Microscopic description of Coulomb and nuclear excitation of multiphonon states in 40 Ca + 40 Ca collisions

Mic oscopic desc ip ion o Coulomb and nuclea exci a ion o mul iphonon s a es in 40Ca¿40Ca collisions M. V. And e ´s Depa amen o de Fı ´sica A o ´mica, Molecula y Nuclea , Uni e sidad de Se illa, Apdo 1065, E-41080 Se illa, Spain F. Ca a a and E. G. Lanza Dipa imen o di Fisica Uni e si a ´di Ca ania and INFN, Sezione di Ca ania, I-95129 Ca ania, I aly Ph. Chomaz GANIL, Boı ˆ e Pos ale 5027, F-14021 Caen Cedex, F ance M. Fallo and J. A. Sca paci Ins i u de Physique Nucle ´ai e, IN2P3-CNRS, F-91406 O say Cedex, F ance 共Recei ed 2 Janua y 2001; published 14 Decembe 2001兲 We calcula e he inelas ic sca e ing c oss sec ions o popula e one- and wo-phonon s a es in hea y ion collisions wi h bo h Coulomb and nuclea exci a ions. S a ing om a mic oscopic app oach based on andom- phase app oxima ion, we go beyond i in o de o ea anha monici ies and nonlinea e ms in he exci ing ield. These anha monici ies and nonlinea i ies a e shown o ha e impo an e ec s on he c oss sec ions bo h in he low ene gy pa o he spec um and in he ene gy egion o he double gian quad upole esonance. By p ope ly in oducing an op ical po en ial, he inelas ic c oss sec ion is calcula ed semiclassically by in eg a ing he exci a ion p obabili y o e all impac pa ame e s. A sa is ac o y ag eemen wi h he expe imen al esul s is ob ained. DOI: 10.1103/PhysRe C.65.014608 PACS numbe 共s兲: 25.70.De, 21.60.E , 21.60.Jz, 24.30.Cz I. INTRODUCTION All heo e ical app oaches used o calcula e he c oss sec- ion o he mul iple exci a ion o gian esonances 共GR兲in hea y ion collisions a e based on a semiclassical desc ip ion o he p ocess 关1兴whe e he exci a ion o one eac ion pa - ne is assumed o be due o he ac ion o he mean ield o he o he and is ea ed quan um mechanically while he ela i e mo ion is de e mined classically. Fo each eigens a e ␣ o he in e nal Hamil onian o one nucleus, one can calcula e i s exci a ion p obabili y P ␣ (b)by pe u ba ion heo y o by sol ing a sys em o coupled equa- ions. This is done by in eg a ing he equa ions o mo ion along he classical ela i e mo ion ajec o y co esponding o he impac pa ame e b. The o al exci a ion c oss sec ion ␴ ␣ is hen e alua ed by in eg a ing he p obabili y o e all he impac pa ame e s s a ing om a minimum one, bmin .In Coulomb exci a ion s udies, he alue o he la e is chosen acco ding o a sys ema ics 关2兴 ollowing some p esc ip ion based on he condi ion ha he con ibu ions om he nuclea ield should be elimina ed. E en so, howe e , some ambigui ies a e p esen since he calcula ed c oss sec ions can a y app eciably o small a ia ions o bmin . Mo eo e , when he bomba ding ene gy is no e y high and he wo nuclei a e no e y hea y, he nuclea exci a ion is he domi- nan p ocess. In his si ua ion one canno apply ha p oce- du e because, in p inciple, one should add mo e in e nal a- jec o y o he de e mina ion o ␴ ␣ . On he o he hand, he ajec o ies co esponding o small impac pa ame e s would no con ibu e oo much o he inelas ic c oss sec ion i he abso p ion due o all o he channels is aken in o accoun . This can be done by in oducing an op ical po en ial as was al eady done in a quali a i e way in Re . 关3兴. In his pape we p esen calcula ions o he exci a ion c oss sec ion o one- and wo-phonon s a es in he 40Ca⫹40Ca eac ion a 50 MeV/m o which expe imen al esul s exis 关4兴. The calcula ions a e done wi hin he ex- ended andom-phase app oxima ion 共RPA兲model desc ibed in ou p e ious wo ks 关5–7兴whe e we ha e in oduced an- ha monici ies in he in e nal Hamil onian and nonlinea e ms in he ex e nal ield. This model has been success ul in he desc ip ion o he exci a ion o he double gian eso- nances 共DGR兲, educing he disc epancy be ween he mea- su ed c oss sec ion and he s anda d heo e ical es ima e. He e he model is ex ended by in oducing an op ical po en- ial in o de o a oid he unce ain y on he in eg a ion o e he impac pa ame