Dynamical polarization potential due to the excitation of collective states

PHYSICAL REVIEW CVOLUME 44, NUMBER 6DECEMBER 1991 Dynamical pola iza ion po en ial dne o he exci a ion o collec i e s a es M. V. And es, '"F.Ca a a, '''and E.G. Lanza' ' "'Depa amen o de Fisica A omica yNuclea , Uni Ue sidad de Se illa, 41080Se illa, Spain ''Dipa imen o di Fisica dell'Uni e si a, 95129Ca ania, I aly ''Is i u o Nazional'e di Fisica Nuclea e, Sezione di Ca ania, 95129Ca ania, I aly (Recei ed 21 Ma ch 1991) Wi hin he Feshbach o malism we calcula e he nucleus-nucleus dynamical pola iza ion po en ial a ising om he coupling o he elas ic channel o he collec i e ib a ional s a es, desc ibed by he andom-phase app oxima ion. Calcula ions o he sys ems '0+ Ca and Ca+ Ca show he impo - ance o he high-lying s a es. They gi e he main con ibu ion o he eal pa o he pola iza ion po en- ial a e e y inciden ene gy, while hey domina e he imagina y pa only a e y high ene gy. A low ene gy he abso p ion is gi en by he low-lying s a es. The eal and imagina y pa s o he po en ial a e shown o obey adispe sion ela ion. Calcula ions o elas ic c oss sec ion gi e agood desc ip ion o he expe imen al da a. I. INTRODUCTION The elas ic sca e ing o nuclei is well desc ibed by he op ical po en ial. I s eal pa is usually cons uc ed by means o he double- olding model [1]. The imagina y pa desc ibes he depopula ion o he elas ic channel due o i s coupling o he nonelas ic ones. This coupling gen- e a es also aco ec ion o he eal pa . Recen ly, a enewed in e es in his so-called dynamical pola ized po- en ial has been aised by he expe imen al e idence [2] o as ong ene gy dependence o bo h eal and imagina y pa s o he op ical po en ial a ene gies close o he Coulomb ba ie , which is known as h eshold anomaly. Adispe sion ela ion has been used [3,4] o ela e he s ong inc ease o he eal pa o he dec ease o he imagina y one. Thus i is in e es ing o s udy he beha - io o he pola iza ion po en ial wi h he ene gy wi hin a mic oscopic app oach. Semiclassical models ha e been p oposed in o de o calcula e he pola iza ion po en ial by including low- lying collec i e ib a ional s a es and one nucleon ans e [5], only one nucleon ans e [6], and bo h low- and high-lying collec i e ib a ional s a es [7]. Al hough he po en ials so ob ained gi e agood desc ip ion o he expe imen al elas ic c oss sec ion, ad awback o hese models is ha he pola iza ion po en ial is no cons uc - ed di ec ly bu only by making app oxima ions on quan i- ies exp essed as in eg als along classical ajec o ies. The pola iza ion po en ial can be calcula ed in acom- ple e quan um way h ough he use o he Feshbach o - malism [8]. This app oach has been exploi ed by Vinh Mau by using he closu e app oxima ion [9]. The main ad an age o his me hod is i s simplici y. One can ob- ain he pola iza ion po en ial wi hou de ailed in o ma- ion on he s uc u e o he nonelas ic channels. A he same ime, one limi a ion o he model is i s inabili y o desc ibe how speci ic channels con ibu e o he o al po- la iza ion po en ial. In his pape we calcula e, ollowing in spi i he ap- p oach o Vinh Mau, he pola iza ion po en ial aking ex- plici ly in o accoun he e ec s o he exci a ions o he collec i e ib a ional s a es, desc ibed wi hin he andom-phase app oxima ion (RPA). An analysis o he eal and imagina y pa s o he pola iza ion po en ial in e ms o he single collec i e modes shows he impo - ance o he gian quad upole esonance (GQR) s a es. They gi e as ong con ibu ion o he imagina y pa only a high inciden ene gies, while he eal pa is dom- ina ed by he high-lying s a es a all ene gies. We show ha he