Global and consistent analysis of the heavy-ion elastic scattering and fusion processes

Global and consis en analysis o he hea y-ion elas ic sca e ing and usion p ocesses L. R. Gasques,1L. C. Chamon,1D. Pe ei a,1M. A. G. Al a ez,1E. S. Rossi, J .,1C. P. Sil a,1and B. V. Ca lson2 1Labo a ó io Pelle on, Ins i u o de Física da Uni e sidade de São Paulo, 05315-970 São Paulo, São Paulo, B azil 2Depa amen o de Física, Ins i u o Tecnológico de Ae onáu ica, Cen o Técnico Ae oespacial, São José dos Campos, São Paulo, B azil (Recei ed 17 Oc obe 2003; published 10 Ma ch 2004) We ha e de eloped a model o he nuclea in e ac ion which is based on he e ec s o he Pauli nonlocali y. In ea lie wo ks, we ha e success ully used his in e ac ion o desc ibe he elas ic sca e ing o se e al sys ems in a e y wide ene gy ange. In he p esen wo k, we ha e checked he alidi y o he same in e ac ion in he desc ip ion o abou 2500 usion c oss sec ion da a o 165 di e en sys ems. By in oducing only one ene gy- and sys em-independen e ec i e pa ame e , he nonlocal model desc ibes he global beha io o he usion p ocess wi h good p ecision. DOI: 10.1103/PhysRe C.69.034603 PACS numbe (s): 24.10.H , 25.60.Pj, 25.70.Jj The hea y-ion usion p ocess has been ex ensi ely s ud- ied o e he las decades [1]. I is well known ha usion c oss sec ions o hea y-ion sys ems ha e shown la ge en- hancemen s a sub-ba ie ene gies in compa ison wi h heo- e ical p edic ions om he ba ie pene a ion model [2]. These enhancemen s can be desc ibed by in oducing e ec- i e ba ie pa ame e s, which ha e been s udied in a global way o a la ge numbe o sys ems [2–4]. On he o he hand, he enhancemen s ha e been explained o se e al pa icula sys ems by conside ing he in e nal s uc u e o he pa ici- pa ing nuclei h ough couped-channel calcula ions (e.g., Re s. [5,6]). A ew models ha e also been p esen ed o de- sc ibe he elas ic sca e ing p ocess and he ene gy and sys- em dependences o he co esponding op ical po en ial [7]. Howe e , he consis ency be ween he models o he usion and elas ic sca e ing p ocesses has only been e i ied in ce - ain pa icula cases (e.g., Re . [8]). We ha e de eloped a model o he nuclea in e ac ion which has been success ul in desc ibing he elas ic sca e ing o se e al sys ems in a wide ene gy ange [9–15]. The model is based on he e ec s o he Pauli nonlocali y and is o ally pa ame e ee. In his wo k, we use his in e ac ion in he desc ip ion o abou 2500 usion c oss sec ion da a [16–53] o 165 di e en sys ems om sub-ba ie o high ene gies. Wi hin he nonlocal model, he ba e in e ac ion VNis con- nec ed wi h he olding po en ial VF h ough [54] VN共R,E兲=VF共R兲e−4 2/c2,共1兲 whe e cis he speed o ligh and is he local ela i e e- loci y be ween he wo nuclei, 2共R,E兲=2 ␮ 关E−VC共R兲−VN共R,E兲兴.共2兲 The olding po en ial 关Eq. 共3兲兴 can be ob ained in wo di e - en ways 关54兴:共i兲using he nucleon dis ibu ions o he nu- clei and an app op ia e o m o he nucleon-nucleon in e - ac ion, and 共ii兲using he ma e dis ibu ions o he nuclei wi h a ze o- ange app oach o 共 ជ 兲. We dis inguish he ma - e densi y om he nucleon one by aking in o accoun he ini e size o he nucleon. Bo h al e na i es a e equi alen in desc ibing he hea y-ion nuclea po en ial 关54兴, and we ha e adop ed he ze o- ange app oach o desc ibe he usion p o- cess: VF共R兲= 冕 ␳ 1共 1兲 ␳ 2共 2兲 共R ជ − ជ 1+ ជ 2兲d ជ 1d ជ 2.