1. Introduction
It is known that in nuclear structure physics there are three fundamental models [1–7] describing nuclear states, which are based on different physical ideas about the structure of the nucleus. These models, in their simplest formulation, are the shell model [8], the cluster model [3, 4, 9] and the collective model [5]. The shell model suggests that the atomic nucleus is something like a small atom, the cluster model suggests that it is like a molecule, and the collective model says that it is like a microscopic liquid drop. Decay properties and nuclear reactions are most naturally interpreted within the framework of different cluster models based on the molecular picture of the structure of atomic nucleus. The most fundamental of these three models is, however, the nuclear shell model representing the atomic nucleus as a proton–neutron nuclear system endowed with microscopic antisymmetric wave functions and described within the framework of standard multifermion quantum mechanics. The shell model, in a wider sense, provides a general microscopic framework, in which other models of nuclear structure can be founded and microscopically interpreted.
The main connections between the three types of nuclear structure models were discovered already in the 1950s. Elliott [10] has shown how quadrupole deformation and collective rotation can be derived from the spherical shell model, in which the collective states of the rotational band are determined by their SU(3) symmetry. Wildermuth and Kanellopoulos [11], in turn, have established a connection between the cluster and shell models based on the Hamiltonian of a harmonic oscillator. In this case, the wave function of the one model is expressed as a linear combination of the wave functions of another model. This connection has been interpreted by Bayman and Bohr [12] in terms of the SU(3) symmetry. As a consequence of this, the cluster states are also selected from the model space of the shell model by their specific SU(3) symmetry. Moreover, when a given SU(3) irreducible representation is present in the cluster model, but does not appear in the shell model, then it must be Pauli forbidden. In this way, the mutual relationship between the three fundamental models of nuclear structure has been established in terms of the SU(3) symmetry for the case of a single shell. Recently, such a connection between the shell, cluster and collective models has been established for the multi-shell case using the Ux(3) ⨂ Uy(3) ⊃ U(3) dynamical symmetry chain [13, 14], which appears to be a common intersection. All this demonstrates the crucial role played by the U(3) (or SU(3)) symmetry of the three-dimensional harmonic oscillator.
There are various groups of cluster models, which can be separated into two types—phenomenological and microscopic. In different cluster models the corresponding degrees of freedom can be divided into two categories, related to: (1) the relative motion of the clusters; and (2) their internal structure. Especially powerful are the algebraic models, based on the use of spectrum generating algebras (SGA) and dynamical groups [15]. Further we will restrict our consideration only to the algebraic nuclear models. The most popular algebraic model of nuclear structure is the interacting boson model (IBM) [16, 17], in terms of which a large body of the experimental data has been analyzed. An interesting relation between the SU(3) IBM spaces and the fermion shell-model subspaces has recently been obtain in [18], which sheds light on the longstanding problem of constructing a certain boson collective subspace that decouples from the full shell-model space.
In algebraic models all model observables, such as Hamiltonian and transition operators, are expressed in terms of the elements of a Lie algebra of observables. An example of a complete algebraic model that is a submodel of the shell model is provided by the Elliott SU(3) model [10] of nuclear rotations for the light sd-shell nuclei. Examples of algebraic phenomenological cluster models are provided by the nuclear vibron model (NVM) [19–23], the algebraic cluster model (ACM) [24–27], whereas the microscopic cluster models are represented, e.g. by the semimicroscopic algebraic cluster model (SACM) [28, 29], the semimicroscopic algebraic quartet model (SAQM) [30]. It is well known that a characteristic feature that distinguishes between the two groups of models (phenomenological and microscopic) is provided by the Pauli principle. The important role of the Pauli principle in the cluster models of nuclei has well been demonstrated recently in [31].