ene gy dependence o he eal and imagina y pa s o he pola iza ion po en ial ollows he end o he expe imen al da a and sa is ies adispe sion ela ion. Finally, calcula ions o elas ic c oss sec ion pe - o med wi h ou mic oscopic op ical po en ial gi e a good desc ip ion o he expe imen al da a. P elimina y esul s o his model ha e been p esen ed elsewhe e [10]. II. THE MODEL An elegan and anspa en way o ake in o accoun he coupling o he elas ic channel o he nonelas ic ones is h ough he use o he Feshbach o malism [8]. In his app oach he e ec i e hea y-ion in e ac ion is w i en as (R R')=(oo I.(R)loo )S(R R') +g(oolu(R)lK, K) EC lE2 &G««, (R,R')(Ic,Ic,lU(R') loo) =VF(R)+hV(R, R') . In he i s e m he nucleon-nucleon in e ac ion U(R) is double olded wi h he g ound-s a e densi ies o he wo nuclei. In he second e m he sum is o e all he none- las ic channels, and U(R) is double olded wi h he ansi- ion densi ies o he wo nuclei: i desc ibes he coupling o he elas ic channel o he nonelas ic ones. This e m is he so-called dynamical pola iza ion po en ial. I s physi- 2709 1991 The Ame ican Physical Socie y 2710 M. V. ANDRES, F.CATARA, AND E. G. LANZA cal meaning is anspa en om Eq. (1): he in e ac ion ac ing a he dis ance R' akes he sys em in o one o he elimina ed channels; hen i is p opaga ed a ano he dis- ance Rwhe e he in e ac ion, ac ing again, b ings he sys em back in o he elas ic channel. Then he pola iza- ion po en ial is nonlocal, and i one o mo e o he elim- ina ed channels is open i is also complex and i s abso p- i e pa desc ibes he loss o Aux om he elas ic chan- nel. The coupling also make b,Vene gy dependen be- cause o he appea ance o he ene gy in he p opaga o Gx. x(R,R'): Vk«)V (R') Gx x(R,R')= dkEE~ — E — i k— /2p+i i whe e E, is he cen e -o -mass inciden ene gy, EK1 and EK a e he exci a ion ene gies o nucleus 1and 2, e- 2 spec i ely, and pis he educed mass. The yk(R)'s a e he ela i e mo ion wa e unc ions o he colliding nuclei. The calcula ion o Eq. (1) is a e y di icul ask be- cause o he p esence o GK K.We calcula e he p opa- 12 ga o in he WKB app oxima ion, which has been shown o be agood app oxima ion o a-nucleus sca e ing [11] a ene gies g ea e han he Coulomb ba ie . In his ap- p oxima ion we ha e a e e ms o he o m a , =y&oolU(R)ll~, o&G~,(R,R')&I~,olU(R')loo& . K1%0 (8) The closu e app oxima ion model o Vinh Mau con- sis s o eplacing he exci a ion ene gies EK and EK by 12 a e age alues E& and E2, espec i ely. In his case he p opaga o Gdoes no depend on he pa icula exci a- ion and i can be aken ou o he summa ion o Eq. (8); hen aclosu e ela ion o e he s a es E, and K2 can be used. This app oxima ion is e y good a e y high in- ciden ene gies, while a low ene gies he alues o M(p) [Eq. (3)] can be qui e di 'e en depending on he alues o EK and EK .Fu he mo e, he e6'ec i e nucleon-nucleon 12 in e ac ion is app oxima ed by asepa able o ce. We wan o calcula e he pola iza ion po en ial a ising om he exci a ion o he collec i e ib a ional s a es and o s udy he ela i e impo ance o low-lying and high- lying ones. Thus we do no make use o he abo e ap- p oxima ions and we explici ly sum o e he ele an s a es, whose ene gies and ansi ion densi ies a e calcu- la ed mic oscopically wi hin he RPA. Wi hin he double- olding app oach he o m ac o s can be w i en as F~,(R)— =&ool U(R) lI~ &o & pexp[i'. x. (p)s] 12 G~ ~(p,s)=- 2~%2 S wi h 2p M~, x,(p) =,[E. Ex, Ex, — ~c(p— )— 1'c(p)] whe e p=-,'(R+R'), The local op ical po en ial VL (p) is gi en by (2) (3) (4) = d d 2px. O( , )U(l , — z+Rl)poo( z) . The explici exp ession o he adial pa o he ansi ion densi y pox ( &) is J'& +i/2 p,o(") g( ) (l —lg —, 'lL o) 4m X(Xpq' — Y„h' )8 ( )Rh ( ), (10) ph ph p 1.5 &L, (p) =&p(p)+ &VL(p), while Vc is he Coulomb po en ial be ween he wo nu- clei. Ap ocedu e o de ine he local pola iza ion po en- ial b,VI will be gi en la e on in his pape . In o de o make mo e anspa en he con ibu ion o he single e ms in Eq. (1), we can sepa a e he summa- ion o e K,,E2 in o h ee pa s, X=X+X+X K1 K2 Kl =0 K2%0 K1%0 K2 0K1XO K2+ which gi e ise o h ee di 'e en con ibu ions o b,V, AV=AV2+hV, +AV)2, 0.0 — 0.50 L= 1 l 0.5 cos 8 I 0.5 cos 8 L=3 I 0.5 cos 8 'i /I. I' I espec i ely. In he i s e m nucleus 1s ays in i s g ound s a e while nucleus 2is exci ed, ice e sa in he second e m, while in he hi d one bo h nuclei a e exci - ed. This las e m is expec ed o gi e asmall con ibu ion and will be neglec ed. Then wha we ha e o calcula e FIG. 1. The quan i y Al, de ined in Eq. (11),as unc ion o cosO o h ee alues o he angula momen um L. In each sec- o he h ee cu es co espond o a ixed alue o p= 8 m and o he ollowing alues o s: 1 m (solid line), 4 m (dashed line), and 8 m (do -dashed line). The esul s co esponding o odd alues o La e mul iplied by — 1. DYNAMICAL POLARIZATION POTENTIAL DUE TO THE. ..2711 whe e R~ (Rh )is he pa icle (hole) wa e unc ion; X~&' and Yza e he o wa d and backwa d RPA ampli udes K) co esponding o he mode K& whose mul ipola i y is L]. The calcula ion o b,V,z(p, s) is epo ed in he Appen- dix. Exp essions o i and o EV(p, s) a e gi en by Eqs. (AS) and (A9), espec i ely. Thei angula pa can be easily wo ked ou i we assume ha he po en ial b,Vis weakly dependen on he angle 0be ween pand s. This has been shown o be ue o nucleon-nucleus sca e ing [12]. The angula pa o b.V, can be w i en as A (p, s,x)=(+p +—, 's +psx Qp +„'s — psx— )pi/2 1/2 2L+1 2L+1 [2(L— A, )+1]'~ [2(L — A, ')+ I]' A, +A, L ()i. X,X=0 A, +A, Xg(— ) W(A, ,L— A, ,A.',LA, ',L — )(L — A, 0L— I,'Ol 0)(A, 0A,'Ol 0)P (x) . whe e x=cos8 and 8is he angle be ween pand s. Nume ical calcula ions o Eq. (11) show ha he a ia ion o A wi h 0may be e y di e en depending on he alues o pand s. In pa icula , we show in Fig. 1 he beha io o Az, o p= 8 m and s=1,4,8 m, as unc ion o cosO o h ee alues o L. We see ha a apid a ia ion wi h cosO is ound o p=s =8 m, while o small alues o s he quan i y A~ is almos independen o cosO. This ange o alues o la ge pand small sis he physically in e es ing domain, whe e he localiza ion p ocedu e ha we will desc ibe la e on also makes sense. So in his ange o alues hV is weakly dependen on he angle 8. Choosing he alue cos8=1, which co esponds o ake he maximum con ibu ion, we ha e A (p,s,x) =( '— — '') x=1 A. =O 2L+1 2P2A, (A, 0LOlL — 10)(— )~ 2L +1 A.'=0 (A,'0LOlL — ~' 0)=(— )c .(12) Then he po en ials AV&z and b.Vi can be w i en as and EV,~(p,s)—6gG»» (p, s)L E qgA»» (p,s)%»» (p, s)(L, 0L~ Ol J0— ) »1212 l~ 2J(13) EVi(p, s)= 5XG», o(P s)+»,o(P s)+» o(p~ — s)L i, 16m g1 (14) whe e %»» (p,s) is de ined in Eq. (A7) o he Appendix. The po en ial calcula ed wi h jus he desc ibed p o- cedu e is nonlocal. Bu , since i is always p e e able o use local po en ials, a leas o he calcula ion o physical obse ables, we use as anda d p ocedu e [13] o ob ain a local po en ial om anonlocal one. This p ocedu e is applicable whene e he ange o nonlocali y o b,Vis small wi h espec o i s adius. Thus, we ha e I has been shown [9] ha he hypo hesis o e he ange o nonlocali y o b,Vis alid, a leas o la ge alues o p. Indeed, in Fig. 2we show he eal and imagina y pa s o Eq. (14) as unc ions o s o wo ixed alues o p. Fo la ge alues o pi is ound ha he ange o nonlocali y is abou 1 m. This alue is consis en wi h p e ious ones gi en in he li e a u e [9,14]. Thus he hypo hesis o in- dependence o b,Von he angle be ween pand s, dis- cussed be o e, is a1so jus i ied. AV(p) =4m jo(ks )RehV(p, s)s ds, (15) III. RESULTS AND DISCUSSION 8'(p) =4m jo(ks) Imb, V.