Random-phase-approximation-based dynamical polarization potential

PHYSICAL REVIEW CVOLUME 39, NUMBER 1JANUARY 1989 Random-phase-app oxima ion-based dynamical pola iza ion po en ial M. V. And es Depa amen o de F&'sica A omica yNuclea , Uni e sidad de Se illa, 41080Se illa, Spain F.Ca a a Is u o di Fisica dell'Un e si a, 98100Messina, I aly and Is i u o Nazionale di Fisica Nuclea e, Sezione di Ca ania, 95129 Ca ania, I aly Ph. Chomaz Di ision de Physique Theo ique, Ins i u de Physique Nucleai e, 91406 O say Cedex, F ance E.G. Lanza Is i u o Nazionale di Fisica Nucleai e, Sezione di Ca ania, 95129Ca ania, I aly (Recei ed 18 July 1988) Adynamical pola iza ion po en ial is de ined aking in o accoun he e ec s on he elas ic chan- nel due o he exci a ion o ib a ional collec i e modes, desc ibed wi hin he andom-phase app ox- ima ion. The p obabili y ampli udes o exci ing hese modes a e e alua ed by in eg a ion along classical ajec o ies de e mined by he eal pa o he nucleus-nucleus po en ial wi h ene gy and angula momen um loss. Calcula ions pe o med o he Ca+ Ca sys em show ha , a high bom- ba ding ene gy (E/A =44 MeV), bo h he eal and he imagina y pa s o he pola iza ion po en ial a ise mainly om he exci a ion o he high-lying modes. A ene gies nea he Coulomb ba ie , he calcula ed elas ic c oss sec ion is in good ag eemen wi h he expe imen al da a. The inclusion o he eal pa o he pola iza ion po en ial imp o es his ag eemen . I. INTRODUCTION An impo an p oblem in he s udy o pe iphe al hea y-ion p ocesses is o connec he nucleus-nucleus po- en ial o he unde lying nucleon-nucleon in e ac ion and o he mic oscopic s uc u e o he wo pa ne s. In he olding model' he eal pa o he op ical po en ial is cons uc ed om he g ound-s a e densi y dis ibu ions o he wo colliding nuclei. On he o he hand, e en in he pe iphe al collision egime, p ocesses like nucleon ans e and exci a ion o collec i e deg ees o eedom ake place. The p oblem hen a ises o cons uc ing apo- en ial which desc ibes he e ec s o such p ocesses on he elas ic channel, he pola iza ion po en ial. The pola iza ion po en ial a ising om he ans e mode was s udied in Re . 2(see also, e e ences quo ed he ein). The modi ica ions o he olding po en ial due o he exci a ion o he collec i e ib a ional s a es we e calcula ed, in he adiaba ic limi , i i Re . 3. I was ound ha he co ec ions a e impo an and gi e as ong enhancemen o he sub-ba ie usion c oss sec ion. In he p esen pape we ex end he p e ious analysis by aking in o accoun he dynamics o he collision. This gi es ise o acomplex pola iza ion po en ial which, o- ge he wi h he olding one, de ines an op ical po en ial while aking in o accoun he coupling o he elas ic channel o he ones co esponding o he exci a ion o collec i e deg ees o eedom. The la e a e desc ibed wi hin he andom-phase app oxima ion (RPA) while he ela i e mo ion is ea ed classically as go e ned by he olding po en ial plus he eal pa o he pola iza ion po- en ial. The ene gy and angula momen um loss a e also app oxima ely aken in o accoun . The op ical po en ial has been used o calcula e he elas ic c oss sec ion. We ha e s udied he sys em Ca+ Ca a