dc ,(14) whe e nis he wid h o he bin o which he ene gy epN belongs. This can be eadily ans o med o a double di e en ial c oss sec ion wi h espec o he ene gy o he ou going co e in he o e all c.m. ame (Ec), using he usual 1No e ha , ea ing he (p,2p) eac ion as a bina y p ocess o he o m p+A→2He +C, he 2 ac o can be ega ded as he spec oscopic ac o o he o e lap 2He|p. 044605-3 A. M. MORO PHYSICAL REVIEW C 92, 044605 (2015) non ela i is ic ela ion o bina y collisions, Ec=m∗ pN m∗ pN +mc Ec.m.,(15) whe e m∗ pN =mp+mN+epN . Thus, d2σj,π(α) dEcdc=m∗ pN mc+m∗ pN d2σj,π(α) depN dc .(16) The inclusi e c oss sec ion will be hen ob ained summing he con ibu ions om all inal jπcon igu a ions, d2σ(α) dEcdc= j,π d2σj,π(α) dEcdc .(17) No e ha Eq. (15), along wi h Eq. (11), es ablishes a one- o-one co espondence be ween he ene gy o he co e, in he c.m. ame, and he in e nal ene gy o he p+Npai . No e also ha Eq. (9) esembles he ansi ion ampli ude o a ans e p ocess, analogous o ha appea ing in he s anda d CCBA me hod o bina y collisions [17]. Taking ad an age o his o mal analogy, one can e alua e his ansi ion ampli ude using s anda d coupled-channels codes. Fo he calcula ions p esen ed in his wo k, he code FRESCO [18] has been used, wi h some sui able modi ica ions desc ibed below. B. Rela i is ic kinema ics Ongoing and planned QFS expe imen s in in e se kinema - ics a e pe o med a ene gies o se e al hund eds o MeV pe nucleon. A hese ene gies, ela i is ic kinema ics is clea ly impo an . Al hough he ea men p esen ed in he p e ious sec ions is non ela i is ic, some ela i is ic co ec ions can be eadily implemen ed o ake in o accoun , a leas app oxi- ma ely, hese e ec s. The o al ene gies o he inciden nucleus and he p o on a ge in he cen e -o -momen um ame a e gi en by (c=1 in his sec ion) εA=s+(mA−mp) 2√s;εp=s−(mA−mp) 2√s,(18) whe e sis he usual Mandels am in a ian , co esponding o he squa e o he o al ene gy. Assuming ha he p o on a ge is ini ially a es , s=(mp+mA)2+2mpTLAB,(19) whe e TLAB is he kine ic ene gy o he p ojec ile. Analogously, o he exi channel, he o al ela i is ic ene gy o he ou going co e can be w i en as εc=s+(mc−m∗ pN ) 2√s,(20) whe e one has exploi ed again he analogy o he p ocess unde s udy wi h a bina y p ocess o he o m A+p→C+ (pN). Because m∗ pN =mp+mN+epN and Ec=εc−mc, Eq. (20) ela es he ela i e ene gy o he ou going p-Npai o he kine ic ene gy o he co e. Using his ela ion, he gene aliza ion o he double di e en ial c oss sec ion, Eq. (16), esul s in d2σj,π(α) dEcdc=√s m∗ pN d2σj,π(α) depN dc ,(21) which educes o Eq. (16) in he non ela i is ic limi . The modi ica ion o he dis o ed wa es owing o ela i is ic kinema ics has been also s udied. Un o una ely, he e is no unique p esc ip ion o inco po a e ela i is ic kinema ics wi hin he Sch ¨ odinge scheme. Two common p esc ip ions used in he con ex o nucleon-nucleus sca e ing a in e medi- a e ene gies ha e been conside ed he e. The i s one (see e.g., Re . [19]) consis s o eplacing he educed mass appea ing in he kine ic-ene gy e m o he Sch ¨ odinge equa ion (5) wi h he so-called educed ene gy. Fo example, o he inciden channel, μi≡μpA →εpA =εpεA εp+εA .