Structure of 8 B from elastic and inelastic 7 Be + p scattering

PHYSICAL REVIEW C 87, 054617 (2013) S uc u e o 8B om elas ic and inelas ic 7Be +psca e ing J. P. Mi chell,1,2,3,*G. V. Rogache ,1,4,†E. D. Johnson,1L. T. Baby,1K. W. Kempe ,1A. M. Mo o,5P. Peplowski,1,6 A. S. Volya,1and I. Wiedenh¨ o e 1 1Depa men o Physics, Flo ida S a e Uni e si y, Tallahassee, Flo ida 32306, USA 2Depa men o de As onom´ ıa y As o ´ ısica, Pon i icia Uni e sidad Ca ´ olica de Chile, Vicu˜ na Mackenna 4860, Macul, San iago, Chile 3A gelande Ins i u ¨ u As onomie, Uni e si ¨ a Bonn, Au dem H¨ ugel 71, 53121 Bonn, Ge many 4Na ional Supe conduc ing Cyclo on Labo a o y, Michigan S a e Uni e si y, Eas Lansing, Michigan 48824, USA 5Depa men o Physics, Uni e si y o Se ille, Spain 6Johns Hopkins Uni e si y Applied Physics Labo a o y, Lau el, Ma yland 20723, USA (Recei ed 1 Ma ch 2013; published 23 May 2013) Backg ound: De ailed expe imen al knowledge o he le el s uc u e o ligh weakly bound nuclei is necessa y o guide he de elopmen o new heo e ical app oaches ha combine nuclea s uc u e wi h eac ion dynamics. Pu pose: The esonan s uc u e o 8B is s udied in his wo k. Me hod: Exci a ion unc ions o elas ic and inelas ic 7Be +psca e ing we e measu ed using a 7Be a e iso ope beam. Exci a ion ene gies anging be ween 1.6 and 3.4 MeV we e in es iga ed. An R-ma ix analysis o he exci a ion unc ions was pe o med. Resul s: New low-lying esonances a 1.9, 2.54, and 3.3 MeV in 8B a e epo ed wi h spin-pa i y assignmen 0+,2 +,and1 +, espec i ely. Compa ison o he ime-dependen con inuum shell (TDCSM) model and ab ini io no-co e shell model/ esona ing-g oup me hod (NCSM/RGM) calcula ions is pe o med. This wo k is a mo e de ailed analysis o he da a i s published as a Rapid Communica ion.J. P. Mi chell, G. V. Rogache , E. D. Johnson, L. T. Baby, K. W. Kempe e al.,[Phys.Re .C82, 011601(R) (2010)]. Conclusions: Iden i ica ion o he 0+,2 +,1 +s a es ha we e p edic ed by some models a ela i ely low ene gy bu ne e obse ed expe imen ally is an impo an s ep owa d unde s anding he s uc u e o 8B. Thei iden i ica ion was aided by ha ing bo h elas ic and inelas ic sca e ing da a. Di ec compa ison o he c oss sec ions and phase shi s p edic ed by he TDCSM and ab ini io no-co e shell model coupled wi h he esona ing g oup me hod is o pa icula in e es and p o ides a good es o hese heo e ical app oaches. DOI: 10.1103/PhysRe C.87.054617 PACS numbe (s): 21.10.−k, 24.30.− , 25.60.− I. INTRODUCTION One o he main goals o mode n nuclea heo y is o combine he nuclea eac ion models wi h nuclea s uc u e calcula ions o p o ide he uni ied amewo k ha allows he calcula ion o le el spec oscopy and eac ion c oss sec ions s a ing om he same es ablished nuclea Hamil onian. Se e al heo e ical app oaches ha e been sugges ed o ad ance his goal. B oadly, wo majo di ec ions can be iden i ied, phenomenological and ab ini io. The i s one uses he well-es ablished shell-model Hamil onian and couples i wi h he co esponding eac ion channels. The ecoil co ec ed con inuum shell model (RCCSM) [1] and he ime-dependen con inuum shell model (TDCSM) [2] a e examples o hese app oaches. The second majo di ec ion is he a emp o calcula e he c oss sec ion s a ing om bo h ba e nucleon- nucleon o ces and h ee-nucleon o ces. One example o his app oach is he no-co e shell model combined wi h he esona ing-g oup me hod (NCSM/RGM) [3]. The e y a ac i e ea u e o hese de elopmen s is ha he exci a ion unc ions o he esonance eac ions, such as elas ic and inelas ic nucleon sca e ing, (p,n) and (p,α) eac ions, e c., can, in p inciple, be calcula ed and di ec ly compa ed o he *[email p o ec ed] †[email p o ec ed] expe imen al da a. This is in addi ion o all known s uc u e da a. Howe e , his compa ison is no as s aigh o wa d as i may appea . Because o model space unca ion, limi a ions om nucleonic deg ees o eedom and nume ical complexi y, i is na u al o expec ha he nuclea spec um a he low exci a ion ene gy is ep oduced be e han he spec um o he highe lying exci ed s a es by any model. The e o e, i is desi able o e i y he heo e ical p edic ions in he egion o low exci a ion ene gy i s and weakly bound nuclei p o ide a good es o hese models. He e, he con inuum appea s a low ene gy, hus pe mi ing examina ion o he s uc u e- eac ion ansi ion. Mo eo e , because o unca ion o he model space, pa ame e s o models a e adjus ed o he well known spec um o s able nuclei, esul ing in unsu p isingly easonable ag eemen wi h he expe imen al da a o hese nuclei. The be e es is p o ided by exo ic, weakly bound nuclei. The neu on de icien bo on iso ope, 8B, is o pa icula in e es . I s p o on sepa a ion ene gy is only 137 keV and all o i s exci ed s a es a e in he con inuum, as can be seen in i s le el s uc u e in Fig. 1. In addi ion, his nucleus has been a subjec o nume ous heo e ical s udies. In he ecen ab ini io NCSM/RGM analysis [3]o 8B he p o on +7Be elas ic sca e ing phase shi s as well as he c oss sec ion o he 7Be(p,p) and he 7Be(p, γ ) eac ions we e calcula ed. Di ec compa ison o he expe imen al esul s on he 7Be(p,p) and 7Be(p,p) eac ions wi h hese calcula ions and also he analysis o he expe imen al da a using he TDCSM app oach is he main objec i e o his wo k. 054617-1 0556-2813/2013/87(5)/054617(13) ©2013 Ame ican Physical Socie y J. P. MITCHELL e al. PHYSICAL REVIEW C 87, 054617 (2013) Li 8 0.98 + 1 2.26 + 3 3.21 + 1 + 2 B 8 + 2 0.77 + 1 2.32 + 3 3.2 - 2 (a) (b) 3.3 + 1 2.55 + 2 1.9 + 0Li+n 7 2.03 Li*+n 7 2.51 Be+p 7 0.14 Be*+p 7 0.57 FIG. 1. (Colo online) The le el schemes o 8B (a) and i s mi o 8Li (b). S a es om ou p e ious wo k [8] a e in ed. The dashed- do