Elastic Scattering and Breakup Reactions of

Bulg. J. Phys. 44 (2017) S49–S59 Elastic Scattering and Breakup Reactions of 8 B Proton-Halo Projectile on Nuclear Targets at Energies 20 < E < 170 MeV M.K. Gaidarov1 , V.K. Lukyanov2 , D.N. Kadrev1 , E.V. Zemlyanaya2 , K.V. Lukyanov2 , A.N. Antonov1 , K. Spasova1 1 Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria 2 Joint Institute for Nuclear Research, Dubna 141980, Russia Received 31 July 2017 Abstract. A microscopic analysis of the optical potentials (OPs) and cross sections of elastic scattering of 8 B on 12 C, 58 Ni, and 208 Pb targets at energies 20 < E < 170 MeV is carried out. The real part of the OP is calculated by a folding procedure and the imaginary part is obtained on the base of the highenergy approximation (HEA). The density distributions of 8 B obtained within the variational Monte Carlo (VMC) model and the three-cluster model (3CM) are used to construct the potentials. In the hybrid model, the only free parameters are the depths of the real and imaginary parts of OP obtained by fitting the experimental data. The use of HEA to estimate the imaginary OP at energies just above the Coulomb barrier is discussed. The analysis of the behavior of 3CM and VMC densities and the corresponding OPs in comparison with the fitted Woods-Saxon OP gives additional information on the decisive role of the nuclear surface on the elastic scattering mechanism in the particular example of 8 B+58 Ni cross sections measured in a wide range of angles and energies of 20.7, 23.4, 25.3, 27.2, and 29.3 MeV. In addition, cluster model, in which 8 B consists of a p-halo and the 7 Be core, is applied to calculate the breakup cross sections of 8 B nucleus on 9 Be, 12 C, and 197 Au targets, as well as momentum distributions of 7 Be fragments. A good agreement of the theoretical results with the available experimental data is obtained. PACS codes: 21.10.Gv, 24.10.Ht 1 Introduction The development of the radioactive ion beams has allowed studies of nuclei far from stability. This technical headway led to the discovery of halo nuclei on the neutron-rich side of the valley of stability [1, 2]. These weakly bound nuclei 1310–0157 c 2017 Heron Press Ltd. S49 M. K. Gaidarov, V. K. Lukyanov, D. N. Kadrev,... have a strongly clusterized structure [3–6]. In a simple model, they are seen as a core that contains most of the nucleons to which one or two neutrons are loosely bound. The latter is related to the tunneling of the valence neutrons far outside the classically allowed region which form a sort of halo around the core [7]. Although less probable, proton halos are also possible. In recent years, the short-lived radioactive nucleus 8 B with a very low breakup threshold energy (0.137 MeV), adjacent to the proton drip line, has attracted much attention because it may present a proton halo structure and is valuable for astrophysical reasons [8, 9]. The narrow momentum distributions of 7 Be fragments in the breakup of 8 B measured in C, Al, and Pb targets at 1471 MeV/nucleon with full width at half maximum (FWHM) of 81 ± 6 MeV/c in all targets have been interpreted in terms of a largely extended proton distribution for 8 B and have implied a radius of 2.78 fm [10]. Here we should mention also the results of the experiments at lower energies for the breakup of 8 B in the collisions with Be and Au targets at 41 MeV/nucleon (81 ± 4 and 62 ± 3 MeV/c FWHM for Be and Au targets, respectively) [11] and for C target at 36 MeV/nucleon [12] with FWHM 124 ± 17 and 92 ± 7 MeV/c for the stripping and diffraction components, correspondingly. Indeed, these experimental results reflect the large spatial extension of the loosely bound proton in 8 B. The halo nature of 8 B nucleus through studies of its breakup has been mostly tested with cluster models presuming simple two-cluster structure that consists of 7 Be core and valence proton (for instance, Refs. [10, 13]). The aims of the present work (see also [14]) are as follows. First, we analyze the differential elastic cross sections for the scattering of 8 B on 12 C at 25.8 MeV [15], 8 B on 58 Ni at 20.7, 23.4, 25.3, 27.2, and 29.3 MeV [16], and 8 B on 208 Pb at 170.3 MeV [17] within the microscopic model of the respective OP and compare the results with the available experimental data. Such a study could lead to a minimization of the ambiguities in the fitted OPs. As in our previous works [18–22], where processes with neutron-rich He, Li, and Be isotopes were considered, we use the hybrid model of OP [23], in which the real part (ReOP) is calculated by a folding of a nuclear density and the effective nucleon-nucleon (NN) potentials [24] and includes direct and exchange isoscalar and isovector parts. The imaginary part (ImOP) is obtained on the base of the high-energy approximation method developed in Refs. [25, 26]. There are only two fitting parameters in the hybrid model, which are related to the depths of the ReOP and ImOP. In the present work we use the density distribution of 8 B nucleus obtained within the variational Monte Carlo model [27] and also the density obtained within the framework of the microscopic three-cluster model of Varga et al. [28]. Second, for a more consistent description of the possible halo structure of 8 B we calculate the momentum distributions of 7 Be fragments from the breakup reactions 8 B+9 Be, 8 B+12 C, and 8 B+197 Au for which experimental data are available. Such a complex study based on the microscopic method to obtain the OPs with a minimal number of free parameters and by testing density S50 Elastic Scattering and Breakup Reactions of 8 B Proton-Halo... distributions of 8 B which reflect its proton-halo structure (in contrast, e.g., to the Hartree-Fock density used in Ref. [17]) would lead to a better understanding of the 8 B structure and to a reduction of the inconsistency of describing the available data. 2 Elastic Scattering of 8 B on 12 C, 58 Ni, and 208 Pb The microscopic volume OP used in our calculations contains the real part (V DF ) including both the direct and exchange terms and the HEA microscopically calculated imaginary part (W H ). It has the form U (r) = NR V DF (r) + iNI W H (r). (1) The parameters NR and NI entering Eq. (1) renormalize the strength of OP and are fitted by comparison with the experimental cross sections. The real part V DF consists of the direct (V D ) and exchange (V EX ) double-folding integrals that include effective N N potentials and density distribution functions of colliding nuclei. The V D and V EX parts of the ReOP have isoscalar (IS) and isovector (IV) contributions. The IS ones of both terms are Z D D (2) VIS (r) = d3 rp d3 rt ρp (rp )ρt (rt )vN N (s), Z EX VIS (r) = d3 rp d3 rt ρp (rp , rp + s)ρt (rt , rt − s) EX × vN N (s) exp h iK(r) · s i M , (3) where s = r + rt − rp is the vector between two nucleons, one of which belongs to the projectile and another one to the target nucleus. In Eq. (2) ρp (rp ) and ρt (rt ) are the densities of the projectile and the target, respectively, while in Eq. (3) ρp (rp , rp +s) and ρt (rt , rt −s) are the density matrices for the projectile and the target that are usually taken in an approximate form. The effective N N D EX interactions vN N and vN N have their IS and IV components in the form of M3Y interaction obtained within g-matrix calculations using the Paris NN potential [24]. The expressions for the energy and density dependence of the effective N N interaction are given, e.g., in Ref. [22]. Concerning the ImOP, it corresponds to the full microscopic OP derived in Refs. [23, 29] within the HEA [25, 26]: Z ∞ 1 E H W (r) = − 2 σ̄N j0 (qr)ρp (q)ρt (q)fN (q)q 2 dq. (4) 2π k 0 In Eq. (4) ρ(q) are the corresponding form factors of the nuclear densities, fN (q) is the amplitude of the NN scattering and σ̄N is the averaged over the isospin S51 M. K. Gaidarov, V. K. Lukyanov, D. N. Kadrev,... of the nucleus total NN scattering cross section that depends on the energy and accounts for the in-medium effect. To apply the microscopic OPs to scattering of 8 B on nuclei we used realistic density distributions of 8 B calculated within the VMC model [27] and from the 3CM in Ref. [28]. In our case, within