(22) Inse ing his de ini ion in o he Sch ¨ odinge equa ion and mul iplying he ull equa ion by εpA/μpA, one inds an equi alen Sch ¨ odinge equa ion wi h he usual educed mass in he kine ic-ene gy ope a o , bu wi h he po en ial scaled by he ac o γ=εpA/μpA.(23) The second p esc ip ion conside ed he e (see, e.g., Re . [20]) assumes ha he mo ion o he hea y nucleus in he c.m. ame can be ea ed non ela i is ically, whe eas he ligh pa icle ( he p o on) is ea ed ela i is ically. A ela i is ic Sch ¨ odinge - ype equa ion is hen gene a ed by educing he Di ac equa ion o a massi e ene ge ic e mion o mass mpand ela i is ic wa e numbe kp,mo ingina cen al po en ial U( ), e i ying Umpand ∇Ukp, bo h well sa is ied o in e media e-ene gy p o on sca e ing. In he educed wo-body p oblem wi h ela i is ic p ojec ile bu non ela i is ic a ge he la ge componen o he wa e unc ion e i ies a Sch ¨ odinge -like equa ion wi h a po en ial eno malized by he ac o γ=2(E−MA) E−MA+mp .(24) No e ha , in bo h p esc ip ions, he momen um o be used is he ela i is ic one. Finally, he exp ession o he di e en ial c oss sec ion (13) is eplaced wi h i s ela i is ic coun e pa (see, e.g., Re . [21]), dσn,j,π (α) dc=1 (2sp+1)(2JA+1) 1 (2π)2 i ×K 2c2(1/εc+1/εpN ) σT3b i, (α) 2,(25) whe e iis he ini ial p ojec ile- a ge ela i e eloci y and pN =√s−c. Expe imen al da a usually consis o ans- e se o longi udinal momen um dis ibu ions o he esidual co e. These a e eadily ob ained om he p e ious di e en ial c oss sec ions by applying he ans o ma ion d2σ dEcdc dEcdc=d3σ dp3d3p, (26) 044605-4 THREE-BODY MODEL FOR THE ANALYSIS OF . . . PHYSICAL REVIEW C 92, 044605 (2015) gi ing ise o d3σ dp3=c2 p 1 (pc)2+(mc2)2 dσ dEcdc .(27) Decomposing he momen um in i s ans e se (p ) and longi udinal (pz) componen s, he di e en ial olume becomes d3p=p dp dφdpz(wi h φ he azimu hal angle), and dσ dp =πp ∞ ∞ d3σ dp3dpz,(28) dσ dpz=2πd3σ dp3p dp .(29) W i ing px=p cos(φ) and py=p sin(φ) he ans e se momen um dis ibu ion can be also decomposed in o i s x and ycomponen s, esul ing in dσ dpx=1 2πp ∞ −∞ dσ dp dpy,(30) and likewise o dσ/dpy. C. Nucleon-nucleon and nucleon-nucleus in e ac ions In he p esen app oach, he nucleon-nucleon (NN) in e - ac ion appea s in he ansi ion ope a o [Eq. (2)], as well as in he equa ion o he h ee-body (CDCC) wa e unc ion (5). Fo his in e ac ion, he Reid93 pa ame iza ion [22], an upda ed egula ized e sion o he pionee ing Reid so -co e po en ial [23] de eloped by he Nijmejen g oup, has been adop ed. This po en ial con ains cen al, spin-o bi and enso componen s and ep oduces accu a ely he p o on-p o on and p o on-neu on phase-shi s up o an ene gy o 350 MeV (χ2/Nda a =1.03). The nucleon-nucleus po en ials a e calcula ed mic oscopi- cally, by olding an e ec i e NN in e ac ion wi h he nucleus g ound-s a e densi y. Sys ema ic s udies o nucleon-nucleus sca e ing a in e media e ene gies show ha his p ocedu e p o ides a good desc ip ion o he elas ic and inelas ic da a, p o ided ha he e ec i e NN in e ac ion con ains some ene gy and densi y dependence. Al hough q-space olding is usually p e e able in olding calcula ions, he p esence o his densi y dependence