共3兲 Wi h he aim o p o iding a global desc ip ion o he nuclea in e ac ion, we p oposed 关54兴an ex ensi e sys ema iza ion o nuclea densi ies, based on expe imen al cha ge dis ibu ions and heo e ical densi ies calcula ed h ough he Di ac- Ha ee-Bogoliubo model. In ha wo k, we adop ed he wo-pa ame e Fe mi 共2pF兲dis ibu ion o desc ibe he nuclea densi ies. The adii o he 2pF dis ibu ions a e well desc ibed by R0= 1.31A1/3 − 0.84 m, 共4兲 whe e Ais he numbe o nucleons o he nucleus. The ma - e densi ies p esen an a e age di useness alue a =0.56 m. Owing o speci ic nuclea s uc u e e ec s 共single pa icle and/o collec i e兲, he pa ame e s R0and a show small a ia ions a ound he co esponding a e age alues h oughou he pe iodic able. This sys ema iza ion o he nuclea dis ibu ions is essen ial o ob ain a pa ame e - ee in e ac ion, since he olding po en ial de- pends on he densi ies o he pa ne s in he collision. In he p esen wo k, we use he nonlocal model in he con- ex o his sys ema ics, i.e., assuming he a e age di use- ness alue and Eq. 共4兲 o he adii o he dis ibu ions. The e o e, he in e ac ion does no con ain any ee pa- ame e and is qui e app op ia e o connec expe imen al esul s o di e en sys ems in a ealis ic manne . In he p esen wo k, we ha e also calcula ed he Coulomb po en- ial h ough a olding p ocedu e using he ealis ic cha ge densi ies o Re . 关54兴. In he con ex o he ba ie pene a ion model (BPM), he e ec i e po en ial is a sum o he Coulomb, nuclea , and cen i ugal pa s: Ve 共R,E兲=VC共R兲+VN共R,E兲+ᐉ共ᐉ+1兲ប2 2 ␮ R2.共5兲 The usion c oss sec ion is associa ed wi h he ansmi ed lux h ough PHYSICAL REVIEW C 69, 034603 (2004) 0556-2813/2004/69(3)/034603(5)/$22.50 ©2004 The Ame ican Physical Socie y69 034603-1 ␴ BPM共E兲= ␲ k2兺共2ᐉ+1兲Tᐉ.共6兲 In ou calcula ions, he sum in Eq. 共6兲is pe o med up o a maximum ᐉwa e, which is he g ea es ᐉ alue ha esul s a pocke 共and a ba ie 兲in he co esponding e ec i e po en- ial. Fo ᐉwa es wi h e ec i e ba ie heigh s below he cen e o mass ene gy, we ha e app oxima ed he e ec i e po en ial by a pa abola wi h cu a u e ប ␻ ᐉ. In such cases, he ansmission coe icien s can be ob ained h ough he Hill- Wheele o mula 关55兴: Tᐉ= 再 1 + exp 冋 2 ␲ 共VBᐉ−E兲 ប ␻ ᐉ 册 冎 −1,共7兲 ប ␻ ᐉ=冏ប2 ␮ d2Ve dR2冏RBᐉ 1/2 ,共8兲 whe e VBᐉand RBᐉa e he ba ie heigh and he co espond- ing adius, espec i ely. On he o he hand, o ᐉwa es wi h e ec i e ba ie s abo e he cen e o mass ene gy, ins ead o he Hill-Wheele o mula we ha e used he mo e app op ia e WKB me hod: Tᐉ=关1 + exp共Sᐉ兲兴−1,共9兲 Sᐉ= 冕 R1 R2冑8 ␮ ប2关Ve 共R,E兲−E兴dR,共10兲 whe e R1and R2a e he classical u ning poin s. A low ene gies, he WKB me hod esul s in alues o he ansmis- sion coe icien s qui e di e en om hose o he Hill- Wheele o mula. In his case, we ha e de ined he ba ie cu a u e connec ing exp essions 共7兲and 共9兲as ប ␻ ᐉ=2 ␲ 共VBᐉ−E兲 Sᐉ .