Tensor analyzing powers for Li7 breakup

PHYSICAL REVIEW CVOLUME 52, NUMBER 6 Tenso analyzing powe s o Li b eakup DECEMBER 1995 N. J. Da is, C. H. Shephe d-Themis ocleous, *A. C. Sho e , T. Da inson, D. G. I eland, ~K. Li ings on, ~ E. W. Macdonald, ~R. D. Page, ~P. J. Sellin, and P. J. Woods Depa men o Physics and As onomy, Uni e si y o Edinbu gh, May ield Road, Edinbu gh EH9 3JZ, Sco land N. M. Cla ke, G. Tunga e, J.A. R. G i i h, S. J. Hall, O. Ka ban, ~I. Ma el-B a o, **and J. M. Nelson School o Physics and Space Resea ch, Uni e si y o Bi mingham, Edgbas on, Bi mingham B152TT, England K. Rusek Sol an Ins i u e o Nuclea S udies, Zaklad I, Hoza 69, 00 68I Wa saw, Poland J. Gomez-Camacho Depa men o de FAMN, Facul ad de Fisicas, Uni e sidad de Se illa, Ap do. 1065, 41080 Se illa, Spain (Recei ed 5July 1995) Di e en ial c oss sec ions and Tzo and T20 analyzing powe s ha e been measu ed o 70 MeV Li b eakup in o he apa icle plus i on channel, on a'Sn a ge . Measu emen s we e made o bo h con inuum b eakup and sequen ial b eakup ia he 4.63 MeV s a e in Li. The T20 da a o he con inuum b eakup do no ag ee wi h asemiclassical Coulomb model, indica ing ha he b eakup a small angles does no p oceed solely ia a Coulomb o ce. The da a gene ally show asomewha be e ag eemen wi h con inuum disc e ized coupled channels calcula ions, indica ing he impo ance o he nuclea o ce and channel coupling in he eac ion mechanism. PACS numbe (s): 25.70.Mn, 24.10.Eq, 24.70.+s I.INTRODUCTION The nplus i on clus e s uc u e o Li esul s in ala ge b eakup yield o hese agmen s. Ade ailed s udy o he b eakup is o in e es because wo mechanisms ha e been obse ed [1— 3], sequen ial b eakup ollowing exci a ion o he Li, in pa icula o he 4.63MeV 7/2 s a e, and b eakup in o he apa icle plus i on ene gy con inuum. In addi ion o he nuclea physics in e es , b eakup eac- ions may also be used o in e low ene gy pho ocap u e c oss sec ions necessa y o an unde s anding o he c ea ion o Li in he big bang s anda d model. I is di icul o ob ain he pho ocap u e da a di ec ly a he low ela i e ene gies which a e mos impo an because Coulomb epulsion be- ween he upa icle and he i on esul s in e y small c oss sec ions. ACoulomb b eakup eac ion may be used ins ead, bu i is essen ial o de e mine he impo ance o he nuclea o ce in he eac ion mechanism, as any con ibu ion om i may in alida e he in e ence o pho ocap u e c oss sec ions P esen add ess: CERN, CH-1211, Gene a, Swi ze land. ~P esen add ess: Depa men o Physics, Uni e si y o Glasgow, Glasgow, UK. ~P esen add ess: Gene al Acciden Insu ance, Pe h, UK. ~P esen add ess: Depa men o Physics, Uni e si y o Li e pool, Li e pool, UK. P esen add ess: Depa men o Physics, Uni e si y o SheNeld, She ield, UK. ~P esen add ess: Velice 26, 37351 D i en Czech Republic. P esen add ess: Depa men o de FAMN, Facul ad de Fisicas, Uni e sidad de Se illa, Ap do. 1065, 41080 Se illa, Spain. The con inuum b eakup yield is s ong a o wa d angles and alls o apidly a la ge angles [2]. The con inuum b eakup can be explained by he di e en ial s ong nuclea o ce be ween he a ge and agmen s [5], in an npa icle plus i on clus e desc ip ion o Li. Howe e , he Coulomb o ce becomes impo an a ex eme o wa d angles [6]. Di e en ial c oss sec ions o he Li con inuum b eakup on a'Sn a ge ha e p e iously been measu ed and ound o ag ee a small angles wi h acalcula ion which assumes a pu e Coulomb o ce [2,3,6,7]. The c oss sec ions a e calcu- la ed using asemiclassical app oxima ion [8] in which he mo ion o he p ojec ile along ahype bolic ajec o y is pa- ame ized in he amewo k o i s o de pe u ba ion heo y by dimensionless o bi al in eg als. In o de o de e mine he di e en ial c oss sec ion he educed ansi ion p obabili y as a unc ion o ela i e ene gy is equi ed. This may be ob- ained [9] om da a on he in e se usion eac ion [10].Thus he semiclassical Coulomb calcula ions o di e en ial c oss sec ion equi e inpu based on o he expe imen al da a. Semi- classical calcula ions o analyzing powe s, howe e , do no ely on addi ional expe imen al da a. This is because he e- duced ansi ion p obabili y is independen o spin subs a e and he e o e cancels. Semiclassical Coulomb calcula ions ha e been success ully applied o Li elas ic and quasielas ic sca e ing Tzo analyzing powe s [11].In he cu en wo k semiclassical Coulomb calcual ions a e de eloped o b eakup analyzing powe s. The analyzing powe s may be mo e sensi i e o con ibu ions om di e en o ces han he di e en ial c oss sec ion because hey depend on in e e - ence e ms be ween sca e ing ampli udes. Acompa ison o calcula ions wi h he da a he e o e p o ides an addi ional es o he impo ance o he di e en o ces. In pa icula , i he Coulomb o ce domina es, de ia ion o he con inuum 0556-2813/95/52(6)/3201(11)/$06. 