Astrophysical S17(0) factor extraction from breakup of 8B on 58Ni at energies near the Coulomb barrier

PHYSICAL REVIEW C 80, 064617 (2009) Astrophysical S17(0) factor extraction from breakup of 8Bon58Ni at energies near the Coulomb barrier T. L. Belyaeva,1E. F. Aguilera,2E. Martinez-Quiroz,2A. M. Moro,3and J. J. Kolata4 1Universidad Aut´ onoma del Estado de M´ exico, Codigo Postal 50000, Toluca, M´ exico 2Departamento del Acelerador, Instituto Nacional de Investigaciones Nucleares, A.P. 18-1027, Codigo Postal 11801 Mexico, Mexico 3Departamento de F´ ısica At´ omica, Molecular y Nuclear, Universidad de Sevilla, Post Office Box 1065, E-41080 Sevilla, Spain 4Physics Department, University of Notre Dame, Notre Dame, Indiana 46556, USA (Received 8 October 2009; published 31 December 2009) We have performed continuum-discretized coupled channels (CDCC) calculations of the breakup of 8Bon58Ni and direct proton transfer for the 8B+58Ni system at laboratory energies of 20–28.4 MeV. The influence of the 7Be core-target optical potential (OP) on the breakup cross section was investigated. Elastic scattering angular distributions for the 7Be +58Ni and 8B+58Ni systems at five different energies around the Coulomb barrier were studied, and a reasonable energy-independent OP for each system was obtained. Using these OPs and two different 7Be-prelative motion wave functions, and summing breakup and direct proton transfer contributions, we were able to fit the experimental cross section at a 8B laboratory energy of 25.75 MeV. We calculated the excitation function for the 7Be emission in the 8B+58Ni reaction, where 7Be products were measured at the forward angle θlab =45◦in the energy interval Elab =20–28.4 MeV. In view of the peripheral character of the 8B breakup reaction at near-barrier energies, we could extract the asymptotic normalization coefficient for the 7Be-psystem, which was found to be C2 Be-p,p3/2=0.543 ±0.027 fm−1. Finally, the astrophysical S17(0) factor was found to be S17(0) =20.8±1.1eVb. DOI: 10.1103/PhysRevC.80.064617 PACS number(s): 25.60.Gc, 25.60.Dz, 21.10.Jx, 25.60.Bx I. INTRODUCTION The radiative capture 7Be(p,γ )8B reaction plays a major role in the production of high-energy neutrinos in the sun from the βdecay of 8B. At solar energies (∼20 keV in the center of mass frame of the 7Be +psystem), E1 radiative capture is dominant and the σpγ (Ec.m.) cross section is directly related to the neutrino flux. It is conventional to determine it in terms of the astrophysical factor S17(Ec.m.)= σpγ (Ec.m.)Ec.m.exp(2πη), where ηis the Sommerfeld parameter. The overall knowledge about S17 for this reaction has improved considerably because of a new direct technique for the proton capture reaction with radiative 7Be targets as well as new radioactive beam facilities for indirect measurements of Coulomb excitation or Coulomb breakup cross sections. The S17(0) value of 21.4±0.5(exp) ±0.6(theor) eV b recommended from the analysis of all the available low-energy data, including recent direct measurements at Ec.m.=116–1244 keV [1] and Ec.m.=302–1078 keV [2], and the extrapolation of σpγ (Ec.m.)toEc.m.=0, was reported in Ref. [1]. This value is consistent with the extrapolated value of S17(0) =21.2±0.7 eV b obtained in Ref. [2]from the entire set of measurements. The S17(0) values extracted from the indirectly measured data, which were obtained from the Coulomb breakup, are estimated to be within the range 16.7⩽S17(0) ⩽20.6eVb[3]. Comparing the S17(0) values following from the direct and indirect experiments, one concludes that the results are slightly technique-dependent. Theoretical calculations have to be used to extrapolate the measured S-factor to the astrophysically relevant energies. The recent status of different cluster models used in the field of nuclear astrophysics is presented, for example, in the review of Descouvemont [4]. Indirect measurements related to the Coulomb dissociation of 8B in the field of heavy nuclei proceed predominantly via E1 transitions, but a contribution of higher multipolesand the nuclear interaction is also possible because the very first breakup measurement [5], a contribution of E2 photons, was intensively studied in a number of experimental and theoretical works [6–8]. A discussion of this problem can be found in Ref. [9]. Important information can be obtained from the study of nuclear reactions with light exotic nuclei, which is related to their structure. It is well known that 8B is an exotic nucleus with a weakly bound proton (0.137 MeV), and the elastic scattering, fusion, and dissociation of 8B in the field of heavy nuclei (in particular, on 58Ni) are of special interest and were widely studied both experimentally [10–14] and theoretically [15–22]. In this article, we report the results of the 8B+58Ni system analysis with the method of continuum-discretized coupled channels (CDCC) [23–25] in the energy interval 20–28.4 MeV in laboratory system, which are on and slightly above the Coulomb barrier (VB=20.8 MeV). We carry out the CDCC calculations of the breakup of 8Bon58Ni and direct proton transfer for the 8B+58Ni system and compare the results with the differential cross sections measured in Refs. [12,13]. The first aim of our analysis is to study the influence of the 7Be core-target interaction potential on the breakup cross section, for which we employ an optical model fit to the recent data on the elastic scattering angular distributions for the 7Be +58Ni and 8B+58Ni systems measured at five different energies around the Coulomb barrier [14]. Second, applying a reasonable 7Be-target OP and two different 7Be-prelative motion wave functions, we fit the experimental cross section at a 8B laboratory energy of 25.75 MeV [12], with the aim 0556-2813/2009/80(6)/064617(8) 064617-1 ©2009 The American Physical Society T. L. BELYAEVA et al. PHYSICAL REVIEW C 80, 064617 (2009) of extracting an experimental spectroscopic factor (SF) Sexp for the 7Be-psystem. Then, this value of SF can be used to calculate the excitation function of the 7Be emission in the 8B+58Ni reaction, in comparison with a recent experiment [13], in which the emission of the 7Be products was measured for the forward angle θlab =45◦at 25.0, 26.9, and 28.4 MeV bombarding energies. The result of the analysis is then used to extract the astrophysical S17(0) factor by means of the asymptotic normalization coefficient method. II. ELASTIC SCATTERING ANALYSIS Recently, 8B and 7Be +58Ni elastic scattering angular distributions were measured [14] in the angular range θc.m.= 25◦–158◦for energies near the Coulomb barrier. The scattering data correspond to 8B laboratory energies of 20.7, 23.4, 25.3, 27.2, and 29.3 MeV and 7Be laboratory energies of 15.1, 17.1, 18.5, 19.9, and 21.4 MeV. The angular distributions for both projectiles show a prominent reduction for θc.m.>50◦because of the flux going to other channels that are associated mainly with breakup. The primary aim of the experiment and its analysis was to compare the elastic and the total reaction cross sections for the proton-halo nucleus 8B and for the “normal,” weakly bound 7Be nucleus using well-adjusted OPs. OPs of Woods-Saxon shape with a shallow real part and a very deep imaginary part for the 8B+58Ni system as well as the Sao Paulo potential [26]forthe7Be +58Ni system were used. The parameters of these OPs were slightly energy-dependent. Lubian et al. [22] have also investigated the effects of inelastic excitations of the target and continuum-continuum couplings on the angular distributions by comparison of the experimental 8B+58Ni angular distributions measured in Ref. [14] with results of the respective CDCC model calculations. To describe the 7Be-target interaction, the authors of Ref. [22] adopted the OP of Moroz et al. [27], which fit the data of 7Li +58Ni elastic scattering at Elab =20.3 MeV. It was shown that the simplest calculations involving a single-channel problem, without any type of couplings, are consistent with the data only at the lowest energies of 8B projectiles, Elab =20.7 and 23.4 MeV. On the other hand, Lubian et al. [22] have found that coupling to breakup greatly improves the agreement with experiment, whereas the inelastic excitations of the target do not have an appreciable influence on the elastic angular distributions. In this article, we carry out a new analysis of the data of Ref. [14] with the aim of finding a single energy-independent set of potential parameters. We apply an OP formalism with the standard complex nuclear OP of the form V(r)=VCoul(r)−V01−Fs.o.