(p,s)s ds, whe e p[&c.m. VF(p) Vc(p)] . 2p g2 We ha e done calcula ions o he sys ems 'O+ Ca and Ca+ Ca a se e al inciden ene gies. The le els used in he calcula ions a e epo ed in Table I. They ha e been ob ained wi h asel -consis en RPA code using an SGII o ce [15]. We ha e included all he s a es which exhaus a leas 10% o he ene gy-weigh ed sum ule 2712 M. V. ANDRES, F.CATARA, AND E.G. LANZA 30 I 20- 150 +40C El,b=104 MeV 10- — 10- — 200s( m) 10- leO +4oc Es ab =104 MeV -100s( m) FICx. 2. Real and imagina y pa s o he nonlocal pola iza- ion po en ial AV, (p,s), o he sys em '0+ Ca a E&,b=104 MeV, as a unc ion o s o wo ixed alues o p: 7 m (dashed line) and 8 m (solid line). (EWSR). The RPA calcula ions ha e been done wi h a la ge numbe o pa icle-hole con igu a ions (-350 o he 3o "Ca) in o de o ge agood desc ip ion o he low-lying s a es [16]. Bo h ene gies and ansi ion densi- ies compa e well wi h he expe imen al da a. The e ec i e M3Y nucleon-nucleon in e ac ion has been used Nuclei 1 1 2+ 2+ 2+ 3 3 3 E* (MeV) 19.096 19.996 19.945 20.620 21.280 7.220 32.870 34.610 %EWSR 27 21.4 44 13 12.6 8.8 8.4 8.9 "Ca 1 1 1 1 2+ 3 3 16.302 16.765 17.372 17.883 18.296 18.632 16.741 4.830 31.281 9.8 15.2 10.6 10 10.6 16 80 15 13~5 TABLE I. P ope ies o he RPA s a es used in he calcula ions. in o de o cons uc he double- olding po en ial V~ o Eq. (1) and he o m ac o s o Eq. (9). In o de o s udy he in e play be ween he low-lying and high-lying s a es we ha e made an analysis in e ms o he single ib a ional collec i e s a es. In he le pa o Fig. 3we show he eal and imagina y pa s o he po- la iza ion po en ial o he sys em Ca+ Ca a E&,b=240 MeV. Each line co esponds o he con ibu- ion due o di e en mul ipola i ies as indica ed in he igu e; he o al po en ial is gi en by he solid line. The ela i e con ibu ion o he single collec i e s a es can be be e seen in he igh pa o Fig. 3, whe e o he same sys em we show he eal and imagina y pa s o he pola - iza ion plo ed in pe cen age o he o al po en ial as unc ion o he ela i e dis ance R. Each line co e- sponds o he ela i e con ibu ion o he di e en mul- ipola i ies, as indica ed in he igu e. We no e ha he abso p i e pa is gi en almos comple ely by he low- lying 3s a e. Con e sely, he con ibu ion o low-lying and GQR s a es o he eal pa b, Vis o he same magni- ude. The high-lying 3s a es and he gian dipole eso- nance s a es gi e essen ially ze o con ibu ion. Fo he dipole iso ec o s a e his con ibu ion comes only h ough he small mix u e o T=O componen s due o he Coulomb in e ac ion. This esul is no peculia o he pa icula sys em: in gene al, a low ene gy he ab- so p i e pa o he op ical po en ial is gi en by he low- lying s a es, while bo h low and GQR s a es con ibu e o he eal pa . In ac , also o he sys em '0+ Ca a E»b =104 MeV (see uppe pa o Fig. 4) asimila beha - io is ound. In o de o in es iga e he pola iza ion po en ial a en- e gies much highe han he Coulomb ba ie we ha e calcula ed 6Vand 8' o he sys em '0+ Ca a se e al inciden ene gies. In Fig. 4we show he ela i e con i- bu ion o he single modes o he eal (le ) and imagina y ( igh ) pa o he pola iza ion po en ial as a unc ion o Rand o h ee di e en alues o he inciden ene gy. Again, he mul ipola i ies a e indica ed o each cu e. We no e wo s iking ea u es. (1) The eal pa is dom- ina ed by he GQR s a es which gi e, a high ene gy, 80% o he po en ial. The con ibu ion o he low-lying 3s a e, compa