makes he use o -space olding necessa y, which, o he cen al pa , eads U( )=d [ρgs( ) D(s,ρgs)+j0(ks) X(s,ρ)ρ( , )], (31) whe e Dand Xco espond o he di ec and exchange e ms o he e ec i e NN in e ac ion, kis he local wa e numbe ,  and  a e he p ojec ile (nucleon) and a ge (nucleus) posi ions, s=| − |, and ρgs is he g ound-s a e densi y, e alua ed a he midpoin posi ion | − |/2. The mixed ansi ion densi y is ep esen ed by ρ( , )=ρgs( )C(kFs),(32) whe e Cis a co ela ion unc ion desc ibing he exchange nonlocali y and kF he Fe mi momen um. These olding calcula ions ha e been pe o med wi h he code LEA [24]. In he calcula ions p esen ed in his wo k, he co ela ion unc ion has been aken as uni y, and he g ound-s a e densi ies a e ob ained om Ha ee-Fock (HF) calcula ions. The la e we e compu ed wi h he code OXBASH [25], using he Sky me Sk20 in e ac ion. I is wo h no ing ha hese Sky me HF densi ies ep oduce well he nuclea adii ex ac ed om elec on sca e ing [26] and a e o common use in he analysis o knockou expe imen s (see, e.g., Re s. [4,5]). D. Bound-s a e wa e unc ions Acco ding o Eq. (7), he ansi ion ampli ude con ains he o e lap unc ion be ween he ini ial (A) and he inal (C) nuclei. As is usually done in he analysis o ans e and knockou eac ions, his o e lap unc ion has been app oxi- ma ed by he single-pa icle wa e unc ion ob ained as he solu ion o a Woods-Saxon po en ial, wi h he dep h adjus ed o ep oduce he e ec i e sepa a ion ene gy o he inal s a e αo he co e, namely, S∗ α=Sn,p +Eα, whe e Sn,p is he g ound-s a e o g ound-s a e nucleon sepa a ion ene gy and Eα he exci a ion ene gy o he co e. Following Re . [5], he di useness pa ame e o he po en ial well was ixed a a0=0.7 m and he adius pa ame e 0was adjus ed so he .m.s. o he calcula ed o bi al coincided wi h ha ob ained om a HF calcula ion, i.e., √ 2 sp=[A/(A−1)]1/2√ 2 HF. Because he sepa a ion ene gy p edic ed by he HF calcula ion does no necessa ily coincide wi h he expe imen al one, he HF sepa a ion ene gy is used o his i . III. RESULTS To illus a e he me hod, some calcula ions o he eac ions 12C(p,pN) and 22O(p,pn)a ∼400 MeV pe nucleon a e p esen ed and discussed in his sec ion. These calcula ions a e pe o med wi h a modi ied e sion o he code FRESCO [18], which inco po a es he Reid93 NN in e ac ion, and he ela i is ic kinema ics co ec ions discussed in Sec. II B. Al hough Eq. (9), wi h he disc e e expansion (12) o he inal h ee-body wa e unc ion, p o ides a nume ically ac able o m, he calcula ions can be signi ican ly simpli ied, wi h a small loss o accu acy, making use o he addi ional app oxima ion o neglec ing couplings among p-Ncon inuum s a es wi h di e en jπ. The omission o hese couplings will al e o some ex en he dis ibu ion o lux be ween he inal s a es. Howe e , o an inclusi e si ua ion, in which he con ibu ion om all he included jπmus be added oge he , his edis ibu ion o lux is no expec ed o a ec signi ican ly he co e obse ables. As an addi ional simpli ica ion, he spin-o bi pa o he nucleon-nucleus po en ials in Eq. (9) has been neglec ed. A. Applica ion o 12C(p,pN) The 12C(p,2p)11B eac ion a 400 MeV/nucleon, which coincides wi h he