(l·s)r0 r d drf0(r) −iW−4WD d drfW(r), fi(r)={1+exp [(r−Ri)/ai]}−1,R i=riA1/3 p+A1/3 T. The Coulomb interaction is represented by the VCoul(r) potential of a uniformly charged sphere of radius RC= rC(A1/3 p+A1/3 T). We performed the coupled-channel calculations using the computer code FRESCO [28]. In our calculations, the coupledchannel equations have been reduced to a single-channel problem, corresponding to the elastic scattering in the entrance channel. We studied a sensitivity of the calculated cross sections to the OP parameters. In particular, we note a strong sensitivity to the parameters that describe a behavior of the potential in the nuclear surface region: the real potential radius and real diffuseness parameter and a very weak sensitivity to the real and imaginary strengths. These features of the OPs explain a possibility of using potentials with the shallow real strengths [14] as well as with deep ones [22].Atthesametime, the calculated cross sections depend drastically on the choice of the radial parameter r0and can change a few times with r0 variation of a few (4%–5%) percentage. Finally, we found the energy-independent OPs producing a good fit to the angular distributions for all energies in this interval for both systems (see Table I). In Figs. 1and 2,the calculated cross sections for the 7Be +58Ni and 8B+58Ni elastic scattering are compared with the experimental data [14]. These OPs have traditional Woods-Saxon shapes and a depth ratio V0/W =2.5. The OP describing the 8B+58Ni system has some specific features: a small real diffuseness parameter a0and a large imaginary diffuseness parameter aW.The corresponding long-range imaginary part of the interaction is responsible for the stronger absorption in this system. The optical model and interaction potential parameters compiled in Table Iare used in all the calculations carried out in this article. TABLE I. Optical model and interaction potential parameters.a Channel V0Fs.o.r0a0WW DrWaWrC (MeV) (fm) (fm) (fm) (MeV) (MeV) ×(fm) (fm) (fm) (fm) 8B+58Ni 150.01.170 0.30 60.01.20 0.75 1.20 7Be +58Ni 150.01.190 0.50 60.01.15 0.62 1.20 p+58Ni (OP) 42.61.170b0.75 7.24 2.59 1.26b0.58 1.25b p+58Ni (bound) 1.25b0.75 1.25b p+7Be (EB) 44.675 0.351 1.25b0.52 1.25b p+7Be (Kim) 32.120 0.666 1.54b0.52 1.54b a“EB” is the Esbensen and Bertsch potential from Ref. [6]; “Kim” is the Kim et al. potential from Ref. [29]. bThe radii of the potentials are defined as Ri=riA1/3 T,i=0,W,C. 064617-2 ASTROPHYSICAL S17(0) FACTOR EXTRACTION ... PHYSICAL REVIEW C 80, 064617 (2009) FIG. 1. (Color online) The elastic angular distributions for 7Be +58Ni scattering calculated within the optical model (curves) in comparison with the experimental data [14]. III. ASYMPTOTIC NORMALIZATION COEFFICIENTS AND SPECTROSCOPIC FACTOR RELATION IN CDCC In a direct reaction, it is convenient to represent the projectile nucleus pas a cluster structure consisting of a core cand a valence fragment v. Thus the main nuclear structure information is contained in the overlap function of cluster configurations φcv(r)=ψcψv|ψp[30], which is a model-independent characteristic of the nucleus. This implies that the projectile wave functions ψp(ζp,r), where ζpdenotes all the intrinsic coordinates of the nucleons in the projectile and ris the relative coordinate between a valence fragment v and a core c, may be expanded in terms of the wave functions of the complex subsystems or clusters. A normalization of the overlap functions defines the spectroscopic factor for the vertex p→c+v: S(p→c+v)=|φcv(r)|2dr.(1) The spectroscopic factor Sis a measure of the overlaps of the cluster wave functions and gives the probability of the FIG. 2. (Color online) The elastic angular distributions for 8B+ 58Ni scattering calculated within the optical model (curves) in comparison with the experimental data [14]. wave function ψp(ζp,r) of the nucleus pbeing composed of the wave functions of the clusters cand vwith a relative orbital angular momentum land total angular momentum j. Consequently, the overlap function is not normalized to unity. The asymptotic behavior for r>R N(RNis the channel radius) of the radial part φcv(r) of the overlap function is determined by the binding energy, εcv, for the vertex p→c+v(see, e.g., Ref. [31]): φcv(r)→Ccv,lj W−η,l+1/2(2kcvr)/r (2) φcv(r)→C cv,lj 2k rKl+1/2(kcvr),(3) where k2 cv =2µcvεcv/¯h2is the relative linear momentum and η=ZcZve2µcv/k2 cv is the Coulomb parameter. Equation (2), with the Whittaker function, corresponds to the separation of a charged cluster, and Eq. (3), with the modified Bessel function, corresponds to the separation of a neutron. Here Ccv,lj and C cv,lj are the asymptotic normalization coefficients (ANC). For the virtual p→c+vvertex, the ANC is related to the nuclear vertex constant (NVC) Gcv,lj : Gcv,lj =−il√π µcv Ccv,lj .