Core excitations in exotic nuclei

The role of core excitations in exotic nuclei is discussed in the framework of a microscopic cluster model. This cluster approach is complemented by the R-matrix theory to take account of the long-range part of the wave functions. We briefly describe the model, and present two recent examples: the neutron-rich nucleus 16B, described by a 15B+n structure, and the proton-rich nucleus 17Na, described by a 16Ne+p structure. In both cases core excitations are shown to play an important role.


Introduction
Clustering is a well known effect in light nuclei (see Ref. [1] and references therein). This property is at the basis of cluster models [2], where the nucleus is described in terms of two (or more) cluster wave functions [3,4,5]. Several variants exist according to whether they exactly include antisymmetrization or not. A natural extension of cluster models to exotic nuclei is the description of halo states, where external nucleons are simply considered as clusters. Typical examples are the 6 He and 11 Li nuclei, which can be described by α+n+n [6] and 9 Li+n+n [7] configurations, respectively.
Simple variants of cluster models only include the ground state of each cluster and, in particular, of the core nucleus. However this approximation may be, in some cases, too simple, and the importance of core excitations needs to be addressed. Here we use a microscopic cluster model with an A-body hamiltonian. In the two-cluster variant, an approximate wave function of the system reads, in a schematic notation, where A is the antisymetrization operator. In this definition, φ 1 and φ 2 are the internal wave functions of the two clusters and g(ρ) the radial function depending on the relative coordinate ρ. This model has been applied to many light nuclei and nucleus-nucleus reactions, including reactions of astrophysical interest (see references in Refs. [8,5]). Definition (1) is the standard RGM (Resonating Group Method) wave function [9,3], and is fully antisymmetric against exchanges of nucleons. Further simplification can be performed by neglecting the internal structure of the clusters, and by choosing an appropriate nucleus-nucleus potential [10]. Wave function (1) can be generalized to multicluster systems [5]. It is well adapted to exotic nuclei [11], where the breakup threshold is low. In that case, the long-range part of the wave function must be accurately determined.
Here we focus on extensions of (1) to core excitations; in other words the total wave function can be generalized as where c represents the excitation level of cluster 1 (cluster 2 is assumed to remain in its ground state). Previous applications to weakly bound nuclei and to unbound systems have shown that core excitations may play an important role [12,13,14]. This phenomenon was recently confirmed by an experimental study of 18 Ne+p elastic scattering, performed in parallel with the 18 Ne(p,p') 18 Ne(2 + ) inelastic scattering [15]. This experiment shows a clear evidence for 19 Na states with a dominant 18 Ne(2 + )+p structure. The importance of 6 He * +n configurations in the 7 He spectrum is also well established [16,17]. The paper is organized as follows. In section 2, we briefly present the microscopic cluster model. Sections 3 and 4 are devoted to the recent examples of 16 B and 17 Na, respectively. These systems are typical neutron-and proton-rich nuclei. Concluding remarks are presented in section 5.