00 3201 0& 1995 The Ame ican Physical Socie y 3202 N. J. DAVIS e al. 52 b eakup da a om he semiclassical Coulomb calcula ions could occu o e en asmall con ibu ion om he nuclea o ce in he b eakup eac ion mechanism. Con inuum disc e ized coupled channels (CDCC) calcula- ions, which include anuclea o ce, may also be pe o med o compa e wi h he da a. Such calcula ions ha e been ound o p o ide a e y good desc ip ion o c oss sec ions and ana- lyzing powe s o elas ic and inelas ic sca e ing o Li [12— 14].CDCC calcula ions, in ol ing anuclea o ce only, ha e p e iously been compa ed wi h di e en ial c oss sec ions o Li con inuum and sequen ial b eakup on a'Sn a ge [12]. Reasonable ag eemen was ob ained excep a he o wa d angles. An in es iga ion o he e ec o he Coulomb o ce on he sequen ial b eakup was pe o med by including a Coulomb in e ac ion de i ed by olding he Coulomb po en- ial be ween anucleon and he a ge wi h he ansi ion den- si ies be ween Li s a es. The Coulomb o ce was ound o con ibu e signi ican ly a o wa d angles. Asimila in es i- ga ion o he e ec o he Coulomb o ce on he con inuum b eakup was no , howe e , pe o med. In he cu en wo k CDCC calcula ions a e applied o b eakup analyzing powe s, p o iding a e y impo an new es o he CDCC app oach. Any b eakup p ocess esul ing in wo agmen s is a h ee-body eac ion wi h wo eac ion planes de ined by he inciden pa icle and he de ec ed agmen s. Consequen ly he choice o coo dina e sys em is gene ally no ob ious, al hough i is impo an o he heo e ical in e p e a ion o he da a ha i be as simple as possible, pa icula ly when pola iza ion e ec s a e in ol ed. Clea ly he simples si ua- ion co esponds o an expe imen al a angemen when he wo eac ion planes nea ly coincide (see Sec. IV) and as an- da d wo-body eac ion coo dina e sys em can be de ined. I should be explained ha in he semiclassical and CDCC cal- cula ions he unde lying h ee-body eac ion is in e p e ed by means o a wo-body eac ion. In o de o do his assump- ions abou he basic eac ion mechanism ha e o be made. These a e ha he inciden Li wi h spin s=3/2 and p ojec- ion nis exci ed o acon inuum o esonan s a e cha ac e - ised by spin s'and p ojec ion yand ha his decays spa ially in o an npa icle and a i on wi h a ela i e angula mo- men um L, whe e L+1/2=s', and asubs a e mz, whe e mz+m, =yand m, is he i on spin subs a e. Only in he case o J=0, when he b eakup is iso opic, can he analyz- ing powe s calcula ed by ei he inelas ic sca e ing heo y be compa ed di ec ly wi h he expe imen al da a. Gene ally, he measu ed analyzing powe s ha e o be compa ed wi h p e- dic ed quan i ies in ol ing pola iza ion ans e coe icien s and co ela ion unc ions (see Sec. III). The aim o he cu en wo k is ameasu emen o he Tpp and Tzp analyzing powe s o Li b eakup, in o de o de e mine he impo ance o he Coulomb and nuclea o ces in he con inuum b eakup eac ion mechanism and in es i- ga e he applicabili y o semiclassical and CDCC calcula- ions. II.SEMICLASSICAL COULOMB CALCULATIONS Tkq (s s Xp~ssk( — 1)' ~.p,slF,sPI' F~P*F~'P y6 y6 whe e n' =u+ qand he usual no a ion k= /2k+ 1is used. Fo aspin ze o a ge and esidual nucleus, his exp ession is simpli ied because P= 8=0. I he mo ion and Coulomb in e ac ion be ween he p o- jec ile and a ge can be desc ibed semiclassically, i may be pa ame ized in he amewo k o i s o de pe u ba ion heo y by dimensionless o bi al in eg als Rz (H, j) [8].The po en ial used in he ime-dependen Sch odinge equa ion o he pa icle mo ion may be expanded in e ms o compo- nen s o di e en mul ipoles kwi h p ojec ions p, .The o - bi al in eg als a e ela ed o he coe icien s o his expansion and depend only on he sca e ing angle 0and he dimen- sionless adiaba ici y pa ame e j.The la e depends on he ene gy Fo he ansi ion and he beam ene gy Ez i bo h a e in uni s o MeV by ZZA F( A~ 1+— I, 12.7E~' (A, ]'(2) whe e Zis he a omic numbe , Ais he mass numbe , and he su ixes pand indica e p ojec ile and a ge , espec i ely. Using he semiclassical app oach, he sca e ing ampli ude o Coulomb exci a ion o as a e wi h spin s', may be ex- p essed in e ms o he o bi al in eg als by [16] (sZs') F=— i(— 1)' kX, ,R„(~.C) (3) ( whe e y, ,is he s eng h pa ame e o mul ipole exci a- ion o o de k o a inal ejec ile s a e o spin s'. The a ge emains in i s