Analysis of the capture reaction using the shell model embedded in the continuum
Introduction
A microscopic description of weakly bound exotic nuclei close to the drip-lines such as, e.g., or in their ground state (g.s.), or nuclei close to the β-stability line in the excited configurations like, e.g., in the first excited state Jπ=1/2+, is an exciting theoretical challenge due to the proximity of the particle continuum. Closeness of the scattering continuum implies that virtual excitations to continuum states cannot be neglected as they modify the effective interactions and cause the large spatial dimension of nuclear density distribution and, in particular, the existence of halo structures. Even basic concepts of the nuclear collective model hardly apply for those weakly bound configurations as the particle motion in the weakly bound orbit is presumably decoupled from the rest of the system [1]. The microscopic description of such exotic configurations should treat properly the residual coupling of discrete configurations and the scattering continuum. Recently, it was proposed to approach this difficult problem in the quantum open system formalism which does not separate the subspaces of (quasi-)bound (the Q subspace) and scattering (the P subspace) states [2], [3]. Such a formalism can be provided by the Continuum Shell Model (CSM) [4] which in the restricted space of configurations generated using a finite-depth potential has been studied for the giant resonances and for the radiative capture reactions probing the microscopic structure of these resonances [4], [5], [6], [7].
In this context, the capture reaction is interesting for several reasons. First of all, precise experimental data in the energy range from 200 keV to 3750 keV are now available [8] and the decays to the g.s. (Jπ=5/2+) and to the first excited state (Jπ=1/2+) of have been accurately resolved [8]. The strikingly different behavior of the astrophysical S-factor for the proton capture into the 5/2+ state and into the 1/2+ weakly bound state (Q=105 keV) has been explained by the existence of proton halo in the 1/2+ state of [8]. Different theoretical approaches, including the potential model [8], [9], the model based on the Generator Coordinate Method (GCM) [10], [11], or the K- and R-matrix analysis [12], have been tried to describe this reaction. Moreover, an expected simplicity of the low energy wave functions in , and , allows one to test certain salient features of models such as the effective interactions between Q and P subspaces and the possible quenching of matrix elements of the residual coupling which depends on the model space used in the calculations.
Secondly, the exact knowledge of the rate for the reaction is necessary for modelling of nucleosynthesis process in the hydrogen-burning stars. Explosive hydrogen burning is believed to occur at various sites in the Universe, including novas, X-ray bursts, or the supermassive stars [13], [14], [15], [16], [17]. Hydrogen burning of second-generation stars proceeds mainly through the proton–proton chain and CNO cycle. The changeover from the pp chain to the CNO cycle happens near T≃2·107 K. The reaction is of particular interest in this context as it provides a link to the higher branches of the CNO cycle. In particular, it starts the side branch: . The contribution of CNO cycles to the total amount of energy produced in the Sun is small and CNO neutrinos account only for about 0.02 of the total neutrino flux [18]. Moreover, most of them are coming from the decay of and in the main CNO cycle CNO-I. The flux of `-neutrinos', which is again two orders of magnitude smaller than the flux of neutrinos from and in CNO-I reactions, is controlled by the reaction in the CNO-II cycle. Hence, the measurement of CNO neutrinos coming from different sources provides an accurate handle on the thermonuclear reaction process in stars like the Sun and allows in principle to distinct between different branches of the CNO cycle.
The Shell Model Embedded in the Continuum (SMEC) model, in which realistic Shell Model (SM) solutions for (quasi-)bound states are coupled to the one-particle scattering continuum, is a recent development of the Continuum Shell Model (CSM) [4], [5], [6], [7] for the description of complicated low energy excitations of weakly bound nuclei. The SMEC approach is based on the realistic SM which is used to generate the N-particle wave functions. This deliberate choice implies that the coupling between SM states and the one-particle scattering continuum has to be given by the residual nucleon–nucleon interaction. In SMEC, like in the CSM, the bound (interior) states together with its environment of asymptotic scattering channels form a quantum closed system. Using the projection operator technique, one separates the P subspace of asymptotic channels from the Q subspace of many-body localized states which are built up by the bound single-particle (s.p.) wave functions and by the s.p. resonance wave functions. P subspace is assumed to contain (N−1)-particle states with nucleons on bound s.p. orbits and one nucleon in the scattering state. Also the s.p. resonance wave functions outside of the cutoff radius Rcut are included in the P subspace. The resonance wave functions for r<Rcut are included in the Q subspace. The wave functions in Q and P are then properly renormalized in order to ensure the orthogonality of wave functions in both subspaces. The application of the SMEC model for the description of structure for mirror nuclei: , , and capture cross sections for mirror reactions: , has been published recently [2], [3].