ed line indica es ha he s a e is en a i e. The exci a ion unc ion o 7Be +phas been p e iously measu ed in se e al expe imen s [4–7]. Howe e , he hick a ge in e se kinema ics expe imen al me hod used in all o hese measu emen s did no allow o sepa a ion be ween elas ic and inelas ic sca e ing excep o he da a om [6], whe e measu emen s we e pe o med a ene gies below he inelas ic sca e ing h eshold. In Re . [7], an a emp has been made o use γ-p o on coincidence o iden i y he inelas ic sca e ing e en s, howe e , he elas ic exci a ion unc ion s ill appea s o be con amina ed wi h inelas ic e en s (see Sec. II o addi ional commen s). The expe imen desc ibed he e does no su e om such de iciency because he in e media e a ge hickness app oach has been applied. This app oach allowed o measu emen o a signi ican ac ion o he 7Be +p exci a ion unc ion, while simul aneously de ec ing he 7Be ecoil in coincidence wi h p o ons in o de o dis inguish be ween elas ic and inelas ic sca e ing e en s kinema ically. The e o e, we did no use expe imen al da a om he p e ious highe ene gy measu emen s [4,5,7] in he analysis bu we included he low ene gy 7Be(p,p) elas ic sca e ing da a be ween 0.3 and 0.75 MeV measu ed in Re . [6]. The subse o he da a epo ed he e was i s published in [8]. This pape con ains a mo e de ailed desc ip ion o he expe imen al esul s and also ex ends he p e iously measu ed exci a ion ene gy egion o highe ene gies. A desc ip ion o he expe imen al me hod ha was used o measu e he exci a ion unc ions o 7Be +pelas ic and inelas ic sca e ing be ween 1.6 o 3.4 MeV is gi en in Sec. II. The analysis o he expe imen al da a was pe o med using he mul ichannel mul ile el R-ma ix app oach and is desc ibed in Sec. III. Sec ion IV con ains a discussion o his inding and i s consis ency wi h he p e ious expe imen al da a on he 8B and 8Li nuclei and discusses whe he i is possible o explain he new expe imen al da a wi hou in oducing he new esonances in 8B. Analysis o he new expe imen al da a in he amewo k o he ime-dependen con inuum shell model is p esen ed in Sec. V. De ailed compa ison o he phase shi s ex ac ed om he analysis o he p+7Be expe imen al da a o he p edic ions o he ab ini io NCSM/RGM model is gi en in Sec. IV. Conclusions a e gi en in Sec. VII. II. EXPERIMENT The exci a ion unc ion o p+7Be elas ic and inelas ic sca e ing be ween 1.6 and 3.4 MeV in he cen e o mass sys em (c.m.s.) was measu ed a he John D. Fox Supe con- duc ing Accele a o Labo a o y a Flo ida S a e Uni e si y. A adioac i e beam o 7Be was p oduced using he 1H(7Li,7Be)n eac ion. A p ima y 7Li beam was accele a ed by a 9 MV Supe FN Tandem Van de G aa accele a o ollowed by a LINAC boos e . The p ima y a ge was a 4 cm long hyd ogen gas cell wi h 2.5 μm Ha a en ance and exi windows. The gas cell was cooled by liquid ni ogen and had a gas p essu e o 390 mBa . The in- ligh p oduc ion a e iso ope beam acili y RESOLUT was used o sepa a e 7Be om o he eac ion p oduc s and he p ima y beam. RESOLUT is a se o wo supe conduc ing solenoids, dipole and quad upole magne s and a supe conduc ing esona o . Th ee 7Be beam ene gies we e used in his expe imen : 27.2, 22.0, and 18.5 MeV. The ypical in ensi y o he 7Be beam was 105pps. The composi ion o he beam was 70% 7Be and 30% 7Li con aminan . Diagnos ics o he seconda y beam we e pe o med using a posi ion sensi i e mic ochannel pla e de ec o ins alled be ween he dipole magne and he second solenoid (2.7 m be o e he C2H4 a ge ) and he E-E elescope consis ing o an ioniza ion chambe (used as E de ec o ) backed by a 50 ×50 mm216 ×16 silicon s ip de ec o posi ioned 66 cm downs eam om he seconda y a ge . A ske ch o he expe imen al se up is shown in Fig. 2.A solid polye hylene (C2H4) a ge o hickness op imized o he gi en beam ene gy (see desc ip ion below) was used. A se o h ee annula mic on semiconduc o silicon s ip de ec o s (S2 design) o he p o on ecoils we e posi ioned 5, 6, and 7 cm downs eam om he a ge , espec i ely. Ano he S2 de ec o o he 7Be ecoils was posi ioned 24.5 cm om he a ge . The Li 7 Be, 7 RESOLUT Ta ge 2 CH 5 cm 6 cm 7 cm 24.5 cm Si Ion De ec o BEAM 21.7 MeV p p Be 7 Be 7 2 1.96 mg/cm FIG. 2. (Colo online) The expe imen al se up. The 7Be beam was deli e ed by he RESOLUT acili y (on he le ). The C2H4 a ge s o a ious hicknesses we e used. The p o ons we e de ec ed in an a ay o h ee mic on semiconduc o s S2 de ec o s and he 7Be’s we e measu ed in an S2 downs eam. (The inse p o ides a mo e de ailed iew o he de ec o a angemen .) 054617-2 STRUCTURE OF 8B FROM ELASTIC AND ... PHYSICAL REVIEW C 87, 054617 (2013) S2 de ec o has annula geome y and consis s o 16 segmen s and a side o ings ha allow o he sca e ing angle o he p oduc s o be de e mined. The i s in he se o h ee p o on de ec o s was a E de ec o o 65 μm, while he o he wo and he 7Be de ec o we e 500 μm each. The a ge hickness was op imized o maximum ene gy losseso he7Be ions in he a ge while ensu ing ha all 7Be ecoils make i ou o he a ge wi h enough kine ic ene gy le o be de ec ed in he downs eam S2 de ec o . Kinema ic coincidence be ween p o ons in he a ay o h ee S2 de ec o s and he 7Be ecoils in he downs eam S2 de ec o we e hen used o iden i y he sca e ing e en s. The 65 μm E S2 de ec o was used only in he ini ial s age o he expe imen o e i y ha kinema ic coincidence be ween ligh and hea y ecoils a e enough o clean iden i ica ion o he p+7Be elas ic and inelas ic sca e ing e en s. This de ec o was hen emo ed. Measu emen s a he beam ene gies o 22 and 18.5 MeV we e pe o med wi hou he 65 μm de ec o , while ha o he 27.2 MeV ene gy included he 65 μm E de ec o . Time be ween he e en s in he p