Precise nuclear matter densities from heavy-ion collisions

Precise nuclear matter densities from heavy-ion collisions M. A. G. Alvarez,1E. S. Rossi, Jr.,1C. P. Silva,1L. R. Gasques,1L. C. Chamon,1D. Pereira,1M. N. Rao,1B. V. Carlson,2 C. De Conti,2R. M. Anjos,3P. R. S. Gomes,3J. Lubian,3S. Kailas,4A. Chatterjee,4and P. Singh4 1Laborato ´rio Pelletron, Instituto de Fı ´sica da Universidade de Sa ˜ o Paulo, 05315-970 Sa ˜ o Paulo, SP, Brazil 2Departamento de Fı ´sica, Instituto Tecnolo ´gico de Aerona ´utica, Centro Te ´cnico Aeroespacial, Sa ˜ o Jose ´dos Campos, SP, Brazil 3Instituto de Fı ´sica, Universidade Federal Fluminense, Av. Litoranea, Nitero ´i 24210-340, RJ, Brazil 4Nuclear Physics Division, Bhabha Atomic Research Centre, Bombay 400 085, India 共Received 3 April 2001; published 30 November 2001兲 An unfolding method is proposed to extract ground-state nuclear matter densities from heavy-ion elastic scattering data analyses at low 共sub-barrier兲and intermediate energies. The consistency of the results is fully checked. The method should be of value in determining densities for exotic nuclei. DOI: 10.1103/PhysRevC.65.014602 PACS number共s兲: 25.70.Bc, 21.10.Gv, 21.10.Ft, 24.10.Ht A long-standing question of nuclear structure concerns the determination of heavy-ion neutron densities, which are far from being as well known as the proton densities that have been extracted from electron scattering experiments. It is worth mentioning the importance of the determination of nuclear densities to distinguish among different nuclear structure theoretical approaches. Several probes 共pion, proton, alpha, etc.兲have been used in order to determine nuclear matter densities, with different sorts of limitations 关1兴. For instance, the use of the strong interacting probes ␲ ⫹and ␲ ⫺ is usually accompanied by the need to ‘‘calibrate’’ the method, which means that only average radii and differences in densities are the most reliable results. In a more fundamental philosophy, the possibility of extracting information on nuclear distributions from heavy-ion elastic scattering is a question of using the folding model for the interaction, including all the important effects from first principles and avoiding the use of adjustable parameters as much as possible. In the present work, a method of determining matter densities from heavy-ion elastic scattering data at sub-barrier and intermediate energies is proposed. It is based on the parameter-free nonlocal energy-independent bare potential 共NLM3Y potential兲, recently developed 关2–5兴for the real part of the nucleus-nucleus interaction. The NLM3Y potential has been tested for several systems 关3,4兴and gives excellent reproductions of measured elastic and inelastic cross sections in a large energy range, particularly at intermediate energies where the refractive elastic data are very sensitive to the real part of the interaction 关6兴. The model 共for details see 关3兴兲 takes into account the Pauli nonlocality involving the exchange of nucleons between the target and the projectile. The energy-independent real part of the interaction is given 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 where b⫽b0m0/ ␮ is the range of the Pauli nonlocality, b0 ⫽0.85 fm, m0and ␮ are the nucleon mass and the reduced mass of the system, respectively. The nonlocal interaction is connected to the usual folding potential 关7兴through VNL共R兲⫽ 冕 ␳ 1共r1兲 ␷ 共R ជ ⫺r ជ 1⫹r ជ 2兲 ␳ 2共r2兲dr ជ 1dr ជ 2,共2兲 where ␳ 1(r1) and ␳ 2(r2) are the ground-state nuclear densities of the colliding partners, and ␷ (r ជ ) is the M3Y effective nucleon-nucleon interaction. The corresponding energydependent local equivalent potential is given 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 with ␥ ⫽ ␮ b2/2ប2. We mention in passing that other approaches for the finite range exchange term 共for example see 关8–10兴兲 are more complicated to calculate and therefore less suitable for extensive studies of nuclear densities. Within the model above, the central idea of the method proposed is to extract ground-state nuclear distributions from elastic scattering data analyses, with the densities as the result of best fits in