Precise nuclear matter densities from heavy-ion collisions

P ecise nuclea ma e densi ies om hea y-ion collisions M. A. G. Al a ez,1E. S. Rossi, J .,1C. P. Sil a,1L. R. Gasques,1L. C. Chamon,1D. Pe ei a,1M. N. Rao,1B. V. Ca lson,2 C. De Con i,2R. M. Anjos,3P. R. S. Gomes,3J. Lubian,3S. Kailas,4A. Cha e jee,4and P. Singh4 1Labo a o ´ io Pelle on, Ins i u o de Fı ´sica da Uni e sidade de Sa ˜ o Paulo, 05315-970 Sa ˜ o Paulo, SP, B azil 2Depa amen o de Fı ´sica, Ins i u o Tecnolo ´gico de Ae ona ´u ica, Cen o Te ´cnico Ae oespacial, Sa ˜ o Jose ´dos Campos, SP, B azil 3Ins i u o de Fı ´sica, Uni e sidade Fede al Fluminense, A . Li o anea, Ni e o ´i 24210-340, RJ, B azil 4Nuclea Physics Di ision, Bhabha A omic Resea ch Cen e, Bombay 400 085, India 共Recei ed 3 Ap il 2001; published 30 No embe 2001兲 An un olding me hod is p oposed o ex ac g ound-s a e nuclea ma e densi ies om hea y-ion elas ic sca e ing da a analyses a low 共sub-ba ie 兲and in e media e ene gies. The consis ency o he esul s is ully checked. The me hod should be o alue in de e mining densi ies o exo ic nuclei. DOI: 10.1103/PhysRe C.65.014602 PACS numbe 共s兲: 25.70.Bc, 21.10.G , 21.10.F , 24.10.H A long-s anding ques ion o nuclea s uc u e conce ns he de e mina ion o hea y-ion neu on densi ies, which a e a om being as well known as he p o on densi ies ha ha e been ex ac ed om elec on sca e ing expe imen s. I is wo h men ioning he impo ance o he de e mina ion o nuclea densi ies o dis inguish among di e en nuclea s uc u e heo e ical app oaches. Se e al p obes 共pion, p o- on, alpha, e c.兲ha e been used in o de o de e mine nuclea ma e densi ies, wi h di e en so s o limi a ions 关1兴. Fo ins ance, he use o he s ong in e ac ing p obes ␲ ⫹and ␲ ⫺ is usually accompanied by he need o ‘‘calib a e’’ he me hod, which means ha only a e age adii and di e ences in densi ies a e he mos eliable esul s. In a mo e unda- men al philosophy, he possibili y o ex ac ing in o ma ion on nuclea dis ibu ions om hea y-ion elas ic sca e ing is a ques ion o using he olding model o he in e ac ion, in- cluding all he impo an e ec s om i s p inciples and a oiding he use o adjus able pa ame e s as much as pos- sible. In he p esen wo k, a me hod o de e mining ma e den- si ies om hea y-ion elas ic sca e ing da a a sub-ba ie and in e media e ene gies is p oposed. I is based on he pa ame e - ee nonlocal ene gy-independen ba e po en ial 共NLM3Y po en ial兲, ecen ly de eloped 关2–5兴 o he eal pa o he nucleus-nucleus in e ac ion. The NLM3Y po en- ial has been es ed o se e al sys ems 关3,4兴and gi es ex- cellen ep oduc ions o measu ed elas ic and inelas ic c oss sec ions in a la ge ene gy ange, pa icula ly a in e media e ene gies whe e he e ac i e elas ic da a a e e y sensi i e o he eal pa o he in e ac ion 关6兴. The model 共 o de ails see 关3兴兲 akes in o accoun he Pauli nonlocali y in ol ing he exchange o nucleons be ween he a ge and he p ojec ile. The ene gy-independen eal pa o he in e ac ion is gi en by V共R ជ ,R ជ ⬘兲⫽VNL 冉 R⫹R⬘ 2 冊 1 ␲ 3/2b3e⫺( 兩 R ជ ⫺R ជ ⬘ 兩 /b)2,共1兲 whe e b⫽b0m0/ ␮ is he ange o he Pauli nonlocali y, b0 ⫽0.85 m, m0and ␮ a e he nucleon mass and he educed mass o he sys em, espec i ely. The nonlocal in e ac ion is connec ed o he