e . Since he op ical po en ial akes in o accoun he abso p ion due o all channels, we ha e in o- duced a p ocedu e in o de o a oid double coun ing he e ec s o he channels explici ly included in ou calcula ions. In he ollowing sec ion we will ecall b ie ly ou model and ex ensi ely desc ibe i s imp o emen s. In Sec. III we p esen ou esul s and he quan i a i e compa ison wi h he expe i- men al indings. We hen d aw ou conclusions and discuss some pe spec i es. II. THE MODEL The bes mic oscopic heo y o desc ibe collec i e exci a- ions in nuclei is he RPA whose Hamil onian can be w i en as PHYSICAL REVIEW C, VOLUME 65, 014608 0556-2813/2001/65共1兲/014608共7兲/$20.00 ©2001 The Ame ican Physical Socie y65 014608-1 HRPA⫽兺 ␯ E ␯ Q ␯ †Q ␯ ,共2.1兲 whe e he phonon c ea ion ope a o is Q ␯ †⫽兺 p,h共Xph ␯ Bph †⫺Yph ␯ Bph兲.共2.2兲 The bosonic ope a o s Ba e he lowes -o de e ms o he bosonic expansion o he e mionic ope a o s 关8兴 ap †ah→Bph †⫹共1⫺&兲兺 p⬘h⬘ Bp⬘h⬘ †Bp⬘h †Bph⬘⫹¯.共2.3兲 He e, he index p(h) labels he pa icle 共hole兲s a es wi h espec o he Ha ee-Fock 共HF兲g ound s a e. The o he e ms a e he i s one co ec o he Pauli p inciple. In he ha monic RPA Hamil onian 共2.1兲only he Vph,p⬘h⬘ and Vpp⬘,hh⬘ e ms o he esidual in e ac ion a e aken in o accoun . I we conside also he o he e ms Vpp⬘,p⬙,p⵮, Vhh⬘,h⬙h⵮,Vpp⬘,p⬙h, and Vph,h⬘,h⬙and in oduce he map- pings 关8兴 ap †ap⬘→共ap †ap⬘兲B⫽兺 hBph †Bp⬘h, 共2.4兲 ahah⬘ †→共ahah⬘ †兲B⫽兺 pBph †Bph⬘. We end up wi h a Hamil onian con aining cubic, qua ic, e c. e ms in he phonon c ea ion and annihila ion ope a o s. In he space spanned by one- and wo-phonon s a es he bosonic Hamil onian is H⫽兺 ␯ E ␯ Q ␯ †Q ␯ ⫹ 冋 兺 ␯ 1 ␯ 2 ␯ 3 V ␯ 1 ␯ 2 ␯ 3 21 Q ␯ 1 †Q ␯ 2 †Q ␯ 3 † ⫹兺 ␯ 1 ␯ 2 ␯ 3 ␯ 4 V ␯ 1 ␯ 2 ␯ 3 ␯ 4 22 Q ␯ 1 †Q ␯ 2 †Q ␯ 3Q ␯ 4 册 ⫹H.c., 共2.5兲 whe e V21 (V22) a e he ma ix elemen s connec ing one- pho on s a es wi h wo-phonon s a es 共 wo-pho on s a es wi h wo-phonon s a es兲. The eigens a es o he Hamil onian 共2.5兲 a e 兩 ⌽ ␣ 典 ⫽兺 ␯ c ␯ ␣ 兩 ␯ 典 ⫹兺 ␯ 1 ␯ 2 d ␯ 1 ␯ 2 ␣ 兩 ␯ 1 ␯ 2 典 共2.6兲 and he co esponding eigen alues do no o m a ha monic spec um. In he semiclassical models o g azing ion-ion collisions he exci a ion o one o he wo nuclei is due o he mean ield o he o he . Since he mean ield is a one-body ope a- o , he exci a ion ope a o has he ollowing o m: W共 兲⫽兺 ␣␤ 具 ␣ 兩 UB„R共 兲… 兩 ␤ 典 a ␣ †a ␤ ,共2.7兲 whe e UBis he mean ield o he o he nucleus. The ime dependence comes in h ough he ela i e dis ance Rbe- ween he wo nuclei. In he s anda d app oach W( ) is linea in he phonon ope a o s because only he ph e ms o Eq. 共2.7兲a e conside ed and he lowes -o de boson expansion is aken. I we include also he pp and hh e ms, hei mapping 关Eq. 共2.4兲兴 leads o a quad a ic o m in Q W⫽W00⫹兺 ␯ W ␯ 10Q ␯ †⫹H.c.⫹兺 ␯␯ ⬘ W ␯␯ ⬘ 11 Q ␯ †Q ␯ ⬘ ⫹兺 ␯␯ ⬘ W ␯␯ ⬘ 20 Q ␯ †Q ␯ ⬘ †⫹H.c. 共2.8兲 The i s e m in Eq. 共2.8兲 ep esen s he in e ac ion o he wo colliding nuclei in hei g ound s a e; in he p esen case i has also an imagina y pa ha desc ibes he abso p ion due o he nonelas ic channels. The W10 pa connec s s a es di e ing by one phonon, he W11 e m couples exci ed s a es wi h he same numbe o phonons, while W20 allows ansi- ions om he g ound s a e o wo-phonon s a es. All he o m ac o s Wa e calcula ed by double olding he Coulomb and nuclea nucleon-nucleon in e ac ions wi h he Ha ee- Fock g ound-s a e densi y o he p ojec ile and wi h he g ound s a e densi y o he ansi ion densi ies o he consid- e ed exci ed s a es o he a ge . In he space o he g ound s a e and he 兩 ⌽ ␣ 典 s a es we can cas he Sch o ¨dinge equa ion in o a se o linea di e - en ial coupled equa ions o he ime-dependen ampli ude p obabili ies A ␣ ( ). Then he c oss sec ion is calcula ed non- pe u ba i ely as desc ibed in Re . 