o on and 7Be de ec o s was measu ed wi h esolu ion o abou 3 ns in o de o elimina e andom coincidence backg ound. Elas ic and inelas ic sca e ing p ocesses can be dis inguished, because comple e kinema ics o he e en s a e measu ed. Mo e speci ically, e en s ha ha e wo equal ene gy p o ons would ha e di e en ene gy o 7Be ecoils i hey o igina e om di e en (elas ic/inelas ic) p ocesses. This is due o di e en eac ion Q- alue and kinema ics, and also e ec i e a ge hicknesses (and hence ene gy losses) expe ienced by he hea y ecoils. The inelas ic e en s ha p oduce p o ons wi h he same kine ic ene gy as elas ic e en s ake place ea lie (ups eam) in he a ge , whe e a nega i e eac ion Q- alue is compensa ed by he highe ene gy o he 7Be p ojec ile. (See also Re . [9] o de ails on his expe imen al echnique.) The 2D sca e plo o he kinema ic coincidence be ween p o ons and 7Be is shown in Fig. 3. The kinema ic loci which co espond o elas ic and inelas ic sca e ing p ocesses a e labeled and ou lined wi h con ou s. Kinema ically comple e measu emen s allow o unambiguous iden i ica ion o he e en s and do no equi e pa icle iden i ica ion. We con- side ed he possibili y o nonbina y p ocesses con ibu ion. Fo example, exci a ion o 7Be on hyd ogen o ene gies Be Ene gy (MeV) 7 67891011 P o on Ene gy (MeV) 4 4.5 5 5.5 6 6.5 7 7.5 Be(p,p) 7 Be(p,p’) 7 FIG. 3. (Colo online) Sca e plo o kinema ic coincidence be ween p o ons and 7Be ions. Regions which co espond o elas ic and inelas ic sca e ing a e labeled. abo e α+3He decay h eshold would esul in h ee pa icle con inuum p+α+3He wi h αand 3He possibly p oducing signals in he wo S2 de ec o s simul aneously. De ailed Mon e Ca lo simula ion ha akes in o accoun he geome y o he expe imen and o he expe imen al pa ame e s was pe o med. I was ound ha α+3He coincidence would esul in a 2D ene gy- s-ene gy co ela ion ha is e y di e en om well de ined ene gy- s-ene gy co ela ion o p+7Be bina y p ocess. Polye hylene a ge hicknesses used in his expe imen we e 2.6, 2.5, and 1.5 mg/cm2 o he 27.2, 22, and 18.5 MeV beam ene gies, espec i ely. In addi ion, a sepa a e un a 18.5 MeV o 7Be beam ene gy was pe o med wi h a sligh ly hicke (2 mg/cm2) a ge , o ex end he measu ed exci a ion unc ion o lowe ene gies wi hou changing he ene gy o he beam. Unde his condi ion coincidence be ween he highes ene gy p o ons and he 7Be ecoils a e los ( he hea y ecoils p oduced a he beginning o he a ge do no make i h ough). Only he lowe ene gy pa o his spec um was used in he analysis. Figu e 4shows exci a ion unc ions o esonance elas ic and inelas ic sca e ing o 7Be +pmeasu ed in ou di e en uns. Ene gy bins a e 30 keV. The e ex up iangles co espond o he 7Be una 18.5MeVwi h he2mg/cm2 a ge , he squa es a e 18.5 MeV 7Be wi h he 1.5 mg/cm2 a ge da a, he ci cles a e 22 MeV 7Be wi h 2.5 mg/cm2da a, he e ex down iangles a e om he 27.2 MeV un wi h he 2.6 mg/cm3 a ge , and in all cases, he illed ma ke s a e o elas ic sca e ing and he hollow ma ke s o inelas ic sca e ing. The angula esolu ion o he expe imen al se up, as de e mined by he pi ch o he ings in he S2 de ec o , dis ance om he a ge and he size o he beam spo on he seconda y a ge , was 1.25◦. We used binning o 4◦in he labo a o y ame, combining e en s eco ded by 12 ings o he S2 de ec o in o one spec um. Exci a ion unc ions (MeV) exc E 1.5 2 2.5 3 3.5 (mb/s )Ω /dσd 0 20 40 60 80 100 120 o Be(p,p) 148 7 o Be(p,p’) 146 7 FIG. 4. The exci a ion unc ion o 7Be +pelas ic and inelas ic sca e ing a 148 ±4◦and 146 ±4◦deg ees, espec i ely. Resul s om uns a h ee di e en ene gies o 7Be beam a e shown. The squa es co espond o he un a 18.5 MeV o 7Be wi h a 1.5 mg/cm2 a ge , he e ex up iangles a e da a aken a 18.5 MeV wi h he 2 mg/cm2 a ge , he ci cles a e he 22 MeV da a wi h he 2.5 mg/cm2 a ge , and he e ex down iangles a e om he 27.2 MeV un wi h a 2.6 mg/cm2wi h solid ma ke s ep esen ing 7Be +pelas ic sca e ing and open ma ke s he inelas ic p(7Be,p)7Be(1 2 −) sca e ing exci a ion unc ions. 054617-3 J. P. MITCHELL e al. PHYSICAL REVIEW C 87, 054617 (2013) Exci a ion Ene gy (MeV) 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 (mb/s )Ω/dσd 20 30 40 50 60 70 80 90 FIG. 5. (Colo online) The exci a ion unc ion o 7Li +pelas ic sca e ing a 148 ±4◦is shown wi h solid ci cles. This exci a ion unc ion was measu ed simul aneously wi h 7Be +p( he a e iso ope beam composi ion was 70% 7Be and 30% 7Li) and used o absolu e no maliza ion. The same exci a ion unc ion om [10,11]isshown o compa ison wi h open ci cles. a h ee sca e ing angles we e ob ained his way. These angles a e 148 ±4◦, 140 ±4◦132 ±4◦in he c.m.s. o elas ic sca e ing and 146 ±4◦, 138 ±4◦130 ±4◦ o inelas ic sca e ing. Absolu e no maliza ion o he c oss sec ion was pe o med using he known exci a ion unc ions o 7Li +p elas ic sca e ing. These exci a ion unc ions we e ex ac ed om he expe imen al da a using he same p ocedu e as o he 7Be +pelas ic sca e ing, he e o e, by no malizing he 7Li +pda a o he known 7Li +pc oss sec ion and aking in o accoun he a io o he 7Be ions o he 7Li ions in he seconda y beam (as measu ed by he 0 deg ee ioniza ion chambe and silicon s ip de ec o ), accu a e no maliza ion is achie ed. No e ha his no maliza ion p ocedu e au oma ically akes in o accoun he e iciency o he expe imen al se up. A sample o he 7Li +pexci a ion unc ion measu ed in his expe imen is shown in Fig. 5(solid ci cles) and compa ed o he expe imen al da a om [10,11]. Exci a ion unc ions ex ac ed om ou da a ag ee well wi h he di e en ial c oss sec ion o elas ic and inelas ic sca e ing o 7Be +pmeasu ed a se e al ene gies o 7Be using a hin a ge app oach and epo ed by G ei