an unfolding procedure involving expressions 共2兲and 共3兲. The data analyses at intermediate energies give information about the total 共neutron ⫹proton兲distributions in a region close to the root-mean-square radius (rrms), while at sub-barrier energies the surface is the region sensitive to the data fits. To characterize the absorption from reaction channels, at intermediate energies we have used an imaginary potential based on the Lax-type interaction 关11兴: W共R;E兲⫽⫺ E kN ␴ T NN共E兲 冕 ␳ 1共 兩 R ជ ⫺r ជ 兩 兲 ␳ 2共r兲dr ជ ,共4兲 where ␴ T NN(E) is the average nucleon-nucleon total cross section with Pauli blocking. For the sub-barrier case, we have selected elastic scattering experimental angular distributions at energies sufficiently below the Coulomb barrier, that couplings to reaction channels are very small. In this case, we have used an inner imaginary potential with WoodsSaxon shape, which takes into account the small internal absorption from barrier penetration. The values adopted for the parameters of this potential result in small strengths at the surface region. This procedure must be used in the subbarrier data analyses due to the small cross sections of peripherical reaction channels. No sensitivity in the cross secPHYSICAL REVIEW C, VOLUME 65, 014602 0556-2813/2001/65共1兲/014602共4兲/$20.00 ©2001 The American Physical Society65 014602-1 tion predictions has been detected related to depth variations of this absorptive potential. We point out that the polarization potential that arises from reaction channel couplings 共Feshbach nonlocality兲has been estimated 关12–14兴through extensive coupled channel calculations for the sub-barrier data set, and represents less than 10% in comparison with the bare 共folding兲interaction. We have chosen 16O as a test case, due to the extensive experimental and theoretical information available about this nucleus, and, as discussed in Ref. 关12兴, because different approaches give quite different results for the 16O nuclear density, particularly at the surface region. In the analyses, we have assumed a two-parameter Fermi model 共2PF兲for the 16O density, with diffuseness 共a兲and radius (R0) searched for the best data fits, and with the ␳ 0parameter determined by the normalization condition 4 ␲ 冕 0 ⬁ ␳ 0 1⫹exp 冉 r⫺R0 a 冊 r2dr⫽16. 共5兲 In Fig. 1 is presented, as an illustration of our method, the determination of the total 共neutron ⫹proton兲density for the 16O nucleus at the rrms radius and surface regions, by using elastic scattering data analyses at intermediate and subbarrier energies, respectively. For each angular distribution, we have found a family of densities which give equivalent data fits. These densities cross 共Fig. 1 top兲at a particular radius rs, hereafter referred to as the sensitivity radius. Similar behavior has been observed in the determination of bare potentials from sub-barrier data analyses 关12–15兴, but in that case only one crossing was detected for each angular distribution. In the density case, two crossings are observed 共Fig. 1兲due to the particular shape and normalization condition imposed on the nuclear density. Thus, the determination of the sensitivity radius is also accompanied by a notch test 共Fig. 1 bottom兲, in which a spline with Gaussian shape is included in the 16O density, and the variation of the chisquare is studied as a function of the position of this perturbation. The notch test guarantees that rsis in a density region important for the data fit, and does not arise from spurious crossing. Since the data fits depend only on the density in a small range of nuclear radii, the determination of the sensitivity radius and corresponding density value is rather independent of the shape assumed for the nuclear distribution 共2PF, harmonic oscillator — see example in Fig. 2兲. For the 16O⫹16O system at the energy of 1120 MeV, besides the Lax interaction we have also used a three free parameter imaginary potential, with Woods-Saxon shape, with the aim of evaluating any possible change in the sensitivity radius. The rsand corresponding density values obtained in this case are quite similar to those from the Lax interaction 共see Fig. 2兲. The 16O experimental density values at the sensitivity radii obtained from heavy-ion data analyses are shown in Fig. 2. For the sub-barrier energies, the elastic scattering data 关12–15兴are from 40 angular distributions of 11 systems like 16O⫹A, where Ais a magic or semimagic target nucleus with mass number ranging from 58 共Ni兲to 208 共Pb兲.Inthe data analyses, we have used Hartree-Fock, Dirac-HartreeFIG. 