g ound s a e. Subs i u ing Eq. (3) in o Eq. (1) gi es A. Gene al o malism Analyzing powe s may be exp essed in e ms o sca e ing ampli udes Fps whe e pand 6a e he spin subs a es o he a ge and esidual nucleus. The analyzing powe s a e gi en by [15] (s P. skX „(— 1)' (a+q k~( sZs' ~( s' sR~ R), a— qp, +q a— p) TENSOR ANALYZING POWERS FOR Li BREAKUP 3203 whe e p'=/L+q and he (O, j) dependence o he o bi al in eg als has been omi ed o he sake o b e i y. The s eng h pa ame e is independen o spin subs a e and he e- o e cancels in he exp ession o he analyzing powe s. The analyzing powe s o aCoulomb p ocess consequen ly de- pend only on he ini ial and inal spins, he mul ipole o he ansi ion, and he o bi al in eg als. Coo dina e Sys em A: B.Applica ion o Li con inuum b eakup Fo applica ion o he calcula ions o he con inuum b eakup o Li in o he upa icle plus i on channel, he ela i e mo ion o he agmen s mus be assumed o ha e a single angula momen um. The mos signi ican Coulomb mul ipole e m leading o b eakup is an E1 ansi ion, co e- sponding o L=O and leading o an npa icle plus i on inal s a e which has spin pa i y 1/2+. The e o e s=3/2, s'=1/2, and X=1. Using Eq. (4) he second ank analyzing powe s a e calcula ed in e ms o he o bi al in eg als as Coo dina e Sys em H: in in ou IR»l'+ IRi-il' — 2IRio ' "2(IR»l'+IRi- l'+IRiol') 'FIG. 1. Coo dina e sys ems Aand Hused in he semiclassical Coulomb calcula ions o analyzing powe s. +3(R ~R o — R ]R~p) 2(IRzz +IR& — l +IRiol )'(6) powe in coo dina e sys em Hmay be exp essed in e ms o he second ank analyzing powe s in coo dina e sys em Aas 3R )R) P~(IR»l'+ IR1—1I'+IR101 ) In o de o p oceed u he wi h he analyzing powe cal- cula ions, exp essions mus be ound o he o bi al in eg als; hese a e coo dina e sys em dependen . The helici y coo di- na e sys em [17],H, was used o he T2p measu emen s. Howe e , o bi al in eg als a e mo e eadily calcula ed in a coo dina e sys em [8],A, wi h he zaxis no mal o he eac- ion plane and he xaxis bisec ing inciden , k;„, and ou go- ing, k,„„wa e ec o s. The coo dina e sys ems Aand Ha e shown in Fig. 1.Analyzing powe s in Hmay be de e mined by i s calcula ing hem in Aand hen o a ing o H. The o bi al in eg als in Acan be exp essed in e ms o Coulomb exci a ion unc ions [8] Ix, which a e eal and coo dina e sys em independen , such ha Tk q=( — 1)~Tk*, (13) esul s in AA 2— 2220 (14) and consequen ly Eq. (12) becomes — [T2p+ +6T22cos( 8)] T20 2(15) whe e he o a ion ma ix elemen s D o~ (( +8)/2, /2, 37 /2) a e unc ions o he Eule angles [19] o he o a ion. I should be no ed ha he gene al p ope y o ana- lyzing powe s [15], +Ii=i R) 2+ ' Ri0— 0. (9) Subs i u ing Eqs. (10) and (11)in o Eq. (15) gi es 6I),I) ) cos( 8)— I, — I 22 To4(I2 +I2 )(16) Subs i u ing Eqs. (8) and (9) in o Eqs. (5) and (7) gi es sec- ond ank analyzing powe s in coo dina e sys em A: A T20 2' (10) — +3I»Ii T22= +2(I ( I+, ) Analyzing powe s a e sphe ical enso s which ans o m in a simple manne [18]unde o a ions. Thus he T2p analyzing T20 is simply T20 e e ed o an axis no mal o he eac- ion plane [20]. Consequen ly T2p= T2p, and so T2p is gi en by Eq. (10). In o de o calcula e angula dis ibu ions o T20 he Cou- lomb exci a ion unc ions mus be calcula ed. This canno be done analy ically bu may be done nume ically [21].Values o he adiaba ici y pa ame e , de e mined om Eq. (2), a e used in he calcula ions. Fo 70MeV Li beam on a'Sn a ge , Z„=3,Z, =50, A~=7, A, =120, and E~=70 in Eq. (2). Emay ake a ange o alues om 2.47, he b eakup h eshold in MeV, upwa ds, and so calcula ions may be pe - o med using di e en adiaba ici y pa ame e s co esponding 3204 N. J. DAVIS e al. 52 TABLE I. Woods-Saxon op ical po en ial pa ame e s used in he CDCC calcula ions. and aindica e adius and di useness pa ame e s espec i ely and suNces , i, and Cindica e eal, imagina y, and Cou- lomb po en ials, espec i ely. Vis he eal po en ial dep h and Wand Wd a e imagina y and de i a i e imagina y po en ials, espec i ely. +120S a + 120S b Li= + o. (MeV) 58.0 145.0 90' 1.46 1.25 1.39 a„ 0.708 0.690 0.700 W (MeV) 28.0 0.0 Wd (MeV) 0.0 27.5 ( m) 1.46 1~11 a; 0.708 0.920 ( m) 1.4 1.4 1.3 'C. M. Pe cy and F. G. Pe cy, A . Da a Nucl. Da a Tables 17, 1(1976). "R. P. Wa d and P. R. Hayes, A . Da a Nucl. Da a Tables 49, 315 (1991). 