ac ual ene gy used in a ecen expe imen pe o med a GSI o his eac ion [27], is conside ed i s . Only hose p ocesses leading o bound s a es o 11Ba e conside ed, which co espond o he emo al o p o ons om he 1p3/2and 1p1/2o bi als, because emo al om he deepe 1s1/2o bi al will lead o unbound s a es o 11B. 044605-5 A. M. MORO PHYSICAL REVIEW C 92, 044605 (2015) 0246 -30 -20 -10 0 10 U( ) HF 2pF 0246 ( m) -40 -20 0 20 40 U( ) Real Imag Real Imag (a) p+12C po en ial a Ep=200 MeV (b) p+12C po en ial a Ep=400 MeV FIG. 2. (Colo online) p+12C mic oscopic op ical po en ial gene a ed wi h he PH densi y-dependen NN in e ac ion [28,29], olded wi h he g ound-s a e densi y o 12C,a Ep=200 MeV ( op) and 400 MeV (bo om). Solid and dashed lines co espond, espec- i ely, o HF (Sk20 in e ac ion) and empi ical densi ies ob ained om elec on sca e ing ( he la e pa ame ized wi h a 2pF dis ibu ion). The equi ed nucleon-nucleus op ical po en ials a e p+ 12C in he inciden channel and p+11B in he exi channel. These po en ials we e gene a ed wi h he mic oscopic olding app oach desc ibed in Sec. II C, using he Pa is-Hambu g (PH) G-ma ix e ec i e in e ac ion [28,29]. This in e ac ion is ene gy and densi y dependen . The p+12C po en ial was cal- cula ed using he inciden ene gy (Ep=400 MeV/nucleon). Fo he exi channel, he choice is less clea , because he ou going p o ons will eme ge wi h a b oad ange o ene gies. Fo a pu e QFS collision, one expec s an a e age alue o abou Elab/2 o each nucleon, and so he ou going op ical po en ials we e e alua ed a Ep=200 MeV in he p esen case. The g ound-s a e densi ies o he 12Cand 11Bnuclei we e ob ained om a HF calcula ion, using he Sky me Sk20 in e ac ion. To es he sensi i i y o he calcula ed po en ials wi h espec o his inpu , a wo-pa ame e Fe mi (2pF) pa ame iza ion o he 12Cdensi y, ex ac ed om elec on sca e ing [30], was also conside ed. 0 1020304050 10-2 100 102 104 106 dσ/dΩ (mb/s ) PH+Sk20: Rel I PH+Sk20: Rel II 0 1020304050 θc.m. (deg) 10-4 10-2 100 102 104 106 dσ/dΩ (mb/s ) PH+Sk20: Rel I PH+Sk20: Rel II p+12C @ Ep=398 MeV p+12C @ Ep=200 MeV (a) (b) FIG. 3. (Colo online) Di e en ial elas ic c oss sec ion o p+ 12C a 200 MeV ( op) and 398 MeV (bo om), using a mic oscopic olding po en ial gene a ed wi h he PH e ec i e NN in e ac ion and he Sky me HF densi y. The dashed and solid lines co espond o he ela i is ic scaling p esc ip ions gi en by Eqs. (23)and(24), espec i ely. The da a a e om Re . [31]. Figu e 2shows he eal and imagina y pa s o he p+12C cen al po en ial a 200 ( op) and 400 MeV (bo om). The solid and dashed lines a e o he HF and 2pF densi ies, espec i ely. I is in e es ing o no e how, o he highe ene gy, he po en ial becomes highly abso p i e and he eal pa , which is mainly a ac i e a low ene gies, de elops a s ong epulsi e co e. The 2pF densi y gi es a quali a i ely simila beha io . The calcula ed po en ials show some di e ences a small dis ances (<2 m) bu a e e y simila a la ge dis ances. Al hough (p,pN) eac ions a e expec ed o explo e sho e dis ances as compa ed o knockou eac ions, hey ha e s ill a pe iphe al na u e. The e o e, one expec s ha bo h po en ials gi e simila (p,pN) c oss sec ions. The quali y o he calcula ed po en ials has been assessed, compa ing he calcula ed di e en ial elas ic c oss sec ion o p+12C a 200 and 398 MeV wi h he expe imen al da a om Re . [31]. Fo hese calcula ions, he olding po en ials compu ed wi h he HF densi ies ha e been used. The compa ison is shown in Fig. 3, whe e he solid and 044605-6 THREE-BODY MODEL FOR THE ANALYSIS OF . . . PHYSICAL REVIEW C 92, 044605 (2015) dashed lines co espond o he ela i is ic scaling ac o s o Eq. (23) (p esc ip ion I he ea e ) and Eq. (24) (p esc ip ion II), espec i ely. Bo h p esc ip ions ep oduce ai ly well he da a a bo h ene gies, wi h p esc ip ion II p o iding a sligh ly be e ag eemen . The e o e, he la e has been also adop ed o he (p,pN) calcula ions p esen ed below. Fo he (p,2p) calcula ions, he ansi ion ampli ude (9)is e alua ed wi h he CDCC expansion o he inal h ee-body wa e unc ion in e ms o p-pbins [Eq. (12)]. Fo his expansion, pa ial wa es j⩽5 and ela i e ene gies (epp ) up o he maximum allowed by ene gy conse a ion we e used. No e ha , in he (p,2p) case, T=1 o he ou going pai and, hence, he gene alized Pauli p inciple es ic s he allowed inal con igu a ions o +S=e en, whe e is he o bi al angula momen um o he p-ppai and Si s o al spin ( S=s1+s2). This excludes some jπcon igu a ions, such as 1+and 3+, which a e pu ely T=0. The bound-s a e wa e unc ion o he s uck nucleon in he p ojec ile is gene a ed wi h a Woods-Saxon po en ial, wi h di useness a=0.70 m and he adius adjus ed o ep oduce he .m.s. p edic ed by a HF calcula ion, as explained in Sec. II D. This p ocedu e yields he po en ial adius R0=2.953 m. Finally, he po en ial dep h is adjus ed o ep oduce he g ound-s a e o g ound-s a e p o on sepa a ion ene gy (Sp=15.96 MeV). Be o e p esen ing he (p,2p) c oss sec ions, he co espon- dence be ween he ela i e ene gy o he ou going p-ppai and he kine ic ene gy o he co e in he c.m. ame, gi en by Eqs. (11) and (15), is discussed. This is illus a ed in Fig. 4. The op panel shows he con ibu ion o he dominan jπp-pcon igu a ions as a unc ion o he ela i e ene gy epp . Fo simplici y, he spec oscopic ac o has been se equal o uni y. I is seen ha a wide ange o ene gies, om epp =0 oepp ≃300 MeV, a e popula ed, wi h a maximum a ound ∼180 MeV. I is also seen ha mos o he c oss sec ion comes om he j<3 con igu a ions, wi h j=1− gi ing he dominan con ibu ion. I is he e o e e y impo an ha he unde lying NN in e ac ion used in he e alua ion o he sca e ing ampli ude ep oduces co ec ly a leas hese pa ial wa es. In he bo om panel, he same dis ibu ions a e plo ed as a unc ion o he co e c.m. ene gy, acco ding o he ela ion (20). Al hough his exp ession is ela i is ic, i is clea om his igu e ha he co e ene gies, in he o e all c.m. ame, can be ea ed non ela i is ically, as assumed in he ela i is ic p esc ip ion II discussed in he p e ious sec ion. Once he double di e en ial c oss sec ions ha e been ob ained, he ans e se and longi udinal momen um dis- ibu ions a e compu ed by means o Eqs. (30) and (29), espec i ely. These quan i ies a e shown in Figs. 5(a) and 5(b), espec i ely. No e ha he longi udinal (pz) momen um dis- ibu