关6兴whe e we in eg a ed he p obabili y o exci ing he s a e 兩 ⌽ ␣ 典 s a ing om a minimum impac pa ame e . In he calcula ion p esen ed he e we in eg a e o e all impac pa ame e s since we ha e in o- duced in W00 he op ical po en ial, which, in an e ec i e way, akes ca e o he mos inne ajec o ies. The imagina y pa Wim o he op ical po en ial is usually de e mined by i ing he expe imen al elas ic c oss sec ion. This po en ial desc ibes he abso p ion due o all nonelas ic channels. The e o e, i canno be inse ed di ec ly in W00 关Eq. 共2.7兲兴 since he abso p ion due o he inelas ic channels explici ly included in he coupled equa ions would be coun ed wice. Le us i s discuss how o sol e his p oblem when no anha monici ies a e p esen and he e o e, he s a es 兩 ⌽ ␣ 典 a e pu e mul iphonon s a es. In such a case one can sol e he Sch o ¨dinge equa ion in a semiclassical app oach by in e- g a ing i along each classical ela i e mo ion ajec o y. The s a e o he sys em 兩⌿典 is a cohe en s a e and he p obabili y o exci e m ␯ imes a phonon ␯ is 关9兴 P ␯ ,m ␯ 0⫽共N ␯ 兲m ␯ m ␯ !Pg.s. 0,共2.9兲 whe e N ␯ is he a e age numbe o ␯ phonons in 兩⌿典. In he abo e equa ion, as well as in he ollowing discussion, he dependence on he impac pa ame e bis unde s ood. The M. V. ANDRE ´Se al. PHYSICAL REVIEW C 65 014608 014608-2 supe sc ip ‘‘0’’ e e s o he ac ha only he abso p ion due o he mul iple exci a ion o phonons is aken in o accoun . In such a case Pg.s. 0⫽e⫺N,共2.10兲 whe e N⫽⌺ ␯ N ␯ . We s ess ha he same su i al p obabil- i y o he g ound s a e appea s as a ac o in all he p obabili- ies in Eq. 共2.9兲. The su i al p obabili y associa ed wi h he imagina y op ical po en ial Wim is calcula ed as Pg.s. W⫽exp 再 2 បc 冕 ⫺⬁ ⫹⬁Wim共 兲d 冎 ,共2.11兲 whe e he in eg al is again done along a classical ajec o y. The depopula ion o he g ound s a e due only o he ne- glec ed channels can be, in p inciple, calcula ed as in Eq. 共2.11兲bu wi h an auxilia y imagina y po en ial W ¯ , which does no con ain he abso p ion due o he adop ed ones. Then Pg.s. W⫽Pg.s. 0⫻Pg.s. W ¯ 共2.12兲 and P ␯ ,m ␯ ⫽P ␯ ,m ␯ 0⫻Pg.s. W ¯ .共2.13兲 When anha monici ies a e aken in o accoun , he s a e o he sys em is no mo e a cohe en s a e. The p obabili y o exci e he s a e 兩 ⌽ ␣ 典 is equal o P ␣ 0⫽ 兩 A ␣ 兩 2,共2.14兲 whe e A ␣ is solu ion o he coupled equa ions o mo ion wi hou any imagina y po en ial. The e o e, P ␣ 0con ains only he abso p ion due o all he adop ed channels. The emain- ing pa , due o all he o he channels, can be in oduced by w i ing, in analogy wi h Eq. 共2.13兲, P ␣ ⫽P ␣ 0⫻Pg.s. W ¯ .共2.15兲 This equa ion can be o mally de i ed by assuming ha he abso p ion due o he excluded channels is he same in all he adop ed ones. This is ce ainly an app oxima ion, howe e , we would like o emphasize ha many impo an inelas ic channels a e explici ly aken in o accoun in he coupled equa ions and ha we sol e he la e exac ly. The e o e, he co esponding abso p ion is calcula ed co ec ly, including he Q- alue e ec s. The unknown auxilia y imagina y po en- ial W ¯ can be elimina ed by inse ing Eq. 共2.12兲in Eq. 共2.15兲, P ␣ ⫽P ␣ 0⫻Pg.s. W Pg.s. 0,共2.16兲 which is he exp ession we ha e used in o de o calcula e he inelas ic c oss sec ion. We would like o s ess ha he pa o he nuclea abso p ion ha co esponds o noninelas- ic channels is o en aken in o accoun as a sha p cu o ansmission coe icien . So he in oduc ion o he imagina y po en ial can be seen as an impo an imp o emen . III. RESULTS AND DISCUSSION The abo e desc ibed model has been applied o he eac- ion 40Ca on 40Ca a E/nucleon⫽50MeV. The one-phonon basis has been ob ained wi h a sel -consis en HF⫹RPA cal- cula ion wi h Sky me in e ac ion SGII 关10兴. Only he mos collec i e one-phonon s a es, exhaus ing a leas 5% o he ele an ene gy-weigh ed sum ule 共EWSR兲, a e aken in o accoun . They a e lis ed in Table I. We hen ha e conside ed all possible wo-phonon s a es ha can be cons uc ed ou o hem, wi h all possible alues o he o al angula momen um L, and in his space we ha e diagonalized he Hamil onian 共2.5兲 o ge he s a es 共2.6兲. In Table II we ha e epo ed some p ope ies o he quad upole s a es, each one labeled wi h he name o i s main componen and whose unpe u bed ene gy is gi en in he second column. In he hi d column he e a e he ene gy shi s due o he anha monici ies. Thei o e laps wi h he single and double isoscala gian quad u- pole esonance 共ISGQR兲s a es a e shown in he las wo columns. Simila ables o he DGR s a es a e epo ed in 关6兴. The elemen a y nuclea o m ac o s W o pu e one- and wo-phonon con igu a ions 关Eq. 