able in he pe iphe al egion o he 2+ s a es, is dec easing when he ene gy inc eases. The high-lying 3s a es and he GDR s a es beha e in an op- posi e ashion. Howe e , hey ne e each impo an alues. (2) Con e sely, he beha io o he abso p i e pa changes d as ically, going om 104 o 640 MeV. In ac , as we ha e seen be o e, a low ene gy he imagina y po en ial is essen ially due o he 3low-lying s a es, while as he ene gy inc eases he dominan abso p ion comes om he GQR o bo h nuclei. This beha io can be unde s ood in semiclassical e ms [7); highe ene gies co espond o sho e in e ac ion imes; hence, by he un- ce ain y ela ions, he p obabili y o exci a ion o high- lying s a es is highe . This mechanism does no wo k in he case o he eal pa because he p ocess in ol ed in he o ma ion o 6Vis a i ual one. Adeepe analysis shows ha he majo pa o he o- al 3con ibu ion comes om he low-lying s a e o he Ca. This can be seen in Fig. 5, whe e o he sys em DYNAMICAL POLARIZATION POTENTIAL DUE TO THE. ..2713 0+ Ca a E]b=104 MeV we show how he pola iza- ion po en ial due only o he low-lying s a es is dis ibu - ed be ween he wo pa ne s o he eac ion. We see ha he hea ie pa ne is esponsible o mos o he e ec . This end does no change a highe ene gies; he only di e ence is ha , o ins ance a EI,b=640 MeV, he wo cu es ge close o la ge R. On he con a y, o he high-lying s a es i is he ligh e pa ne which gi es he g ea e con ibu ion, al hough he di e ence is no as big as in he p e ious case. As an example, we show in Fig. 6 he pola iza ion po en ial due o he GQR s a es o he case o '0+ Ca a E),b=640 MeV. In o de o check he ange o alidi y o he p e ious esul s wi h espec o he ela i e impo ance o GQR and low-lying s a es, we ha e done calcula ions o labo- a o y ene gies om 10 o 320 MeV o he sys em '0+ Ca a a ixed dis ance R=9 m. The esul s a e epo ed in Fig. 7, whe e he eal and imagina y pa s o he pola iza ion po en ial a e plo ed as unc ions o he ene gy. Thei ene gy dependence esembles e y much he expe imen al indings o Re . [2]. The dashed line is a esul o acalcula ion done including only he low-lying 3s a es o he wo nuclei; he solid line has been ob- ained by using all he s a es o Table I. As was expec ed, a low ene gy he abso p i e pa is gi en only by he 3 s a es, he con ibu ion o he high-lying s a es inc easing wi h he ene gy un il eaching amaximum a ound 650 MeV (no shown in he igu e) and hen dec easing e y slowly o 0. Su p isingly enough, o he eal pa we ha e acom- ple ely di e en in e play be ween low- and high-lying s a es: he la e a e gi ing abig con ibu ion e en a e y low ene gy. The di e en beha io depends on he physical p ocess gi ing ise o he eal and imagina y pa s o he pola iza ion po en ial. The la e is due o eal exci a ion o he nuclei, and hus i anishes a low inciden ene gy when he channels a e closed. The eal pa is due o i ual exci a ion o he nuclei, so i is p esen a all ene gies. P e ious calcula ions o he pola - iza ion po en ial ha e no aken in o accoun he GQR s a es, missing in his case a leas 50%%uo o he e ec . The ene gy dependence o he pola iza ion po en ial, shown in Fig. 7, has a wo old o igin. The dynamic ene - gy dependence is due o he appea ance o he cen e -o - mass ene gy in he de ini ion o he p opaga o . The second one can be hough o as aspu ious one [17]be- cause i is due o he localiza ion p ocedu e desc ibed in Eqs. (15) and (16) and i comes h ough he local momen- u n k. The ene gy dependence o he eal and imagina y pa s o he pola iza ion po en ial a e go e ned by he dispe - sion ela ion. This ela ion is deduced om he gene al p inciple o causali y, so e e y pola iza ion po en