'Fo bound s a es he po en ial dep h was adjus ed in o de o ob ain he co ec binding ene gy. o di e en agmen ela i e ene gies. The analyzing powe s a e hen de e mined by using he calcula ed Coulomb exci- a ion unc ions in Eq. (16). III. CONTINUUM DISCRETIXED COUPLED CHANNELS CALCULATIONS The CDCC calcula ions we e pe o med using he com- pu e code FREsco [22]. Clus e olding po en ials we e in- co po a ed. The Coulomb as well as he nuclea in e ac ion was included in he CDCC calcula ions. Bo h diagonal po- en ials and coupling in e ac ions we e aken in o accoun . The nuclea and Coulomb po en ials we e ob ained using he same me hod, by olding po en ials be ween upa icle clus- e and a ge and be ween i on clus e and a ge . The po- en ials o he channel coupling a e de i ed om empi ical op ical model po en ials o a'Sn a ge . These a e Woods- Saxon in shape and a e lis ed in Table I.Ideally he po en ials should be o 40MeV npa icles [23], as is used, and 30MeV i ons, o co espond o beam eloci y agmen s om 70 MeV Li b eakup. Ho~e e a20 MeV i on po en- ial [24] was adop ed because none was a ailable a 30 MeV. The Li 3/2 g ound s a e and 1/2 i s exci ed s a e clus e wa e unc ions we e calcula ed in aWoods-Saxon po en ial well ha ing geome y pa ame e s as lis ed in Table I. These we e chosen so as o ep oduce he empi ical alue o he educed ansi ion p obabili y B(E2;3/2 ~1/2 ) [25].The wa e unc ion o he 7/2 second exci ed s a e was calcula ed using aweak binding ene gy app oxima ion [26], in which a e y small binding ene gy is assumed o his s a e. The 5/2 hi d exci ed s a e was ea ed as an ene gy bin o 5MeV wid h. Tes calcula ions we e pe o med in o de o es ablish he bes disc e iza ion and unca ion o he model space. The Li con inuum was ini ially disc e ized as in ap e ious CDCC s udy by Saku agi e al. [12],wi h alues o he ela- i e o bi al angula momen um L o he a+ clus e s lim- i ed o L= 1,3. The es calcula ions we e pe o med o he Li+ Pb sys em since expe imen al da a o he elas ic, inelas ic, and b eakup channels exis o his sys em [1,6]. Fou channel calcula ions, including he g ound s a e and he i s h ee exci ed s a es, p oduced an angula dis ibu ion o he di e en ial c oss sec ion o he 7/2 s a e simila in shape o he calcula ions pe o med by Saku agi e al. [12], bu o abou 30% lowe magni ude. This may be because he la e we e pe o med wi h double olding po en ials. The inclusion o he non esonan con inuum educed his by less han 10%. The es calcula ions e ealed ha he in luence o he wa e numbe k=(0.75— 1.00) m 'bin on he inal esul s is e y small and ha he model space can in ac be limi ed o he ange k=(0.25 — 0.75) m '. The esul s o he low- es bin k=(0.00— 0.25) m 'unde es ima ed he expe imen- al alues [1] especially a he mos o wa d angles. This ene gy ange was domina ed by he bin wi h L= 1and spin pa i y I=3/2 .Saku agi e al. [12]a gued ha he coupling o he e en Lb eakup s a es mus be much weake han o he odd Ls a es and he e o e he L alues can be es ic ed o L= 1,3. In o de o in es iga e his u he , he L=0bins we e included in he calcula ions. I was ound ha , al hough he esul s o he elas ic and inelas ic sca e ing and o he 7/2 s a e we e no a ec ed by he inclusion o he L=O s a es, he c oss sec ion calcuIa ed o he lowes L=O, k=(0.00— 0.25) m ', con inuum bin was la ge a sca e ing angles anging om 10' o 35'. I o e es ima ed he mea- su ed [I] alues. The la ge c oss sec ion o he L=0bin is in ag eemen wi h adia i e cap u e s udies [27] which e- ealed ha F.1con ibu ions domina e. The same was shown in an analysis o 63 MeV b eakup da a [28].I was he e o e conside ed impo an o include e en Lb eakup s a es in he CDCC calcula ions, especially because he o wa d angle da a a e o pa icula in e es . The a gumen by Saku agi e al. [12] o neglec ing e en L elies on a h ee-body model o npa icle, i on, and a ge in which he npa icle and i on po en ials wi h espec o he a ge a e simila . In he p esen s udy he upa icle and i on op ical po en ials used a e e y di e en , which may explain he la ge L=O c oss sec ion. CDCC calcula ions o 70MeV Li b eakup on 'Sn we e pe o med using he model space shown in Fig. 2. The wid h o he lowes bin was se o 0.38 m ' o co espond o he con inuum b eakup da a measu ed in he expe imen . The model space was unca ed o amaximum o 0.8 m 'and o L=0,1,2,3. Fo L=2, only he lowes k=(0.0— 0.38) m bin was aken in o accoun , in o de o educe he numbe o channels. This limi a ion is no expec ed o signi ican ly a - ec he analysis, since calcula ed con inuum c oss sec ions we e gene ally ound o dec ease wi h inc easing ela i e en- e gy. Fo acompa ison wi h he con inuum b eakup da a i was necessa y o sum he con ibu ions o he calcula ions 52 TENSOR ANALYZING POWERS FOR Li BREAKUP 3205 k( m-& ) 0.8 0.75-- 6.08 6.08 Ene gy ela i