共11兲 Wi hin he con ex o pa abolic ansmission coe icien s, and conside ing RBᐉ⬇RB0and ប ␻ ᐉ⬇ប ␻ 0, Wong has dem- ons a ed 关56兴 ha ␴ Wong共E兲=RB0 2ប ␻ 0 2Eln 再 1 + exp 冋 2 ␲ 共E−VBᐉ兲 ប ␻ 0 册 冎 .共12兲 Wi h he aim o ob aining sys em-independen quan i ies, we ha e de ined he ollowing educed c oss sec ion and ene gy: ␴ ed =2E RB0 2ប ␻ 0 ␴ us,共13兲 E ed =E−VB0 ប ␻ 0.共14兲 By using hese adimensional quan i ies, Eq. 共12兲can be e- cas in he ollowing sys em-independen o m: ␴ ed Wong =ln关1 + exp共2 ␲ E ed兲兴.共15兲 Howe e , we ha e e i ied ha , in some cases, he esul s ob ained om he Wong exp ession 关Eq. 共12兲兴 p esen sig- ni ican di e ences in compa ison wi h hose om he ull BPM calcula ions 关Eq. 共6兲兴. Thus, in o de o compa e ex- pe imen al esul s o e y di e en sys ems, we ha e de- ined he expe imen al educed usion c oss sec ion h ough he ollowing i ial equa ion: ␴ ed exp = ␴ us ␴ BPM ␴ ed Wong.共16兲 The educed usion c oss sec ion da a, acco ding o Eq. (16), a e p esen ed in Fig. 1, as a unc ion o he educed ene gy [Eq. (14)]. The da a se includes 165 qui e di e en sys ems anging in educed mass om abou 4 o 40 amu. A e y good ag eemen be ween da a and heo e ical p edic- ions is ob ained o ene gies abo e he s-wa e ba ie 共E ed艌0兲 o he whole se o sys ems, while sub-ba ie da a p esen la ge enhancemen s in compa ison wi h he BPM cal- cula ions. The sub-ba ie de ia ion is negligible o ␮ 艋8 and inc eases as a unc ion o he educed mass o he sys em in a ela i ely smoo h manne . A simila beha io o he sub- ba ie usion da a has al eady been e y well es ablished [2]. An inspec ion o he slopes o he da a in Fig. 1 indica es ha he enhancemen s a e mo e closely connec ed wi h he ba - ie cu a u e han wi h he ba ie heigh i sel . Taking in o accoun all hese conside a ions, we p opose a simple model o desc ibe he enhancemen s by in oducing e ec i e ba ie cu a u es. This e ec i e pa ame e is assumed o be a simple linea unc ion o he educed mass o he sys em, as gi en in Eq. (17). The i o he da a esul s in ␭=0.1/amu: ប ␻ ᐉe = 再 ប ␻ ᐉ o ␮ 艋8 amu ប ␻ ᐉ关1+␭共 ␮ −8兲兴 o ␮ 艌8 amu. 共17兲 Using Eq. 共17兲wi h ␭=0.1, he ba ie pene a ion model desc ibes wi h good p ecision he global beha io o he 2500 expe imen al usion c oss sec ion da a in he whole ene gy ange 共see Fig. 2兲. In ac , i is possible o desc ibe FIG. 1. The educed usion c oss sec ion as a unc ion o he educed ene gy, using ba ie pa ame e s ob ained wi h he nonlocal model. The igu e shows he esul s o di e en anges o he e- duced mass o he sys em. The solid lines ep esen he educed Wong equa ion. L. R. GASQUES e al. PHYSICAL REVIEW C 69, 034603 (2004) 034603-2 he abo e-ba ie da a wi hin 20% p ecision 共s anda d de- ia ion兲and he sub-ba ie da a wi hin an a e age ac o o abou 3. We conside his dispe sion a he small be- cause ou analysis has been pe o med o a la ge numbe o di e en sys ems and o e a e y wide ene gy ange 共 om abou 18 MeV below he ba ie o 120 MeV abo e i 兲wi h only one sys em- and ene gy-independen ee pa- ame e ␭. Wi h he aim o including he whole da a se , Figs. 1 and 2 ha e been made wi h educed scales. Figu es 3–5 p esen he esul s in usual scales o a ew pa icula sys ems, in o de o p o ide u he examples o