ial should obey i . The possible iola ion o he dispe sion ela ion due o he ex a ene gy dependence in oduced by he localiza ion p ocedu e has been ound o be small [9,18,19]. In o de o check i in ou case we use he linea Ca +Ca, EI g~240 MeV 0.0100 -2.580 -2+ -5.0K4 60 40 -7.5 / / -10.08R( m) 10 20- 3ll 1 0910 11 R( m) 0100 60- -10 40- 20 -15 8R( m) I 10 08R( m) I 10 FIG. 3. Real and imagina y pa s o he local pola iza ion po en ial o Ca+ Ca (E& b=240 MeV) as a unc ion o he ela i e dis ance R. Each line co esponds o he con ibu ion o di Fe en mul ipola i ies as indica ed in he igu e. On he le pa he o al po en ial is gi en by he solid line. On he igh pa he po en ials a e plo ed in pe cen age o he o al one. 2714 M. V. ANDRES, F. CATARA, AND E. G. LANZA schema ic model gi en by Mahaux e al. [4]. In pa icu- la , we use he model o Eq. (3.1S) o Re . [4], whe e we ha e changed he misp in ed plus sign o he las e m o aminus. In his model he imagina y pa is segmen ed in o h ee pa s, and his makes i possible o ind an alge- b aic exp ession o he co esponding eal pa h ough he sub ac ed dispe sion ela ion. By using Eq. (3.1S) o Re . [4] we ha e hen calcula ed he eal pa o he po- la iza ion po en ial o bo h o he imagina y pa s o Fig. 7; namely, he one due o only he 3low-lying s a es and he one calcula ed wi h all he s a es o Table I. In bo h cases he schema ic po en ial has been no mal- ized o he calcula ed one a E ,b=320 MeV. The squa es in Fig. 8a e he esul o his calcula ion. The ag eemen wi h he cu es calcula ed by means o Eq. (1S) is e y good. This implies ha he localiza ion p o- cedu e used he e is agood one in he sense ha i does no iola e he dispe sion ela ion, a leas a he dis ance conside ed in he calcula ions. Op ical model analysis [1,20,21] on elas ic sca e ing da a is a ailable o he sys em '0+ Ca. Ou imagi- na y po en ial shows abeha io in quali a i e ag eemen wi h he empi ical one, namely, i dec eases in he E&,„&80MeV domain and goes o 0a E&,b-40 MeV. In he same ene gy egion we ha e aco esponding in- c ease o 6V. Un o una ely, he empi ical alues show aconside able sca e which makes aquan i a i e co n- pa ison ha d. As poin ed ou by he au ho s o Re . [4],a consis en eanalysis o all he da a would be in e es ing. In o de o ha e a u he es o ou mic oscopic op i- cal po en ial we ha e calcula ed he elas ic c oss sec ion o he sys ems '0+ Ca and Ca+ Ca a a ious en- 100 100 80 -2+ 60- 40- 80- 60'3 g, E) b~104 MeV 1 03hl ~as+~~~,~, ,~1» 789 R( m) I 8 R( m) 100 + 80- 320 MeU E&b ~320 McV 60- 40- I 20 -3 3hl P' 0~y &4O 20- 0 ~~~~ ~W~~~ 10 789 R( m) 100 80 = 100 80- 60- 40 " E~b ~640 McU 60 =2 0 RID ~640 MeU ~~MM~M~~~ 20-3 D '3 hl le 0I 89 R( m) 20- 78 R(~m~ ~K 'g pso he Pola iza ion po en ial o '~+ Ca d - plo ed in pe cen age o he o al po en ial and as a unc ion o he ela i e dis ance R. TENTIAL DUE TOLARIZA TION PO™&T DYNAMICAL P 4o( aR,=9 0+i 2715 100) 104 MeV. & pep +"Ca, E)b = I04 80- 4DCa 6p 40 ~0.1- on&y $0 06 I 8 R( m) I 910 p.0 0.15 yes 100 80 Ca 4D 60 . 40- o.&o I005- 0.00 800 El b{MeV) oh' 3 I 300 80- 10 I 9 78 6R( m) he pola iza ion p no en- lo di a s oeeo E«04 MeV lab o1b }1 lo he po en ial gi en on y nuclei. he pola iza ion po 7. Real adim gi p. he inciden en a s oe FIG. 7. unc ioil oaeal- l16O+40Ca plo ed as shed cu es ~e e ial o R=9 m. The das 3s a es o a ixed alue «.l. he low-lying o a dbincluding on y acalcula ion cua i,h, 1, olid lines a e uclei. eso ' done wi a h ll he s a es oa =640 MeV, 8GQR 0+ Ca, Ei~b =, R I100 80- 60- ieO 40 ."Ca di6'e en ial c oss g' .Fi .9a e epo ed he elas ic i =104 MeV (uppe e gies. In ig. +Ca a Ei,b= he cases 0+ = 'h he da a is e y goo h h l he ola iza ion po en ia 1 a e no a all exhaus i e. n 06 100 I 8 R( m) 10 0.4Ca R= 9 y 0+ a, 80 60 40- 20- 4DC 0.1— ~i 010 786R( m) =640 MeV. He e h p he ola iza- as Fig. 5a Elab = FIG. 6. Same as 'g. GQR s a es. ion po o en ial is due y o'onl o he 300 I 100 200 0E|,b(Me ) Fi .7. The squa es co e- he lowe pa o Fig. .co e Fddispe sion e ai spon o d o asub ac e i 2716 M. V. ANDRES, F.CATARA, AND E. G. LANZA 1O0 -„. 