(4) The theory and methods for their calculation for light nuclei were developed over 30 years ago and are summarized in the review of Blokhintsev et al. [32]. New methods for their evaluation were proposed in Refs. [33–35]. In the CDCC method, the single-nucleon bound state or the state in the continuum of the projectile nucleus pof spin Ipis described by the core +nucleon wave function and is introduced through the parentage expansion, ψp(ζc,r)= α Aα[ψIc(ζc)ϕα(r)]IpMp,(5) in terms of the wave functions ψIc(ζc) of the core nucleus, of spin Ic,with (A−1) nucleons. Here Aαis the coefficient of fractional parentage and ϕα(r) is the wave function for the relative c+vmotion. The sum includes all states of the core cand valence nucleus vwith the angular momentum coupling scheme j=l+sj+Ic=Ip,(6) which we label by α={l,s,j}, where land jdenote the relative orbital and total angular momenta of core and valence particle in the projectile and sis the spin of the valence nucleon. In the case of inelastic excitations in the core-valence particle system from the ground state to excited states in the continuum, Eq. (5) describes the bin wave function calculated within a momentum interval, or bin, [ki−1,k i], labeled by i.The number; the boundaries, ki; and the widths (ki−ki−1)ofbins are defined by the conditions of an optimal discretization of the continuum for every specific case. The relative wave functions ϕα(r) for bound and unbound states differ by their asymptotic behavior. The radial bin wave functions uα(r)are square integrable and are calculated as a superposition, within the momentum bin interval [ki−1,k i], of the eigenfunctions 064617-3 T. L. BELYAEVA et al. PHYSICAL REVIEW C 80, 064617 (2009) fα(k,r)ofthec+vinternal Hamiltonian; this asymptotic behavior for r→∞, fα(k,r)→[cos δα(k)Fl(kr)+sin δα(k)Gl(kr)], is described by the regular FLand irregular GLCoulomb functions and k∈[ki−1,k i]. To achieve effective normalization of the radial bin wave functions uα(r), certain weight factors gα(k)[17] are introduced into the integral over a momentum bin [ki−1,k i]. The asymptotic behavior of the projectile ground-state bound wave function uα(r)forr>R N[cf. Eq. (2)], uα(r)→bcv,lj W−η,l+1/2(2kr)/r, (7) is proportional to the “single-particle” ANC bcv,lj .The radial parts uα(r) of the relative c+vmotion-bound wave functions ϕα(r) are found as the eigenfunctions of a given Vcv(r) interaction potential. Commonly, it is generated via a Woods- Saxon potential, the depth of which is adjusted to fit the valence particle separation energy. In the CDCC method as well as in the distorted wave Born approximation (DWBA), the overlap function φcv(r) and the single-particle wave function uα(r) of the relative motion of c and vfragments describing the bound state of nucleus pare related by φcv(r)=S(p→c+v)1/2uα(r)(8) because uα(r) is normalized to unity. From the comparison of Eqs. (1)–(4), Eq. (7), and Eq. (8), one can deduce, following Ref. [32], that for peripheral direct nuclear reactions, the model-independent NVC and ANC are related to the product of a model-dependent spectroscopic factor (SF) and the “singleparticle” ANC bcv,lj : Gcv,lj =−il√π µcv Ccv,lj ≈−il√π µcv Sexp(p→c+v)1/2bcv,lj . (9) From the first developments of the direct nuclear reaction theory (see, e.g., Refs. [30,32], and references therein), it was noted that the combination Sexpb2, which can be found from DWBA or CDCC analysis of the experimental data, only weakly depends on model parameters, contrary to the behavior of empirical values of SF, Sexp, determined by a normalization of the calculated cross section to the experimental data and to the “single-particle” ANC b, separately. We follow the results that were obtained recently by Capel and Nunes [19]. The authors have shown that the elastic breakup of loosely bound projectile, which is described as a two-body system—a spherical, structureless core cto which a pointlike fragment vis loosely bound—is peripheral in the sense that the breakup cross section is not sensitive to the internal part of the bound wave function for the relative c+vmotion. In Ref. [19], breakup calculations were performed for 8B and 11Be projectiles at intermediate energies (40–70 MeV/A)on208Pb and 12C targets as well as at low energy (26 MeV) on a 58Ni target, and it was demonstrated that only the asymptotic properties of the projectile, namely, the ANC of the bound state and the phase shifts in the continuum, influence the breakup cross section. We emphasize that the 8B breakup and transfer reactions on 58Ni at energies around the Coulomb barrier are peripheral in the sense noted previously, with no contribution from the inner part of the 8B bound-state wave function [8,19], because the valence proton is distant from the 7Be by more than the range RN, where RNis about 4 fm. In fact, because φcv(r) and uα(r) may have quite different forms in the inner region, Sexp may be quite different from the true spectroscopic factor, S(p→c+v), and depend strongly on the choice of uα(r). Nevertheless, the ANC, Ccv,lj = Sexp(p→c+v)1/2bcv,lj , can be well determined from a