Structure effects on the Coulomb dissociation of B at relativistic energies
Introduction
The B isotope produced in the Sun via the radiative capture reaction Be(p,γ)B is the principal source of the high energy neutrinos detected in the Super-Kamiokande and Cl detectors [1], [2]. In fact the calculated rate of events in the former detector (and also in the SNO experiment [3]) is directly proportional to the rate of this reaction measured in the laboratory at low energies (∼ 20 keV) [3]. Unfortunately, the measured cross sections (at relative energies (ECM) of [p–] > 200 keV) disagree in absolute magnitude and the value extracted by extrapolating the data in the region of 20 keV differ from each other by 30–40 %. This makes the rate of the reaction Be(B the most poorly known quantity in the entire nucleosynthesis chain leading to the formation of B [4].
The Coulomb dissociation (CD) method provides an alternative indirect way to determining the cross sections for the radiative capture reactions at low energies [5], [6], [7], [8], [9], [10]. In this method, the radiative capture is reversed by the dissociation of the projectile (the fused system) in the Coulomb field of the target by assuming that the strong interaction between the nuclei is absent and the electromagnetic excitation process is dominated by a single multipolarity (see, e.g., Refs. [6], [9]). However, in the CD of B, the contributions of E2 and M1 multipolarities as well as nuclear breakup can be disproportionately enhanced in certain kinematical regimes [11], [12], [13], [14] and a careful investigation [10], [15] is necessary to isolate the conditions in which these terms have negligible effect on the calculated breakup cross sections.
For the CD measurements of B performed at RIKEN [16], [17], [18] at around 50 MeV/nucleon, it was found in a theoretical model [10] that the data are free from nuclear and E2 contributions if measurements are limited to ECM < 0.70 MeV and to angles of Be c.m. with respect to the beam direction (θCM) <4°. On the other hand, for measurements at subCoulomb energies [19], these contributions turned out to be dominant everywhere [10], [20], [21]. Furthermore, the multi-step breakup effects were also found to be quite important at these energies [20], [21], [22].
Recently, the Coulomb dissociation of B on a Pb target has been performed at GSI-Darmstadt at the beam energy of 250 MeV/nucleon [23], [24]. The advantages of the CD process at this high energy are (i) measurements at ECM lower than RIKEN are possible due to certain experimental advantages [23], [24]. (ii) the multi-step breakup processes (e.g. the Coulomb post-acceleration) are negligible [22], [25], and (iii) the nuclear breakup processes are expected to be negligible in comparison to the Coulomb one due to the strong enhancement in the virtual photon spectrum [9], [25], an advantage that existed already in the experiments performed at the RIKEN energies in the kinematical regime as mentioned above [10]. At the beam energy of 250 MeV/nucleon, the E2 component is expected to be about one third of that found at RIKEN energies although, at the same time, the E1 component is also reduced by about one half. But more importantly, the M1 component is expected to be enhanced particularly in the first resonance region (0.5 MeV < ECM < 0.65 MeV). This provides an alternative opportunity to test various models of the B structure, as they differ in their treatment of the resonance structure of this nucleus [26], [27], [28], [29], [30].
The aim of this paper is to perform calculations of the CD of B on a Pb target at the beam energy of 250 MeV/nucleon, using the cross section for the capture reaction Be(p,γ)B calculated within a recently developed realistic shell model which includes coupling between many-particle quasi-bound states and the continuum of one-particle scattering states (to be referred as the Shell Model Embedded into the Continuum (SMEC) approach) [31], [32]. In order to investigate the sensitivity of the CD process to the structure of B, we also perform calculations within a single-particle potential model using the parameters given by Esbensen and Bertsch (EB) [33]. We compare the results of our calculations with the preliminary data on the angle integrated cross sections for this reaction taken recently in an experiment performed at the GSI, Darmstadt [23], [24]. We discuss the kinematical regime where the breakup measurements performed at these energies can be reliably used to extract the astrophysical S-factor (S17) for the Be(B capture reaction.
The remainder of this paper is organized in the following way. The formalism of the SMEC approach is described in Section 2. The formulas used in the calculation of the Coulomb dissociation cross sections are also described here. The results of our numerical calculations and their discussions are presented in Section 3, while the summary and conclusions of our work are given in Section 4.
Section snippets
The shell model embedded in the continuum (SMEC)
In the next section we shall study the CD process , using the Shell Model Embedded in the Continuum (SMEC) which has been applied recently to examine the structure for mirror nuclei, , , and capture cross sections for mirror reactions , [31], [32]. The SMEC model, in which realistic Shell Model (SM) solutions for (quasi-)bound states are coupled to the one-particle scattering continuum, is a development of the Continuum Shell Model (CSM) [35], [36], [37], [38]
Results and discussion
In Fig. 2 we show the results of the calculations for the energy differential cross sections for the reaction B + Pb → B + Pb at the beam energy of 250 MeV/nucleon, using the capture cross sections obtained with versions I, II, III, IV of SMEC.
The results shown in Fig. 2 have been obtained by integrating Eq. (18)) for angles from 0.01°–1.87°, which is the range of the angle integration in the preliminary GSI data as reported in Refs. [23], [24]. The cross sections for E1, E2 and
Summary and outlook
In this paper, we used the cross sections for the radiative capture reaction Be(p,γ)B, calculated within the shell model embedded into the continuum approach for the structure of B, to study the Coulomb dissociation of B on a Pb target at the beam energy of 250 MeV/nucleon. Cross sections obtained with four versions of SMEC were used. Calculations were also performed with the capture cross sections obtained in a single-particle model using the potential parameters given by Esbensen and
Acknowledgements
We wish to express our gratitude to S. Drożdż, E. Caurier and I. Rotter for many discussions. We thank also F. Nowacki for stimulating collaboration in the course of development of the SMEC model. This work was partly supported by KBN Grant No. 2 P03B 097 16 and the Grant No. 76044 of the French–Polish Cooperation. One of the authors (R.S.) would like to thank Abdus Salam International Center for Theoretical Physics, Trieste, for an associateship award.
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