1O-' j10 16p +40C 104 MeV 0.8 I I 0.4 '0+ Ca, R= 9 m l 20 I 40 60 (deg ees) I.I. 80 100 180 0.0 104 10~ 102 E Cl b10' 40C +40C E...=1.43 MeV 0.4 0.2 0.0 / I I I I I Il. . . .I 100 800 Eg b(Me ) I 300 100 40 I 50 60 70 8,I(deg ees) 80 FIG. 9. Elas ic sca e ing angula dis ibu ion o 'O+ Ca and Ca+ Ca. The la e is plo ed in absolu e uni s. The da a a e om Re s. [21]and [22], espec i e1y. side ed he ans e channels which a e hough o be im- po an a leas o asymme ic sys ems. This may be he easons why we ha e s ong oscilla ions in he elas ic c oss sec ion a backwa d angles o he '0+ Ca case. In ac , as is shown in Re . [5], he imagina y po en ial due o he ans e channels ha ing alonge ange may smea ou he oscilla ions. Ano he poin o be in es iga ed is how much he e- sul s depend on he alues o he exci a ion ene gies o he collec i e s a es used in he calcula ions. In o he wo ds, is he use o an a e age exci a ion ene gy in he p opaga o and hen he use o he closu e app oxima ion jus i ied, o is i impo an o use he p ope ene gies o low- and high-lying s a es'? To answe his ques ion we ha e done acalcula ion including aH he s a es o Table I, whe e he ene gies ha e been a bi a ily pu equal o a Axed alue. In pa icula , we ha e chosen E160 6.5 MeV and E4, =5.0MeV (Re [9]). Th.e esul s a e e- 4'Ca po ed in Fig. 10 as dashed lines, while he solid lines e- sul om he RPA ene gies epo ed in Table I. The be- ha io wi h he ene gy o he imagina y po en ial is qui e di Fe en in he wo cases. In he calcula ion co espond- ing o he solid line he e ec o he high-lying s a es is e iden : he po en ial goes up as he inciden ene gy in- c eases. The dashed line esembles he one in Fig. 7, FIG. 10. Same as Fig. 7. The dashed cu es e e o acalcu- la ion pe o med wi h cons an exci a ion ene gy E&6 =6.5 0 MeV and E4O =5.0MeV, while he do -dashed ones co e- Ca spond o he alues EI6 =20 MeV and E4o =17 MeV (see ex ). The solid lines a e he same as he ones plo ed in Fig. 7. which was calcula ed wi h only he wo low-lying s a es. Thus all he s a es now gi e an equally impo an con i- bu ion, a leas a low ene gies. The magniAca ion ac o be ween he wo cu es is no equal o he numbe o s a es used in he calcula ion because we ha e kep he RPA o m ac o s in bo h cases. Fo comple eness we ha e done acalcula ion wi h he a e age ene gies close o he ene gies o he gian quad upole esonance s a es, i.e., E,6=20 MeV and E4o =17 MeV. The esul s (do - dashed line in Fig. 10) show an enla gemen o he ene gy scale: he apid inc ease and hen dec ease o he eal pa o he pola iza ion po en ial is now sp ead ou o e a much la ge ange o inciden ene gy. As expec ed, he imagina y pa is impo an a high ene gies and anishes a E„b— 100 MeV (see also lowe pa o Fig. 7). The eal pa s a e go e ned by he dispe sion ela ion: di e en beha io s o he abso p ion will p oduce di e en beha - io s o he eal pola iza ion po en ial. In pa icula , ac- co ding o he simple o m o he linea schema ic model o Re . [4] [Eq. (3.17)], he posi ions o he maxima o he eal pa s a e de e mined by he ene gy alue co espond- ing o hal he in e al whe e he imagina y po en ial goes apidly o 0. The ac ha he eal po en ial co e- sponding o he do -dashed line has amaximum nea 200 MeV conA ms his simple model. We ha e shown, a leas o he wo ex eme cases p esen ed in Fig. 10, ha he use o a e age exci a ion en- e gies in he p opaga o o Eq. (2) gi es ise o pola iza- DYNAMICAL POLARIZATION POTENTIAL DUE TO THE. . . 