ion has been calcula ed in he p ojec ile es ame. As in he case o knockou eac ions be ween composi e sys ems, he shape o hese dis ibu ions is mos ly de e mined by he wa e unc ion o he s uck nucleon. I is obse ed ha he longi udinal momen um dis ibu ion exhibi s some asymme y and is somewha shi ed o nega i e alues o pz. This is mainly a consequence o he nega i e Q alue o he eac ion, which educes he a ailable kine ic ene gy in he inal channel. This phenomenon has been analyzed in de ail in a ecen wo k in e ms o he DWIA o malism [32]. The calcula ions we e 0 100 200 300 epp (MeV) 0.00 0.01 0.02 dσ/depp (mb/MeV) 0+ 0- 1- 2+ 2- 3- 0 102030405060 Ec (MeV) 0 0.1 0.2 dσ/dEc (mb/MeV) 12C(p,2p) @ Ep=400 MeV (a) (b) FIG. 4. (Colo online) (Top) Rela i e ene gy dis ibu ion o he ou going p o ons, ollowing he p ocess 12C(p,2p), o se e al jπ con igu a ions o he p-ppai . (Bo om) Co esponding ene gy dis ibu ion o he co e, in he c.m. ame, calcula ed om Eq. (21). The calcula ions a e done wi h mic oscopic dis o ing po en ials (PH +Sk20), he ela i is ic p esc ip ion II, and uni spec oscopic ac o o he 12C→11B+pdecomposi ion. See ex o de ails. epea ed using he 2pF densi y o he 12Cnucleus, bu he esul s a e almos iden ical, so hey a e no shown he e. Simila calcula ions o he 12C(p,pn)11C eac ion ha e been pe o med. The ing edien s a e almos he same as in he (p,2p) case, excep o he ac ha one needs also he n+11C po en ial o he exi channel, which was also ob ained om he mic oscopic olding model, wi h he PH e ec i e nucleon-nucleon in e ac ion and he HF (Sk20) densi y o he 11Cnucleus. The bound-s a e wa e unc ion was ob ained wi h a Woods-Saxon well wi h di useness a=0.7 m, adius R0=2.850 m (ex ac ed om he HF calcula ion), and he dep h adjus ed o ep oduce he expe imen al neu on sepa a ion ene gy (Sn=18.72 MeV). Figu e 6shows he calcula ed (p,pn) c oss sec ion, as a unc ion o he ela i e ene gy p-nene gy, epn ( op), and he co e ene gy in he c.m. ame, Ec(bo om). As in he (p,2p) case, he c oss sec ion is domina ed by he 1−pa ial wa e o 044605-7 A. M. MORO PHYSICAL REVIEW C 92, 044605 (2015) -200 0 200 px (MeV/c) 0 0.02 0.04 dσ/dpx (mb/(MeV/c)) -200 0 200 px (MeV/c) 0 0.02 0.04 -200 0 200 pz (MeV/c) 0 0.02 dσ/dpz (mb/(MeV/c)) -200 0 200 pz (MeV/c) 0 0.02 0.04 12C(p,2p)11B12C(p,pn)11C (a) (b) (c) (d) FIG. 5. T ans e se ( op) and longi udinal (bo om) momen um o he esidual co e nucleus o he 12C(p,2p)(le )and 12C(p,pn) ( igh ) eac ions a 400 MeV/nucleon. In all cases, he spec oscopic ac o has been se o uni y. 050 epn 0 50 100 150 200 250 300 epn (MeV) 0 0.01 0.02 dσ/depn (mb/MeV) 0 102030405060 Ec (MeV) 0.04 0.08 0.1 dσ/dEc (mb/MeV) 0+ 0- 1+ 1- 2+ 2- 12C(p,pn) @ Ep=400 MeV (a) (b) FIG. 6. (Colo online) (Top) Rela i e ene gy dis ibu ion o he ou going nucleons, ollowing he p ocess 12C(p,pn), o some jπ con igu a ions o he ou going p-npai . (Bo om) Co esponding ene gy dis ibu ion o he co e in he c.m. ame. See ex o de ails. he p+nsys em, al hough he con ibu ions o he 1+,2 +, and 2−wa es a e also sizable. The epn dis ibu ion co esponding o he 1+con igu a ion exhibi s a low-ene gy ail, wi h a sha p inc ease nea epn =0. This peculia