usual olding po en ial 关7兴 h ough VNL共R兲⫽ 冕 ␳ 1共 1兲 ␷ 共R ជ ⫺ ជ 1⫹ ជ 2兲 ␳ 2共 2兲d ជ 1d ជ 2,共2兲 whe e ␳ 1( 1) and ␳ 2( 2) a e he g ound-s a e nuclea densi- ies o he colliding pa ne s, and ␷ ( ជ ) is he M3Y e ec i e nucleon-nucleon in e ac ion. The co esponding ene gy- dependen local equi alen po en ial is gi en by 关3兴 VLE共R;E兲⫽1⫺ 冑 1⫺4 ␥ VNL共R兲e⫺ ␥ [E⫺VC(R)] 2 ␥ ,共3兲 wi h ␥ ⫽ ␮ b2/2ប2. We men ion in passing ha o he ap- p oaches o he ini e ange exchange e m 共 o example see 关8–10兴兲 a e mo e complica ed o calcula e and he e o e less sui able o ex ensi e s udies o nuclea densi ies. Wi hin he model abo e, he cen al idea o he me hod p oposed is o ex ac g ound-s a e nuclea dis ibu ions om elas ic sca e ing da a analyses, wi h he densi ies as he e- sul o bes i s in an un olding p ocedu e in ol ing exp es- sions 共2兲and 共3兲. The da a analyses a in e media e ene gies gi e in o ma ion abou he o al 共neu on ⫹p o on兲dis ibu- ions in a egion close o he oo -mean-squa e adius ( ms), while a sub-ba ie ene gies he su ace is he egion sensi- i e o he da a i s. To cha ac e ize he abso p ion om eac ion channels, a in e media e ene gies we ha e used an imagina y po en ial based on he Lax- ype in e ac ion 关11兴: W共R;E兲⫽⫺ E kN ␴ T NN共E兲 冕 ␳ 1共 兩 R ជ ⫺ ជ 兩 兲 ␳ 2共 兲d ជ ,共4兲 whe e ␴ T NN(E) is he a e age nucleon-nucleon o al c oss sec ion wi h Pauli blocking. Fo he sub-ba ie case, we ha e selec ed elas ic sca e ing expe imen al angula dis i- bu ions a ene gies su icien ly below he Coulomb ba ie , ha couplings o eac ion channels a e e y small. In his case, we ha e used an inne imagina y po en ial wi h Woods- Saxon shape, which akes in o accoun he small in e nal abso p ion om ba ie pene a ion. The alues adop ed o he pa ame e s o his po en ial esul in small s eng hs a he su ace egion. This p ocedu e mus be used in he sub- ba ie da a analyses due o he small c oss sec ions o pe- iphe ical eac ion channels. No sensi i i y in he c oss sec- PHYSICAL REVIEW C, VOLUME 65, 014602 0556-2813/2001/65共1兲/014602共4兲/$20.00 ©2001 The Ame ican Physical Socie y65 014602-1 ion p edic ions has been de ec ed ela ed o dep h a ia ions o his abso p i e po en ial. We poin ou ha he pola iza ion po en ial ha a ises om eac ion channel couplings 共Fesh- bach nonlocali y兲has been es ima ed 关12–14兴 h ough ex en- si e coupled channel calcula ions o he sub-ba ie da a se , and ep esen s less han 10% in compa ison wi h he ba e 共 olding兲in e ac ion. We ha e chosen 16O as a es case, due o he ex ensi e expe imen al and heo e ical in o ma ion a ailable abou his nucleus, and, as discussed in Re . 关12兴, because di e en app oaches gi e qui e di e en esul s o he 16O nuclea densi y, pa icula ly a he su ace egion. In he analyses, we ha e assumed a wo-pa ame e Fe mi model 共2PF兲 o he 16O densi y, wi h di useness 共a兲and adius (R0) sea ched o he bes da a i s, and wi h he ␳ 0pa ame e de e mined by he no maliza ion condi ion 4 ␲ 冕 0 ⬁ ␳ 0 1⫹exp 冉 ⫺R0 a 冊 2d ⫽16. 