Symplectic symmetry and clustering in atomic nuclei

A new symplectic-based shell-model approach to clustering in atomic nuclei is proposed by considering the simple system $^{20}$Ne. Its relation to the collective excitations of this system is mentioned as well. The construction of the Pauli allowed Hilbert space of the cluster states with maximal permutational symmetry is given for the $^{16}$O+$^{4}$He $\rightarrow$ $^{20}$Ne channel in the case of one-component many-particle nuclear system. The equivalence of the obtained cluster model space to that of the semi-microscopic algebraic cluster model is demonstrated.

Probably the first authors who have pointed that the collective excitations are related with the symplectic groups as phenomenological dynamical groups are Goshen and Lipkin, which have considered in detail the one- [1] and two-dimensional [2] cases. After them, on the phenomenological level the group Sp(6, R) has been proposed as a dynamical group of collective excitations in nuclear system by P. Raychev [3,4] and later considered in detail on microscopic level by G. Rosensteel and D. Rowe [5,6].
From the other side, it was shown that the collective effects are associated with operators that are scalar in O(m) and the collective Hamiltonian is obtained by projecting the many-particle Hamiltonian on a definite O(m) irrep associated with the m relative Jacobi vectors in the configuration space R 3m , where m = A − 1 and A is equal to the total number of nucleons in the system [7][8][9][10][11]. By considering the many-particle system as consisting of A − 1 Jacobi quasiparticles one readily avoids the problem of separation of the center-of-mass motion from the very beginning. Asherova et al. [12] have shown within the framework of the generalized hyperspherical functions method (GHFM) that in its minimal approximation, in which one restricts itself to a single O(m) irreducible representation (ω 1 , ω 2 , ω 3 ) of states of hyperspherical function within the lowest harmonic oscillator shell, is equivalent to restricting to a single SU (3) representation (ω 1 − ω 2 , ω 2 − ω 3 ). They also showed, by means of an important Sp(6, R) ⊗ O(m) complementarity theorem of Moshinsky and Quesne [13], that the diagonalization of the GHFM Hamiltonian in the minimal approximation is equivalent to its diagonalization in the collective space of a single Sp(6, R) irreducible representation (irrep). In this way the equivalence of the O(m)based and symplectic-based theories of nuclear collective motion was quickly realized by many authors, e.g. [9-11, 14, 15]. Further, it was shown in detail, e.g., by Fil-ippov and collaborators [9] that the group O(m) in the Sp(6, R) ⊗ O(m) is related to the intrinsic motion of the relative Jacobi quasiparticles with respect to the intrinsic principal axes frame of the mass quadrupole tensor, on which the collective excitations are built up. The group O(m) is also of importance because, as we will see further, it allows one to ensure the proper permutational symmetry of the nuclear wave functions.
The above considerations can be reformulated more generally. We can say that the set of basis states of the full dynamical symmetry group Sp(6m, R) of the whole many-particle nuclear system contains different kinds of possible motions − collective, intrinsic, cluster, etc. However, often, one restricts himself to a certain type of dominating excitation modes in the process under consideration. Thus, by reducing the group Sp(6m, R) one performs the separation of the 3m nuclear many-particle variables {q} into kinematical (internal) and dynamical (collective) ones, i.e. {q} = {q D , q K }. The choice of the reduction chain depends on the concrete physical problem we want to consider. As we mentioned, e.g., the group Sp(6, R) plays an important role in the treatment of the collective excitations in the onecomponent many-particle nuclear system. The reduction Sp(6m, R) ⊃ Sp(6, R) ⊗ O(m) thus turns out to be of a crucial importance in the microscopic nuclear theory of collective motions. In this way the considered reduction corresponds to the splitting of the microscopic manyparticle configuration space R 3m , spanned by the relative Jacobi vectors, into kinematical and dynamical submanifolds, respectively. According to this, the many-particle nuclear wave functions can be represented respectively as consisting of collective and intrinsic components where χ η ≡ |0 = |N 0 (λ 0 , µ 0 ); KLM determines the bandhead structure, and the collective function Θ η , in second quantized form, can be written as a polynomial in the Sp(6, R) raising operators.