peripheral reaction and can serve for the determination of the astrophysical Sfactor [8]. Thus the cross section from any reaction in which a proton is separated from 8B is proportional to C2 cv,lj , as in both the DWBA and the CDCC method. IV. 8B BREAKUP AND STRIPPING CALCULATIONS AND ASTROPHYSICAL S17(0) FACTOR ESTIMATION A. 8B breakup A theoretical analysis within the CDCC model was made for breakup of 8B using the same model space as used in Ref. [17]. We suppose that the proton within 8B has an orbital angular momentum lrelative to 7Be and a total angular momentum j=l+s=l+1/2. Inelastic excitations in the 7Be-proton system from the ground state to excited states with orbital angular momenta l=0−4 and energies up to 8 MeV in the continuum were taken into account, and 24, 24, 17, 15, and 15 discretized-continuum states were taken for the l=0, 1, 2, 3, and 4 states, respectively. The continuum bins were integrated up to Rbin =60 fm, and the CC equations were solved with Rmax =500 fm and up to Lmax =1000 partial waves using the code FRESCO [28]. The multipoles for the nuclear and Coulomb coupling potentials were included up to λ=4. The model space described earlier turns out to give good convergence of the resulting breakup cross sections. The cross sections calculated using FRESCO in the center of mass frame of the projectile and the target were transformed to the laboratory coordinate system related to a measurement of the reaction products with an elastic Jacobian. For the case of the inelastic (breakup) differential cross sections, this transformation gives practically the same result in comparison with the exact method proposed by Tostevin et al. [17]. Calculations were performed with the nuclear interaction and OPs shown in Table I. The OP for the p-58Ni scattering was taken from the global parametrization of Becchetti and Greenlees [36]. For the 7Be-58Ni interaction, the OPs obtained from the elastic scattering analysis were used. The results of the CDCC calculations for the 7Be angular distributions and the experimental data obtained at the University of Notre Dame [12]atElab =25.75 MeV are shown in Fig. 3. This figure also shows a comparison of the CDCC calculations made using two sets of optical parameters for the core-target interaction: the 7Be +58Ni and 8B+58Ni OPs from Table I. Two values of spectroscopic factors (Sexp =1 and Sexp =1.1025) and the EB bound-state wave function were used. One can see that the calculated breakup differential cross 064617-4 ASTROPHYSICAL S17(0) FACTOR EXTRACTION ... PHYSICAL REVIEW C 80, 064617 (2009) FIG. 3. (Color online) The CDCC calculations for the 7Be differential cross sections from the 8B+58Ni breakup reaction at Elab =25.75 MeV, using the 7Be +58Ni and 8B+58Ni OPs from Table Iand for two values of spectroscopic factors: Sexp =1and Sexp =1.1025, in comparison with the experimental data [12]. sections are less sensitive to the choice of the core-target OP at the maximum of the angular distribution, as it has been noted in Ref. [17], but the main difference between the calculations carried out with these different OPs is seen for the middle angles 40◦<θ lab <100◦. The spectroscopic factor Sexp(8B→7Be +p) was estimated by a comparison of the experimental and calculated differential cross sections. As mentioned earlier, the normalization of the calculated cross section depends on the choice of the p+7Be bound-state wave function. The variation of the geometric parameters of the p+7Be interaction potential can significantly change the normalization of the cross section and thus the spectroscopic factor. In particular, an increase of the p+7Be interaction radius leads to a decrease in the Sexp(8B→7Be +p) value. The “single-particle” asymptotic normalization coefficient bis obviously model-dependent because it is well characterized by the geometry of the potential. Nevertheless, as shown in Refs. [37,38], if one varies both geometric parameters, the radius and diffuseness, of the single-particle potential, leaving the single-particle ANC bconstant, the breakup cross section does not change. In Ref. [38], this method was applied to determine the correct ANC in the 7Be(p,γ )8B reaction and to estimate the astrophysical S17 factor. In our analysis, two normalized single-particle 7Be +p wave functions generated by two Woods-Saxon potentials (see Table I) and adjusted to fit the proton separation energy of EB=0.137 MeV were used. The first was tested in Ref. [6]for the radiative capture calculations of the 7Be(p,γ )8B reaction cross sections and the second was applied in Ref. [29]. These wave functions are very different