a ious ene gies. Ou esul s can be sum- ma ized as ollows. The heo e ical c oss sec ion is in good ag eemen wi h he exis ing expe imen al da a, which shows ha ou model is able o gi e, wi hou any adjus able pa ame e , a easonable desc ip ion o he op- ical po en ial. A low ene gies, bo h he eal and imagi- na y pa s o he pola iza ion po en ial a e needed in o - de o ep oduce he da a. On he o he hand, a sligh ly highe ene gy (E/A =6 MeV), once he abso p ion has been aken in o accoun , he heo e ical esul s wi h and wi hou he eal pa o he pola iza ion po en ial a e equi alen ly good wi h espec o he a ailable expe i- men al da a. A low ene gies (e.g.,a E/A ~10 MeV), he con ibu ion o he low-lying collec i e s a es o he pola iza ion po en ial is ound o be dominan . In o de o analyze o which ex en he exci a ion o he gian esonances plays a ole in he de e mina ion o he op ical po en ial, we ha e also made acalcula ion a E/2 =44 MeV, whe e hei e ec s a e expec ed o be la ge . Due o he absence o expe imen al da a in his case, we only make acompa ison be ween heo e ical e- sul s ob ained wi h o wi hou hem. We ind ha he in- clusion o he gian quad upole esonance (GQR) s ong- ly modi ies he elas ic sca e ing c oss sec ion a angles nea he g azing. 39 99 1989 The Ame ican Physical Socie y M. V. ANDRES, F.CATARA, PH. CHOMAZ, AND E.G. LANZA 39 II. THE MODEL The basic assump ions o ou model a e ha o g az- ing collisions, he ela i e mo ion can be ea ed classical- ly and he mu ual exci a ion o he wo pa ne s can be desc ibed as due o he mean ield o one nucleus ac ing on he o he one. In pa icula , we will concen a e on he exci a ion o collec i e ib a ional s a es, which we desc ibe mic oscopically wi hin he RPA. Unde he abo e-men ioned assump ions, he in insic Hami1 onian o he sys em is he sum o wo e ms, one o he a ge and one o he p ojec ile, each o hem ha ing he o m H;„,„=gEkB), "B),+gVko( )Bk kk +H.c.+gV)k.( )B„"Bk kk' whe e he ime dependence comes in h ough he ela i e dis ance R( ), which obeys classical equa ions o mo ion. The eedback o he exci a ion on he ajec o y is aken in o accoun by de ining he ene gy and angula - momen um loss as equal o he mean alues o he ene gy and angula momen um s o ed in he collec i e deg ees o eedom. In addi ion o ha , a each in eg a ion s ep in ime, we modi y he nucleus-nucleus olding po en ial by adding o i he eal pa o he pola iza ion po en ial. In Eq. (1), B&(B&)a e boson ope a o s co esponding o he RPA ope a o s Qk(Q„) whe e N( ) =glI„( )l' . In he amewo k o he semiclassical app oach, we hen = de ine he complex pola iza ion po en ial as (ol p), &=exp — i d '[b, V( ')+iW( ')], (12) whe e AV and 8' depend on ime h ough he ela i e dis ance R( ) and he in eg a ion is made along aclassi- cal ajec o y. F om Eqs. (10) and (12) i ollows: 0V( )=P( )= Im QI„*( )I„( ) k(13) whe e P( ) =Im g F„*( ')d ' F„( ")d ", I„( )=— i Fk( ')d ', iF.