It is well known that within the microscopic shell-model approaches to nuclear structure, the so-called explosion dimensionality problem arises with the increase of the number of particles and the number of available fermion states. This requires the usage of different truncation schemes. Among them the algebraic models appear to be very efficient in reducing the huge many-particle shell-model space of the nucleus into a direct sum of tractable irreducible collective subspaces of the Hilbert space by using symmetry-adapted bases, which already capture many of the important many-body correlations. In this respect, a microscopic algebraic cluster model based on the symplectic symmetry has recently been proposed [32]. The present work extends the early attempts of Hecht and Suzuki [33–35] to unify the cluster and symplectic Sp(6, R) collective-model degrees of freedom. The present approach treats in a natural way both the relative intercluster and internal cluster excitations on equal footing by considering the group structure Sp(6, R)R ⨂ Sp(6, R)C that governs the more complete cluster dynamics and which is embedded in the full dynamical group Sp(6(A − 1), R) of the whole many-nucleon nuclear system. Actually, as will be demonstrated later, the essential physics is represented by the subgroup Sp(6, R)R ⨂ Sp(6, R)C ⨂ O(A − 2) ⊂ Sp(6(A − 1), R) with A = A1 + A2, where the direct-product group O(A − 2) ≡ O(A1 − 1) ⨂ O(A2 − 1) allows us to ensure the proper permutational symmetry of the two clusters. In the limiting case when there is no clustering, the group structure Sp(6, R)R ⨂ Sp(6, R)C ⨂ O(A − 2) ⊂ Sp(6(A − 1), R) transforms to Sp(6, R) ⨂ O(A − 1) ⊂ Sp(6(A − 1), R) with the identification Sp(6, R) ≡ Sp(6, R)C, i.e. to that of the Sp(6, R) microscopic collective model [36].
In [32] by considering the 16O + α → 20Ne two-cluster system, the equivalence of the new approach to the SACM has been demonstrated. The approach of [32] has not been, however, yet practically applied to the description of cluster states in specific nuclear system. Thus, it is the purpose of the present work to test the new modified approach, which we term the symplectic symmetry approach to clustering (SSAC), to a real nuclear system. As such a system, we choose the two-cluster system 20Ne + α → 24Mg. In contrast to the case of 16O + α → 20Ne considered in [32], one of the clusters here is characterized by a non-scalar UC(3) representation. This is the reason to choose 20Ne + α → 24Mg instead of the 16O + α → 20Ne two-cluster nuclear system, since the cluster dynamics in the latter case is reduced to the dynamics of the R-subsystem only. In addition, 24Mg exhibits in the experimentally observed low-lying spectrum well developed ${K}^{\pi }={0}_{1}^{+}$, ${K}^{\pi }={2}_{1}^{+}$ and ${K}^{\pi }={0}_{1}^{-}$ rotational bands. The nucleus 24Mg has been studied within the framework of different algebraic nuclear structure models. For instance, it was studied a long time ago within the nuclear SU(3) shell model [37]. In [38, 39] the low-lying structure of the ground and γ rotational bands in 24Mg has been investigated within the framework of the microscopic Sp(6, R) symplectic model (sometimes referred to as a microscopic collective model), whereas in [40] this nucleus has been studied within the SACM considering the single 12C + 12C → 24Mg channel, including both the low-lying and the high-energy quasimolecular resonance states. For different cluster model approaches to 24Mg see, e.g., the detailed list of relevant references given in [40]. The 20Ne + α cluster structure of 24Mg has been studied in a microscopic cluster model [41, 42] using the generator-coordinate method (GCM), producing, however, significantly underestimated B(E2) transition probabilities for the ground band. In the present work we consider only the low-energy excited states of the lowest ${K}^{\pi }={0}_{1}^{+}$, ${K}^{\pi }={2}_{1}^{+}$ and ${K}^{\pi }={0}_{1}^{-}$ bands, observed in the experimental spectrum of 24Mg. Although the 20Ne + α threshold is high (9.32 MeV), this kind of structure is also expected to be relevant to the low-lying states of 24Mg [43]. We point out also that 24Mg can be considered as a system of 6α particles. An interesting mode of nα cluster excitations near the nα breakup threshold in the light 4n self-conjugate nuclei has been proposed to have a gas-like (Bose–Einstein condensate) structure at low density, which is described by the so-called Tohsaki–Horiuchi–Schuck–Röpke (THSR) [44] wave function of α particles siting in a wide cluster-mean-field-like potential (container). This excitation mode is a novel type of nuclear clustering and the search for such nα condensate states is the focus of the experimental nuclear structure physics. For instance, the possible candidates for 5α [45] and 6α [46] condensate states have recently been reported, extending the previous studies of 3α and 4α condensate states.