e e al.,[12]. The exci a ion unc ions o Yamaguchi e al. [7] howe e , di e om ou s, especially in he inelas ic channel whe e hey ound he exci a ion unc ion o be ai ly la ac oss hei en i e ene gy ange measu ed, while ou esul s ha e a la ge peak a an exci a ion ene gy o 2.5 MeV. This disc epancy may be ela ed o he backg ound in he NaI scin illa o de ec o s used in Re . [7] ha could ha e p e en ed a clean γ-p o on coincidence spec um o be ex ac ed. III. R-MATRIX ANALYSIS The exci a ion unc ions o elas ic 1H(7Be,p)7Be(g.s.) and inelas ic 1H(7Be,p)7Be(1/2−; 0.43 MeV) sca e ing we e analyzed using a wo channel, mul ile el R-ma ix app oach. The na u al s a ing poin o he analysis is o in oduce only he exci ed s a es o 8B ha we e iden i ied in p e ious expe imen s [13], he 1+a 0.77 MeV, he 3+a 2.32 MeV, and he b oad 2−a ∼3 MeV. These h ee s a es ep oduce he exci a ion unc ion o p+7Be elas ic sca e ing be ween (mb/s )Ω/dσd 20 40 60 80 100 120 140 o Be(p,p) 148 7 (a) B Exci a ion Ene gy (MeV) 8 1.6 1.8 2 2.2 2.4 2.6 2.8 (mb/s )Ω/dσd 0 5 10 15 20 25 30 35 40 o Be(p,p’) 146 7(b) a 2.50 MeV + 1 a 3.0 MeV + 1 - and 2 + Only 3 added - and 1 + 1 coupled channel FIG. 6. (Colo online) R-ma ix i o he elas ic and inelas ic 7Be +psca e ing wi h known 3+and 2−s a es, a second exci ed 1+seen in 8Li, and he “backg ound” 1−s a e in oduced a highe ene gy. The solid cu e co esponds o only 3+and 2−s a es a 2.3 and 3.5 MeV, espec i ely. The ed sho -dashed cu e includes he con ibu ion o he highe lying 1+s a es assumed a 3.0 MeV. Dash-do ed pu ple cu e shows he 1+s a e shi ed o 2.5 MeV and he long dashed g een cu e also includes he 1−s a e in oduced a 5MeV. 0.5 and 3.5 MeV easonably well, as shown in Fig. 6(a) by he solid line. Howe e , i is no possible o explain 30 mb/s inelas ic c oss sec ion a 2.5 MeV i only known s a es in 8B a e conside ed [Fig. 6(b)]. This ailu e can be unde s ood om he ollowing simple conside a ions. The i s exci ed 1+s a e a 0.77 MeV is oo na ow o ha e any signi ican impac on he exci a ion unc ions a ene gies abo e 1.6 MeV. The second exci ed s a e, 3+a 2.32 MeV, can only decay o he 3/2−g ound s a e o 7Be because decay o he 1/2− i s exci ed s a e equi es angula momen um o =3. The e o e, e en i he co esponding educed wid h is la ge he inelas ic pa ial p o on wid h, p=2P(kR)γ2, would s ill be small compa ed o he elas ic pa ial p o on wid h due o a small pene abili y ac o o high angula momen um decay. Hence, he c oss sec ion o popula ion o he i s exci ed s a e in 7Be due o he 3+ esonance in 8B, de e mined by he pp/2 o a io, is small. The same is ue o he b oad 2−s a e in 8B a ≈3 MeV as i can only decay o he i s exci ed s a e in 7Be wi h angula momen um =2 while decay o he g.s. p oceeds wi h =0. Figu e 6shows he esul s o an R-ma ix calcula ion wi h only p e iously known 1+,3 +, and 2−s a es a 0.77, 2.32, and 3.7 MeV wi h educed wid h pa ame e s e alua ed using he TDCSM (mo e de ails on TDCSM calcula ions a e gi en in Sec. V) and known o al wid hs o hese s a es. (Exci a ion ene gy and wid h o he 2−we e adjus ed sligh ly o p oduce a be e i .) I is clea ha while he elas ic sca e ing da a is well ep oduced, he inelas ic sca e ing da a canno be explained by he known s a es. Based on he le el scheme o 8Li (Fig. 1) i is na u al o in oduce he second 1+s a e in 8B a an exci a ion ene gy a ound 3 MeV. Reduced wid hs o his s a e we e chosen acco ding o TDCSM calcula ions ca ied ou wi h 054617-4 STRUCTURE OF 8B FROM ELASTIC AND ... PHYSICAL REVIEW C 87, 054617 (2013) he Cohen-Ku a h CKI in e ac ion [14]. I was e i ied ha hese educed wid hs ep oduce he known wid h o his s a e in 8Li (∼1 MeV). The sho dashed cu e ( ed) in Fig. 6 shows he e ec o he 1+s a e on he i . While he elas ic exci a ion unc ion is i ed well, he inelas ic c oss sec ion is s ill unde es ima ed. E en i his s a e is shi ed o 2.5 MeV, whe e inelas ic sca e ing has i s maximum c oss sec ion, i s ill unde es ima es he da a [dash-do ed (blue) cu e in Fig. 6]. Finally, in an a emp o inc ease he inelas ic c oss sec ion wi hou using new s a es below 3 MeV we in oduced a1 −“backg ound” s a e. This s a e can decay o he i s exci ed s a e o 7Be wi h =0, he e o e i may con ibu e signi ican ly o he inelas ic c oss sec ion. The educed wid hs o he 1−s a e we e e alua ed using he shell model, and he s a e was in oduced a 5 MeV. As expec ed, he 1−backg ound s a e inc eased he inelas ic c oss sec ion o e all (long-dashed g een cu e in Fig. 6). Bu e en wi h his s a e included he inelas ic c oss sec ion canno be ep oduced. The ab ini io calcula ions o 8B[3,15–17] p edic h ee mo e posi i e pa i y (p-shell) s a es a low exci a ion ene gy. Thesea e he0 +1,1 +2, and 2+2. The exci a ion ene gies o hese s a es a y be ween 2 and 6 MeV depending on he h ee-body o ce pa ame iza ion and he speci ics o he calcula ions. Simila esul s a e ob ained in shell model calcula ions (exci a ion ene gies o hese “missing” s a es a y be ween 2 and 6 MeV in he shell model as well, depending on he esidual in e ac ion used). The e o e, i is na u al o in oduce hese s a es in an a emp o ep oduce he la ge inelas ic sca e ing c oss sec ion. The 1+2s a e has al eady been in oduced. Tha lea es only he 0+1and 2+2. In oduc ion o a new 2+s a e placed a 2.5 MeV, ep oduces bo h he magni ude and angula dependence o he obse ed peak in he inelas ic c oss sec ion while keeping he elas ic exci a ion unc ion in ag eemen wi h he expe imen al da a (blue dashed cu e in Fig. 7). Howe e , e en wi h his new s a e he c oss sec ion o inelas ic sca e ing below 2.3 MeV is s ill unde es ima ed. The 2+s a e should ha e a ela i ely 0 20 40 60 80 100 120 