Microscopic cluster theories
Microscopic models are based on fundamental principles of quantum mechanics, such as the treatment of all nucleons, with exact antisymmetrization of the wave functions. Neglecting three-body forces, the Hamiltonian of an A-nucleon system is written as where T i is the kinetic energy of nucleon i, and V ij an effective nucleon-nucleon interaction [3]. Effective forces, such as the Volkov [18] or the Minnesota [19] interactions are adapted to the model wave functions. Both contain one adjustable parameter which can be tuned to reproduce an important property of the nucleus, such as a threshold breakup energy. In practice, the Schrödinger equation associated with this Hamiltonian cannot be solved exactly when A > 4. For very light systems (A ∼ 4 − 5) efficient methods [20] exist, even for continuum states [21]. Recent developments of ab initio models (see for example Refs. [22,23,24]) are quite successful for spectroscopic properties of nuclei up to A ≈ 12. These models make use of realistic interactions, which are fitted to many properties of the nucleonnucleon system, and include three-body forces. However, a consistent description of bound and scattering states of an A-body problem remains a very difficult task [21], in particular for transfer reactions.
In cluster models, it is assumed that the nucleons are grouped in clusters [3,13]. We present here the specific application to two-cluster systems. The internal wave functions of the clusters are denoted as φ I i π i ν i i (ξ i ), where (I i , ν i ) and π i are the spin/projection and parity of cluster i, and ξ i represents a set of their internal coordinates. A channel function is defined as where different quantum numbers show up: the channel spin I, the relative angular momentum ℓ, the total spin J and the total parity π = π 1 π 2 (−) ℓ . The total wave function (1) of the A-nucleon system is therefore written in a more explicit way as where index α refers to different two-cluster arrangements. In most applications, the internal cluster wave functions φ I i π i ν i i are defined in the shell model. The relative wave functions g Jπ αℓI (ρ) are to be determined from the Schrödinger equation which, in the RGM, is transformed into an integro-differential equation involving a non-local potential [9]. In most applications, this relative function is expanded over Gaussian functions [9,3,5], which corresponds to the Generator Coordinate Method (GCM). The wave function (5) is therefore rewritten as where Φ JM π αℓI (R) is a projected Slater determinant, and f Jπ αℓI (R) the generator function, which must be determined. The GCM is equivalent to the RGM, but is better adapted to numerical calculations, as it makes uses of projected Slater determinants (see Refs. [9,3,5] for detail).
The main advantage of cluster models with respect to other microscopic theories is their ability to deal with reactions, as well as with nuclear spectroscopy. As mentioned before, the RGM radial wave functions are expanded over a Gaussian basis. The GCM is well adapted to numerical calculations, and to a systematic approach, but the Gaussian behaviour is not physical at large distances, and must be corrected. We use the Microscopic R-matrix Method (MRM) [25,26] which is a direct extension of the standard R-matrix technique [27], based on the existence of two regions: the internal region (with channel radius a), where the nuclear force and the nucleus-nucleus antisymmetrization are important, and the external region where they can be neglected. In the external region, the Gaussian expansion of the RGM radial function is replaced by Coulomb functions. Matching the internal and external components provides either the collision matrix (for scattering states) or the binding energy (for bound states).

Application to the nucleus 16 B
The unbound nature of 16 B has been first proposed by Bowman et al. [28] and by Langevin et al. [29], and later confirmed by Kryger et al. [30] in an analysis of the 17 C breakup. The low-lying structure has been investigated by Kalpachieva et al. [31] in a heavy-ion multi-nucleon transfer reaction. The existence of a narrow peak above the 15 B+n threshold (E = 40 ± 40 keV) with a width lower than 100 keV, together with a higher level at E = 2.40 MeV (Γ = 0.15 MeV) have been reported. More recently, the 16 B spectrum has been studied by single-proton removal from a 35 MeV/nucleon 17 C beam by Lecouey et al. [32]. In that experiment, a narrow resonant structure at E = 85 ± 15 keV above the 15 B+n threshold, with a width of Γ ≪ 100 keV, has been observed and confirms the experiment of Kalpachieva et al. This peak is interpreted as a narrow resonance which decays by d-wave neutron emission.
In this section, we investigate the 16 B nucleus in a 15 B+n model, involving several 15 B excited states (see detail in Ref. [13]). Shell-model wave functions are built from all configurations with 3 protons in the p shell, and 2 neutrons in the sd shell. This shell-model description involves 1320 Slater determinants and provides 15 B states from the diagonalization of several operators: the total angular momentum J 2 (and its projection J z ), the orbital momentum L 2 and the intrinsic spin S 2 [5]. The lowest states can be considered as physical, and high-energy levels correspond to pseudostates, which simulate the 15  The resonance energies are clearly sensitive to the size of the variational basis. In particular, the 0 − level, assumed to be the 16 B ground state [31,32], is strongly sensitive to excited configurations. In the multichannel calculation, the 1 − and 2 − states converge near the 15 B+n threshold. The 1 − 1 state becomes the 16 B ground state, and is even slightly bound, when the full basis is considered. All these results support the importance of a multichannel framework to describe the 16 B nucleus. Figure 3 provides the full GCM spectrum, compared with the available experimental data. The shell-model results of Ref. [32] are also shown for comparison. The existence of a narrow 0 − (ℓ = 2) narrow resonance just above the threshold is confirmed by the GCM. The GCM predicts 1 − and 2 − states at low energies with a dominant component in the 15 B(g.s)+n channel. We find that the 1 − level is even slightly bound, although it presents a large reduced width. Of course a microscopic model cannot be expected to provide a precision of a few keV, but the existence of this low-lying state is likely. A similar s-wave resonance was predicted by the GCM in 13 Be at low energies [34], and found experimentally later [35]. As a general conclusion the 16 B spectrum is predicted to have several additional states, which have not been observed experimentally yet. Exp.