because the Kim interaction potential has a larger radius. We assume that the 2+ground state (g.s.) of 8B is described by a p3/2proton wave function coupled to the 3/2−g.s. of the 7Be core. The 8B(g.s) wave function generated by the EB potential has a single-particle ANC b=0.7005 fm−1/2, and the 8B(g.s.) wave function generated by the Kim potential has a single-particle ANC b=0.8013 fm−1/2. We obtained the best fits to the experimental data at Elab =25.75 MeV [12] by the CDCC breakup calculations with a spectroscopic factor for the 8B→7Be +pvertex Sexp(8B→7Be +p)=1.1025 using the EB wave function and Sexp(8B→7Be +p)=0.85 using the Kim wave function. The calculated curves are practically indistinguishable (see Fig. 3). The main result is that the ANC CBe-p,p3/2was found to be essentially the same for both wave functions used in the calculations: C2 Be-p,p3/2=0.541 fm−1for the EB wave function and C2 Be-p,p3/2=0.545 fm−1for the Kim wave function. The uncertainty of the spectroscopic factor estimation is within 5%, and the average value of the ANC that we accept for the following calculations is C2 Be-p,p3/2=0.543 ±0.027 fm−1; CBe-p,p3/2=0.737 ±0.018 fm−1/2. This value is in perfect accord with the theoretical analysis of the latest direct S- factor determination given in Ref. [38]: (Cexp p3/2)2=0.558 ± 0.015 fm−1. It also coincides with the CBe-p,p3/2=0.740 fm−1/2obtained in Ref. [8] from a CDCC analysis of the 8B breakup on 208Pb at 52 MeV/nucleon. B. Proton transfer In the recent article of Aguilera et al. [14], the 8B+58Ni reaction cross section was studied, and interesting results about the proton halo properties of 8B were obtained. The total reaction cross section found for the 8B+58Ni system shows a very large enhancement with respect to the predictions typically expected for weakly bound “normal” nuclei, for instance, Li and Be projectiles. This behavior of the 8B+58Ni cross section is very similar to the cross section enhancement observed for the interaction of the neutron halo nucleus 6He (6He +209Bi, 6He +64Zn). In contrast to the 6He-induced reactions, where most of the reaction yield comes from twoneutron transfer to neutron-unbound levels, the enhancement in the 8B-induced total reaction cross section comes mainly through the long-range Coulomb plus nuclear breakup force, and proton transfer is not expected for a proton halo system. Nevertheless, we would like to estimate the contribution from the direct proton transfer to the bound states of 59Cu because this process can contribute to the total 7Be yield. In accordance with the aim of this work, to obtain a complete description of the experimental data and spectroscopic information, we performed calculations of the differential cross sections of the direct proton transfer to the bound states in 59Cu in the exact finite-range DWBA using the FRESCO computer code. The shell-model studies of Cu isotopes [39] showed that 59Cu can be rather well described by an inert core with one extra proton particle and two valence neutrons outside the core. The ground state of 59Cu is 3/2−, and it conforms to the occupation of the 2p3/2oscillator orbital. The low-lying states in near-semimagic nuclei usually carry a major part of the proton single-particle strength. We consider the direct proton transfer to some states of 59Cu with spin and parity quantum numbers of the valence proton above the Z=28 shell: 2p3/2,1f5/2,2p1/2,1g9/2, and some other states for which the experimental spectroscopic strength, following from the analysis of (d,n) reactions [40], are most important. We use 064617-5 T. L. BELYAEVA et al. PHYSICAL REVIEW C 80, 064617 (2009) TABLE II. Total j2, orbital l2angular momenta, and a number of nodes N2of the bound-state wave functions for the 59Cu →58Ni +p vertex, and the proton spectroscopic factors of the lowest states in 59Cu. Ex(MeV) Iπ 59Cu j2l2N2 Sa 0.0 3/2−3/21 2 1.85 0.491 1/2−1/21 2 0.84 0.914 5/2−5/23 1 2.5 1.398 7/2−7/23 1 0.4 2.319 1/2−1/21 2 0.1 3.043 9/2+9/24 1 2.4 a S=(2If+1)C2S;hereC 2is the isospin Clebsch-Gordan coefficient. the same OPs (Table I) as for the scattering calculations. The normalized single-particle 7Be +pwave function generated by a EB Woods-Saxon potential (Table I) is used to describe the bound-state relative motion of 7Be +pin 8B. We take into account the standard angular-momentum coupling scheme (see, e.g., Ref. [30]) Ia=Ib+j1,If=IA+j2, j1=sp+l1,j2=sp+l2(10) L=j1+j2=l1+l2, to describe the direct stripping mechanism 8B(Ia)+ 58Ni(IA)→[7Be +p(sp)] +58Ni →7Be +[p(sp)+58Ni] →7Be(Ib)+59Cu(If) of the reaction 58Ni (8B,7Be)59Cu. Here Lis the transferred angular momentum; l1and