As will see further, by considering another reduction chain of the Sp(6m, R) group one is able to isolate the cluster degrees of freedom within the unified framework of the symplectic-based shell-model approach to nuclear excitations. To be more specific, we restrict ourselves to the case of two-cluster system only and consider the simplest case of 20 Ne system, which consists of two structureless (closed-shell) 16 O and α clusters. Thus, for example, the intrinsic wave function of maximal space symmetry which determines the intrinsic motion in 20 Ne has the O(m) symmetry (12,4,4). The collective excitations are then generated by acting with the Sp(6, R) raising generators on the complementary Sp(6, R) symplectic bandhead with SU (3) symmetry (8, 0). The SU (3) basis states of the so obtained Sp(6, R) irreducible representation for 20 N e are given in Table I.  Using the simple 16 O+α cluster system 20 Ne, in particular, we will prove the equivalence of the semimicroscopic algebraic cluster model (SACM) [16,17] and the one-component symplectic-based scheme in the classification of the cluster states in the many-particle nuclear Hilbert space.

A. The semi-microscopic algebraic cluster model
The semi-microscopic algebraic cluster model [16,17] was proposed as an approach to the cluster structure of light nuclei. In the SACM, the relative motion of the clusters is described by the vibron model [18], whereas their internal structure is treated in terms of the Elliott shell model, having a symmetry group is the symmetry group of the three-dimensional harmonic oscillator [19] and U ST (4) is the Wigner spinisospin group [20]. The model space is constructed in a microscopic way by respecting the Pauli principle, but the interactions are expressed in terms of algebra generators. The states within the SACM for two-cluster system are then classified by the following reduction chain [16,17]: The spin-isospin irreducible representations for 16   At this point we want to point out once again that the cluster excitations, together with the collective and intrinsic motions, are naturally contained in the full dynamical group Sp(6m, R) of the many-nucleon nuclear system. In the case of two-cluster system (A = A 1 + A 2 ), the well-known anzatz [21] can be related to the symplectic scheme by considering the following reduction chain: where the groups Sp(6(A 1 − 1), R) and Sp(6(A 2 − 1), R) describe the intrinsic state of the fist and second cluster, respectively. One of the (A − 1) Jacobi vectors, denote it by q 0 (from the set A 1 ), will describe the relative motion of the two clusters, whereas the rest (A−2) Jacobi vectors will be related to the intrinsic states of the clusters. Thus, the group Sp(6, R) 0 in (5) will describe the "cluster excitations", related to the relative distance vector q 0 . The group Sp(6(A 1 −1), R), describing the first cluster states, can be further reduced to Sp To relate the present classification scheme to that of SACM we consider further the reduction of the subgroups in Eq.(5), i.e. the complete reduction chain: Then the permutational symmetry of the combined systems will be f = {4, 4, 4, 4, 4}. Because of the full antisymmetry of the total wave function, the spin-isospin content of the combined system is given by conjugate Young scheme f = [5,5,5,5]. The cluster model space for the 20 Ne two-cluster system is then spanned by the even and odd Sp(6, R) 0 irreps respectively, i.e. by the sets of SU (3) irreps: (8, 0), (10, 0), (12, 0), . . . and (9, 0), (11, 0), (13, 0), . . ., which constitute a representation of the double covering metaplectic group M p(6, R). Alternatively, the two Sp(6, R) 0 irreps could be unified into a single irrep of the semidirect product group [HW (3) 0 ]Sp(6, R) 0 . Note that because the Sp(6, R) 0 irreps are built up by means of a single Jacobi vector q 0 , corresponding to the intercluster distance, only onerowed irreps of the subgroup SU (3) 0 ⊂ Sp(6, R) 0 are allowed of the type (n 0 + 2n, 0). Thus, the positiveparity cluster state space in the 16 O+ 4 He → 20 Ne channel within the one-component symplectic-based scheme will coincide with the Sp(6, R) irreducible collective space that is spanned by the fully symmetric SU (3) irreducible representations only, given in Table I in red. Correspondingly, the full cluster model space with maximal permutational