1. Top: Examples of the determination of the sensitivity radii, rs, and the corresponding experimental values for the 16O nuclear matter density, ␳ (rs), using two-parameter Fermi distributions which give equivalent elastic scattering data fits for angular distributions of the 16O⫹16O(Elab⫽1120 MeV) and 16O⫹92Mo (Elab⫽49 MeV) systems. Bottom: The sensitivity regions for the 16O nuclear matter density characterized by notch tests. FIG. 2. Experimental nuclear density values for the 16O共semiclosed symbols兲and 18O共open symbols兲nuclei, as obtained from sub-barrier elastic scattering data analyses for different systems and bombarding energies. The closed symbols represent density values (16O) from intermediate energy data analyses (16O⫹16O, Elab ⫽1120 MeV), using different models for the shape of the 16O density 共2PF or HO兲and for the imaginary potential 共WS or Lax兲. The lines correspond to theoretical Dirac-Hartree-Bogoliubov 共DHB兲calculations for the 16O nucleus, and a two-parameter Fermi distribution 共2PF兲with or without a damped oscillatory correction. M. A. G. ALVAREZ et al. PHYSICAL REVIEW C 65 014602 014602-2 Bogoliubov, and shell-model densities for the target nuclei 共see Refs. 关12–16兴兲. In this sub-barrier region the position of the sensitivity radius is energy-dependent, with variation connected to the classical turning point of the effective potential. This fact allows us to characterize the 16O nuclear distribution 共semiclosed symbols in Fig. 2兲in a large and superficial region. The data 共from Ref. 关17兴兲 analyses at Elab⫽1120 MeV for the 16O⫹16O system have provided information of the 16O density in a much inner region 共closed symbols in Fig. 2兲. A theoretical prediction for the 16O density derived from the Dirac-Hartree-Bogoliubov 共DHB兲model 关18兴using NL3 potential parameters 关19兴is also shown in Fig. 2. In the surface region, the experimental 16O density is much greater than the theoretical prediction. An analysis of the single-particle levels of the theoretical calculation shows, as one might expect, that the falloff of the density in the surface region is determined by the least bound levels. Although the NL3 parameter set was adjusted to reproduce binding energies and charge and neutron radii across the periodic table, it did not take into account single-particle properties, which suggests a direction for future improvements in such a parameter set. For the purpose of comparison and demonstration of the sensitivity of the method, we have also shown in Fig. 2 the experimental density values for the 18O nucleus 共open symbols兲obtained with the same method through optical model analyses of sub-barrier elastic scattering data for the 18O ⫹58,60Ni systems. As theoretically expected 关20兴and clearly demonstrated by our results, the two extra neutrons of the 18O (2s1/2 ,1d3/2 , and 1d5/2 orbitals兲increase the 18O density at the surface region in comparison to that for the 16O nucleus. In our method, the experimental density values have been extracted based on very fundamental grounds. The parameter-free real part of the interaction contains as basic inputs just the well-known M3Y effective nucleon-nucleon interaction and the model for the Pauli nonlocality, which has been tested extensively 关2–5兴. Also the imaginary part of the interaction has been based on general assumptions: the lack of superficial absorption at sub-barrier energies and the parameter-free Lax-type interaction 共for the 16O⫹16O system at Elab⫽1120 MeV), which is known to be appropriate for high energies 关11兴. The adjustable parameters of the method (R0and a) are connected only with the quantity to be determined: the projectile nuclear density, and the results obtained are rather insensitive to the 共realistic兲shape assumed for the distribution. We