l2are orbital angular momenta of relative motion of the proton in the nuclei 8B and 59Cu, respectively; and j1and j2are total transferred angular momenta. For the case of the 58Ni(8B,7Be)59Cu reaction, the angular momentum and parity values are Iπ A=0+,Iπ a=2+, Iπ b=3/2−,j1=3/2, and l1=1.The experimental proton spectroscopic factors for these states [39,40], which were used in our calculations, are shown in Table II. Figure 4shows the contribution of the direct proton transfer mechanism (dotted line) to the cross section for 7Be production in the 58Ni(8B,7Be) reaction at Elab =25.75 MeV. The solid curve represents the incoherent sum of the breakup and stripping mechanisms. This calculation shows that proton stripping provides approximately 5% of the total 7Be emission cross sections. This result confirms the intuitive expectation declared in Ref. [14]. Nevertheless, the maximum contribution to the cross section falls in the angular interval from 50◦to 80◦and provides a slightly improved description of the local maximum in the experimental differential cross section at θlab ≈50◦. C. Excitation function The integrated breakup cross section obtained in Ref. [14] from the CDCC calculation for the 8B+58Ni system reproduces the measured breakup yield at Elab =25.8MeV quite well. There is a slightly enhanced energy dependence in the energy interval from Elab ≈20–29 MeV. In this article, FIG. 4. (Color online) The contribution of the direct proton transfer mechanism (dotted line) comparing with the breakup cross section (dashed line). The incoherent sum of both cross sections and the experimental data of Kolata et al. [12] are shown by the solid line and the data points, respectively. we present a detailed analysis of the energy dependence of the differential cross sections for this system. Figure 5shows the results of CDCC calculations at Elab =20.7, 23.4, 25.0, 25.75, 26.9, and 28.4 MeV. For comparison purposes, the experimental differential cross sections for the 58Ni(8B,7Be) reaction reported in Ref. [12] at a beam energy of 25.75 MeV are also shown. The theoretical curves represent the incoherent sum of the breakup and direct transfer mechanisms. It is observed that the calculations predict a sufficient increase with energy of the cross sections for breakup at their maxima near θlab ≈12◦. A relative increase of the cross section near θlab ≈45◦is also seen in Fig. 5, especially at Elab =28.4MeV. The energy dependence of the cross section of the 7Be product emission at θlab =±45◦was measured in a recent experiment, performed using the TwinSol facility at the University of Notre Dame [13]. 8B beams with Elab =25.0, 26.9, and 28.4 MeV were scattered on a 58Ni target. The 45◦ excitation function calculated at six energies, mentioned earlier for the breakup and proton-transfer channels, are shown in the lower part of Fig. 6, in comparison with the data for the 8B+58Ni reaction. The calculations performed without any energy dependence of the potential parameters demonstrate a growth of the cross section with energy, which can also be seen in the experimental data. The ∼12◦(maximum of the angular distribution) excitation function and integrated cross sections are also shown in Fig. 6. It is observed that the magnitude of the cross section at the maximum of the angular distribution increases more rapidly with energy than the second maximum near 45◦and the integrated cross section as a whole. The integrated cross section (the upper curve in Fig. 6) is practically the same as the one indicated by the dotted line (CDCC calculations) in Ref. [14]. D. S17(0) factor estimation Let us now apply the results obtained from the 8B breakup and stripping calculations to estimate the astrophysical S17(0) factor based on the potential model developed in Refs. [41,42]. 064617-6 ASTROPHYSICAL S17(0) FACTOR EXTRACTION ... PHYSICAL REVIEW C 80, 064617 (2009) (a) (b) (c) FIG. 5. (Color online) Breakup plus direct proton transfer cross sections calculated at (a) Elab(8B) =28.4 and 26.9 MeV, (b) Elab(8B) =25.75 MeV, and (c) Elab(8B) =25.0, 23.4, and 20.7 MeV. The data points are from Ref. [12]. At very low incident energies, the proton capture reaction 7Be(p, γ )8B leading to the ground state of 8Bisextremely peripheral because of the Coulomb barrier. The amplitude of the reaction, therefore, depends on the overlap between the ground-state wave functions of 7Be and 8B only in the tail region, where the radial part of the overlap has the form of Eq. (2). The cross sections σpγ(0), and consequently, S17(0), are proportional to C2 Be-p. An accurate approximation of the S17(E) at very low