共5兲 In Fig. 1 is p esen ed, as an illus a ion o ou me hod, he de e mina ion o he o al 共neu on ⫹p o on兲densi y o he 16O nucleus a he ms adius and su ace egions, by using elas ic sca e ing da a analyses a in e media e and sub- ba ie ene gies, espec i ely. Fo each angula dis ibu ion, we ha e ound a amily o densi ies which gi e equi alen da a i s. These densi ies c oss 共Fig. 1 op兲a a pa icula adius s, he ea e e e ed o as he sensi i i y adius. Simi- la beha io has been obse ed in he de e mina ion o ba e po en ials om sub-ba ie da a analyses 关12–15兴, bu in ha case only one c ossing was de ec ed o each angula dis i- bu ion. In he densi y case, wo c ossings a e obse ed 共Fig. 1兲due o he pa icula shape and no maliza ion condi ion imposed on he nuclea densi y. Thus, he de e mina ion o he sensi i i y adius is also accompanied by a no ch es 共Fig. 1 bo om兲, in which a spline wi h Gaussian shape is included in he 16O densi y, and he a ia ion o he chi- squa e is s udied as a unc ion o he posi ion o his pe u - ba ion. The no ch es gua an ees ha sis in a densi y egion impo an o he da a i , and does no a ise om spu ious c ossing. Since he da a i s depend only on he densi y in a small ange o nuclea adii, he de e mina ion o he sensi- i i y adius and co esponding densi y alue is a he inde- penden o he shape assumed o he nuclea dis ibu ion 共2PF, ha monic oscilla o — see example in Fig. 2兲. Fo he 16O⫹16O sys em a he ene gy o 1120 MeV, besides he Lax in e ac ion we ha e also used a h ee ee pa ame e imagina y po en ial, wi h Woods-Saxon shape, wi h he aim o e alua ing any possible change in he sensi i i y adius. The sand co esponding densi y alues ob ained in his case a e qui e simila o hose om he Lax in e ac ion 共see Fig. 2兲. The 16O expe imen al densi y alues a he sensi i i y a- dii ob ained om hea y-ion da a analyses a e shown in Fig. 2. Fo he sub-ba ie ene gies, he elas ic sca e ing da a 关12–15兴a e om 40 angula dis ibu ions o 11 sys ems like 16O⫹A, whe e Ais a magic o semimagic a ge nucleus wi h mass numbe anging om 58 共Ni兲 o 208 共Pb兲.In he da a analyses, we ha e used Ha ee-Fock, Di ac-Ha ee- FIG. 1. Top: Examples o he de e mina ion o he sensi i i y adii, s, and he co esponding expe imen al alues o he 16O nuclea ma e densi y, ␳ ( s), using wo-pa ame e Fe mi dis ibu- ions which gi e equi alen elas ic sca e ing da a i s o angula dis ibu ions o he 16O⫹16O(Elab⫽1120 MeV) and 16O⫹92Mo (Elab⫽49 MeV) sys ems. Bo om: The sensi i i y egions o he 16O nuclea ma e densi y cha ac e ized by no ch es s. FIG. 2. Expe imen al nuclea densi y alues o he 16O共semi- closed symbols兲and 18O共open symbols兲nuclei, as ob ained om sub-ba ie elas ic sca e ing da a analyses o di e en sys ems and bomba ding ene gies. The closed symbols ep esen densi y alues (16O) om in e media e ene gy da a analyses (16O⫹16O, Elab ⫽1120 MeV), using di e en models o he shape o he 16O densi y 共2PF o HO兲and o he imagina y po en ial 共WS o Lax兲. The lines co espond o heo e ical Di ac-Ha ee-Bogoliubo 共DHB兲calcula ions o he 16O nucleus, and a wo-pa ame e Fe mi dis ibu ion 共2PF兲wi h o wi hou a damped oscilla o y co ec ion. M. A. G. ALVAREZ e al. PHYSICAL REVIEW C 65 014602 014602-2 Bogoliubo , and shell-model densi ies o he a ge nuclei 共see Re s. 关12–16兴兲. In his sub-ba ie egion he posi ion o he sensi i i y adius is ene gy-dependen , wi h a ia ion connec ed o he classical u ning poin o he e ec i e po- en ial. This ac allows us o cha ac e ize he 16O nuclea dis ibu ion 共semiclosed symbols in Fig. 2兲in a la ge and supe icial egion. The da a 共 om Re . 