140 dσ/dΩ (mb/s ) 7Be(p,p) 148° 7Be(p,p) 140° 7Be(p,p) 132° 7Be(p,p’) 146° 7Be(p,p’) 138° 7Be(p,p’) 130° 7Be(p,p) 148° 7Be(p,p) 140° 7Be(p,p) 132° 7Be(p,p’) 146° 7Be(p,p’) 138° 7Be(p,p’) 130° 7Be(p,p) 148° 7Be(p,p) 140° 7Be(p,p) 132° 7Be(p,p’) 146° 7Be(p,p’) 138° 7Be(p,p’) 130° 0 5 10 15 20 25 30 35 40 1.5 2 2.5 dσ/dΩ (mb/s ) Eexc (MeV) 7Be(p,p) 148° 7Be(p,p) 140° 7Be(p,p) 132° 7Be(p,p’) 146° 7Be(p,p’) 138° 7Be(p,p’) 130° 1.5 2 2.5 Eexc (MeV) 7Be(p,p) 148° 7Be(p,p) 140° 7Be(p,p) 132° 7Be(p,p’) 146° 7Be(p,p’) 138° 7Be(p,p’) 130° 1.5 2 2.5 3 3.5 Eexc (MeV) 7Be(p,p) 148° (a) 7Be(p,p) 140° (b) 7Be(p,p) 132° (c) 7Be(p,p’) 146° (d) 7Be(p,p’) 138° (e) 7Be(p,p’) 130° ( ) FIG. 7. (Colo online) Elas ic and inelas ic exci a ion unc ions o 7Be +psca e ing. The blue dashed cu e is a i wi h he p e iously known 1+and 3+s a es as well as a 2+a 2.54 MeV o ep oduce he peak in he inelas ic da a. The 2−and 1−phase shi s we e a ied. The ed solid cu e is he bes i wi h he R-ma ix pa ame e s om Table I. 100 150 200 250 300 dσ/dΩ (mb/s ) 7Be(p,p) 123.6° 7Be(p,p) 163.8° (a) (b) 50 100 150 200 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 dσ/dΩ (mb/s ) Eexc (MeV) 7Be(p,p) 123.6° 7Be(p,p) 163.8° (a) (b) FIG. 8. (Colo online) Exci a ion unc ion o 7Be +pelas ic sca e ing a low ene gies om [6] a 123.6◦(a) and 163.8◦(b). The bes i is a solid black cu e. The calcula ed c oss sec ion was con olu ed o accoun o 30 keV expe imen al esolu ion epo ed in Re . [6]. Sys ema ic e o s we e included in o he e o ba s. The dashed ed cu e is he R-ma ix i wi h he ab ini io 1+phase shi s om [17] o he1 + 1s a e (channel spins 1 and 2 con ibu e abou equally). small wid h (270 ±40 keV) o i he obse ed peak-like s uc u e in he inelas ic exci a ion unc ion a 2.5 MeV and i s in luence below 2.3 MeV is small. In oducing he 0+s a e a an exci a ion ene gy o 1.9±0.1 MeV wi h a wid h o 530+600 −100 keV allows he inelas ic sca e ing da a o be i below 2.3 MeV wi hou des oying he i o he elas ic sca e ing da a (solid ed line in Fig. 7). I was e i ied ha a 1+spin-pa i y assignmen o his s a e would esul in sligh ly wo se χ2,bu mos impo an ly he elas ic educed wid h ampli ude o he 1+a 2.0 MeV would ha e o be e y small (∼0.1) o i he elas ic and inelas ic da a simul aneously. This would make i impossible o ep oduce he o al 7Li(n,n) c oss sec ion, o which he 1+ 2s a e plays dominan ole (see Fig. 10). The low-ene gy da a om [6] we e used (Fig. 8) o p o ide addi ional cons ain on he beha io o he phase shi s a low ene gy. I p o ed o be pa icula ly impo an o he nega i e pa i y phase shi s. We used he p edic ions o he ab ini io calcula ions [17] o he2 −and 1−phase shi s as he s a ing poin , bu he bes i was achie ed wi h he nega i e pa i y phase shi s di e en om [17]. (I is discussed in mo e de ail in Sec. VI.) The bes i ha included he low ene gy da a om [6] and da a om his expe imen was achie ed using R-ma ix pa ame e s gi en in Table I. The no malized χ2 o he bes i was 0.89. S a es shown in pa en hesis in Table Ia e he b oad “backg ound” s a es ha a e used in R-ma ix o malism o p oduce he desi ed beha io o he co esponding phase shi s. IV. NEW STATES IN LIGHT OF PREVIOUS EXPERIMENTAL DATA He e we s udy/assess i he low lying 0+ 1and 2+ 2s a es a e consis en wi h he a ailable expe imen al da a on 8B and 8Li nuclei. The s uc u e o 8B has been ex ensi ely s udied in p+7Be esonance elas ic sca e ing expe imen s [4–7]. In Re . [6] he7Be +pexci a ion unc ion o elas ic sca e ing was measu ed in he c.m. ene gy ange om 0.3 o 0.75 MeV. 054617-5 J. P. MITCHELL e al. PHYSICAL REVIEW C 87, 054617 (2013) TABLE I. Pa ame e s o esonances in 8B om he R-ma ix bes i . S a es in pa en hesis a e ou side o he measu ed exci a ion ene gy ange bu p o ide essen ial “backg ound” h ough low ene gy ails. The ene gy eigen alue and he educed wid hs ampli udes o 7Be(p,p) and 7Be(p,p)7Be(1/2−) sca e ing wi h channel spins 1 and 2 o he o me and 0 and 1 o he la e used in he R-ma ix i a e shown in columns 6–10. We used 4.20 m as he channel adius o bo h he elas ic S=1,2 and inelas ic S=0,1 channels. JπEex (MeV)  o (MeV) p(MeV) p(MeV) Eeigen γel S=1γel S=2γ1/2−S=0γ1/2−S=1 2+0–––−0.657 −0.793 −0.531 0.000 0.430 1+0.768(4) 0.027(6) 0.026(6) 0.001 0.276 0.718 0.130 −0.875 −0.335 0+1.9(1) 0.53+0.6 −0.10.06+0.3 −0.02 0.47+0.4 −0.12.102 0.353 0.000 0.000 1.303 3+2.31(2) 0.33(3) 0.33(3) 0.0 2.305 0.000 0.607 0.000 0.000 2+2.50(4) 0.27(4) 0.05 0.22 2.471 0.224 0.000 0.000 0.534 1+3.3(2) 3.2(9) 2.8 0.4 4.740 0.937 −1.179 0.029 0.664 (1−) – – – – 5.548 1.664 0.000 0.000 2.827 (2−) – – – – 12.059 0.000 3.15 0.000 0.000 The new s a es a e a 1.9, 2.5, and 3.3 MeV exci a ion ene gies and hei in luence on he low ene gy pa o he exci a ion unc ion is e y small. In gene al, he i o he elas ic sca e ing da a does no equi e he low lying 0+ 1and 2+ 2s a es. The expe imen al da a in Re s. [4,5,7] we e i ed wi h only 1+and 3+s a es a 0.77 and 2.32 MeV and a 2−s a e a ∼3MeV. (P esence o a 1+s a e a ∼3 MeV was sugges ed in Re . [4].) Howe e , he new s a es ha e li le in luence on he exci a ion unc ion o elas ic sca e ing. The e o e, he absence o hese s a es in he R-ma ix analysis o he elas ic sca e ing da a canno be used as an a gumen agains hese s a es. I is in e es ing o no e ha in all h ee p e ious measu emen s [4,5,7] he c oss sec ion a he esonance ene gy o he 3+ s a e was measu ed o be ≈190 mb/s a 180◦.TheR-ma ix i o ou elas ic sca e ing da a p oduces a lowe c oss sec ion a 180◦,≈160 mb/s . This is no su p ising because he expe imen al echnique used in p e ious measu emen s did no sepa a e elas ic om inelas ic sca e ing. P o ons om inelas ic sca e ing