共2.8兲兴 we e calcula ed by double olding he M3Y nucleon-nucleon in e ac ion 关11兴 wi h he RPA ansi ion densi ies. The ansi ion ma ix ele- men s be ween mixed s a es 兩 ⌽ ␣ 典 we e compu ed by mixing he elemen a y o m ac o s acco ding o he uni a y ans- o ma ion 共2.6兲. The same p ocedu e was used wi h he Cou- lomb in e ac ion o calcula e he Coulomb o m ac o s. The ela i e mo ion ajec o ies we e de e mined by sol ing he classical equa ion o mo ion in he p esence o bo h he Cou- lomb ield and he eal pa o he nuclea po en ial. The eal pa o he op ical po en ial was ob ained by double olding he M3Y nucleon-nucleon po en ial wi h he Ha ee-Fock densi ies o he wo nuclei while i s imagina y pa was chosen wi h he same geome y and mul iplied by a scale ac o whose alue 共0.627兲was de e mined by a i o he expe imen al elas ic c oss sec ion o he collision 40Ca TABLE I. RPA one-phonon basis o he nucleus 40Ca. We ha e wo gian monopole esonance 共GMR兲s a es as well as wo dipole 共GDR兲and wo quad upole: isoscala 共ISGQR兲and iso ec o 共IVGQR兲. Fo he oc upole esonance we ha e he low ene gy 共LEOR兲and he high ene gy one 共HEOR兲. Fo each s a e i s spin, pa i y, isospin, ene gy, and pe cen age o he EWSR a e epo ed. Phonons J ␲ TE共MeV兲EWSR 共%兲 GMR10⫹0 18.25 30 GMR20⫹0 22.47 54 GDR11⫺1 17.78 56 GDR21⫺1 22.03 10 ISGQR 2⫹0 16.91 85 IVGQR 2⫹1 29.59 26 3⫺3⫺0 4.94 14 LEOR 3⫺0 9.71 5 HEOR 3⫺0 31.33 25 MICROSCOPIC DESCRIPTION OF COULOMB AND . . . PHYSICAL REVIEW C 65 014608 014608-3 on 40Ca a E/nucleon⫽50MeV o Re . 关12兴. In hese calcula ions bo h he nuclea and Coulomb exci- a ions we e included. Ac ually, he Coulomb exci a ion alone does no p oduce a sizable c oss sec ion because he colliding nuclei a e no e y hea y, bu when i is conside ed oge he wi h he nuclea exci a ion i p oduces an in e e - ence e ec , which can be impo an . This is due o he ac ha on one hand we ha e a coupled channel e ec and, on he o he hand, some wo-phonon s a es a e exci ed only when bo h ields a e ac ing. This was clea ly demons a ed in ou p e ious wo k 关7兴. Since ou calcula ions a e based on a disc e e RPA we ge a disc e e exci a ion spec um and a c oss sec ion ␴ ␣ co e- sponding o each s a e 兩 ⌽ ␣ 典 . The ene gy di e en ial c oss sec ions p esen ed in Fig. 1 a e ob ained by summing up all he con ibu ions coming om he s a es 兩 ⌽ ␣ 典 a e a smoo hing o each indi idual line by a Lo en zian wi h a 3 MeV wid h. The dashed line e e s o a calcula ion whe e he in e nal Hamil onian is ha monic and he ex e nal ield is linea . The solid line co esponds o a calcula ion whe e he anha monici y and nonlinea i y we e in oduced, which p o- duce a sizable inc ease wi h espec o he s anda d case. In he igu e we can clea ly dis inguish h ee ene gy egions. The c oss sec ions gi en in Tables III–V a e ob ained by summing up he ␴ ␣ ’s o he disc e e s a es 兩 ⌽ ␣ 典 lying in each egion. As al eady obse ed in Re s. 关6,7兴, he inc ease a low ene gies is due bo h o he anha monici ies and non- linea i ies. In pa icula , he anha monici ies a e impo an because he low-lying wo-phonon s a es can be exci ed by he W10 pa o he ex e nal ield h ough hei la ge one- phonon componen . A high ene gies he main con ibu ion comes om he nonlinea i ies because hei p esence in- c eases he numbe o exci a ion ou es. This is seen be e in Table IV whe e he exci a ion c oss sec ion in he double gian quad upole esonance ene gy egion is epo ed. Fo each mul ipola i y we ha e summed he exci a ion c oss sec- ion in he ene gy egion be ween 28 and 38 MeV, and his is done o ou di e en cases as shown in he able. The L ⫽3 con ibu ion is due o he HEOR a 31.33 MeV, while he L⫽0, 2, and 4 con ibu ions a e domina ed by he double exci a ion o he double ISGQR. As we can see in Table V, he nonlinea e ms a e also esponsible o he inc ease o he c oss sec ion in he ISGQR egion, especially o he L ⫽2 s a e whose main componen is he ISGQR. This is a a iance wi h he ela i is ic Coulomb exci a ion s udied in Re . 