~I F„( )=V„o( )e and all he in eg als a e e alua ed along aclassical ajec- o y. F om Eq. (6) i ollows ha he p obabili y ampli- ude o he sys em o emain in i s g ound s a e is gi en by (()l@ &— i(0( )— i/2A( ) Qk'= gI&k(Ii)[I il~+I"~(Ih)[h Il~l, ph W( )= ,'N( )= — Re— —QI, *( )I„( ) k(14) lq &=Q'lo&, Qlo&=o, whe e lo& is he RPA g ound s a e and l&Ip), & he collec- i e s a es we a e in e es ed in. The ampli udes Xand Y and he ene gies Ek a e solu ions o he RPA equa ions. The ma ix elemen s (4) a e calcula ed wi hin he RPA. The ope a o U ep esen s he mean ield o he o he nucleus. In wha ollows we ha e neglec ed he nonlinea e ms in he in- insic Hamil onian o Eq. (1) since hey a e no impo - an o he p esen calcula ion. Indeed, as i was shown in Re . 4, a low-bomba ding ene gy, hei inclusion enhances he popula ion o he collec i e high-lying s a es which, howe e , emains much smalle han he popula ion o he low-lying modes. Then, hese mixing e ms a e impo an only when one is s udying aselec i e p ocess like he inelas ic sca e ing. A high-bomba ding ene gy hey a ec e y li le, e en he inelas ic sca e ing c oss sec ion. The s a e o he sys em a any ime is he p oduc o wo cohe en s a es [I ( )] k(B1') k k whe e he Ik( ) a e de ined in Eq. (8). The so-de ined unc ions 6Vand 8'canno di ec ly be in e p e ed as he eal and he imagina y pa s o apo en ial. Indeed, hey depend on he whole his o y o he sys em up o he ime . In pa icula , hey ha e di e en alues when he sys- em eaches he same ela i e dis ance in he app oaching and he ou going phase. They wi11 also depend on he conside ed classical ajec o y and hen on he angula momen um as well as on he ene gy. In he nex sec ion we will discuss ap ocedu e o de ine alocal-pola iza ion po en ial s a ing om Eqs. (13) and (14). Adi e en p ocedu e is conside ed in Re . 7whe e he a en ion is mos ly concen a ed on he ela i e impo ance o he low-lying s a es and high-lying s a es in de e mining he ail o he imagina y pa o he op ical po en ial a high- inciden ene gy. III. THE POLARIZATION POTENTIAL The ba e nucleus-nucleus po en ial is calcula ed by olding he M3Y e ec i e in e ac ion wi h he Ha ee- Fock densi ies o he wo nuclei. The collec i e s a es a e calcula ed wi hin he consis en RPA wi h he Sky me in- e ac ion SGII. The esul s we will show la e we e ob- ained by including he s a es epo ed in Table I, i.e., hose exhaus ing a leas 5% o he ene gy weigh ed sum ules (EWSR). In o de o discuss how o ex ac he pola iza ion po- en ial om he quan i ies i)), V( ) and W( ), le us exam- ine in de ail aspeci ic case. %'e conside he Ca+ Ca lPl DYNAMICAL NgASED D ROXIMATIO pHASE-AP ANDOM 0 39 E(MeV) J7T NucleI 5031 16.561 16 970 17 488 17.667 18.20& 18 399 19.219 22.166 23 554 32 716 11 24 17 8 10 14 14 17 10 3 2+ 2+ 2' 4+ 4+ 0+ 0+ 0+ 0 3 4oCa calcula Ion. ic da a use Spec oscopIc &E~SR TABI-E 0 -2 O 4 -6 III 1O 11 n y 3&2 pn~y 3 91O 11 12]O 11 Me& bomb s unc 1ons jn ene gy he 44 Me gp'and Vas ugIn he sys em d2we hw 5O and 26 In F1gs R o L=o he esul s ~]an ob a1ne ela i e d1 h igu e we Pl he lpw-ly' g s P eac e Pn yj.e., he ee sec o jgh o 2+ s a es ~ om h ee pnucl s a ~ ll he s a p each c GQ 'ho he (nPi h)n e a&&e . o in- R) and ae Pa edue o app oa p' inc ease, They con he ach eac jn abso n jnue yand 's a es. eaches i ies ~he collec 1 e he sys em ion o he ime ~ e dec ease, a gi a ol o so ed hen h y ease s ead1 yach an . +Van c o closes