we e con ibu ing o he “elas ic” exci a ion unc ions which esul ed in highe c oss sec ion alues o he measu ed “elas ic” exci a ion unc ions. Mo e expe imen al in o ma ion is a ailable ega ding he s uc u e o he mi o nucleus, 8Li. One and wo neu on ans e eac ions, 7Li(d,p)[18] and 6Li( ,p)[19]we eused o popula e s a es in 8Li. I is e y unlikely ha bound s a es in 8Li could ha e been missed in hese expe imen s. The e o e, he 0+ 1and 2+ 2s a es a e p obably abo e he neu on decay h eshold (2.03 MeV) in 8Li. The exci a ion unc ion o he 7Li(n,γ )8Li eac ion was measu ed a low c.m. ene gies (up o 1 MeV) [20–23]. Only he 3+s a e a 2.25 MeV (0.22 MeV abo e he neu on decay h eshold) has been obse ed. In p inciple, lack o e idence o he 0+and he 2+s a es in he 7Li(n,γ )8Li exci a ion unc ion canno be conside ed as a decisi e a gumen agains hei p esence. I he pa ial γwid h (γ) o hese s a es is small hen hey can be ha d o iden i y wi hin he backg ound om di ec neu on cap u e and esonance cap u e due o he 3+s a e. Figu e 9shows TDCSM calcula ions o he (n,γ ) exci a ion unc ion wi h he known 3+and 1+s a es and he new 0+and 2+s a es a 2.4 and 2.5 MeV ( op panel) and a 2.8 and 3.3 MeV (bo om panel). I is clea om his igu e ha obse a ion o he new s a es in he 7Li(n,γ ) eac ion is di icul . Resonances in 8Li a exci a ion ene gies o up o 9.0 MeV ha e been s udied in elas ic and inelas ic n+7Li sca e ing and analyzed using he R-ma ix app oach in Re . [24], whe e he new low-lying s a es we e sugges ed. Fo example, he 0+ s a e a 3.02 MeV was in oduced. Un o una ely, he n+7Li exci a ion unc ion is ela i ely ea u eless, which makes R-ma ix analysis ambiguous. The con empo a y ( o 1987) shell model p edic ions we e used in Re . [24] as guidance o he i . We pe o med ou own R-ma ix analysis o he n+7Li exci a ion unc ions and a emp ed o inco po a e he new low-lying 0+and 2+s a es in o he n+7Li i . I appea s ha he low ene gy n+7Li exci a ion unc ion o elas ic sca e ing can be ep oduced wi h he 0+,1 +, and 2+s a es i hey a e placed a exci a ion ene gies abo e 2.8 MeV wi hou any modi ica ions o hei educed wid hs. The o al c oss sec ion o he 7Li(n,n)7Li(g.s.) eac ion is shown in Fig. 10. The solid line is he R-ma ix i wi h he s a es men ioned 0 5 10 15 20 25 30 35 40 2.2 2.4 2.6 2.8 3 3.2 3.4 c oss sec ion [μb] Exci a ion ene gy [MeV] 3+1+ 0+2+ 0 5 10 15 20 25 30 35 40 3+1+ 0+2+ 7Li(n,γ) s-wa e FIG. 9. (Colo online) The 7Li(n,γ) eac ion exci a ion unc ion calcula ed using TDCSM app oach. The known 3+and 1+s a es and he new 0+and 2+s a es a 2.4 and 2.5 MeV ( op panel) and a 2.8 and 3.3 MeV (bo om panel) a e shown. 054617-6 STRUCTURE OF 8B FROM ELASTIC AND ... PHYSICAL REVIEW C 87, 054617 (2013) 1 10 0.2 0.4 0.6 0.8 1 1.2 1.4 σ (b) Ec.m. (MeV) FIG. 10. (Colo online) To al 7Li(n,n)7Li eac ion exci a ion unc ion om [24]. The solid line is he R-ma ix i wi h he known 3+s a e a 2.25 MeV and he new 0+,1 +,and2 +s a es a 2.8, 3.1, and 3.3 MeV. The dashed line shows he e ec o shi ing he 0+s a e down by 200 keV. abo e, he backg ound s a es om Table I, and he known 3+ a 0.22 MeV. No e ha i he 0+is shi ed down by as li le as 200 keV i would appea as ela i ely na ow peak, which is no obse ed expe imen ally (do ed line in Fig. 10). F om he conside a ions abo e we can conclude ha exis ence o he new low lying 0+and 2+s a es in 8Li does no , in p inciple, con adic a ailable n+7Li elas ic sca e ing expe imen al da a. Howe e , hese s a es ha e o be shi ed up in exci a ion ene gy by ∼800 keV compa ed o hei sugges ed loca ion in 8B. The e is also s ong e idence agains degene acy o he new s a e(s) wi h he 3+s a e. I such degene acy exis s hen he expe imen al c oss sec ion a he maximum o he 3+peak (0.22 MeV) would be highe han can be accoun ed o by he 3+s a e alone. Ou R-ma ix i shows ha his is no he case. I is mo e di icul o econcile he new s a es in 8B and he a ailable 7Li(n,n)7Li(1/2−) expe imen al da a. I educed wid hs pa ame e s om Table I o hese s a es a e used hen he 7Li(n,n)7Li(1/2−) c oss sec ion is o e es ima ed due o oo s ong con ibu ion om he 2+s a e (do ed blue cu e in Fig. 11). The elas ic educed wid h ampli ude o he 2+s a e has o be educed om 0.276 o <0.1 in o de o p oduce a good i o he 7Li(n,n)7Li(1/2−) da a ( ed solid cu e in Fig. 11). All o he pa ame e s o he 2+and also all pa ame e s o he 0+and 1+do no equi e any modi ica ion. The 7Be(g.s.) +p spec oscopic ac o o he 2+s a e is al eady small (4%) in 8B bu i appea s ha i needs u he educ ion o less han 1% in 8Li o ep oduce he 7Li(n,n)7Li(1/2−) da a. We do no ha e a good explana ion o his si ua ion. The exci a ion ene gy shi o 800 keV be ween s a es in mi o nuclei (Thomas-Eh man shi [25,26]) is e y la ge. While no unique ( o example, he shi be ween he 1/2+ second exci ed s a e in 19O and 19Na is 730 keV [27,28]) i is gene ally associa ed wi h single pa icle s uc u e, whe e he alence nucleon is in he s-wa e s a e. A la ge Thomas-Eh man shi esul s om di e en asymp o ic beha io o he alence nucleon wa e unc ion be ween bound and unbound s a es in mi o nuclei (Nolen-Schi e e ec [29]). The 0+and 2+ a e p-shell s a es, he e o e, a la ge Thomas-Eh man shi is no expec ed. Realizing ha his is an unusual si ua ion, 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.4 0.6 0.8 1 1.2 1.4 σ (b) Ene gy in cms (MeV) FIG. 11. (Colo online) To al 7Li(n,n)7Li(1/2-) eac ion exci a- ion unc ion om [36]. The dashed blue cu e is he R-ma ix i wi h he educed wid h pa ame e s o Table I.The2 + 2 om he able o e es ima es he c oss sec ion a a ound 1.2 MeV in he c.m.s. The solid ed cu e is he same R-ma ix i , wi h he elas ic componen o he 2+ 