关6兴because he Coulomb in e ac ion e y selec i ely popula es dipole ansi ions and he e o e canno exci e he mos impo an wo-phonon componen s o he ISGQR, which a e buil wi h monopole and quad upole phonons 共see Table II兲. The ob ained a io be ween he c oss sec ion in he gian TABLE II. Cha ac e is ics o he 兩 ⌽ ␣ 典 quad upole 2⫹s a es whose majo componen s a e in he i s column. In he second column we show he ene gies o he majo componen s in he ha monic app oach. The shi in he ene gy p oduced by he anha monici ies is indica ed by ⌬E共in KeV兲. We can compa e hese alues wi h he diagonal ma ix elemen s o he esidual in e ac ion, ⌬E0共in KeV兲. In he las columns we epo he ampli ude wi h which he single and double ISGQR componen s appea in he mixed s a es. Quad upole S a es E0共MeV兲⌬E(⌬E0)cISGQR cISGQR-ISGQR ISGQR 16.910 ⫺402 0 0.985 ⫺0.014 IVGQR 29.594 ⫺506 0 ⫺0.005 0.017 GMR1丢ISGQR 35.155 87 ⫺11 ⫺0.073 ⫺0.028 GMR1丢IVGQR 47.845 ⫺42 ⫺187 ⫺0.000 0.002 GMR2丢ISGQR 39.378 246 ⫺31 ⫺0.108 ⫺0.014 GMR2丢IVGQR 52.067 190 ⫺178 ⫺0.002 0.003 GDR1丢GDR135.560 ⫺464 ⫺505 0.034 0.087 GDR1丢GDR239.814 ⫺436 ⫺439 0.009 0.006 GDR1丢3⫺22.722 ⫺31 ⫺35 0.029 ⫺0.000 GDR1丢LEOR 27.486 ⫺444 ⫺442 ⫺0.013 ⫺0.007 GDR1丢HEOR 49.110 ⫺278 ⫺288 ⫺0.005 0.006 GDR2丢GDR244.058 ⫺435 ⫺436 0.004 0.002 GDR2丢3⫺26.976 ⫺6 7 0.003 0.001 GDR2丢LEOR 31.740 ⫺307 ⫺309 0.600 ⫺0.007 GDR2丢HEOR 53.364 ⫺212 ⫺217 0.000 0.000 ISGQR 丢ISGQR 33.819 0 4 ⫺0.020 0.995 ISGQR 丢IVGQR 46.508 39 40 0.002 0.002 IVGQR 丢IVGQR 59.198 ⫺247 ⫺250 ⫺0.007 ⫺0.004 3⫺丢3⫺9.884 750 776 ⫺0.045 ⫺0.005 3⫺丢LEOR 14.648 ⫺267 ⫺241 0.086 0.001 3⫺丢HEOR 36.272 ⫺104 ⫺120 0.025 ⫺0.003 LEOR 丢LEOR 19.413 ⫺271 ⫺269 ⫺0.021 ⫺0.000 LEOR 丢HEOR 41.037 ⫺192 ⫺197 ⫺0.005 0.002 HEOR 丢HEOR 62.660 ⫺212 ⫺215 ⫺0.006 ⫺0.001 M. V. ANDRE ´Se al. PHYSICAL REVIEW C 65 014608 014608-4 esonance egion and ha in he wo-phonon egion a ies om 3.7 in he anha monic and nonlinea case o 4.6 in he ha monic and linea calcula ion. I we only conside he c oss sec ion up o he single and double isoscala gian quad upole esonance, hese a ios inc ease o 6.5 and 9.6, espec i ely. Those alues a e smalle han he ones epo ed in Re . 关3兴 o he c oss sec ions a he g azing angle. This di e ence can be aced back o he p esen a ailabili y o he expe imen al elas ic c oss sec ion needed o ix he imagina y pa o he op ical po en ial and o he ac ha he heo e ical app oach has been imp o ed in se e al aspec s, especially in he calcula ion o he o m ac o s. Ou calcula ion can be compa ed wi h he expe imen al da a o Re . 关4兴whe e he eac ion 40Ca⫹40Ca a 50 MeV/ nucleon has been s udied. Le us esume he impo an e- sul s o Re . 关4兴and he mos c i ical poin s. We discuss i s he inclusi e spec um and la e on we will analyze he one ob ained in coincidence wi h backwa d emi ed pa icles. The inelas ic spec um was ex ac ed o ejec iles sca e ed be- ween 3.4° and 10° in he cen e o mass ame. The GR con ibu ion was ob ained om he inclusi e inelas ic spec- um by decon olu ion o he angula dis ibu ions in o in- elas ic exci a ions and a noninelas ic backg ound. Fo he inelas ic exci a ion, a dis o ed-wa e Bo n app oxima ion 共DWBA兲p edic ion was used. As o he backg ound, i s angula dis ibu ion was assumed o be simila o he one o he ene gy egion loca ed immedia ely abo e he GRs. This p ocedu e ga e 113 mb/s be ween 12 and 22 MeV o he inelas ic exci a ion co esponding o 40% o he quad upole EWSR. Howe e , i should be no iced ha he es ima e o he noninelas ic backg ound unde lying he GR is no unam- biguous. Indeed, i inelas ic exci a ion is s ill p esen in he egion abo e he esonance as expec ed om Fig. 1, he as- sumed backg ound is o e es ima ed. In his case he ex- ac ed alue should be unde s ood as a