mention that other experimental data for the 16O density in the region 2⭐r⭐4.5 fm could be found through the analyses of other angular distributions in energies above the barrier, but in this case the imaginary potential must have adjustable parameters and the reliability of the results for the density should be studied very carefully 关21兴. Thus, we consider the theoretical densities for the target nuclei 共in the sub-barrier data analysis兲as the only assumption of our method that needs to be checked. The good agreement among the results for the 16O density obtained using different target nuclei indicate that any possible deviation in such theoretical calculations would be systematic. Thus, as a test of the consistency of the assumed hypothesis of the method, we compare in Fig. 3 the data 共from Refs. 关22,23兴兲 with predictions for electron scattering cross sections. We have used charge distributions obtained by folding the proton density of the nucleus with the intrinsic charge distribution of the proton. For the doubly-magic 16O nucleus, the proton density is quite close to one-half of the total density 共see the theoretical neutron and proton distributions in Fig. 2兲. The electron scattering cross sections have been calculated in the plane-wave Born approximation, which, for light nuclei such as 16O, should produce cross sections close to the exact phase-shift method, except for momentum transferred near a minimum of diffraction. Considering a best fit 2PF distribution (R0⫽2.49 fm and a⫽0.55 fm — solid line in Fig. 2兲to describe the 16O density, a reasonable description of the electron scattering 共solid line in Fig. 3 top兲is obtained, with some discrepancies in the momentum transferred region 1.5⭐q⭐3.0 fm⫺1. Based on the theoretical calculations for the 16O density 共see Fig. 2兲, such discrepancies are understood considering the decreasing contribution of the 1p3/2 and 1p1/2 components for the nuclear density in the inner radius region. We have taken this into account by adding a damped oscillatory function to the 2PF distribution 共2PF⫹correction in Fig. 2兲, resulting in a better overall description of the electron cross section 共dashed line in Fig. 3 top兲. As reported earlier 关22兴, a similar procedure has been adopted to improve 16O electron scattering data fits that have been obtained by using fenomenological charge densities. These fits 关22兴have precision comparable to those of the present work. We point out that the disagreement between predicted and measured cross sections near the minima of diffraction (q⬇1.5 fm⫺1and q ⬇3fm ⫺1) is due to the use of the Born approximation in the cross section calculations. Thus, for the first time, it is FIG. 3. Experimental electron scattering cross sections for the 共top兲16O and 共bottom兲58Ni nuclei as a function of the momentum transferred. The dotted lines in the figure are theoretical predictions using charge distributions from Dirac-Hartree-Bogoliubov 共DHB兲 calculations in the plane-wave Born approximation. The other lines 共top兲are the results for charge distributions derived from experimental nuclear matter densities, using 2PF shapes 共as shown in Fig. 2兲with 共dashed line兲or without 共solid line兲a damped oscillatory correction. PRECISE NUCLEAR MATTER DENSITIES FROM . . . PHYSICAL REVIEW C 65 014602 014602-3 possible to describe electron scattering cross sections from an experimental 16O nuclear density obtained through heavyion elastic scattering data analyses. The theoretical DiracHartree-Bogoliubov 共DHB兲charge distribution predicts electron scattering cross sections which are in disagreement with the data for large qvalues 共see Fig. 3 top兲. We point out that, as a further test of the consistency of the assumed hypothesis of the method, the theoretical DHB distributions for the target nuclei used in this work predict electron scattering cross sections in agreement with the data 共as illustrated in Fig. 3 bottom for the 58Ni). In conclusion, using the progress reached in the last 20 years to describe heavy-ion elastic scattering, it is possible to determine ground-state nuclear matter densities. 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