2 educed o be e i he da a. we a emp ed o ep oduce he obse ed p+7Be inelas ic sca e ing exci a ion unc ion wi hou in oducing he new esonances in 8B bu assuming a di ec exci a ion mechanism o he 7Be i s exci ed s a e in p+7Be sca e ing. These calcula ions we e pe o med using he coupled- channels app oach. The po en ial o Kim e al. [30]was i s conside ed o he ba e pa o he 7Be in e ac ion. The ansi ion po en ial o he coupling be ween he g ound and he i s exci ed s a e was gene a ed de o ming he ba e po en ial and assuming ha hese wo s a es o he 7Be nucleus a e membe s o a K=1/2 o a ional band wi h a quad upole de o ma ion leng h o δ2=2 m. Besides he ansi ion po en ial, his p ocedu e gi es ise also o eo ien a ion e ms, which we e also aken in o accoun in he calcula ions. The coupled equa ions we e sol ed o all o de s using he compu e code FRESCO [31]. In Fig. 12(a) we show he exci a ion unc ion o a θc.m.=146◦as a unc ion o he p+7Be c.m. ene gy. FIG. 12. (Colo online) Inelas ic sca e ing di e en ial c oss sec ion om he 7Be(p,p) eac ion calcula ed wi hin he coupled- channels app oach, assuming a di ec mechanism. The le panel uses ap+7Be po en ial which does no con ain esonances wi hin his ene gy in e al. The igh panel shows he esul o he calcula ion using a po en ial ha con ains a esonance. 054617-7 J. P. MITCHELL e al. PHYSICAL REVIEW C 87, 054617 (2013) Clea ly, he con ibu ion o he di ec mechanism is e y small in his ene gy ange, sugges ing ha he magni ude o he measu ed inelas ic c oss sec ion a hese ene gies canno be explained by a pu e di ec eac ion mechanism. We pe o med a second coupled-channels calcula ion using a po en ial ha p oduces a esonance a hese ene gies. This po en ial was pa ame ized using a Woods-Saxon shape, wi h adius R= 2.23 m (deduced om he ma e adius o he 7Li nucleus), di useness a=0.65 m and he dep h adjus ed o p oduce a esonance a ound Ec.m.=2 MeV. The calcula ed inelas ic exci a ion unc ion ob ained wi h his po en ial is gi en by he solid line in Fig. 12(b). The p esence o he esonance p oduces a p onounced maximum abou 2 MeV and a signi ican inc ease o he magni ude o he c oss sec ion. So, based on he coupled-channels analysis we conclude ha he high inelas ic sca e ing c oss sec ion canno be ep oduced unless eso- nance(s) is(a e) in oduced in he co esponding ene gy ange. Finally, we ha e o make an impo an dis inc ion be ween he 0+and he 2+s a es. While exis ence o he 2+s a e is ha d o dismiss, he case o he 0+s a e is somewha weake . In spi e o he ac ha wi hou his esonance he inelas ic c oss sec ion a 2.0 MeV due o di ec exci a ion o he i s exci ed s a e in 7Be is 3 imes smalle han he expe imen al alue, one should be ca e ul making he inal call based on such e idence. Fu he in es iga ion is wa an ed. Speci ically, accu a e measu emen o he p+7Be exci a ion unc ion o inelas ic sca e ing in he ene gy ange om 0.7 o 2.0 MeV and in a b oad angula ange should p o ide a de ini i e answe on he exis ence o he 0+. A his poin we can only ega d his s a e as en a i e. V. THE CONTINUUM SHELL MODEL ANALYSIS OF THE p+7BE DATA. The ime-dependen con inuum shell model [2] was used as an al e na i e and mo e mic oscopically cons ained way o analyze he p+7Be da a. This model ex ends he adi ional shell model in o he domain o eac ion physics. I inco po a es he many-body dynamics wi h all essen ial s uc u e and eac ion componen s, and allows one o p edic he eac ion obse ables. Some ea u es, such as he angula dependence o c oss sec ions and in e e ence be ween esonances a e pa icula ly sensi i e o he many-body s uc u e. The TDCSM is buil upon one o he well-es ablished Hamil onians o he adi ional shell model coupled o eac ion channels, whe e a Woods-Saxon shaped po en ial is aken om a global Woods-Saxon pa ame iza ion [32]. This heo e ical ea men o 8B using he WBP shell model Hamil onian [33] is epo ed in Re . [2]. The WBP Hamil onian was selec ed because unlike mos in e ac ions i esul s in low-lying 1+ 2,0+ 1,and 2+ 2s a es in 8B, a exci a ion ene gies below 3 MeV. To conside a ull spec um o possible Hamil onians in his in es iga ion, in addi ion o WBP, we use PWT [33] and CKI [14] shell model in e ac ions. The compa ison o he expe imen al spec oscopic ac o s o he posi i e pa i y s a es in 8B o he p edic ions o he shell model wi h di e en in e ac ions is gi en in Table II. The expe imen al spec oscopic ac o s we e calcula ed as he a io be ween he pa ial wid h and he single pa icle wid h calcula ed using a Woods-Saxon po en ial wi h a global Woods-Saxon pa ame iza ion [32]. In Table II TABLE II. Expe imen al spec oscopic ac o s o posi i e pa i y s a es compa ed o he shell model p edic ions. Jπ1+0+3+2+1+ Eex (MeV) 0.768 1.9 2.31 2.50 3.3 S7Be(g.s.)+pa0.38 0.05 0.20 0.04 ≈1 S7Be(1/2−)+pa– 0.94 – 0.19 0.14 1+ 10+ 13+ 12+ 21+ 22+ 3 ECKI 1.08 4.95 1.69 4.24 2.77 5.15 CKI el. 0.44 0.34 0.33 0.56 1.10 0.06 CKI in. 0.87 0.90 – 0.04 0.14 0.27 EPWT 1.54 4.01 2.14 4.39 3.80 6.06 PWT el. 0.45 0.27 0.30 0.55 0.95 0.11 PWT in. 0.84 0.96 – 0.03 0 0.22 EWBP 0.55 1.75 1.99 2.40 1.73 3.25 WBP el. 0.40 0.48 0.37 0.13 1.0 0.40 WBP in. 0.77 0.84 – 0.40 0.14 0.03 aExpe imen al alues. i can be seen ha all h ee esidual in e ac ions a e in good ag eemen wi h he expe imen al spec oscopic ac o s o he 1+ 1and 3+ 1 esonances. All h ee in e ac ions ep oduce he inelas ic spec oscopic ac o o he 0+ 1, and bo h he elas ic and inelas ic componen s o he 1+ 2. The WBP in e ac ion is he only in e ac ion ha p edic s a 2+ 2 ha is domina ed by an inelas ic componen , as is seen expe imen ally, while he PWT and CKI in e ac ions bo h p edic a 2+s a e wi h a simila inelas ically