symmetry of the SACM given by the SU (3) irreps (n 0 + n, 0) with n = 0, 1, 2, . . . is obtained by considering both the even and odd irreducible collective spaces of Sp(6, R) 0 , in which the states of the three-dimensional harmonic oscillator of even and odd number of oscillator quanta fall. Thus, the cluster model spaces of the SACM and the one-component symplectic-based approach to the cluster states are identical. It is then clear that, based on the equivalence of the microscopic model spaces, the usage of the same (algebraic) Hamiltonian in both the SACM and one-component symplectic-based schemes to the clustering in atomic nuclei will produce identical spectra. We note also that the role of the Wigner supermultiplet group SU ST,i (4) (i = 1, 2), which is important in the construction of the Pauli allowed model space, is played in the symplectic-based scheme by the orthogonal The present work also concerns the important question of the mutual interrelations between the shell model, symplectic Sp(6, R) model and cluster model states. A microscopic cluster model has been introduced in 1958 by K. Wildermuth and T. Kanellopoulos [22] who have showed its relation to the shell model. Soon this relation has been reformulated in terms of the SU (3) symmetry by B. F. Bayman and A. Bohr [23]. In this way it has been recognized very early the important role played by the SU (3), and complementary to it SU (4), symmetry in clustering phenomena in atomic nuclei. The connection of the shell model and the cluster model states has later been given in algebraic terms by K. T. Hecht [24] using SU (3) and SU (4) coupling and recoupling techniques. The relation of the Sp(6, R) and α-cluster model states has been done in Refs. [25][26][27]. Recall that the symplectic classification of the nuclear states organizes the Hilbert space of the nucleus vertically into vertical cones or slices (called irreducible collective spaces), in contrast to the conventional shell model in which the Hilbert space is organized horizontally into different shells or layers. In Ref. [25,26] it has been demonstrated that the α-cluster and Sp(6, R) states are essentially complementary with decreasing overlap with the increase of the oscillator quanta excitations 2n ω and may both be needed for a meaningful microscopic description of light nuclei. The cluster states obtained in the present paper actually coincide with the so called stretched SU (3) states [14] of the type (λ 0 + 2k, µ 0 ) with k = 0, 1, 2, . . . of the Sp(2, R) ⊂ Sp(6, R) submodel [28] representing the core collective excitations along the z-direction only. The connection of the α-cluster and Sp(2, R) states has been investigated in Refs. [29,30] and [31] for the case of 8 Be and 12 C, respectively. Recently, the no-core symplectic model (NCSpM) with the Sp(6, R) symmetry has been used to study the many-body dynamics that gives rise to the ground state rotational band together with phenomena tied to alpha-clustering substructures in the low-lying states in 12 C [32,33]. The intersection of the shell, collective and cluster models of the atomic nuclei has also been given in a similar algebraic perspective in Refs. [34,35] through the consideration of the common dynamical algebraic structure SU x (3) ⊗ SU y (3) of all the three fundamental models of nuclear structure, where x stands for the bandhead (valence shell), whereas y refers to the major-shell excitations.

III. CONCLUSIONS
In the present paper, a new symplectic-based shell model approach to clustering in atomic nuclei is proposed. The cluster degrees of freedom are isolated by an appropriate separation of the full set of relative Jacobi many-particle variables into dynamical (collective) and kinematical (intrinsic) ones by reducing the full dynamical symmetry group Sp(6m, R) of the whole onecomponent (no distinction is made between the proton and neutron degrees of freedom) many-particle nuclear system. According to this, the nuclear wave functions are represented as having cluster (collective) and intrinsic components. The kinematical part allows to ensure all the integrals of motion of the considered nuclear system, including the proper permutational symmetry. The symplectic symmetry thus provides the nuclear cluster systems with fully microscopic shell-model wave functions that respect the Pauli principle.