the VMC method the proton and neutron densities have been computed with the AV18+UX Hamiltonian, in which the Argonne v18 two-nucleon and Urbana X three-nucleon potentials are used [27]. Urbana X is intermediate between the Urbana IX and Illinois-7 models (the latter was used by us in Ref. [22] for the densities of 10 Be nucleus). As far as the 3CM densities of Varga et al. [28] are concerned, the 8 B nucleus has been studied in a microscopic α+h+p three-cluster model (h = 3 He) using the stochastic variational method, where a Minnesota effective two-nucleon interaction composed from central and spin-orbit parts was used. It has been shown in [28] that the proton separation energy of 8 B is reasonably reproduced, but the calculated point matter radius exceeds the "empirical" one. The VMC and 3CM densities are given in Figure 1. It can be seen that they have been calculated with enough accuracy up to distances much larger than the nuclear radius. In both methods the total densities of 8 B occur quite similar up to r ∼ 4 fm and a difference between them is seen in their periphery. Due to the cluster-structure model of 8 B, where the proton is considered as a single cluster [28], the tail part of the point-proton distribution of 8 B is significantly larger than that of the neutron one, causing considerable difference in the corresponding rms radii (see Ref. [14]). The calculated within the hybrid model elastic scattering cross sections of 8 B+12 C at energy E = 25.8 MeV in the laboratory frame are given in Figure 2 and compared with the experimental data [15]. It can be seen that in both cases of calculations with VMC or 3CM densities the results are in good 1 1 8 8 B ρ(r) [fm-3] ρ(r) [fm-3] 10-2 10-4 proton 3CM neutron 3CM proton VMC neutron VMC 10-6 10 2 4 6 r [fm] 10 -4 10 -6 B 3CM VMC -8 0 10 -2 8 10 10 -8 0 2 4 6 8 10 r [fm] Figure 1. Point-proton (normalized to Z = 5), point-neutron (normalized to N = 3) (left panel) and the total densities (right panel) of 8 B (normalized to A = 8) obtained in the VMC method [27] and in the 3CM [28]. S52 Elastic Scattering and Breakup Reactions of 8 B Proton-Halo... agreement with the available data. The differential cross section obtained with VMC density demonstrates more developed diffraction pattern. It would be desirable to measure the elastic scattering in the angular range beyond 55◦ , where the differences between the theoretical results start, in order to determine the advantage of using VMC or 3CM microscopic densities of 8 B. Complementary measurements at smaller steps of scattering angle would also allow one to observe some possible oscillations of the cross section. Our next step is to study 8 B elastic scattering on a lead target at 170.3 MeV incident energy. The same Figure 2 shows a fair agreement of our microscopic calculations with the experimental data for the cross section. Both VMC and 3CM densities used in the calculations are able to reproduce the data that are restricted in a range of small angles. Similarly to the case of 8 B+12 C process, the reasonable agreement of our model with the data on 8 B+208 Pb elastic scattering is in favor of the very 10 (dσ/dΩ)/(dσR/dΩ) 8 B + 12C @ 25.8 MeV 1 10-1 -2 10 10-3 10-4 10 20 30 40 50 θc.m. [deg] 60 70 10 (dσ/dΩ)/(dσR/dΩ) 8 B + 208Pb @ 170.3 MeV 1 10 -1 10 -2 5 10 15 θc.m. [deg] 20 Figure 2. (Top) 8 B+12 C elastic scattering cross sections at E = 25.8 MeV. Solid line: calculations with 3CM density of 8 B; dashed line: calculations with VMC density of 8 B. Experimental data are taken from Ref. [15]; (Bottom) 8 B+208 Pb elastic scattering cross sections at E = 170.3 MeV. Solid line: calculations with 3CM density of 8 B; dashed line: calculations with VMC density of 8 B. Experimental data are taken from Ref. [17]. S53 M. K. Gaidarov, V. K. Lukyanov, D. N. Kadrev,... weak contribution from other reaction mechanisms, which is supported by the results from CDCC calculations [15, 17]. In what follows, we present in Figure 3 (left panel) our results for 8 B+58 Ni elastic scattering cross sections at energies 20.7, 23.4, 25.3, 27.2, and 29.3 