Application to the nucleus 17 Na
Our aim here is to show that narrow states can also exist in the spectrum of the proton rich nucleus 17 Na, which is the mirror analogue of 17 C famous for its peculiar structure (see details in Ref. [12]). The neutron binding energy in the ground state 17 C(3/2 + ) is only 728 keV, typical of halo nuclei. However, knockout experiments have shown that the weakly bound 16 C(0 + )+n configuration is suppressed in 17 C(3/2 + ) and that this state is mainly based on the 16 C(2 + )+n configuration [36] where the neutron binding energy is 2.5 MeV. A similar structure should be expected in the mirror nucleus 17 Na(3/2 + ). Therefore, the decay branch 17 Na(3/2 + ) → 16 Ne(0 + )+p could be suppressed and, if energetically allowed, the main decay mode would be 17 Na(3/2 + ) → 16 Ne(2 + )+p. If its decay energy is below the Coulomb barrier then its width may be small. Since the decay product 16 Ne is unstable with respect to 2p emission, 17 Na should be a three-proton (3p) emitter. The 17 C and 17 Na are studied in parallel with the Volkov and Minnesota interactions. The two 16 C valence neutrons occupy the 0d 5/2 , 1s 1/2 and 0d 3/2 orbitals, which gives many excitations in 16 C. We only consider the 2 + 1,2 , 3 + 1 and 4 + 1 states, motivated by neutron knockout experiments where they are strongly populated. The 17 C spectrum is fairly well known from experiment, and will be used to estimate the precision of the model applied to the mirror 17 Na nucleus.
Let us first discuss the 17 C spectrum shown in Fig. 4, compared to experiment. The parameters involved in the nucleon-nucleon interaction have been chosen to reproduce both the 16 C+n threshold in 17 C and the excitation energy of 17 C(1/2 + 1 ). With these parameters, the energies of the three low-lying states are in very good agreement with experiment. In particular the level ordering is correctly reproduced. For comparison, the shell model spectrum is also shown in Fig. 4.
The decay scheme of the lowest part of the 17 Na spectrum is shown in Fig. 5. We find that the 17 Na(7/2 + 1 ) level should be very narrow. It decays into the d-wave 16 Ne(2 + 1 )+p and s-wave 16 Ne(4 + 1 )+p channels with partial widths of Γ(2 + 1 ) = 25 keV and Γ(4 + 1 ) = 98 keV respectively. It should be noted that the theoretical value of the latter threshold is underestimated by 630 keV and, therefore, the energy in this channel is too large. Decreasing this energy by tunning the Majorana parameter m, we obtain a partial width Γ(4 + 1 ) = 4 keV. Thus, the 17 Na(7/2 + 1 ) state should be as narrow as 17 Na(3/2 + 1 ). A similar situation occurs for the 17 Na(9/2 + 1 ) state. The partial width for the decay into 16 Ne(2 + 1 )+p is predicted to be 211 keV. A similar width is expected for the decay into the 16 Ne(4 + 1 )+p channel. Tuning the energy of this channel to reproduce the position of the 17 C(9/2 + 1 ) state with respect to the 16 Ne(4 + 1 )+p threshold, similar to what has been done in the case of 7/2 + 1 , we get a partial width Γ(4 + 1 ) = 78 keV.   There should be at least four narrow states, 3/2 + 1 , 5/2 + 1 , 7/2 + 1 and 9/2 + 1 in the 17 Na spectrum. The decay product of these states, 16 Ne, is unstable with respect to two-proton emission. Therefore, 17 Na is in fact a three-proton emitter with a decay path 17 Na → 16 Ne * +p → 14 O+2p+p. Consequently, 17 Na states can be identified by detecting 14 O+p+p+p events in coincidence. A multichannel algebraic study [38] confirms the importance of core excitations in 17 C and 17 Na, but finds some differences for the resonance properties of 17 Na.

Conclusion
Microscopic cluster models represent a powerful tool to investigate the structure of light nuclei. With the help of the R-matrix method, unbound systems can be studied with a rigorous treatment of the asymptotic wave function. This is particularly important for broad resonances showing up in many nuclei close to, or beyond, the driplines. Another advantage is that core excitations can be introduced without further parameters. The couplings between the channels are directly determined from the nucleon-nucleon interaction.
In most exotic nuclei, core excitations play an important role. This property is supported by