2. Theoretical approach
Consider an A-body (one-component) nuclear system, which can be described in terms of m = A − 1 translationally invariant relative Jacobi coordinates qis. This allows us to avoid the problem of the center-of-mass motion from the very beginning. All bilinear Hermitian combinations of the Jacobi position qis and momentum pis (i = 1, 2, 3; s = 1, 2, …, m) coordinates generate Sp(6 m, R) − the full dynamical group of the whole nuclear system. The Sp(6 m, R) group contains different kinds of possible motions − collective, internal, cluster, etc., which can be obtained by reducing it in different ways. By doing this, one performs a separation of the 3m nuclear many-particle fermion variables {q} into kinematical (internal) and dynamical (collective) ones, i.e. {q} = {qD, qK}. According to this, the many-particle nuclear wave functions can be represented, respectively, as consisting of collective and internal components [32]
$\begin{eqnarray}{\rm{\Psi }}(q)=\displaystyle \sum _{\eta }{{\rm{\Theta }}}_{\eta }({q}_{D}){\chi }_{\eta }({q}_{K}).\end{eqnarray}$
For two-cluster systems A = A1 + A2, where A1 and A2 are the mass numbers of the two clusters. The (one-component) symplectic symmetry approach to clustering in atomic nuclei for a two-cluster nuclear system was firstly introduced by considering the chain [32]:
$\begin{eqnarray}\begin{array}{l}Sp(6(A-1),R)\\ \supset \,Sp{(6,R)}_{R}\displaystyle \otimes Sp(6({A}_{1}-1),R)\displaystyle \otimes Sp(6({A}_{2}-1),R)\\ \supset \,Sp{(6,R)}_{R}\displaystyle \otimes Sp{(6,R)}_{{A}_{1}-1}\displaystyle \otimes O({A}_{1}-1)\\ \,\quad \displaystyle \otimes \,Sp{(6,R)}_{{A}_{2}-1}\displaystyle \otimes O({A}_{2}-1)\\ \supset \,{U}_{R}(3)\displaystyle \otimes {U}_{{A}_{1}-1}(3)\displaystyle \otimes {U}_{{A}_{2}-1}(3)\displaystyle \otimes {S}_{{A}_{1}}\displaystyle \otimes {S}_{{A}_{2}}\\ \,[{n}_{R},0,0]\,[{n}_{1}^{{C}_{1}},{n}_{2}^{{C}_{1}},{n}_{3}^{{C}_{1}}]\,[{n}_{1}^{{C}_{2}},{n}_{2}^{{C}_{2}},{n}_{3}^{{C}_{2}}]\quad {f}_{1}\quad {f}_{2}\\ \supset \,{U}_{R}(3)\displaystyle \otimes {U}_{C}(3)\\ \,[{n}_{R},0,0]\quad [{n}_{1}^{C},{n}_{2}^{C},{n}_{3}^{C}]\\ \supset \,U(3)\,\supset \,SU(3)\,\supset \,SO(3).\\ \,[{N}_{1},{N}_{2},{N}_{3}]\,(\lambda ,\mu )\quad \kappa \quad L\end{array}\end{eqnarray}$
The groups Sp(6, R)R, Sp(6(A1 − 1), R) and Sp(6(A2 − 1), R) are related to the intercluster, first and second cluster system degrees of freedom, respectively. The intercluster relative motion of the two clusters is described by one of the (A − 1) relative Jacobi vectors, denote it by qR, whereas the rest (A − 2) Jacobi vectors are related to the internal states of the two clusters. $Sp{(6,R)}_{{A}_{\alpha }-1}$ subgroups with α = 1, 2 in $Sp{(6,R)}_{{A}_{\alpha }-1}\otimes O({A}_{\alpha }-1)\subset Sp(6({A}_{\alpha }-1),R)$ are the dynamical groups of collective excitations of the two clusters (referred also to as internal cluster excitations), while the groups O(Aα − 1) allow to ensure the proper permutational symmetries of both clusters.According to the reduction chain (2 ), corresponding to the case of two-cluster nuclear system (A = A1 + A2), the well known resonating group method (RGM) anzatz [3]1 ) by making the following identifications: χη(qK) = φ1(A1 − 1)φ2(A2 − 1) and Θη(qD) = f(qR). In the present work, however, we consider the following modified coupling scheme:
$\begin{eqnarray}{\rm{\Psi }}(q)={ \mathcal A }\{{\phi }_{1}({A}_{1}-1){\phi }_{2}({A}_{2}-1)f({{\boldsymbol{q}}}^{R})\},\end{eqnarray}$