MeV using the VMC density. These results are obtained with NR and NI which reproduce in a best way the experimental cross sections at considered five energies. One can see that the results are in a good agreement with the data for all energies considered. It is well known that the couplings to non-elastic channels lead to polarization potentials that can considerably modify the bare potential calculated within the double folding formalism. Obviously, for more successful description of cross sections at low energies near Coulomb barrier an inclusion of polarization contributions due to virtual excitations and decay channels of the reactions is necessary to obtain unambiguously the OP renormalization parameters. The good fit obtained for the experimental angular distributions in Ref. [16] with real and imaginary potentials of the Woods-Saxon type and our best fit to the same data using microscopic OP in this work lead to values of the predicted total re- 20.7 MeV 1 |V| [MeV] (dσ/dΩ)/(dσR/dΩ) 10-1 1 3 10 2 8 B+ 25.3 MeV -1 10 -2 10 -3 |W| [MeV] 27.2 MeV 10-1 29.3 MeV 10 3 10 2 10 10 -3 20 40 60 80 100 θc.m. [deg] 120 140 160 B+ Ni @ 25.3 MeV 10 1 10-1 -2 58 0.317VDF - VMC 0.235VDF - 3CM WS 8 1 Ni @ 25.3 MeV 1 10 10-1 1 58 10 23.4 MeV 1 10 10 -1 10 -2 1.030WH - VMC H 0.212W - 3CM WS 0 2 4 6 8 10 12 r [fm] Figure 3. (Left) 8 B+58 Ni elastic scattering cross sections at E = 20.7, 23.4, 25.3, 27.2 and 29.3 MeV calculated using the VMC density of 8 B. Experimental data are taken from Ref. [16]; (Right) The absolute values of the real NR V DF and imaginary NI W H parts of the calculated optical potentials for the 8 B+58 Ni elastic scattering at E = 29.3 MeV obtained using the VMC and 3CM densities of 8 B in comparison with those of the WS potential from Ref. [16]. S54 Elastic Scattering and Breakup Reactions of 8 B Proton-Halo... action cross section σR very close to each other (see [14]), the latter exhibiting a smooth increase with the energy increase. Also, we give in Figure 3 (right panel), as an example (for E = 29.3 MeV), the comparison of the obtained real and imaginary parts of the OPs for both 3CM and VMC densities with the corresponding parts of the fitted Woods-Saxon potential used in [16]. The values of our parameters NR and NI are indicated in the figure. Here we mention that at such energies the surface part of the ImOP plays a decisive role on the behavior of the elastic cross sections. One can see that the use of the VMC density leads to a very good agreement of the imaginary part of our OP with the imaginary part of the fitted WS OP in the surface region. Also, the slope of the real part of OP obtained with the VMC density in this region (8 < r < 10 fm) is similar to that of the real part of WS OP. There exist some differences in the surface region for the real and imaginary parts of the OP obtained with the 3CM density and the corresponding parts of the WS OP. 3 Breakup Reactions of 8 B In this section we consider the characteristics of breakup processes of the 8 B nucleus on the example of the stripping reaction cross sections and the momentum distributions of the fragments. We use a simple model in which 8 B consists of a core of 7 Be and a halo of a single proton (see, e.g., Refs. [10]). In this model the density of 7 Be has to be known. We use that one obtained from the calculations performed by means of the 3CM density of 8 B [28]. For calculations of breakup cross sections and momentum distributions of fragments in the 7 Be+p breakup model within the eikonal formalism (see, e.g. Ref. [30]), one needs the expressions of the S-matrix (as a function of the impact parameter b):   Z ∞ p i (b) 2 2 (5) U ( b + z )dz , S(b) = exp − ~v −∞ where U (b) = V + iW (6) 8 is the OP of the breakup of B in its collision with nuclear targets within the 7 Be+p cluster model. The longitudinal momentum distribution of 7 Be fragments produced in the breakup of 8 B in the case of stripping reaction (when the proton leaves the elastic channel) is (more details can be found in Ref. [14])   Z   dσ 1 ∞ = bp dbp 1 − |Sp (bp )|2 dkL str 3 0 Z X Fm (ρ). (7) × ρdρdϕρ |Sc (bc )|2 m=0,±1 S55 M. K. Gaidarov, V. K. Lukyanov, D. N. Kadrev,... The wave function of the relative motion of the proton and 7 Be clusters in 8 B is obtained by solving the Schrödinger equation with the Woods-Saxon potential for a particle with a reduced mass of two clusters. The parameters of the WS potentials are obtained by a fitting procedure, namely, to reach the proton separation energy Sp = 137 KeV. However, this procedure could provide several sets of potential parameters that satisfy the above condition. They are close to each other leading, at the same time, to different valence proton rms radii. Therefore, in order to understand better the observed widths of the longitudinal momentum distributions of 7 Be fragments formed in the breakup of 8 B and measured at different targets and energies, we consider three cases. The values of WS potential parameters and corresponding rms radii of the cluster formation for 1p state in which the valence proton in 8 B is mainly bound (see Refs. [13, 30]) are listed in Table 1. Table 1. The parameters V0 (in MeV), R (in fm), a (in fm) of the Woods-Saxon potentials and the rms radii of the cluster wave function (in fm) obtained by using of the 3CM density of 7 Be for three cases (see the text). V0 R a rms radii 38.22 38.70 38.77 2.70 2.50 2.48 0.55 0.20 0.50 4.51 5.08 6.24 The stripping cross sections for reactions 8 B+9 Be, 8 B+12 C, and 8 B+197 Au are calculated from Eq. (7). The obtained results are illustrated in Figure 4. The blue dotted, black solid, and red dashed curves in the figures correspond to the three sets of WS parameters given in Table 1. Here we note that due to the arbitrary units of the measured cross sections of the considered processes it was not necessary to renormalize the depths of our OPs of the fragments-target nuclei interactions. It is worth to note the relevance between the rms radii of the wave function of the 7 Be-p relative motion and the obtained FWHMs for the considered three cases. Due to the uncertainty principle the widths become smaller with the increase of the distance between two clusters. We note the good agreement with the experimental data from light and heavy breakup targets. It can be seen from Figure 4 that the best agreement with the experimental data for the parallel momentum distributions of 7 Be fragments in a breakup reaction of 8 B on a 9 Be target at 41 MeV/nucleon and on a 12 C target at 36 MeV/nucleon is achieved when the relative 7 Be-proton distance is 5.08 fm or 4.51 fm, respectively, while in the case of 8 B breakup on a 197 Au target at 41 MeV/nucleon shown in the same figure a larger distance (6.24 fm) is needed to get better coincidence with the data. The values of the theoretical FWHMs in this case are 72.07 MeV/c when describing the stripping mechanism of the 8 B breakup on the 9 Be target, 108.71 MeV/c for the breakup on 12 C target, and 54.86 in the case of 197 Au target. These values are close to the experS56 Elastic Scattering and Breakup Reactions of 8 B Proton-Halo... 500 dσ/dkL [arbitrary units] 8 400 9 B + Be 300 200 100 0 1700 1800 1900 2000 kL [MeV/c] 2100 2200 dσ/dkL [arbitrary units] 70 60 8 B + 12C 50 40 30 20 10 0 1300 1400 1500 1600 1700 kL [MeV/c] 1800 1900 dσ/dkL [arbitrary units] 140 120 8 B+ 197 Au 100 80 60 40 20 0 1800 1900 kL [MeV/c] 2000 2100 Figure 4. Cross sections of stripping reactions in 8 B+9 Be scattering at E = 41 MeV/nucleon, 8 B+12 C scattering at E = 36 MeV/nucleon, and 8 B+197 Au scattering at E = 41 MeV/nucleon. Experimental data are taken from Refs. [11], [12], and [11], respectively. imentally measured widths and are within the range found in other theoretical analyses. S57 M. K. Gaidarov, V. K. Lukyanov, D. N. Kadrev,... 