beha io is a consequence o he inal-s a e in e ac ion (FSI) be ween he ou going pand nnucleons. The con ibu ion o he (p,d) channel, leading o he deu e on g ound s a e, is also included, al hough i was ound o be negligibly small a hese ela i ely high ene gies. This is be e seen in he inse , whe e he low-ene gy egion has been enla ged. The FSI e ec owing o he 1+wa e is also appa en in he co e ene gy dis ibu ion (bo om panel), gi ing ise o a high-ene gy ail. The esul s o he ans e se (px) and longi udinal (pz) momen um dis ibu ions a e shown in Figs. 5(c) and 5(d), espec i ely. As in he (p,2p) case, he s uck neu on is also emo ed om he p3/2o bi al, and so he shapes o he calcula ed momen um dis ibu ions u n ou o be e y simila in bo h cases. Howe e , he magni ude o he (p,pn) case is ound o be ∼20% la ge . Using uni spec oscopic ac o s, one inds σ(p,pn)=6.6 mb and σ(p,2p)=5.4mb and, hence, σ(p,pn)/σ(p,pp)≃1.2, which is close o he ee nucleon-nucleon c oss sec ions a his ene gy, σpn/σpp =1.27. This is consis en wi h a QFS in e p e a ion o his eac ion. The depa u e om hese ee nucleon-nucleon c oss sec ions can be a ibu ed o he di e ence in sepa a ion ene gies, he Coulomb in e ac ion, and he sligh di e ences in he ma e densi ies (and hence in he nucleon-nucleus po en ials). To inish his sec ion, he ole o he di e en pa ial wa es o he ou going p-Npai is discussed. This is shown in Fig. 7, whe e he his og am co esponds o he con ibu ion o each jπ o he (p,pN) c oss sec ion, and he inse shows he o al ans e se (px) momen um dis ibu ion (summed in jπ). I is seen ha mos o he con ibu ion comes om he j=1,2 wa es. I is o be no ed ha he pu e T=0 con igu a ions 1+ and 3+a e absen in he (p,2p) case. I is also seen ha he 0+0- 1+ 1- 2+ 2- 3+3-4+ 4- jπ 0 1 2 3 4 σjπ (mb) 12C(p,pn)11C 12C(p,2p)11B -400 -200 0 200 400 px (MeV/c) 0 0.01 0.02 0.03 0.04 dσ/dpx (mb/(MeV/c)) (p,pn) SF=1 (p,2p) SF=1 12C(p,2p)11B s. 12C(p,pn)11C FIG. 7. (Colo online) Con ibu ion o each angula momen- um/pa i y (jπ) o he 12C(p,2p)11Band 12C(p,pn)11C eac- ions a E=400 MeV/nucleon o a p3/2con igu a ion and uni spec oscopic ac o . The inse shows he co esponding ans e se momen um dis ibu ions, summed o e all he jπcon ibu ions. 044605-8 THREE-BODY MODEL FOR THE ANALYSIS OF . . . PHYSICAL REVIEW C 92, 044605 (2015) con ibu ion o j⩾3 is small and hence con e gence o hese obse ables equi es only a small numbe pa ial wa es. This is in con as o he usual DWIA o mula ion, in which he inal s a es a e w i en in e ms o he dis o ed wa es o he ou going nucleus, and hence a la ge numbe o pa ial wa es is expec ed o be equi ed in bo h wa es o achie e con e gence o inclusi e obse ables. B. Applica ion o 23O(p,pn) The 23O(p,pn) eac ion a 450 MeV/u, popula ing bound s a es o he 22Onucleus, is conside ed now. In a simple mean- ield pic u e, he single-pa icle con igu a ion o he ou e mos neu ons o 23Ois expec ed o be ν(1p1/2)2(1d5/2)6(2s1/2)1. Ha ee-Fock calcula ions wi h he Sky me Sk20 in e ac- ion yield he single-pa icle ene gies ε(p1/2)=−12.6MeV, ε(1d5/2)=−6.1 MeV, and ε(2s1/2)=−4.3 MeV. Because he neu on sepa a ion ene gy o 22Ois Sn=6.85 MeV, one would expec ha only