10Be in the Cluster Shell Model

The Cluster Shell Model describes light nuclei in terms of clusters of k α-particles and x extra nucleons, in which the extra nucleons move in the deformed field generated by geometric configuration of α-particles. We present the first study for the case of x = 2 nucleons in an application to 10Be as a cluster of two α-particles and two neutrons.


Clusters and the light Nuclei
The study of the light nuclei goes all the way back to the pioneering contributions of Wheeler [1] and Hafstad and Teller [2] in the 1930s and Brink in the 1960s [3].A recent review can be found in [4] In the year 2000, an algebraic method called the Algebraic Cluster Model (ACM), was proposed to describe light nuclei as clusters of α-particles [5].As an example, the nucleus 12 C was described successfully in the ACM as a triangular configuration of three α-particles [6], in particular, the L P = 5 − state was predicted more than a decade before its experimental discovery [7].In subsequent years, the Cluster Shell Model (CSM) was introduced to describe the properties of neighboring cluster nuclei of the type kα + x as composed of k α-particles plus x = 1 extra nucleons [8,9,10,11].This contribution aims to present the first application for x = 2 extra nucleons.As an example, we study the nucleus 10 Be as a cluster of k = 2 α-particles and x = 2 neutrons and investigate what extent the cluster structure of two alpha-particles persist under the addition of two neutrons

Cluster Shell Model
The Cluster Shell Model describes configurations of light nuclei of the type kα + x i.e. composed of k α-particles plus x extra nucleons, in which the extra nucleons move in the deformed field generated by the cluster of α-particles [8].The case of one extra nucleon with x = 1 was studied for 9 Be and 9 B (k = 2) [9], 13 C (k = 3) [10], and 21N e and 21 N a (k = 5) [11].

arXiv:2309.14505v1 [nucl-th] 25 Sep 2023
In this contribution, we present the first application for two extra nucleons for the case of the nucleus 10 Be as a cluster of k = 2 α-particles and x = 2 neutrons.The neutron single-particle levels are described by the CSM Hamiltonian which is the sum of the kinetic energy, a central potential, a spin-orbit interaction, and a Coulomb potential in the case of an odd proton.The central potential is obtained by convoluting the density with the nucleon-α interaction [7], The coefficient α is related to the size of the α-particle, and the ⃗ r i represents the coordinates of the α-particles: ⃗ r 1 = (β, 0, −) and ⃗ r 2 = (β, π, −) where β represents the distance of the α particles with respect to the center of mass.
The CSM wave functions are calculated in the intrinsic or body-fixed frame.A system of two identical alpha-particles has axial symmetry, and hence the eigenstates are characterized by K P : the projection of the angular momentum along the symmetry axis K and the parity P .The calculations are carried out on the harmonic oscillator basis.The i-th eigenstate is given by the expansion: ), j, K⟩δ P,(−1) l Figure 1 shows the splitting of the spherical single-particle levels in the deformed field generated by the clusters of two α-particles as a function of β [8,9].For β = 0 we recover the spherical harmonic limit, whereas for β > 0 the harmonic oscillator levels are mixed and split into the levels characterized by K P .For the case of interest, the value of β is determined from the moment of inertia of the rotational band in 8 Be as β = 1.82fm [9].The five lowest eigenstates are listed in table 1.
Table 1.Eigenvalues of CSM in MeV

The nucleus 10 Be
In this section, we discuss the nucleus 10 Be as a cluster of k = 2 α-particles and x = 2 neutrons.In addition to the CSM Hamiltonian of Eq. ( 1) we have to add the residual interaction between the two neutrons.Here we propose a pairing interaction [12] satisfy the SU (2) fermion quasi-spin algebra The two-particle states are given by: The matrix elements of the pairing interaction in the two-particle states are: The relevant single-particle levels are shown in Table 1.The first two levels with K P = 1  The only configurations affected by the pairing interaction are the K P = 0 + states.The eigenvalues can be obtained by diagonalizing the CSM Hamiltonian plus the pairing interaction, H CSM + V pair , with matrix elements.The matrix elements of the pairing interaction are then: Figure 2 shows the behavior of the K P = 0 + eigenstates as a function of the pairing strength G.The value of the pairing strength is determined from the excitation energy of the first exited 0 + state to be G = 2.079Mev the final results for 10 Be are shown in Figure 3.

Summary and conclusions
We presented the first extension of the Cluster Shell Model to a cluster of two α-particles with two extra nucleons with an application to the nucleus 10 Be.The residual interaction between the two extra neutrons was taken as a pairing potential whose strength was adjusted to the excitation energy of the first 0 + state.We find a good agreement with the available experimental data.A more detailed CSM analysis of 10 Be including other forms of residual interaction as well as a study of transition probabilities will be published elsewhere.

Figure 1 .
Figure 1.Excited states of the 9 Be as a function of β

2 + 1 and 1 2 − 1 2 − 1 , 1 2 − 2 and 1 2 + 2
are occupied by the four neutrons of the two α-particles.In our calculation, we consider the next three available levels with K P = 3 for the two extra neutrons.

Figure 2 .
Figure 2. Eigenvalues corresponding of J = 0 + as a function of the intensity of the pairing strength G

Figure 3 .
Figure 3. Eigenvalues given by the Paring interaction and the experimental data (dotted line)