4 Conclusions In the present work (see also [14]) we performed microscopic calculations of the optical potentials and cross sections of elastic scattering 8 B+12 C at 25.8 MeV, 8 B+58 Ni at 20.7, 23.4, 25.3, 27.2, and 29.3 MeV, and 8 B+208 Pb at 170.3 MeV, in comparison with the available experimental data. The direct and exchange isoscalar and isovector parts of the real OP (V DF ) were calculated microscopically using the double-folding procedure and density dependent M3Y (CDM3Y6-type) effective interaction based on the Paris N N potential. The imaginary part of the OP (W H ) was calculated as a folding integral that corresponds to the one in a phase of HEA. Two model microscopic densities of protons and neutrons in 8 B were used in the calculations: the density calculated within the VMC model [27] and from the three-cluster model [28]. We have tested our microscopic model studying the role of the breakup mechanism to analyze properly the whole picture of 8 B scattering. For this purpose, we use another folding approach to consider the 8 B breakup by means of the simple 7 Be+p cluster model for the structure of 8 B. We calculate in HEA the ImOP of the interaction of 7 Be with the target, as well as the p+target interaction. Using them the corresponding S-matrices for the core and proton within the eikonal formalism are obtained. The latter are used to get results for the longitudinal momentum distributions of 7 Be fragments produced in the breakup of 8 B on different targets. This includes the breakup reactions of 8 B on 9 Be and 197 Au at E = 41 MeV/nucleon and 8 B on 12 C at E = 36 MeV/nucleon, for which a good agreement of our calculations for the stripping reaction cross sections with the available experimental data were obtained. The theoretical widths are close to the empirical ones. In general, we can conclude that our microscopic approach can be applied to reaction studies with exotic nuclei such as 8 B. The consistency of our results with the measured elastic cross sections and narrow longitudinal momentum distributions may provide supplemental information on the internal spatial structure of the 8 B nucleus supporting its proton-halo nature. Acknowledgments The authors are grateful to S. C. Pieper for providing with the density distributions of 8 B nucleus calculated within the VMC method and to S. L. Jin for the experimental longitudinal momentum distributions of 7 Be fragments from the breakup of 8 B on a carbon target. Four of the authors (D.N.K., A.N.A., M.K.G. and K.S.) are grateful for the support of the Bulgarian Science Fund under Contract No. DFNI–T02/19 and one of them (D.N.K.) under Contract No. DFNI–E02/6. S58 Elastic Scattering and Breakup Reactions of 8 B Proton-Halo... References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] I. Tanihata et al. (1985) Phys. Lett. B 160 380. I. Tanihata et al. (1985) Phys. Rev. Lett. 55 2676. P.G. Hansen, A.S. Jensen, B. Jonson (1995) Annu. Rev. Nucl. Part. Sci 45 591. I. Tanihata (1996) J. Phys. G 22 157. B. Jonson (2004) Phys. Rep. 389 1. G. Baur, K. Hencken, D. Trautmann (2003) Prog. Part. Nucl. Phys. 51 487. P.G. Hansen, B. Jonson (1987) Europhys. Lett. 4 409. B. Davids et al. (1998) Phys. Rev. Lett. 81 2209. L. Trache et al. (2003) Phys. Rev. C 67 062801. W. Schwab et al. (1995) Z. Phys. A 350 283. J.H. Kelley et al. (1996) Phys. Rev. Lett. 77 5020. S.L. Jin et al. (2015) Phys. Rev. C 91 054617. H. Esbensen (1996) Phys. Rev. C 53 2007. V.K. Lukyanov, D.N. Kadrev, E.V. Zemlyanaya, K.V. Lukyanov, A.N. Antonov, M.K. Gaidarov, K. Spasova (2017) Eur. Phys. J. A 53 31. A. Barioni et al. (2011) Phys. Rev. C 84 014603. E.F. Aguilera et al. (2009) Phys. Rev. C 79 021601(R). Y.Y. Yang et al. (2013) Phys. Rev. C 87 044613. K.V. Lukyanov et al. (2007) Eur. Phys. J. A 33 389. V.K. Lukyanov et al. (2009) Phys. Rev. C 80 024609. V.K. Lukyanov et al. (2010) Phys. Rev. C 82 024604. V.K. Lukyanov et al. (2013) Phys. Rev. C 88 034612. V.K. Lukyanov et al. (2015) Phys. Rev. C 91 034606. K.V. Lukyanov, E.V. Zemlyanaya, V.K. Lukyanov (2006) Phys. At. Nucl. 69 240. D.T. Khoa, G.R. Satchler (2000) Nucl. Phys. A 668 3. R.J. Glauber (1959) in Lectures in Theoretical Physics, Interscience, New York, p. 315. A.G. Sitenko (1959) Ukr. Fiz. J. 4 152. S.C. Pieper, private communication. K. Varga, Y. Suzuki, I. Tanihata (1995) Phys. Rev. C 52 3013. P. Shukla (2003) Phys. Rev. C 67 054607. K. Hencken, G. Bertsch, H. Esbensen (1996) Phys. Rev. C 54 3043. S59