关17兴兲 analyses a Elab⫽1120 MeV o he 16O⫹16O sys em ha e p o ided in o ma ion o he 16O densi y in a much inne egion 共closed symbols in Fig. 2兲. A heo e ical p edic ion o he 16O densi y de i ed om he Di ac-Ha ee-Bogoliubo 共DHB兲model 关18兴using NL3 po en ial pa ame e s 关19兴is also shown in Fig. 2. In he su ace egion, he expe imen al 16O densi y is much g ea e han he heo e ical p edic ion. An analysis o he single-pa icle le els o he heo e ical calcula ion shows, as one migh expec , ha he allo o he densi y in he su ace egion is de e mined by he leas bound le els. Al hough he NL3 pa ame e se was adjus ed o e- p oduce binding ene gies and cha ge and neu on adii ac oss he pe iodic able, i did no ake in o accoun single-pa icle p ope ies, which sugges s a di ec ion o u u e imp o e- men s in such a pa ame e se . Fo he pu pose o compa ison and demons a ion o he sensi i i y o he me hod, we ha e also shown in Fig. 2 he expe imen al densi y alues o he 18O nucleus 共open sym- bols兲ob ained wi h he same me hod h ough op ical model analyses o sub-ba ie elas ic sca e ing da a o he 18O ⫹58,60Ni sys ems. As heo e ically expec ed 关20兴and clea ly demons a ed by ou esul s, he wo ex a neu ons o he 18O (2s1/2 ,1d3/2 , and 1d5/2 o bi als兲inc ease he 18O den- si y a he su ace egion in compa ison o ha o he 16O nucleus. In ou me hod, he expe imen al densi y alues ha e been ex ac ed based on e y undamen al g ounds. The pa ame e - ee eal pa o he in e ac ion con ains as basic inpu s jus he well-known M3Y e ec i e nucleon-nucleon in e ac ion and he model o he Pauli nonlocali y, which has been es ed ex ensi ely 关2–5兴. Also he imagina y pa o he in e ac ion has been based on gene al assump ions: he lack o supe icial abso p ion a sub-ba ie ene gies and he pa ame e - ee Lax- ype in e ac ion 共 o he 16O⫹16O sys- em a Elab⫽1120 MeV), which is known o be app op ia e o high ene gies 关11兴. The adjus able pa ame e s o he me hod (R0and a) a e connec ed only wi h he quan i y o be de e mined: he p ojec ile nuclea densi y, and he esul s ob ained a e a he insensi i e o he 共 ealis ic兲shape as- sumed o he dis ibu ion. We men ion ha o he expe imen- al da a o he 16O densi y in he egion 2⭐ ⭐4.5 m could be ound h ough he analyses o o he angula dis ibu ions in ene gies abo e he ba ie , bu in his case he imagina y po en ial mus ha e adjus able pa ame e s and he eliabili y o he esul s o he densi y should be s udied e y ca e ully 关21兴. Thus, we conside he heo e ical densi ies o he a ge nuclei 共in he sub-ba ie da a analysis兲as he only assump- ion o ou me hod ha needs o be checked. The good ag ee- men among he esul s o he 16O densi y ob ained using di e en a ge nuclei indica e ha any possible de ia ion in such heo e ical calcula ions would be sys ema ic. Thus, as a es o he consis ency o he assumed hypo h- esis o he me hod, we compa e in Fig. 3 he da a 共 om Re s. 