minimum. The maxi- mum inelas ic con ibu ion compa ible wi h he measu ed angula dis ibu ion is 223 mb/s . This co esponds o he o he ex eme when no noninelas ic backg ound is consid- e ed. The e o e he GR c oss sec ion ex ac ed om he in- clusi e spec um is be ween 113 and 223 mb/s depending upon he backg ound hypo hesis. The associa ed EWSR would hus ange be ween 40% and 80% i he whole c oss sec ion is assumed o be coming om quad upole s a es. In o de o ge he o al c oss sec ion one has o ex apo- la e he measu ed di e en ial c oss sec ion beyond he solid angle co e ed by he ejec ile de ec o . This was done by assuming ha he DWBA angula dis ibu ion used o i he measu ed angula dis ibu ion in Re . 关4兴was also alid in FIG. 1. Inelas ic c oss sec ion o he sys em 40Ca⫹40Ca a 50 MeV/nucleon as a unc ion o he exci a ion ene gy. Bo h cu es a e he esul o a smoo hing p ocedu e wi h a Lo en zian o wid h ⌫ ⫽3 MeV. The shadowed a eas a e he ene gy egions o e which we ha e summed he c oss sec ions epo ed in he ables. TABLE III. Coulomb plus nuclea exci a ion c oss sec ion o 40Ca⫹40Ca a 50 MeV/nucleon. Each mul ipola i y con ibu ion is shown o se e al anha monic and nonlinea combina ions. The al- ues o L⫽1 and 5 a e e y small and hey a e no shown. The c oss sec ions 共in milliba ns兲a e summed o e he ene gy egion (0⭐E⭐12 MeV). Phonons Ha m. and lin. Ha m. and nonlin. Anh. and lin. Anh. and nonlin. L⫽0 0.1 0.3 1.3 2.3 L⫽2 0.2 0.4 0.2 0.1 L⫽3 14.2 16.9 14.3 16.8 L⫽4 0.2 0.3 0.2 0.3 L⫽6 0.5 0.7 0.4 0.7 To al 15.2 18.6 16.4 20.2 TABLE IV. Same as Table III bu o he double ISGQR egion. The c oss sec ions 共in milliba ns兲a e summed o e he ene gy e- gion (28⭐E⭐38MeV). The alues in pa en heses co espond o he double ISGQR s a e. Phonons Ha m. and lin. Ha m. and nonlin. Anh. and lin. Anh. and nonlin. L⫽0 0.2 共0.15兲0.3 共0.26兲0.2 共0.14兲0.3 共0.21兲 L⫽2 0.6 共0.33兲1.0 共0.51兲0.6 共0.33兲1.1 共0.53兲 L⫽3 2.2 2.5 2.3 2.5 L⫽4 1.0 共0.90兲1.9 共1.83兲0.9 共0.85兲1.8 共1.73兲 L⫽6 0.2 0.2 0.2 0.2 To al 4.2 共1.38兲5.9 共2.60兲4.2 共1.32兲5.9 共2.47兲 TABLE V. Same as Table III bu o he ISGQR egion. The c oss sec ions 共in milliba ns兲a e summed o e he ene gy egion (14⭐E⭐20 MeV). In his egion he e a e no s a es wi h L⫽3 and 5. The alues in pa en heses co espond o he ISGQR s a e. Phonons Ha m. and lin. Ha m. and nonlin. Anh. and lin. Anh. and nonlin. L⫽0 2.7 2.8 2.2 2.2 L⫽1 3.0 2.9 3.6 3.3 L⫽2 13.3共13.2兲16.1共16.0兲13.8共13.6兲16.0共16.0兲 L⫽4 0.1 0.1 0.1 0.1 L⫽6 0.2 0.3 0.2 0.3 To al 19.3 22.2 19.9 21.9 MICROSCOPIC DESCRIPTION OF COULOMB AND . . . PHYSICAL REVIEW C 65 014608 014608-5 he egion in which no da a a e a ailable. The a io be ween he in eg als o he DWBA c oss sec ion o e he ull angula ange and ha o e he angles co e ed by he de ec o is 3.16. Taking in o accoun he ac ion o he solid angle co - e ed by he spec ome e one ge s a o al compensa ing ac o o 6.67⫻10⫺2. Such a ac o ans o ms he double di e en- ial c oss sec ion in o he ene gy di e en ial one. The esul - ing o al c oss sec ion is hen 7.5 and 15 mb, espec i ely. These alues ha e o be compa ed wi h he heo e ical in- elas ic c oss sec ion, which, in he anha monic and nonlinea case, adds up o 22 mb in he GR egion. Taking in o accoun he unce ain ies o he analysis o he expe imen al da a and he ac ha ou heo e ical esul s a e ob ained wi hou ad- jus ing any pa ame e , he compa ison can be conside ed sa - is ac o y. In o de o d aw quan i a i e conclusions one should elabo a e on di e en issues bo h om he expe i- men al and heo e ical sides. A ecen expe imen on he same eac ion 关13兴using an imp o ed appa a us is expec ed o elimina e mos o he expe imen al unce ain ies. These new da a will allow a mo e eliable de e mina ion o some pa ame e s en e ing in he heo e ical calcula ion, mainly in he op ical po en ial. Coincidences wi h backwa d emi ed pa icles p o ide an unambiguous signal o he inelas ic exci a ions and could, in p inciple, be used o a oid he noninelas ic backg ound p ob- lems. This was he idea o Re . 