aPP The beha ' lybe ween di anc a dis an~~ h he in i ~) o 1s 1n .Phase 1s ga ac e is lc collision ime ~clusjpn o es, he '" he collec om he gn ibu 1o ea en main As i 1s app di es he ucj» an GQR is c a , Eq. (1 }eimagina y p F1bu FIG. a ne as F'g' a h1s enc- e ac ha , apn- ~'due o he Ris la ge. Co xc1 a ' ",. he con ' "' nc 1o s~~ nd ou go1ng p e ec s The unc oaching an he memp y same R.man1 es la'Lcula ion jn he app 's a jpn hjs is aamjca ca .p en 1a lisipn. T jn ou dyna pla iza 1on p h1ch a «e in pon &&es a& dene g» each L) he s1m a aje«o y'loses (i.e., ~~+iW) deP ha each dis ance o i e dis an he pp en hcollec i d ines he hexci a 1P alueo 'np nce 0 e'nce e.a oun .e he d he d1s a oach. S'n ll in e al uld 1nclu. aPP in asma i ion shou be easily plac hsp ese P app oach, h ep ocess- loses le ec s 1n phy qu an 1 he wo q shown +gV( )d p(+ ~)= (&s) and +"w( )d' )~(+m)= O' O +.,pn~y 3 )y 3&2 I 1O 11 12 10 11 -5 g10 E.(i3)] « o en ~al [q he he po as un a iza ion P c ion o Ral pa,l o880 Me 'd he esul he gIll .h o Je ,,07~gR), an edis ance - m igh o' he ela I ~ 1ding, om + a e~ ('e' e o b ained bP, nd he h e in u'ee 2s es e he h ee cu 50, s a e, an hsec o , e.L, =240, 2 s a e, e Table I. In ela mome~™ a»,„ alue. o he s a e he ang h ee di e en a and 260 A. R=R1+R~, 0 whe e he eal jpn conne .The ing ha sen in he poo app o his 1s ah ajec o y ession o i he iza 1on so e jcexP lwe ha e ion p an ana ypp en ja e w1 h In o de o he pola jza u p ocedu a s p-dby us1ng jmag1n lues ob a .9 lc 1on o a i h adius col eon o ms, wWoods-Saxon 102 M. V. ANDRES, F.CATARA, PH. CHOMAZ, AND E. G. LANZA 39 R, =(1.20M,'— 0.09) m .(16) ~~~~ The deduced alues o he dep h and he di usi i y, when all he s a es a e included, a e AVo= — 14.26 MeV, az&=0.491 m o he eal pa and lgo= — 25.34 MeV, a~ =0.495 m o he imagina y pa . The op ical po en- ial is hen ob ained by adding he eal pa AV o he olding po en ial while he imagina y pa is gi en by 'lV. Calcula ions ha e been done a se e al ene gies, in pa - icula , we ha e s udied he beha io o he pola iza ion po en ial nea he Coulomb ba ie . A such ene gies, he main con ibu ion comes om he low-lying 3s a e. All he esul s a e collec ed in Table II, whe e o each ene gy we epo he alues o he dep hs and di usi i ies o AV and lH. The ene gy dependence o he eal pa o he op ical po en ial is be e illus a ed in Fig. 3whe e we compa e he ba e- olding po en ial plus he Coulomb pa (dashed line), wi h he ba ie s ob ained a he ene - gies indica ed in Table II and in he cap ion. As he in- ciden ene gy dec eases, he co ec ion o he olding po- en ial inc eases, eaching e en ually he adiaba ic limi o he pola iza ion po en ial (see Re . 3). This beha io wi h he ene gy can be ela ed wi h he indings o Re . 10 whe e i was shown ha in o de o ge agood i o he expe imen al elas ic c oss sec ion, i is no su hcien o a y he pa ame e s o he abso p i e pa o he op ical po en ial, bu i is also necessa y o mul iply he olding po en ial by an ene gy-dependen ac o g ea e han 1, when he ene gy app oaches he ba ie om abo e. This has been explained, and con i med by explici calcula ions, "as acoupled- channels e ec . The au ho s o Re . 