We aim in SMEC at a microscopic description of low lying, complicated many-body bound and resonance states. For that reason, the description of the particle continuum is restricted to the subset of one-nucleon decay channels. This should be a reasonable approach for describing the structure of mirror nuclei and around , and the capture reactions: and . One expects that at higher excitation energies, e.g., above the α emission threshold, the one-particle continuum approximation is too restrictive and the residual correlations generated in bound state wave functions by the coupling to those channels cannot be described. Effects of such correlations have been seen in the structure of Jπ=3+ resonances in and [3] which strongly couples to the three-particle decay channels. In this case, one should try to employ methods based on the cluster expansion of the wave function and the three-body continuum models [19], [20], [21], [22], [23]. More complicated two-particle channels like, e.g., the α-decay channel, can be treated in SMEC, following the approach of Balashov et al. [24].
The paper is organized as follows. In Section 2 we recall certain elements of the SMEC formalism and, in particular, those features of the S-matrix in SMEC which are involved in the calculation of the elastic cross sections, phase shifts and the capture cross sections. Section 3 is devoted to the discussion of specific properties of and . In particular, the properties of the matrix elements of the effective operator which couple Q and P subspaces in and will be discussed in Subsection 3.1. Features of the self-consistent average potentials in Q subspace for different many-body states of will be presented in Subsection 3.2. Section 4 is devoted to the discussion of capture cross sections for different multipolarities and different final states of . The differential elastic cross section and the elastic phase shifts in scattering will be compared to the experimental data in Section 5. A summary and outlook will be given in Section 6.
Section snippets
The formalism of shell model embedded in the continuum
The full solution of SMEC approach is constructed in three steps. In the first step, one calculates the (quasi-)bound many-body states in Q subspace. For that one solves the multiconfigurational SM problemwhere HQQ≡QHQ is the SM effective Hamiltonian which is appropriate for the SM configuration space used. For solving (1), we use the code ANTOINE [25] which employs the Lanczos algorithm and allows for the diagonalization in large model spaces.
For the coupling between bound and
The effective interactions for the psd SM space
For the purpose of the present study we have constructed SM effective interactions in the cross-shell model space connecting the 0p and 1s0d shells. The interactions have three distinctive parts: the 0p-shell part taken to be the CK(8-16) interaction [29], the 1s0d-shell part taken to be the Brown–Wildenthal interaction [30], [31], [32], [33]. For the cross-shell matrix elements, we use the G matrix of Kahana, Lee and Scott (KLS) [34]. This part of the interaction has to be modified
The astrophysical factor for
In Fig. 6 we show the calculated multipole contributions (E1, E2, M1) to the total capture cross section as a function of the c.m. energy ECM, separately for the transitions to the g.s.5/21+ and to the first excited state 1/21+ in . The SMEC calculation is done with the DDSM1 residual interaction as used in the calculations of spectra shown in Fig. 3, Fig. 4. The parameters of the initial potentials U(5/2+) and U(1/2+) can be read from Table 6. The energy scale is adjusted to reproduce the
Elastic cross-section and phase shifts
Other observable quantities that can be calculated using the solutions of the SMEC are the elastic phase shifts and the elastic cross sections for different proton bombarding energies. Results for the elastic phase shifts, shown in Fig. 9, and those for the elastic cross section, shown in Fig. 10, have been obtained using the SMEC solutions for identical parameters of the initial potentials and the DDSM1 residual coupling between the Q and P subspaces as discussed before in Subsection 3.4.1 for
Summary and outlook
In this work we have applied the SMEC approach for a microscopic description of the and spectra, the low-energy radiative capture cross sections in the reaction , and the elastic cross section for the reaction . In the SMEC model, which is a development of the CSM model [4], [5] for the description of low energy properties of weakly bound nuclei, realistic SM solutions for (quasi-)bound states are coupled to the one-particle scattering continuum. For that reason,
Acknowledgements
We thank E. Caurier for his help in the early stage of development of SMEC model, and P. Descouvement and R. Morlock for helpful information. This work was partly supported by KBN Grant No. 2 P03B 097 16 and the Grant No. 76044 of the French–Polish Cooperation.
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