For simplicity, the proposed algebraic approach is illustrated for the case of two-cluster nuclear systems by considering the simple system 20 Ne. The construction of Pauli allowed Hilbert space of the cluster states with maximal permutational symmetry is worked out for the 16 O+ 4 He → 20 Ne channel in the case of onecomponent many-particle nuclear system. The equivalence of the obtained cluster model space to that of the semi-microscopic algebraic cluster model is demonstrated. Further all the symplectic-based computational machinery can be used in practical applications. In contrast to the semi-microscopic algebraic cluster model, the symplectic Sp(6, R) symmetry allows to build up the required quadrupole collectivity observed in some nuclei without the use of an effective charge. Thus, the present approach can further be tested in obtaining the excitation spectrum, including the microscopic structure of the cluster states and the transition probabilities, not only in 20 Ne, but also in other light nuclei for which the clustering is supposed to play an important role.
The relation of the present symplectic-based shell model approach to the collective excitations in 20 Ne is mentioned as well. We note also that the cluster motion is a relative motion of the one group of nucleons in phase with respect to the another group, i.e. it is also of collective nature. In this regard, the cluster motion appears as a specific kind of collective excitations. Indeed, as we have demonstrated in the present work, the cluster model state space is a restricted part of the corresponding irreducible collective space of the two-cluster system as a whole.
Finally, the algebraic approach presented here could be generalized for more clusters or/and to the case of the two-component proton-neutron cluster nuclear systems. The extension to more clusters, however, requires the proper consideration of the point group symmetries. The role of the different point symmetries in clustering have recently been studied very successfully by R. Bijker, F. Iachello and their collaborators [36][37][38] within the framework of the algebraic cluster model (ACM) with U (ν + 1) dynamical group of ν = 3(k − 1) degrees of freedom for nuclei composed of k α-particles [39]. The point symmetries can be taken into account in the present symplectic-based shell model approach to clustering by considering the reduction of the orthogonal group O(A − 1) and its subgroup S A ⊂ O(A − 1). For the two-cluster systems (A = A 1 + A 2 ), subject of the present work, one needs to consider the following two reduction chains: and S A ⊃ S A1 ⊕ S A2 ⊕ S 2 . In this case, the geometry of the intercluster motion described by a single Jacobi vector q 0 is determined by the reduction O(1) ⊃ S 2 of the trivial orthogonal group O(1), consisting of two discrete elements {±1}. The cluster states of the two-cluster system are therefore classified by the scalar S 2 irrep {2} obtained according to the O(1) ↓ S 2 branching rule. Taking the isomorphism S 2 ∼ Z 2 , one sees that this classification corresponds to a dumbbell configuration with Z 2 symmetry. Similarly, for the case of three-cluster system (A = A 1 +A 2 +A 3 ), the point symmetry can be taken into account by considering the following reduction chains: and S A ⊃ S A1 ⊕ S A2 ⊕ S A3 ⊕ S 3 . Then the geometry of the three-cluster system, which intercluster motion is described by two Jacobi vectors q 01 and q 02 , can be classified by the reduction O(2) ⊃ S 3 associated with these two Jacobi vectors. The branching rules for O(2) ↓ S 3 are as follows: (0) ↓ {3}, (0) ⋆ ↓ {111}, (3a + p) ↓ {3} + {111} for p = 0, and {21} for p = 1, 2. In this way, the threecluster system states can be classified according to the different irreducible representations of S 3 ∼ D 3h , corresponding to an equilateral triangle configuration with D 3h symmetry.