12 ha e in e p e ed his ene gy dependence by using he dispe sion ela ion be ween he eal and imagina y pa o he op ical po en- ial. Ou esul s a e no based on aphenomenological analysis, bu a he on an app oach which akes in o ac- coun he collec i e ib a ional modes in acomple ely mic oscopic way. In he nex sec ion we will see ha he c oss sec ions ob ained, including he eal pa o he po- la iza ion po en ial, a e in be e ag eemen wi h he ex- pe imen al da a han he ones wi hou such co ec ion. IV. ELASTIC SCATTERING CROSS SECTION The op ical po en ial V ,&d+AV+i%', de ined in he p e ious sec ion, has been used o calcula e elas ic O O52- / l I ~ ~o 10 FIG. 3. Ba e double- olding po en ial plus he Coulomb e m (dashed line) compa ed wi h he ba ie s calcula ed a se e al cen e o mass ene gies: 880, 120, 71.8, 64.8, 60.6, and 55.45 MeV ( om abo e o below). di e en ial c oss sec ions o he Ca+ Ca sys em and o se e al alues o he ene gy. The case E/A =44 MeV is in e es ing since a such high ene gy he p obabil- i y o exci ing he GQR is much highe han a lowe en- e gies. The e 'ec s o he GQR, which ha e al eady been shown in he p e ious sec ion o be e y impo an in he calcula ion o he pola iza ion po en ial, show up also in he elas ic c oss sec ion. In o de o ha e abe e insigh , we ha e done calcula ions wi h he op ical po en ial con- s uc ed by including only he low-lying 3s a e and all he s a es o Table I. F om he esul s shown in Fig. 4, we see ha o angles la ge han he g azing one he di e ences a e impo an . This is in ag eemen wi h he indings o Re . 13, whe e i ws shown ha he GQR is s ongly exci ed a his ene gy and a ound he g azing angle. The impo ance o he eal pa o he pola iza ion po- en ial is illus a ed in Fig. 5whe e he esul ob ained, wi hou including i (dashed line), is compa ed wi h he c oss sec ion calcula ed wi h he comple e op ical po en- ial. F om he igu es we see ha in he same angula ange as be o e he inclusion o he I'eal pola iza ion po- en ial s ongly enhances he c oss sec ion. This no el e- TABLE II. Pa ame e s o he Woods-Saxon o ms o he eal and imagina y pa s o he pola iza- ion po en ial o he Ca+ Ca sys em. The alue o Ro (Re . 9) is 8.0279 m o bo h. The o al eac- ion c oss sec ion calcula ed wi h an op ical po en ial whose eal pa is jus V Jd is deno ed by az', he one calcula ed by Sa chle and Lo e (Re . 1) is deno ed by o~". The las line e e s o he calcula ion done including only he low-lying 3 s a e. (MeV) 55.45 60.6 64.8 71.8 120 880 880 ~&O (MeV) — 179.87 — 155.84 — 142.68 — 127.06 — 49.78 — 14.26 — 1.27 ( m) 0.380 0.385 0.386 0.389 0.459 0.491 0.504 lÃo (MeV) — 37.40 — 33.23 — 50.36 — 92.14 — 31.22 — 25.34 — 8.54 ( m) 0.424 0.441 0.411 0.371 0.443 0.495 0.475 (mb) 100 361 534 779 1642 2423 2038 old (mb) 59 302 474 719 1583 2414 2037 SL (mb) 671 939 1893 39 &ANDOM pH ASE-APPRQX IQN-BASED 0 I YNAMICAL I ~~ ~ ioo i0-2 Ig » &~-~ ~ - A--~ %/ x/ l 84 FI 4El .