domina ed componen as he 2+ 3. The bes alida ion o he heo e ical model p edic ions can be pe o med i he measu ed c oss sec ion is calcula ed di ec ly om he model. Un o una ely, he eac ion physics is e y sensi i e o kinema ics and o he exac posi ion o le els in he spec um because o he phase space and ba ie pene abili y. While he adi ional shell model may, in gene al, be good in desc ibing posi ions and o de ing o s a es, o en i s p ecision is no close o wha is equi ed by he eac ion physics. Thus, i is common p ac ice o se he exac eac ion kinema ics based on obse a ion. In ou app oach all known s a es and h esholds a e adjus ed om expe imen al da a and we ea he ene gies o unknown 1+ 2,0+ 1,and 2+ 2s a es as pa ame e s. In ou s udy we a y hese h ee pa ame e s o bes i he obse ed c oss sec ion. The TDCSM p o ides an e ec i e mechanism o modi y he posi ion o any s a e in he Hamil onian while keeping all s uc u al aspec s unchanged. This is done by adding o a shell model Hamil onian a ac o izable e m δE|αα|, whe e |αis he eigens a e o be shi ed and δE is he ene gy shi . The co esponding change in he many-body p opaga o is pe o med exac ly wi h he help o Dyson’s equa ion, o de ails see Re . [2]. In Fig. 13 he inelas ic sca e ing c oss sec ion o 7Be(p,p) ob ained wi h TDCSM is compa ed o expe imen . Panels (a), (b), and (c) co espond o calcula ions wi h WBP, PWT, and CKI in e ac ions, espec i ely. The spin and pa i ies o esonances in he ene gy egion plo ed a e ma ked. The inelas ic c oss sec ion is no sensi i e o he 3+s a e which is seen in he elas ic sca e ing c oss sec ion. All models p edic a simila s uc u e o he 3+s a e and he e o e p oduce 054617-8 STRUCTURE OF 8B FROM ELASTIC AND ... PHYSICAL REVIEW C 87, 054617 (2013) 0 5 10 15 20 25 30 35 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Exci a ion ene gy [MeV] (c) 3+1+ 0+2+ 0 5 10 15 20 25 30 35 C oss Sec ion [mb/s ] (b) 3+ 0+1+ 2+ 0 5 10 15 20 25 30 35 (a) 1+0+3+2+ 129o 138o EXP-129o EXP-138o FIG. 13. (Colo online) Inelas ic sca e ing di e en ial c oss sec ions o 7Be(p,p) eac ion ob ained wi h he TDCSM ha uses h ee di e en Hamil onians is compa ed o he expe imen al da a. (a), (b), and (c) co espond o WBP, PWT, and CKI in e ac ions. a compa able elas ic c oss sec ion which ag ees well wi h expe imen . The elas ic 7Be(p,p) c oss sec ion wi h WBP in e ac ion is demons a ed in Re . [2]. Posi ions o 1+ 2,0+ 1, and 2+ 2 esonances, indica ed in Fig. 13, a e no known ap io i; he e hey a e adjus ed by isual examina ion o bes ep oduce he expe imen al da a. The main peak in he 7Be(p,p)c oss sec ion is due o he 2+ 2 esonance a a ound 2.5 MeV o exci a ion. I was ound ha ag eemen wi h he expe imen al da a is good i he 0+ 1is placed a ound 2 MeV and he 1+ 2is mo ed o highe exci a ion ene gy. (Sensi i i y o he inelas ic c oss sec ion o he posi ion o he 1+ 2s a e is weak. Howe e , he 1+ 2s a e a exci a ion ene gies below 2.3 MeV would p oduce a peak in he elas ic c oss sec ion, which is no obse ed expe imen ally. See discussion in Sec. III.) In he case o he WBP in e ac ion, Fig. 13(a), no posi ion adjus men was made o he 1+ 2and 0+ 1s a es and he 2+ 2is only mo ed down by abou 140 keV. The CKI Hamil onian gi es wo 2+exci ed s a es a 4.2 and 5.1 MeV o exci a ion. Bo h o hese s a es ha e been ied as candida es o he 2.5 MeV esonance and i was de e mined ha he second 5.1 MeV s a e in he CKI Hamil onian has he co ec s uc u e. Ou main conclusion om he calcula ions shown in Fig. 13 is ha he CKI in e ac- ion appea s o be bes in ep oducing he c oss sec ion. The s a es ob ained wi h he CKI appea o ha e s uc u e which ag ees wi h he obse ed in e e ence and angula dependence ea u es. In pa icula , only he CKI in e ac ion is able o ep oduce he obse ed inc ease in he c oss sec ion a 2.5 MeV o highe angle [Fig. 13(c)]. The heigh o he esonance peak a 2.5 MeV is he p ima y di e ence be ween heo y and obse a ion. We a ibu e his di e ence pa ially o he 1− s a e, which was no included in he shell model analysis (only p-shell s a es we e conside ed) and also o he somewha di - e en a io be ween he elas ic and inelas ic pa ial wid hs o he 2+ 2. The ampli udes om he inal R-ma ix i wi h he CKI in e ac ion a e summa ized in Table III ( esonance educed wid h pa ame e s om he R-ma ix i we e ecoupled om heLS o hejj coupling scheme o di ec compa ison wi h TDCSM ampli udes). I should be no ed ha he choice o channel adius in he R-ma ix calcula ions will ha e he a ec o scaling he educed wid h pa ame e s, hus one should no di ec ly compa e absolu e alues, bu a he sign and ela i e alues o he educed wid hs. We no e ha he i only sligh ly modi ies he ampli udes o he 3+ 1and 2+ 2s a es lea ing he gene al ea u es o CKI unchanged. Howe e , he e is a signi ican di e ence be ween expe imen al exci a ion ene gies and CKI p edic ions o he 2+ 2s a e. The 0+ 1also has a signi ican shi be ween he expe imen al and CKI p edic ed ene gy. TABLE III. Decay ampli udes om he inal R-ma ix i and om he CKI based TDCSM. The i s column deno es he spin and pa i y o he esonance, ollowing a e exci a ion ene gy and ou ampli udes om he R-ma ix i . Exci a ion ene gy and ampli udes o s a es in 8B om he CKI Hamil onian a e lis ed in he emaining i e columns. R-ma ix i TDCSM wi h CKI in e ac ion JπE[MeV] 7Be 3/2−g.s. 7Be 1/2−E[MeV] 7Be 3/2−g.s. 7Be 1/2− p1/2p3/2p1/2p3/2p1/2p3/2p1/2p3/2 2+ 10 0.19 −0.94 0.43 0.00 0.23 −0.98 −0.43 1+ 10.768 −0.17 0.71 0.24 −0.91 1.08 −0.35 0.57 0.24 −0.91 0+ 11.9 0.35 1.30 4.95 −0.59 0.95 3+ 12.31 0.61 1.69 0.58 2+ 22.50 −0.17 0.17 0.53 5.15 −0.17 0.16 −0.52 1+ 23.3 −1.46 0.37 0.53 0.41 2.77 0.84 0.62 0.33 0.18 054617-9