e o 7Li~e+ b eakup h eshold (Me ) whe e he A~=(kcm. s'ylAlk s~) (18) 0.6 0.5-- 3.16 -—3.16 -—3.16 0.4 0.38 4.94 2.16 4.21 a e ampli udes calcula ed by he FRESCO code, s and m, a e he spin and p ojec ion o he i on, k, k, ,and k; a e he upa icle, i on, and inciden momen a, espec i ely, Ais an app op ia e o m ac o , and 0.86 0.86 0.86 0.86 0.86 0.86 0.86 c.m. ka+ ~ - (19) Equa ion (17) may be w i en in e ms o asum o e sphe i- cal ha monics o he b eakup angula momen um L=3, p o- jec ion mz.' (k,k, ,s,m, iA ik, ;su) L: 1+ 2 -2.47 3 2 -1.99 I 2 5+ 2 3+ 2 5 2whe e =g Ag(s,m,Lmiis'y)YI (k„), ym~ 4k, — 3k k„= (20) (21) FIG. 2. The disc e iza ion o he Li=a+ b eakup con inuum used in he coupled channels calcula ions. o di e en spin pa i y alues o ob ain o als o he di e - en ial c oss sec ion and analyzing powe s. An incohe en summa ion o e he con ibu ions was aken and analyzing powe con ibu ions we e weigh ed by he co esponding di - e en ial c oss sec ions. In o de o in es iga e he oles o he nuclea and Coulomb o ces in he CDCC calcula ions, addi ional calcula ions we e pe o med wi h anuclea in e - ac ion only. I was ound ha , subjec o limi a ions on he nume ical in eg a ions o he coupled equa ions o 40 m and on he pa ial wa es o 150A,, he esul s we e e y simila , indica ing adominan e ec o he nuclea in e ac ion. To achie e aCDCC esul o consis en compa ison wi h he measu ed analyzing powe s, some conside a ion has o be made o he phase space de ec ed in he expe imen . Fo he con inuum b eakup, he la ge con ibu ion om L=O esul s in he phase space ha ing li le e ec on he calcula- ions; hence, esul s di ec om he FRESCO code a e com- pa ed wi h he da a. Fo he L=3sequen ial b eakup, phase space is a mo e impo an . In o de o ake accoun o his, conside he sca e ing ampli ude (k,k, ,s,m, iAik;;su) =g(k,k, ,s,m, ik, ,s' y)A (17) This ela i e momen um aken as i on ela i e o c pa icle could equally well be aken as apa icle ela i e o i on. In o de o calcula e app op ia e analyzing powe s om he sca e ing ampli udes, he coincidence de ec ion p obabil- i y (0„,$„) o agmen s wi h ela i e momen um di ec- ion gi en by sphe ical pola coo dina es (0„,@„)in he he- lici y coo dina e sys em Hneeds o be inco po a ed. This p obabili y unc ion was calcula ed using aMon e Ca lo simula ion code [29]in which he collima o posi ions o he coincidence de ec ion we e de ined. Li nuclei we e exci ed o he 4.63 MeV s a e wi h aLo en zian dis ibu ion and we e sca e ed iso opically o e asolid angle la ge enough o include he collima o s. The p obabili y unc ion (8„,P„) was ex ac ed om he Mon e Ca lo calcula ion in he o m o a wo-dimensional spec um o numbe o coun s e sus O„and P„, o which he apa icle and i on we e de ec ed in coincidence. Fo his spec um he O„ ange was di ided in o 5' bins and he P„ ange was di ided in o 10' bins. (0„,$„) is an e en unc ion o @„because o he symme y o he coincidence de ec ion sys em abo e and below he plane in which he sca e ing angle is de ined. The double di e en ial c oss sec ion do.(k, ,k„)/ dA, md'„and analyZing POWe S Tkq CO eSPOnding O haV- ing he cen e o mass momen um o he upa icle plus i on in asolid angle 0, and he ela i e momen um in a solid angle A„a e gi en by dc (k, ,k„) 1Lk' (2s+ 1)TI,q+XI,qqiqis L dQ, dA„4, ,q' qI00 L) 0) W(LLs's';k's, )Yg q(k„), (22) whe e he pola iza ion ans e coe icien s a e gi en by Xlq ~q=g (s~&qis~')&(s'y&'q'is'y')&'AP (23) 3206 N. J. DAVIS e al. Howe e , he expe imen does no esol e in de ail he double di e en ial c oss sec ion. Ins ead, hose ele an e en s whe e an upa icle and a i on a e de ec ed in coincidence a e coun ed as sequen ial b eakup (SBU), which happens wi h he p obabili y (8„,P„).Thus (2s+ 1)T c.m. 2do.(k, ,k„) (9„,@„)(2s+1)Tk 'sing„d g„d@„ 0@~=0J8„=0 c.m. 1.,I. gXlq ks'L W(LLs's';k's, )Ik (24) whe e (2m m co ding o Eq. (23). The analyzing powe s a e hen calcu- la ed om Eq. (24), no malizing such ha Tpp= 1. I„= Yl,«(9„,@„) (8„,P„)sing„d O„d@„. qb„=0 39„=p (25) 1.0 0.5 0.0 -0 5" /j' /:/ /:/ 'J' 20 1.0 05" 0.0- — 0.5 — 3.0IIIII 030 60 90 120 150 180 8„(deg ees) FIG. 3. CDCC calcula ed analyzing powe s as a unc ion o cen e o mass ela i e momen um di ec ion, o 15 labo a o y sca e ing angle. The solid, dashed, do dashed, and do ed cu es a e o P„= 0, 30', 60', and 90', espec i ely. This in eg al is necessa y because o he sensi i i y o he calcula ed analyzing powe s o he angles O„and P„.This is illus a ed in Fig. 3, which shows analyzing powe s calcu- la ed assuming single H„and P„ alues o alabo a o y sca - e ing angle