he qual- i y o ou p edic ions. We belie e ha he e ec i e cu a u e simula es, in some way, he a e age e ec o a la ge numbe o coupled chan- nels ha con ibu e o he usion p ocess. The e ec o he couplings depends on he educed mass o he sys em be- cause hea ie sys ems p esen a g ea e numbe o eac ion channels and also la ge coupling ampli udes (which a e con- nec ed wi h he size o he nuclei). The dispe sion obse ed in Fig. 2 is p obably due [57] o pa icula s ong couplings ha we e no included in ou model, and is also connec ed wi h a ia ions o he densi ies, a ising om nuclea s uc- u e e ec s, which ha e an in luence on he nuclea po en ial [54]. Fo example, he g ea iso opic dependence (see Fig. 6— op)o he sub-ba ie usion c oss sec ion o he 16O+144,148,150,152,154Sm sys ems is dec eased by using he e ec i e cu a u e (Fig. 6—bo om). E en so, he di e ence is no o ally elimina ed p obably due o s uc u e e ec s no included in ou model. The e ec o he nonlocali y is no e y signi ican a nea -ba ie ene gies (low eloci ies), whe e Eq. (1)indica es ha VN共R,E兲⬇VF共R兲. The e ec becomes g ea e as he en- e gy inc eases, such ha a ene gies o abou 200 MeV/nucleon VN共R,E兲is abou one o de o magni ude less in ense han he co esponding olding po en ial [9,10]. FIG. 2. The same as Fig. 1, conside ing he co ec ion [Eq. (17) wi h ␭=0.1/amu] o he ba ie cu a u es. FIG. 3. The usion c oss sec ion o he 12C+12C, 12C+16O, 16O+208Pb, and 58Ni+64Ni sys ems. The lines ep esen ull ba ie pene a ion model calcula ions wi h (solid lines)o wi hou (dashed lines)including he e ec o he e ec i e cu a u e. FIG. 4. The same o Fig. 3 o he 16O+144,148,150,154Sm sys- ems. FIG. 5. The usion c oss sec ion o he 12C+27Al, 32S sys ems. The lines ep esen ull ba ie pene a ion model calcula ions wi h (do ed lines)o wi hou (solid lines) he e ec o he Pauli nonlo- cali y. GLOBAL AND CONSISTENT ANALYSIS OF THE HEAVY-ION…PHYSICAL REVIEW C 69, 034603 (2004) 034603-3 Howe e , he e a e ew a ailable usion da a a high ene - gies. Thus, o mos o he cases conside ed in his wo k, he heo e ical esul s o he BPM c oss sec ions ob ained using Eq. (1)di e by less han 5% in compa ison om hose ob ained conside ing VN共R,E兲=VF共R兲. E en so, we ha e ound wo cases in he p esen da a se in which he ene gy is high enough o emphasize such a di e ence (see Fig. 5). He e he dec ease o he c oss sec ion o highe ene gies is connec ed wi h he compe i ion be ween cen i ugal epul- sion and nuclea a ac ion. The ene gy dependence o he nuclea in e ac ion ha a ises om nonlocali y plays an im- po an ole by p o iding u he educ ion o he c oss sec- ion. On he o he hand, he nonlocali y has been undamen- al in ou desc ip ion o he hea y-ion elas ic sca e ing p ocess, in which he ene gy dependence o he po en ial has been success ully aken in o accoun by Eq. (1). The e o e, we conside he majo eason o using he nonlocal in e ac- ion o be he goal o ob aining a uni ied desc ip ion o bo h he elas ic sca e ing and usion p ocesses, wi hin a consis- en model om he sub-ba ie egion o in e media e ene - gies. In summa y, ou pa ame e - ee nonlocal model o he nuclea in