can be related to equation ( $\begin{eqnarray}\begin{array}{l}Sp(6(A-1),R)\\ \supset \,Sp{(6,R)}_{R}\displaystyle \otimes Sp(6({A}_{1}-1),R)\displaystyle \otimes Sp(6({A}_{2}-1),R)\\ \supset \,Sp{(6,R)}_{R}\displaystyle \otimes Sp{(6,R)}_{C}\displaystyle \otimes O(A-2)\\ \,\langle {\sigma }^{R}\rangle \quad \,\,\langle {\sigma }^{C}\rangle \\ \,\cup {n}_{R}{\rho }_{R}\,\,\cup \,{n}_{C}{\rho }_{C}\\ \quad {U}_{R}(3)\quad \displaystyle \otimes \quad {U}_{C}(3)\quad \supset \\ \quad [{E}_{1}^{R},0,0]\quad [{E}_{1}^{C},{E}_{2}^{C},{E}_{3}^{C}]\\ \supset \,U(3)\qquad \quad \supset \qquad SO(3),\\ \,[{E}_{1},{E}_{2},{E}_{3}]\,\kappa \,\,L\end{array}\end{eqnarray}$
which turns out to be more convenient for our further considerations and is directly related to the symplectic symmetry associated with the cluster effects in atomic nuclei. We note that $Sp{(6,R)}_{C}\equiv Sp{(6,R)}_{{C}_{1}}\otimes Sp{(6,R)}_{{C}_{2}}$ and O(A − 2) ≡ O(A1 − 1) ⨂ O(A2 − 1) are direct-product groups, related to the subspaces spanned by the sets ${q}_{{C}_{1}}=({q}_{1},\ldots ,{q}_{{A}_{1}-1})$ and ${q}_{{C}_{2}}=({q}_{1},\ldots ,{q}_{{A}_{2}-1})$ of the Jacobi coordinates of the two clusters. If we consider the (A − 1) system as a whole, not separated into clusters, then the relevant reduction of the full dynamical group Sp(6(A − 1), R) is provided by the chain Sp(6(A − 1), R) ⊃ Sp(6, R) ⨂ O(A − 1), which defines the (one-component) Sp(6, R) symplectic collective model [36] with six microscopic collective (dynamical) and 3(A − 1) − 6 internal (kinematical) degrees of freedom [47–50]. Then the permutational symmetry is ensured by the symmetric group SA through the reduction O(A − 1) ⊃ SA. Similarly, the permutational symmetry of the two clusters is ensured by the chains $O({A}_{1}-1)\supset {S}_{{A}_{1}}$ and $O({A}_{2}-1)\supset {S}_{{A}_{2}}$, respectively.Introducing the standard creation and annihilation operators of three-dimensional harmonic oscillator quanta 5 ) $\mu =\left(\right.\frac{{A}_{1}{A}_{2}}{{A}_{1}+{A}_{2}}\left)\right.M$ is the reduced mass and we chose ${\omega }_{R}\,=\left(\right.\frac{{A}_{1}+{A}_{2}}{{A}_{1}{A}_{2}}\left)\right.\omega $, so that for the oscillator length parameter we obtain ${b}_{0}=\sqrt{\frac{\hslash }{\mu {\omega }_{R}}}=\sqrt{\frac{\hslash }{M\omega }}$. As can be seen, the operators (6 ) create or annihilate a pair of oscillator quanta, whereas the operators (7 ) preserve the number of quanta and generate the subgroup UR(3) ⊂ Sp(6, R)R. In this way, the Sp(6, R)R generators of intercluster excitations can change the number of oscillator quanta by either 0 or 2. Thus, acting on the ground state by the Sp(6, R)R generators one can produce the positive-parity SU(3) cluster-model states of even oscillator quanta only. The negative-parity cluster-model states, in turn, consist of an odd number of oscillator quanta and are associated with the SU(3) basis states of the odd irreps of the group Sp(6, R)R. Alternatively, one may consider the slightly extended semi-direct product group $[HW{(3)}_{R}]Sp{(6,R)}_{R}=\{{F}_{ij}^{R},{G}_{ij}^{R},{A}_{ij}^{R},{b}_{i}^{\dagger R},{b}_{i}^{R},I\}$, consisting of the Sp(6, R)R and $HW{(3)}_{R}=\{{b}_{i}^{\dagger ,R},{b}_{i}^{R},I\}$ generators. In this way, by acting on the ground state by the WSp(6, R)R = [HW(3)R]Sp(6, R)R generators one generates the cluster-model states of both even and odd number of oscillator quanta by changing the number of oscillator quanta by 0, 1 and 2. The latter considerations can be made rigorous by replacing the group Sp(6(A − 1), R) in (4 ) by [HW(3(A − 1))]Sp(6(A − 1), R), which will be the maximal dynamical group for the whole A-nucleon nuclear system. We note also that the WSp(6, R)R = [HW(3)R]Sp(6, R)R group contains, in contrast to the group Sp(6, R)R, the E1 dipole operator among its generators, which actually couples the even and odd Sp(6, R)R irreps.