关22,23兴兲 wi h p edic ions o elec on sca e ing c oss sec- ions. We ha e used cha ge dis ibu ions ob ained by olding he p o on densi y o he nucleus wi h he in insic cha ge dis ibu ion o he p o on. Fo he doubly-magic 16O nucleus, he p o on densi y is qui e close o one-hal o he o al den- si y 共see he heo e ical neu on and p o on dis ibu ions in Fig. 2兲. The elec on sca e ing c oss sec ions ha e been cal- cula ed in he plane-wa e Bo n app oxima ion, which, o ligh nuclei such as 16O, should p oduce c oss sec ions close o he exac phase-shi me hod, excep o momen um ans- e ed nea a minimum o di ac ion. Conside ing a bes i 2PF dis ibu ion (R0⫽2.49 m and a⫽0.55 m — solid line in Fig. 2兲 o desc ibe he 16O den- si y, a easonable desc ip ion o he elec on sca e ing 共solid line in Fig. 3 op兲is ob ained, wi h some disc epancies in he momen um ans e ed egion 1.5⭐q⭐3.0 m⫺1. Based on he heo e ical calcula ions o he 16O densi y 共see Fig. 2兲, such disc epancies a e unde s ood conside ing he dec easing con ibu ion o he 1p3/2 and 1p1/2 componen s o he nuclea densi y in he inne adius egion. We ha e aken his in o accoun by adding a damped oscilla o y unc ion o he 2PF dis ibu ion 共2PF⫹co ec ion in Fig. 2兲, esul ing in a be e o e all desc ip ion o he elec on c oss sec ion 共dashed line in Fig. 3 op兲. As epo ed ea lie 关22兴, a simila p ocedu e has been adop ed o imp o e 16O elec on sca e - ing da a i s ha ha e been ob ained by using enomenologi- cal cha ge densi ies. These i s 关22兴ha e p ecision compa- able o hose o he p esen wo k. We poin ou ha he disag eemen be ween p edic ed and measu ed c oss sec ions nea he minima o di ac ion (q⬇1.5 m⫺1and q ⬇3 m ⫺1) is due o he use o he Bo n app oxima ion in he c oss sec ion calcula ions. Thus, o he i s ime, i is FIG. 3. Expe imen al elec on sca e ing c oss sec ions o he 共 op兲16O and 共bo om兲58Ni nuclei as a unc ion o he momen um ans e ed. The do ed lines in he igu e a e heo e ical p edic ions using cha ge dis ibu ions om Di ac-Ha ee-Bogoliubo 共DHB兲 calcula ions in he plane-wa e Bo n app oxima ion. The o he lines 共 op兲a e he esul s o cha ge dis ibu ions de i ed om expe i- men al nuclea ma e densi ies, using 2PF shapes 共as shown in Fig. 2兲wi h 共dashed line兲o wi hou 共solid line兲a damped oscilla- o y co ec ion. PRECISE NUCLEAR MATTER DENSITIES FROM . . . PHYSICAL REVIEW C 65 014602 014602-3 possible o desc ibe elec on sca e ing c oss sec ions om an expe imen al 16O nuclea densi y ob ained h ough hea y- ion elas ic sca e ing da a analyses. The heo e ical Di ac- Ha ee-Bogoliubo 共DHB兲cha ge dis ibu ion p edic s elec- on sca e ing c oss sec ions which a e in disag eemen wi h he da a o la ge q alues 共see Fig. 3 op兲. We poin ou ha , as a u he es o he consis ency o he assumed hypo hesis o he me hod, he heo e ical DHB dis ibu ions o he a - ge nuclei used in his wo k p edic elec on sca e ing c oss sec ions in ag eemen wi h he da a 共as illus a ed in Fig. 3 bo om o he 58Ni). In conclusion, using he p og ess eached in he las 20 yea s o desc ibe hea y-ion elas ic sca e ing, i is possible o de e mine g ound-s a e nuclea ma e densi ies. The me hod p esen ed in his wo k should be o alue in s udying densi- ies o exo ic nuclei, pa icula ly a he su ace egion h ough sub-ba ie hea y-ion elas ic sca e ing. We poin ou ha he di e ence be ween densi ies o exo ic and neighbo - ing s able nuclei is much emphasized in he su ace egion. This seems o be bo ne ou by p elimina y esul s 关24兴 o he 6He nucleus in compa ison o 4He. 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