2717 ion po en ials which di e in shape and magni ude om he po en ial calcula ed wi h he p ope ene gies o bo h low- and high-lying s a es. Since hese esul s a e no a all exhaus i e, his p oblem should be u he in es iga - ed. IV. CONCLUSIONS Wi hin he Feshbach o malism we ha e calcula ed he dynamical pola iza ion po en ial a ising om he cou- pling o he elas ic channel o he collec i e ib a ional s a es. The la e we e cons uc ed wi hin sel -consis en RPA wi h he SGII o ce. Bo h ene gies and ansi ion densi ies compa e well wi h he a ailable expe imen al da a. The p opaga o was calcula ed in he WKB ap- p oxima ion, which is e y good a inciden ene gies g ea e han he Coulomb ba ie . The ba e nucleus- nucleus po en ial has been cons uc ed by double olding he e ec i e M3Y in e ac ion wi h he HF densi ies o he wo nuclei. We ha e no made use o he closu e ap- p oxima ion bu we ha e summed o e a ini e numbe o ele an s a es, each one wi h i s own ene gy. The locali- za ion p ocedu e we ha e used does no des oy he ene - gy dependence o he pola iza ion po en ial: The eal and imagina y pa s sa is y he dispe sion ela ion. We ha e done an analysis o he eal and imagina y pa s o he pola iza ion po en ial in e ms o he ela i e con ibu ions o he single collec i e s a es o he sys- ems '0+ Ca and Ca+ Ca a se e al inciden ene - gies. As one should expec , a low inciden ene gies he main con ibu ion o he imagina y pa o he pola iza- ion po en ial comes om he 3low-lying s a es. As he ene gy is inc eased he ole played by he 3s a es is ak- en o e by he GQR s a es which gi e he main e ec . Con e sely, he eal pa is domina ed by he high-lying s a es in all he ene gy ange in es iga ed, and hey con- ibu e up o 80% o b, V. This no el esul shows ha he GR s a es canno be dis ega ded in he cons uc ion o adynamical pola iza ion po en ial. We ha e also shown ha he use o he a e age exci a- ion ene gies, in he wo ex eme cases ea ed he e, gi es ise o apola iza ion po en ial which di e s app eciably bo h in shape and magni ude om he one calcula ed wi h he p ope ene gies. The ene gy dependence o he pola iza ion po en ial has been checked by means o he dispe sion ela ion. In pa icula , we ha e used he linea schema ic model o Re . [4]. The esul is e y good. Calcula ions o elas ic c oss sec ion gi e agood desc ip ion o expe imen al da a: hese calcula ions ha e been done wi h no adjus - able pa ame e s. ACKNOW%'LED GMENTS We wish o hank M. A. Naga ajan, N. Van Giai, and N. Vinh Mau o use ul discussion and sugges ions. We also hank N. Van Giai o making his RPA code a ail- able o us. APPENDIX In his Appendix we wan o calcula e he quan i y b,V,2which is de ined as AV,2(R,R')= gFK K(R)GK K(R,R')FK K(R'), K1K2 whe e FK K(R) is gi en by 12 FK K(R)=(00~u(R)~K, Kz) = d , d 2pK o( , )u(~ , — 2+R~)poK ( z) (Al) /dp pu(p)pK o(p)poK (p)i ~1L2( 4~'" 12 Xgj~(PR)(L, M, L, M, ~J M, +M, )(L, 0Lz O~J 0)YJM +M (R), J (A2) whe e we ha e used he ac ha pK, o( i)=pK, o( , )YI. M( i) and he de ini ion o he Bessel-Fou ie ans o m pK 0(p)=4~ " i«iJI.,(p i) K,o( i ). (A.3) (A4) p('s) g(J— AMpA p~J M) YJ iM — „(p)Yq(s) . We ha e deno ed wi h jl and Y& he Bessel and sphe ical ha monic unc ions, espec i ely. The ca e o e he spin a iables deno es J=VZJ+ I, and 8'(p) is he Fou ie ans o m o u. Acco ding o Eq. (4) we can w i e R=p+ —,s; hen in his case we ha e [Eq. (6.50)] o Re . [23]: I/2 2I. +1 RYJM(R)=&4m. g Q=o