e ac ion has been success ul in desc ibing he hea y-ion elas ic sca e ing p ocess om sub-ba ie ene gies up o 200 MeV/nucleon. This in e ac ion has also been suc- cess ully es ed in some cases o inelas ic sca e ing and ans e a sub-Coulomb and in e media e ene gies [11,12,15]. In he p esen wo k, we ha e also ob ained good p edic ions o usion c oss sec ions, using he nonlocal in- e ac ion in he con ex o he ba ie pene a ion model, o a e y la ge numbe o di e en sys ems and o e a wide en- e gy ange ( om he sub-ba ie egion o ene gies abou 15 MeV/nucleon, whe e he e a e s ill a ew exis ing expe i- men al da a). We emphasize ha he heo e ical calcula ions ha e no ee pa ame e s, excep o he sys em- and ene gy- independen one connec ed wi h he e ec i e ba ie cu a- u e. We also emphasize ha ou model p o ides ema kable p edic ions o he ligh hea y-ion sys ems o e nine o de s o magni ude (see Fig. 1 o ␮ 艋8). In his ange o educed mass he e is no need o include any co ec ion in he ba ie cu a u es. The e o e, he pa ame e - ee nonlocal model seems o be a good basis o s udying he usion p ocess in impo an cases o as ophysics, which in ol e mainly ligh sys ems. The model also p o ides a good desc ip ion, again wi hou using any ee pa ame e , o he whole se o sys- ems in ene gies abo e he ba ie (see Fig. 1 o E ed艌0). Clea ly, he e is oom o imp o emen o he model [57] o hea ie sys ems in lowe ene gies, by conside ing wo poin s: (i) he de elopmen o a mo e ealis ic model o he e ec i e ba ie pa ame e s, based on esul s ob ained om coupled-channel calcula ions; and (ii)inclusion o he ex- pe imen al esul s ob ained o usion ba ie dis ibu ions in he analysis. This wo k was pa ially suppo ed by Financiado a de Es udos e P oje os (FINEP), Fundação de Ampa o à Pesquisa do Es ado de São Paulo (FAPESP), and Conselho Nacional de Desen ol imen o Cien í ico e Tecnológico (CNPq). [1]A. B. Balan ekin and N. Takigawa, Re . Mod. Phys. 70,77 (1998). [2]L. C. Vaz and J. M. Alexande , Phys. Re . C 18, 2152 (1978). [3]S. Gil and D. E. DiG ego io, Phys. Re . C 57, 2826 (1998). [4]U. Jahnke e al., Phys. Re . Le . 48,17(1982). [5]R. Vandenbosch, Annu. Re . Nucl. Sci. 42, 447 (1992). [6]A. M. S e anini e al.,inP oceedings o he Wo kshop on Hea y Ion Fusion, Explo ing he Va ie y o Nuclea P ope - ies, Pado a, I aly, 1994, edi ed by A. M. S e anini e al. (Wo ld Scien i ic, Singapo e, 1994). [7]M. E. B andan and G. R. Sa chle , Phys. Rep. 285, 143 (1997). [8]A. Scalin e al., Nuo o Cimen o 37, 129 (1983). [9]M. A. Cândido Ribei o e al., Phys. Re . Le . 78, 3270 (1997). [10]L. C. Chamon e al., Phys. Re . Le . 79, 5218 (1997). [11]L. C. Chamon e al., Phys. Re . C 58, 576 (1998). [12]M. A. Al a ez e al., Nucl. Phys. A656, 187 (1999). [13]C. P. Sil a e al., Nucl. Phys. A679, 287 (2001). [14]L. R. Gasques e al., Phys. Re . C 65, 044314 (2002). FIG. 6. The educed usion c oss sec ion o he 16O+144,148,150,152,154Sm sys ems as a unc ion o he educed en- e gy, wi h (bo om)o wi hou ( op)including he e ec o he e ec i e cu a u e. The same da a se is p esen ed in usual scales in Fig. 4. L. R. GASQUES e al. PHYSICAL REVIEW C 69, 034603 (2004) 034603-4 [15]E. S. Rossi, J . e al., Nucl. Phys. A707, 325 (2002). [16]L. C. Vaz, J. M. Alexande , and G. R. Sa chle , Phys. Rep. 69, 373 (1981), and e e ences he ein. [17]M. Becke man, Phys. Rep. 129, 145 (1985), and e e ences he ein. [18]J. B. Na owi z e al., Phys. Rep., Phys. Le . 