关4兴, bu some o he sou ces o unce ain ies appea . The coincidence a e wi h backwa d emi ed p o ons was con e ed in o a di e en ial c oss sec- ion, co ec ing o he ene gy dependence o he p o on mul- iplici y. A ha ime i was al eady s essed ha his co ec- ion ac o can be subjec o many unce ain ies. Fi s o all, his p o on mul iplici y unc ion was calcula ed wi h a s a is- ical decay code, which does no include any di ec decay componen . Fu he mo e, due o he absence o ou -o -plane de ec o s, he azimu hal angula dis ibu ion was no mea- su ed and was assumed o be uni o m. This p ocedu e gi es a c oss sec ion o he GR ex ac ed om he coincidence da a 共339 mb/s 兲la ge han he one ob ained om he inclu- si e inelas ic spec um 共be ween 113 and 223 mb/s 兲. This shows ha he hypo heses used a e no co ec . The use o a 4 ␲ de ec o in a ecen expe imen 关13兴should sol e hese ambigui ies since i will p o ide he angula dis ibu ion o he emi ed p o ons and he e will be no need o ely on a s a is ical code o in e hei mul iplici y. Tha was no he case in he expe imen o Re . 关4兴. The e o e, only he a io was deduced om he coincidence da a. Two alues o his a io we e epo ed by assuming wo backg ounds o he wo-phonon egion, while he GR peak was conside ed wi h no backg ound sub ac ion in he coincidence spec um. The alues o he second phonon c oss sec ion we e, a e sub- ac ion o he wo backg ounds, 30 and 17 mb/s , espec- i ely, o an ene gy unning om 28 o 40 MeV, while he GR c oss sec ion was 339 mb/s in he coincidence spec um in he ange 12–22 MeV, leading o he a ios 11 and 20 quo ed in Re . 关4兴.1Such alues a e he a ios be ween he single GR c oss sec ion and only a small ac ion o he DGR c oss sec ion. We wan o s ess he e ha he co ec p oce- du e should be no o sub ac any backg ound in he wo- phonon egion. Indeed, on one hand, coincidence wi h back- wa d emi ed pa icles a oids any con ibu ion om noninelas ic backg ound in he expe imen al da a. On he o he hand, in ou heo e ical calcula ion, no only double GQR has been included bu many con ibu ions om di e - en inelas ic exci a ions ha e been aken in o accoun . These wo ema ks plead in a o o a di ec compa ison o he one- and he wo-phonon egions wi h no backg ound sub ac ion. In o de o ha e a mo e di ec compa ison we p esen , in Fig. 2, he expe imen al coincidence inelas ic spec um o Re . 关4兴关Fig. 16共b兲兴 wi h no backg ound sub ac ion. The igh scale is he double di e en ial c oss sec ion while he le scale is he ene gy di e en ial c oss sec ion ob ained wi h he abo e men ioned ac o o 6.67⫻10⫺2. In he igu e we p esen he heo e ical esul s smoo hed by a Lo en zian o 5 MeV wid h, a he han he 3 MeV used in Fig. 1. F om he igu e we see ha wi h his alue he shape o he expe i- men al peak in he GR egion is well ep oduced. I should be no iced ha some con ibu ion o he expe imen al c oss sec ion is p esen jus below 14 MeV. Howe e , due o he p oximi y o he p o on emission h eshold, he co ec ion o he mul iplici y is mo e delica e in ha ene gy egion. Dis- ega ding hese wo poin s, he o e all ag eemen be ween heo y and expe imen is a he sa is ac o y. A ough es ima e o he one-phonon and wo-phonon c oss sec ions can be ob ained by in eg a ing bo h cu es in he ene gy anges shown in Fig. 2 as shadowed a eas. By doing ha one would ge an expe imen al and heo e ical a io o 2.4 and 2.3, e- spec i ely. We wan o s ess ha he expe imen al a io 1One digi -in e sion e o was spo ed in he ex o Figs. 16共c兲 and 16共d兲o Re . 关4兴共e a um o be published兲. FIG. 2. The do s ep esen he expe imen al coincidence inelas- ic spec um o Re . 关4兴关Fig. 16共b兲兴 wi h no backg ound sub ac ion 共 igh scale兲. The solid line is he esul o a smoo hing p ocedu e wi h a Lo en zian wi h a wid h ⌫⫽5 MeV o he heo e ical inelas- ic c oss sec ion o he anha monic and nonlinea case 共le scale兲. The shadowed a eas a e he ene gy egions o e which we ha e in eg a ed he ene gy di e en ial c oss sec ions. The esul ing al- ues in milliba ns a e he numbe s epo ed in he wo a eas. Those abo e he cu es e e o he heo e ical esul s, while he ones below e e o he expe imen al da a. In he inse we epo he a ios be ween he single GR c oss sec ion and he DGR ones o he wo cases. M. V. ANDRE ´Se al. PHYSICAL REVIEW C 65 014608 014608-6 quo ed abo e is di e en om he one deduced in Re . 