energies was obtained in Refs. [41,42], taking into account the asymptotic behavior of the p+7Be scattering wave function and its derivatives at zero energy. The model-independent analytical expression for S17/C2 Be-p (assuming that the initial scattering wave function at zero energy can be replaced by its asymptotic form) corresponding FIG. 6. (Color online) Energy dependence of the breakup plus direct proton transfer cross sections calculated at the six energies. The upper line shows the integrated cross sections; the middle line corresponds to the maxima of the differential cross sections; the lower line and points illustrate a comparison between calculated and measured in Ref. [13] differential cross sections at an angle of 45◦, respectively. to the external capture process takes the form S17(0)C2 Be-p≈38.0(1 −0.0013a0)eVbfm,(11) where sand dwaves of the relative 7Be +pmotion were incorporated into the overlap integral. The main unknown parameter of the model is the scattering length a0, which was taken to be the average value of the scattering lengths a01 and a02 for the channel spins I=1 and I=2. In recent calculations [8,38], this average value varies between −2 and −7 fm. The theoretical error of S17(0) associated with this range of a0does not exceed 0.3% and is therefore negligible compared with the errors following from the uncertainties in the determination of other parameters. In our calculation, we take a0=−7 fm, following Ref. [38]. Finally, according to Eq. (9) and taking into account the estimated errors of 5.3%, we find that S17(0) =20.8±1.1eVb. This value is in perfect accord with the value of S17(0) = 20.9±2.0(exp) ±1.9(theor) eV b obtained in Ref. [8]. Comparing with the S17(0) approximations of the direct and indirect data, we conclude that our estimation is a bit smaller than those obtained from the direct data [1] (though it lies inside the error bar) and corresponds to the upper limit of the S17(0) values obtained from the indirect data [3]. V. CONCLUSIONS The process of 8B breakup in the field of heavy nuclei is of special interest as it can provide the cross section for the 7Be radiative proton capture at low energies, a reaction that is essential for the understanding of a solar neutrino emission. The astrophysical S17(0) factor extracted from the indirectly measured Coulomb breakup data is estimated to be within the range 16.7⩽S17(0) ⩽20.6 eV b, whereas recent values obtained from the direct measurements are in the range 21.2– 21.4 eV b. 064617-7 T. L. BELYAEVA et al. PHYSICAL REVIEW C 80, 064617 (2009) We have performed CDCC calculations for the breakup of 8Bon58Ni, and direct proton transfer for the 8B+58Ni system, at energies of 20–28.4 MeV, which are on and slightly above the Coulomb barrier. The influence of the 7Be core-target OP on the breakup cross section was investigated. For this purpose, the recent data on the elastic scattering angular distributions for the 7Be-58Ni and 8B-58Ni systems measured at five different energies around the Coulomb barrier were analyzed. The data were fit within the optical model, and a reasonably energyindependent OP for each system, which allows a good fit to the experimental angular distributions in this energy interval, was obtained. After applying these OPs and two different 7Be +p relative motion wave functions, and summing breakup and direct proton transfer contributions, we were able to fit the experimental cross section at a 8B laboratory energy of 25.75 MeV [12] and to extract the spectroscopic factors Sexp for the 7Be-psystem corresponding to the two specific choices for the 7Be +pbound-state wave functions. The Sexpvalue was usedto calculate the excitation function of the breakup of 8Bon58Ni, in comparison with the recent experiment [13], in which 7Be products were measured at the forward angle of θlab =45◦for Elab =25.0, 26.9, and 28.4 MeV. In view of the peripheral character of the 8B breakup reaction at near-barrier energies, the asymptotic normalization coefficient CBe-p,p3/2was calculated by the relation C2=Sexpb2, where bis a single-particle ANC following from the adopted 7Be +psingle-particle wave functions. The ANC was found to be C2 Be-p,p3/2=0.543 ±0.027 fm−1, independently of the choice of the 7Be +pbound-state wave function. Finally, the astrophysical S17(0) factor was found to be S17(0) = 20.8±1.1 eV b. This value lies between the experimental values obtained from the direct and indirect data and is in accordance with the value obtained from the similar CDCC analysis of the 8B breakup on 208Pb data at 52 MeV/A [8]. 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