he emo al o neu ons om he 2s1/2and 1d5/2o bi als o 23Owould lead o bound s a es o 22O. Remo al om deepe o bi s will lead o a esidual 22O sys em wi h an exci a ion ene gy abo e i s neu on sepa a ion h eshold, which will he e o e decay in o 21O. Howe e , shell-model calcula ions, as hose p esen ed below, indica e ha pa o he 1p1/2s eng h in he 23Onucleus co esponds o bound s a es o 22O. Consequen ly, his o bi al has been also conside ed in he calcula ions o his sec ion. As in he p e ious examples, he equi ed nucleon-nucleus po en ials (p+23O o he inciden channel and p/n +22O o he exi channel) we e ob ained by olding he PH NN e ec i e in e ac ion wi h HF (Sk20) g ound-s a e densi ies. Rela i is ic co ec ions we e included acco ding o he p esc ip ions discussed in Sec. II B, wi h he scaling ac o o Eq. (24) o he po en ials. The e ec i e sepa a ion ene gy o each single-pa icle con igu a ion will depend on he conside ed s a e o 22O. In he p esen calcula ions, hese s a es ha e been ob ained om a shell-model calcula ion, pe o med wi h he code OXBASH [25] using he WBT e ec i e in e ac ion o Wa bu on and B own [33]. The esul s a e summa ized in Table I. Guided by hese esul s, o he 2s1/2con igu a ion he e ec i e one-neu on sepa a ion ene gy co esponding o he ansi ion o he 22O g ound-s a e has been conside ed, whe eas o he d5/2and 1p1/2con igu a ions he exci a ion TABLE I. Shell-model spec oscopic ac o s (SF) o 23O→ 22O+p, co esponding o 22Obound s a es. The exci a ion ene gies and angula momen um-spin assignmen a e also hose p edic ed by he shell-model calcula ion. See ex o de ails. Eα(MeV) Iπ αnj SF 0 (g.s.) 0+ 12s1/20.80 3.4 2+ 11d5/22.08 4.8 3+ 11d5/23.08 4.6 0+ 22s1/20.12 5.8 1− 21p1/20.76 6.1 0− 21p1/20.32 6.5 2+ 21d5/20.24 100 200 300 epn (MeV) 0.0 0.1 0.2 dσ/depn (mb/MeV) s1/2 SF=1 p1/2 SF=1 d5/2 SF=1 0 10203040 Ec (MeV) 1 2 dσ/dEc (mb/MeV) (a) 23O(p,pn)22O (b) FIG. 8. (Colo online) (Top) Di e en ial c oss sec ion as a unc- ion o he ou going p o on-neu on ela i e ene gy o he eac ion 23O(p,pn) a 445 MeV/nucleon. Solid, dashed, and do ed lines co espond o he emo al om 2s1/2,1d5/2,and1p3/2con igu a ions in 23O, assuming in all cases uni spec oscopic ac o . (Bo om) Di e en ial c oss sec ion as a unc ion o he kine ic ene gy o he esidual nucleus 22Oin he c.m. ame. ene gies o he 2+ 1and 1− 1s a es we e used, espec i ely. This gi es he alues Sn=2.739 MeV, S∗ n=5.939 MeV, and S∗ n=8.539 MeV o he 2s1/2,d5/2, and 1p1/2con igu a ions, espec i ely. Figu e 8shows he calcula ed (p,pn) di e en ial c oss sec ions as a unc ion o he p-n ela i e ene gy ( op panel) and he co e ene gy in he c.m. ame (bo om). The solid, dashed, and do ed lines co espond o he emo al om 2s1/2,1d5/2, and 1p3/2o bi als in 23O. The ma ked dependence is clea ly seen, wi h he =0 con igu a ion gi ing a much na owe ene gy dis ibu ion. In his case, he p-Npai will be de ec ed wi h a ela i ely na ow dis ibu ion o ene gies cen e ed a ∼210 MeV, which co esponds o hal o he inciden beam ene gy, in acco dance wi h a quasi ee NN sca e ing. Fo o he alues o , he dis ibu ion is also app oxima ely cen e ed a his alue, bu wi h a la ge dispe sion. 044605-9