o 15'. Iki ~ educes o a eal numbe since (8„,$„) is e en in P„and he in eg a ion is aken o e 2q in P„. In o de o calcula e he analyzing powe s, he in eg als Ik qa e calcula ed acco ding o Eq. (25) using he p obabili y unc ion (0„,P„) ob ained om he Mon e Ca lo calcula ion and he pola iza ion ans e coe icien s Xkq p q&a e calcula ed om he FREsm ampli udes Ayac- IV. EXPERIMENT The expe imen was pe o med using 70 MeV pola ized 7Li beams om he pola ized hea y ion sou ce [30] and accele a ed by he andem Van de G aa accele a o , a he Nuclea S uc u e Facili y a Da esbu y Labo a o y in he UK. FOI he T2o measu emen s pola iza ion o he beam was achie ed using ansi ions be ween he 2-8 and 4-6 hype ine a omic le els [31]in amagne ic ield, esul ing in s a es wi h equal magni ude bu opposi e signs o he enso pola iza ion and equal odd ank pola iza ions. This equali y was e i ied by compa ison wi h an unpola ized beam. The T20 measu emen s we e made a ala e da e when op ical pumping [32] was a ailable o pola iza ion o he beam, doubling he heo e ical maximum pola iza ion ob ainable. Ions wi h each o he ou spin subs a es we e selec ed in u n, using ahigh equency ansi ion in amagne ic ield o swi ch be ween subs a es. Fo bo h T20 and T20 measu e- men s, he pola iza ion s a es we e swi ched e e y ew sec- onds, a e aspeci ied in eg a ed beam cu en was measu ed, o minimize sys ema ic e o s due o beam d i o pola iza- ion luc ua ions. AWien il e was used o o ien he pola - iza ion symme y axis along he beam di ec ion a he a ge o he T2O measu emen s and no mal o aplane bisec ing agmen coincidence de ec o cen e s o he T2o measu e- men s. The beam pola iza ion was de e mined om he 'H( Li,n) He eac ion [16] using apu pose buil down- s eam pola ime e [33].The measu ed magni udes o second ank beam pola iza ions we e ypically 2o= 0.4 o he T20 measu emen s and 20=0.6 o he T20 measu emen s. Measu emen s o i s and hi d ank pola iza ions in he op- ically pumped beam esul ed in magni udes no la ge han 0.05 each. Fo he b eakup eac ions a2mg cm 'Sn a ge was used. The yield o con inuum b eakup o Li in o an n pa icle and a i on is la ge o small ela i e ene gies o he agmen s [2].Fo his eason agmen s wi h asmall open- ing angle mus be de ec ed. The de ec ion sys em consis ed o wo pai s o AF. Ede ec o elescopes placed symme i- cally, one pai ei he side o he beam, as shown in Fig. 4. The symme ic a angemen was used so ha da a om bo h sides o he beam could be summed, hus elimina ing sys- ema ic e o s a ising om any shi in posi ion o he beam on a ge and he e ec s o odd ank pola iza ion componen s in he beam. The elescopes comp ised o 230 p,m hick p-n TENSOR ANALYZING POWERS FOR Li BREAKUP 3207 Plan iew 120 S a ge 7Li beam Side iew agmen s e' 6 agmen s AE Ve o Pai o de ec o elescopes Pai o de ec o elescopes Ve o Ti(H) pola ime e a ge Pola ime e ~de ec o elescope I I AE E Ta beam s op Coun s 40 30" 20" 0 0.00.51.01.52.02.53.0 Rela i e ene gy (MeV) l20 S a ge 7Li beam EVe o FIG. 4. De ec o sys em. FIG. 6. F agmen ela i e ene gy spec um o he 'Sn(7Li, n ) 'oSng, eac ion a alabo a o y angle o 15'. b eakup ia he 4.63 MeV s a e in Li and he da a be ween he peaks a ise om con inuum b eakup [2]. The ela i e ene gy ebe ween he i on and npa icle may be calcula ed acco ding o V. RESULTS The as coincidence da a we e ga ed on he g ound s a e o 'Sn by summing i on and npa icle ene gies. An en- e gy esolu ion o 0.4MeV was achie ed. Acoincidence spec um o i on ene gy is shown in Fig. 5. The sha p peaks co espond o he wo kinema ic solu ions o sequen ial Coun s 50 40" 30" 20" 10" ~ ga a I L. . 10 20 30 40 Ene gy (MeV} FIG. 5. Ene gy spec um o i on s om he Sn( Li, ) 'Sng, eac ion a alabo a o y angle o 15'. junc ion silicon AE de ec o s and 4mm hick li hium d i ed silicon Ede ec o s. Simila de ec o s placed behind he E de ec o s ac ed as e os o elimina e high ene gy cha ge 1 pa icles which pass h ough he Ede ec o s. The de ec o collima o s we e 8mm wide and 6mm high wi h he cen e s o agi en pai 12 mm apa . The de ec o s in each pai o elescopes we e moun ed symme ically abo e and below he beam axis and 150mm om he a ge . The de ec o s we e ene gy calib a ed using 5.486MeV npa icles om 'Am sou ces moun ed close o he de ec- o s. Pa icle iden i ica ion was achie ed using he AE and E signals. Fas iming was achie ed by signals gene a ed om he AE p eampli ie s, used o s a and s op a ime o ampli- ude con e e o each pai o elescopes. Da a we e ans- mi ed om analogue o digi al con e e s o aGEC 4190 compu e and eco ded