(d g) IO sys em a E— ' 1c o 88O sec ion ob ained b.oMeV. The do he OCa+ 4o using he edashed line aCa low-lying 3—epo en ial ceshows he a«, wh;l ns uc ed i pie e calcula . '»e he ull ]i 'ncluding onl h a (on co esPonds ye s o he co n 1O3 102 10 ns whe e h by he old; o he op ical men al dgone. The po en jal is g' aa a is im o eag eemen wgi en « he pol P o ed by he 'wi h he exp po a izaiion o 'einclusion o &pe i on he o 1 "po en ial. I s ."0 he eal pa Table II o h ec ion, whose 1 pp eciable also ege wi h ho aues a e e oen jal op ical p« .oecalcula d.'p ed jn sec ion. Th' ob ained b i e . 1by usin in R is e ec is y ing he clas ' a o } ss sec jon ip~a angula pola iza ion 'o sensi i e whe e epo en ial ao pe imen al da leas in he ~e»s (gee F; 10+ 1g. 7) sui shows ha o a a eno maaliza ion o h 1 ia is also e eal a e ec able a igh en In o de j esi h 1 ssec ion. ea e also caacula ed he en al clas po en ial is ic c oss se gyp eima in o e o i We ha e he ec ion nea h eexpe i- dmb ba ie . ee o which d u'ea jo wi h mo el is able h Th eines co es a e e- spon o calcula- L E 10 10 IO'— 10 l0 2= lO ~ 0 Ecm= 880 MeV 6 cm. (d g) Kl U I 8IO 10 I 50 60 c.m. (deg) 80 90 FICx. 5. ~.Same as Fi ~.ig. 4, bu in edashed line is h is e gyp e eal o1 ep esen ed aken in o apo a iza ion co ec- iine. ain ecalcula ion FIG. 6. C Compa ison p e wh' as ed line is h -o -mass ene ie omgies as indica - asned using V old + M. V. ANDRES, F.CATARA, PH. CHOMAZ, AND E.G. LANZA 39 10 10 E10 1 10 c.m. =" 0 10 0I 10 I 20 50 40 50 FIG. 7. Same as Fig. 6bu a E, =120 MeV. In all he conside ed cases, he ag eemen wi h he ex- pe imen al da a is a he good. This is an indica ion ha he op ical po en ial is well es ima ed in ou model and shows he ele ance o he collec i e modes. We s ess ha we do no ha e any adjus able pa ame e . Besides, in gene al, o he deg ees o eedom, such as nucleon ans e and noncollec i e pa icle-hole exci a ion, neglec ed in he p esen calcula ion, a e expec ed o play some ole. V. CONCLUSION We ha e calcula ed adynamical pola iza ion po en ial as due o he coupling o he collec i e ib a ional s a es, desc ibed wi hin he RPA. The dynamics is in oduced by in eg a ing he equa ions o mo ion o he ele an deg ees o eedom along classical ajec o ies go e ned by he eal pa o he op ical po en ial. Ene gy and angula -momen um loss a e aken in o accoun con- sis en ly. The op ical po en ial is hen cons uc ed by adding he pola iza ion po en ial o he olding one. We ha e pe o med calcula ions o elas ic di e en ial c oss sec ion o he Ca+ Ca sys em a se e al ene - gies. Ou esul s a e in good ag eemen wi h he expe i- men al da a, showing ha he op ical po en ial is well es- ima ed. The inclusion o he eal pa o he pola iza ion po en ial imp o es his ag eemen . A ene gies nea he ba ie , his beha io can be ela ed o he indings o Re . 10 whe e a eno maliza ion o he olding po en ial was shown o be necessa y in o de o i he elas ic sca e ing c oss sec ion. Ade ailed analysis o he con i- bu ions coming om he di e en collec i e modes shows ha a low ene gies he pola iza ion po en ial a ises essen ially om he low-lying 3s a e. Con e sely, a high ene gy, namely a E/A =44 MeV, he GQR plays a e y impo an ole in he de e mina ion o he op ical po en ial and has as ong e ec on he elas ic di e en ial c oss sec ion a ound he g azing angle. 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