6, 2133 (1972). [19]S. Pi one e al., Phys. Re . C 64, 024610 (2001). [20]S. Gil e al., Phys. Re . C 31, 1752 (1985). [21]J. F. Ma eja e al., Phys. Re . Le . 47,311(1981). [22]N. Keeley e al., Nucl. Phys. A628,1(1998). [23]M. Lange in e al., Phys. Re . C 14, 152 (1976). [24]M. Benjelloun e al., Nucl. Phys. A560, 715 (1993). [25]R. G. S oks ad e al., Phys. Re . C 21, 2427 (1980). [26]C. Ten ei o e al.,inP oceedings o he Wo kshop on Hea y Ion Collisions a Ene gies Nea he Coulomb Ba ie Held a Da esbu y Labo a o y, England, edi ed by M. A. Naga ajan (IOP, G ea B i ain, 1990). [27]P. H. S elson e al.,inP oceedings o he Wo kshop on Hea y Ion Collisions a Ene gies Nea he Coulomb Ba ie Held a Da esbu y Labo a o y, England, edi ed by M. A. Naga ajan (IOP, G ea B i ain, 1990). [28]S. Landowne, in P oceedings o he Wo kshop on Hea y Ion Collisions a Ene gies Nea he Coulomb Ba ie Held a Da esbu y Labo a o y, England, edi ed by M. A. Naga ajan (IOP, G ea B i ain, 1990). [29]A. M. S e anini, in P oceedings o he Wo kshop on Hea y Ion Collisions a Ene gies Nea he Coulomb Ba ie Held a Da esbu y Labo a o y, England, edi ed by M. A. Naga ajan (IOP, G ea B i ain, 1990). [30]J. O. Fe nandez Niello e al., Phys. Re . C 43, 2303 (1991). [31]M. J. Smi hson e al., Nucl. Phys. A505,84(1989). [32]A. M. S e anini e al., Nucl. Phys. A456, 509 (1986). [33]R. J. Tighe e al., Phys. Re . C 42, 1530 (1990). [34]M. Dasgup a e al., Nucl. Phys. A539, 351 (1992). [35]S. J. Sko ka e al., Z. Phys. A 328, 355 (1987). [36]J. J. Vega e al., Phys. Re . C 42, 947 (1990). [37]S. Gil e al., Phys. Re . C 51, 1336 (1995). [38]B. Siko a e al., Phys. Re . C 20, 2219 (1979). [39]R. Pengo e al., Nucl. Phys. A411, 255 (1983). [40]P. Da id e al., Nucl. Phys. A287, 179 (1977). [41]R. Schicke e al., Phys. Le . B 206,9(1988). [42]P. F öb ich and J. Ma en, Z. Phys. A 339, 171 (1991). [43]M. L. Halbe e al., Phys. Re . C 40, 2558 (1989). [44]F. L. H. Wol s e al., Phys. Le . B 196,113(1987). [45]M. Dasgup a e al., Annu. Re . Nucl. Pa . Sci. 48, 401 (1998). [46]C. P. Sil a e al., Phys. Re . C 55, 3155 (1997). [47]L. C. Chamon e al., Phys. Le . B 275,29(1992). [48]J. R. Leigh e al., Phys. Re . C 52, 3151 (1995). [49]J. O. New on e al., Phys. Re . C 64, 064608 (2001). [50]D. J. Hinde e al., Nucl. Phys. A654, 864 (1999). [51]A. M. S e anini e al., Phys. Re . C 62, 014601 (2000). [52]C. R. Mo on e al., Phys. Re . C 60, 044608 (1999). [53]H. Timme s, Ph.D. hesis, The Aus alian Na ional Uni e si y, Canbe a, 1996. [54]L. C. Chamon e al., Phys. Re . C 66, 014610 (2002). [55]D. L. Hill and J. A. Wheele , Phys. Re . 89, 1102 (1953). [56]C. Y. Wong, Phys. Re . Le . 31, 766 (1973). [57]L. R. Gasques e al. (unpublished). GLOBAL AND CONSISTENT ANALYSIS OF THE HEAVY-ION…PHYSICAL REVIEW C 69, 034603 (2004) 034603-5