e en by e en on ape. T2p and "T2p da a we e ob ained o a ange o angles o he beam di ec ion, 8, shown in Fig. 4, om 9 o 25 in he labo a o y ame. 4E,+3E 4$3E— E cos(@) 7(26) whe e E, and Ea e he i on and npa icle ene gies e- spec i ely and @is he angle be ween hem. Figu e 6shows a ela i e ene gy spec um, whe e Phas been aken o be he angle be ween he de ec o collima o cen es. The peak is om he sequen ial b eakup and he da a a lowe ela i e ene gies a e om con inuum b eakup. Yields we e ob ained o each pola iza ion s a e o he beam. This enabled da a o he con inuum b eakup o be ob ained om a h eshold o 100keV o 1.7MeV in ela i e ene gy. The measu ed analyzing powe s o he 'oSn( Li,n ) 'Sng, con inuum b eakup a e compa ed wi h semiclassical Coulomb and CDCC calcula ions in Fig. 7.The small ela i e ene gy dependence o he semiclassical Cou- lomb calcula ions o T2p is shown by he dashed cu e which co esponds o ze o ela i e ene gy and he do ed cu e which co esponds o a ela i e ene gy o 2.16 MeV, equi a- len o he alue o sequen ial b eakup ia he 4.63 MeV s a e in Li. The co esponding calcula ion o T2p is no ela i e ene gy dependen and is shown by he dashed cu e. The "T2p da a ag ee well wi h he semiclassical calcula ion a small angles bu de ia e om i a he la ge angles. The T2p da a clea ly do no ag ee wi h hese calcula ions. The angula end o he da a opposes ha o he calcula ions and la ge magni ude T2p alues a e measu ed, in disag eemen wi h he small magni udes p edic ed by he calcula ions. This esul is in con as wi h he esul s o di e en ial c oss sec- ion [2,3,6] measu emen s which indica ed ha he Coulomb o ce is esponsible o he con inuum b eakup a small angles. Analyzing powe s a e mo e sensi i e han di e en ial c oss sec ions o he con ibu ions o di e en o ces in he eac ion mechanism and hus will be a ec ed by anuclea o ce p esen in he eac ion mechanism e en i he Coulomb o ce also plays ala ge ole. The ex en o he disag eemen be ween he T2p da a and he calcula ions indica es, howe e , ha he nuclea con ibu ion o he con inuum b eakup is signi ican . The T2p measu emen p o ides amo e igo ous es o he semiclassical Coulomb model han he T2p mea- su emen . This is because he calcula ion o T2p depends di- 3208 N. J. DAVIS e al. 52 20 1.0 0.5 T~ Q4 20 0.2 0.00.0 20 1.0 05" — 0.5" 20 — 0.2" 0.2 01I~ L=O, I=1/2 L=2, I=3/2 L=2, I=5/2 — 0.5'— 0.1" 0510 15 20 25 30 Cen e o mass angle (deg ees) 20 — 0.2 0.4510 15 20 25 30 FIG. 7. Tzo and Tqo o 'Sn( Li,n ) 'Sns, con inuum b eakup o e~ 1.7MeV. The dashed cu es show semiclassical cal- cula ions o Tzp and o Tpp wi h a=0 MeV and he do ed cu e shows he semiclassical calcula ion o Tzp wi h e=2.16MeV. The solid cu es show he CDCC calcula ions wi h all L alues om 0 o 3included. ec ly on he Coulomb exci a ion unc ions whe eas he cal- cula ion o Tzp does no . The semiclassical calcula ion o Tpp only depends on he assump ion o mul ipola i y 1, which is good o aCoulomb o ce bu is no necessa ily so o anuclea o ce. Ne e heless, Tpp can no dis inguish he mul ipola i y 1componen o anuclea o ce om aCou- lomb o ce. The Tzp da a hus expose he disag eemen wi h he semiclassical Coulomb model be e han he "Tzp da a. I aCoulomb o ce we e esponsible o he con inuum b eakup asmall de ia ion o he da a om he calcula ions migh be expec ed due o he app oxima ions used in he semiclassical model. Howe e , he ex en o he de ia ion o Tpp indica es ha he disag eemen is mo e undamen al and asigni ican nuclea o ce con ibu ion o he con inuum b eakup eac ion mechanism is he e o e expec ed o be p esen . The poo ag eemen o he Tzp da a wi h he semi- classical Coulomb calcula ions, obse ed oge he wi h he good ag eemen o he Tpp da a wi h he calcula ions a small angles sugges s ha he nuclea o ce con ibu ion o he con inuum b eakup eac ion mechanism is no only sig- ni ican , bu also has ala ge mul ipola i y 1componen . CDCC calcula ions o ~oSn(7Li, n ) 'oSn, con inuum b eakup analyzing powe s a e shown o he di e en Lcon- ibu ions in Fig. 8. The L=O con ibu ions a e he mos impo an , because o he la ge associa ed di e en ial c oss sec ion, wi h he L=2and L=3con ibu ions insigni ican . The L=0 componen o he CDCC calcula ion o Tpp akes alues be ween 0.4and 0.6o e an angula ange o 0'— 25, and so is e y simila o he semiclassical calcula- ion which gi es a alue o 0.5. The L=1con ibu ions ha e — 0.2" 0.2 20 0.1' L=1, I=1/2 L=1, I=3/2 L=3, I=5/2 L=3, I=7/2 0.0 — 0.1" R 0 5 10 15 20 25 30 Cen e o mass angle (deg ees) FIG. 8. CDCC calcula ions o Tzp and T~p o Sn( Li,n ) 'Sns, con