关4兴 because he la e is he a io be ween he ull peak o he single GR and he DGR wi h backg ound sub ac ion, while he o me one is ob ained wi hou backg ound sub ac ion in bo h single GR and DGR. Fu he mo e, he i s wo expe i- men al poin s in Fig. 2 we e no included as explained be- o e. Finally, we would like o commen on he dependence o he heo e ical a ios upon he smoo hing wid h. This a io is dec easing wi h he inc easing wid h, due o he ac ha while he in eg al o he single GR is dec easing, ha o e he egion o he DGR emains almos unchanged. This is ela ed o he ac ha in he single GR ene gy egion he peaks o he single ⌽ ␣ s a es a e qui e sepa a e while he densi y o s a es in he DGR egion is e y high. In any case he dependence on he wid h is no e y s ong; by a ying ⌫ om 3 o 6 MeV he a io changes om 2.75 o 2.20. These alues canno be di ec ly compa ed wi h he alues epo ed in Tables III–V because he la e ha e been ob ained jus by summing he c oss sec ions associa ed wi h each disc e e s a e. IV. CONCLUSIONS We ha e calcula ed he inelas ic sca e ing c oss sec ions o one- and wo-phonon s a es o he 40Ca⫹40Ca collision a E/nucleon⫽50MeV. Se e al e ec s ha e been e idenced. In pa icula , we ha e analyzed he ole played by anha mo- nici ies in he exci a ion spec um and nonlinea i ies in he ope a o desc ibing he mu ual in e ac ion o he collision pa ne s. The anha monici ies a e pa icula ly impo an a ela i ely low ene gy whe e he exci a ion comes h ough he one-phonon componen o he mixed s a es. The nonlinea i- ies gi e hei main con ibu ion a high ene gy, in pa icula , in he egion o he double quad upole gian esonance. Namely, in he in e al be ween 28 and 38 MeV, hey gi e an inc ease o abou 40% wi h espec o a ha monic and linea calcula ion. This inc ease is due o he exci a ion o o he wo-phonon s a es ha a e popula ed because o he p esence o he anha monici ies and nonlinea i ies. Wi h all he p e i- ously discussed ca ea s, he compa ison o he smoo hed he- o e ical esul wi h he expe imen al coincidence inelas ic spec um o Re . 关4兴is sa is ac o y. The inclusion o h ee- phonon s a es in he calcula ion will inc ease he inelas ic c oss sec ion a highe exci a ion ene gies. A he same ime, a ac ion o he popula ion o he wo-phonon s a es will mo e o highe ene gies. In Re . 关14兴i has been shown, wi hin a simple model, ha he spec um calcula ed by di- agonalizing he Hamil onian ob ained by a boson expansion unca ed a he qua ic o de in a space including up o h ee phonons is in easonable ag eemen wi h he exac one. A simila calcula ion is easible also in a ealis ic case. This, oge he wi h he esul s shown he e, encou ages us o p o- ceed in he di ec ion o calcula ing he h ee-phonon exci a- ion c oss sec ion o he sys em 40Ca⫹40Ca a E/nucleon ⫽50MeV o which expe imen s ha e al eady been done 关13兴. ACKNOWLEDGMENTS This wo k has been pa ially suppo ed by he Spanish DGICyT unde Con ac No. PB98-1111, by he Spanish- I alian ag eemen be ween he CICyT and he INFN, and by he Spanish-F ench ag eemen be ween he CICyT and he IN2P3. 关1兴H. Emling, P og. Pa . Nucl. Phys. 33, 729 共1994兲; Ph. Chomaz and N. F asca ia, Phys. Rep. 252, 275 共1995兲; T. Au- mann, P. F. Bo ignon, and H. Emling, Annu. Re . Nucl. Pa . Sci. 48, 351 共1998兲. 关2兴C. Benesh, B. Cook, and J. Va y, Phys. Re . C 40, 1198 共1989兲. 关3兴F. Ca a a, Ph. Chomaz, and A. Vi u i, Nucl. Phys. A471, 661 共1987兲. 关4兴J. A. Sca paci e al., Phys. Re . C 56, 3187 共1997兲; Phys. Re . Le . 71, 3766 共1993兲. 关5兴C. Volpe, F. Ca a a, Ph. Chomaz, M. V. And e ´s, and E. G. Lanza, Nucl. Phys. A589, 521 共1995兲;A599, 347c 共1996兲. 关6兴E. 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