inuum b eakup, o di e en L alues. mos e ec o he la ge angles, 20' — 30 .The esul o incohe en ly combining all componen s o he CDCC calcu- la ion gi es he solid cu es shown in Fig. 7.Agood desc ip- ion o he Tzp da a is ob ained o e he angula ange mea- su ed, wi h he signi ican con ibu ion om he L=1 componen s a la ge angles enhancing ag eemen wi h he da a. The CDCC and semiclassical calcula ions o Tzp a e qui e simila , wi h he CDCC gene ally ali le lowe han he semiclassical esul o 0.5. Howe e , he oscilla ions in he CDCC calcula ion, which a e no p esen in he semiclassical calcula ion, imp o e he ag eemen wi h he da a. The CDCC calcula ion is wi hin e o ba s o h ee o he i e da a poin s while he semiclassical calcula ion only exhibi s his quali y 52 TENSOR ANALYZING POWERS FOR Li BREAKUP 3209 3 10 „ 10 (mbs )10'. 10 ' 20 1.0 05 0.0 Q,o. (mbs )10'; 10. 10 '" '— —— Sum o e L= 0 o 3 No malised by 0.187 L=O, l=1/2 —-—-L=2, l=3/2 20 1.0 05" 0.0-- — 0.5" 10' — 0.5 10', dQ, 10'... {mbs )10'. 10; 10'; l=1/2 l=3/2 l=5/2 l'=7/2 -1.00 5 10 15 20 25 30 Cen e o mass angle (deg ees) FIG. 10. T20 and T2O o 'Sn( Li,a ) 'Sns, sequen ial b eakup ia he Li 4.63 MeV s a e. The do ed cu es show he CDCC calcula ions ob ained di ec ly om FREsCo and he solid cu es ha e phase space e ec s included. 10 0510 15 20 25 30 Cen e o mass angle (deg ees) FIG. 9. Di e en ial c oss sec ions o 'Sn( Li,a ) 'Sns, con inuum b eakup o m~1.7MeV. The cu es show CDCC cal- cula ions o di e en L alues. o ag eemen o he smalles angle da a poin . The o al CDCC calcula ion o T2p shown in Fig. 7 ep oduces he da a a he smalle angles, la gely om he L=0con ibu ion. A he la ge angles whe e he L= 1, I"=3I2 componen has a signi ican e ec and is o opposi e sign o he da a, he ag eemen is poo . De ec ion phase space e ec s would be mo e impo an a he la ge angles bu a e imp ac ical o inco po a e in o he con inuum b eakup calcula ions. The CDCC calcula ion does, howe e , gi e be e ag eemen wi h he small angle T2p da a han he semiclassical Coulomb cal- cula ion does. The CDCC calcula ion ep oduces wi hin e - o s he end be ween he wo smalles angle da a poin s, while he semiclassical calcula ion me ely c osses his end a abou 12'. I was no he aim o he expe imen o measu e absolu e di e en ial c oss sec ions o he b eakup and no maliza ion di icul ies p e en a eliable esul . Howe e , angula dis i- bu ions o di e en ial c oss sec ions, o which he end is accu a e, we e ob ained and i is in e es ing o compa e hese wi h CDCC p edic ions. Di e en ial c oss sec ion da a o Sn( Li,a ) 'Sns, con inuum b eakup a e compa ed wi h he esul s o CDCC calcula ions in Fig. 9.I is in e es ing o no e ha L=0gi es he dominan con ibu ion o he calcu- la ed di e en ial c oss sec ion o 7— 19 .Beyond 19' he L=1, I=3)2 con ibu ion domina es. The o al CDCC cal- cula ion exhibi s e y good ag eemen wi h he da a i no - malized by a ac o o 0.187. The e a e se e al explana ions o why his no maliza ion is necessa y. The da a may ha e sys ema ic unce ain ies due o inaccu acies in he a ge hickness and in he solid angle which had o be de e mined by aMon e Ca lo calcula ion [29]in o which some assump- ion o he con inuum b eakup yield a ia ion wi h ela i e ene gy had o be inco po a ed, he esul o asemiclassical Coulomb calcula ion being used. Acalcula ion using he FRESCO code could no easonably be pe o med because o he small ene gy bins which would be equi ed. Some da a we e also los due o de ec o dead egions and elec onic h esholds. The CDCC calcula ions may no gi e he co ec absolu e magni ude o he di e en ial c oss sec ions because o hei clus e model basis. The ag eemen o he angula dis ibu ion o he CDCC calcula ions wi h he di e en ial c oss sec ion da a is, howe e , excellen . Analyzing powe da a o 'Sn( Li,n ) 'Sns, sequen- ial b eakup ia he 4.63 MeV s a e in Li a e compa ed wi h CDCC calcula ions in Fig. 10. The esul s ob ained di ec ly om he FRESCO code, shown by he do ed cu es, a e no su icien o ep oduce he da a. Howe e , when he phase space e ec s a e included, as shown by he solid cu es, he calcula ions exhibi excellen ag eemen wi h he da a, wi h he excep ion o he p edic ed 13 maximum in T2p which is no seen in he da a. This is mos p obably because he da a a e smea ed ali le o e a ange o angles because o ini e de ec o sizes. The p edic ed maximum in T2p co esponds o aminimum in he di e en ial c oss sec ion, as shown in Fig. 11.This means such smea ing will ha e ala ge e ec in his case, b inging he measu ed T2p owa ds alues o nea by angles.