$\begin{eqnarray}\begin{array}{rcl}{b}_{i}^{\dagger R} & = & \sqrt{\frac{\mu {\omega }_{R}}{2\hslash }}\left(\right.{q}_{i}^{R}-\frac{i}{\mu {\omega }_{R}}{p}_{i}^{R}\left)\right.,\\ {b}_{i}^{R} & = & \sqrt{\frac{\mu {\omega }_{R}}{2\hslash }}\left(\right.{q}_{i}^{R}+\frac{i}{\mu {\omega }_{R}}{p}_{i}^{R}\left)\right.,\end{array}\end{eqnarray}$
the group of intercluster excitations Sp(6, R)R can be represented by means of the following set of generators $\begin{eqnarray}{F}_{ij}^{R}={b}_{i}^{\dagger R}{b}_{j}^{\dagger R},\quad {G}_{ij}^{R}={b}_{i}^{R}{b}_{j}^{R},\end{eqnarray}$
$\begin{eqnarray}{A}_{ij}^{R}=\frac{1}{2}\left(\right.{b}_{i}^{\dagger R}{b}_{j}^{R}+{b}_{j}^{R}{b}_{i}^{\dagger R}\left)\right.\end{eqnarray}$
in the form $Sp{(6,R)}_{R}=\{{F}_{ij}^{R},{G}_{ij}^{R},{A}_{ij}^{R}\}$. In equation (Similarly one obtains the oscillator realizations for the dynamical group of collective (internal cluster or major shell) excitations of the two clusters Sp(6, R)C$\,\equiv \{{F}_{ij}^{C}$ $=\displaystyle {\sum }_{s=1}^{A-2}{b}_{is}^{\dagger }{b}_{js}^{\dagger }$, ${G}_{ij}^{C}=\displaystyle {\sum }_{s=1}^{A-2}{b}_{is}{b}_{js}$, ${A}_{ij}^{C}=\frac{1}{2}\displaystyle {\sum }_{s=1}^{A-2}\left(\right.{b}_{is}^{\dagger }{b}_{js}+{b}_{js}{b}_{is}^{\dagger }\left)\right.\}$ in terms of the harmonic oscillator creation and annihilation operators
$\begin{eqnarray}\begin{array}{rcl}{b}_{is}^{\dagger } & = & \sqrt{\frac{M\omega }{2\hslash }}\left(\right.{q}_{is}-\frac{i}{M\omega }{p}_{is}\left)\right.,\\ {b}_{is} & = & \sqrt{\frac{M\omega }{2\hslash }}\left(\right.{q}_{is}+\frac{i}{M\omega }{p}_{is}\left)\right.,\end{array}\end{eqnarray}$
where i, j = 1, 2, 3, s = 1, 2, …, A − 2 and M is the nucleon mass. Actually, this realization of the $Sp{(6,R)}_{C}\,\equiv Sp{(6,R)}_{{C}_{1}}\otimes Sp{(6,R)}_{{C}_{2}}$ group corresponds to the case when its ${K}_{1}{X}_{{C}_{1}}+{K}_{2}{X}_{{C}_{2}}$ generators (with ${X}_{1}\in Sp{(6,R)}_{{C}_{1}}$ and ${X}_{2}\in Sp{(6,R)}_{{C}_{2}}$) are taken in the form ${X}_{{C}_{1}}+{X}_{{C}_{2}}$, i.e. when K1 = K2 = 1. This particular case corresponds to the reduction $Sp{(6,R)}_{{C}_{1}}\otimes Sp{(6,R)}_{{C}_{2}}\supset Sp{(6,R)}_{{C}_{1}+{C}_{2}}$. If required, one can consider the symplectic groups related separately to each cluster, i.e. $Sp{(6,R)}_{{C}_{\alpha }}\equiv \{{F}_{ij}^{{C}_{\alpha }}$ $=\displaystyle {\sum }_{s=1}^{{A}_{\alpha }-1}{b}_{is}^{\dagger }{b}_{js}^{\dagger }$, ${G}_{ij}^{{C}_{\alpha }}\,=\displaystyle {\sum }_{s=1}^{{A}_{\alpha }-1}{b}_{is}{b}_{js}$, ${A}_{ij}^{{C}_{\alpha }}=\frac{1}{2}\displaystyle {\sum }_{s=1}^{{A}_{\alpha }-1}\left(\right.{b}_{is}^{\dagger }{b}_{js}+{b}_{js}{b}_{is}^{\dagger }\left)\right.\}$ with α = 1, 2.The symplectic basis states of the R- or C-subsystem are determined by the Sp(6, R)α (α = R, C) lowest-weight state ∣σα⟩, defined by 4 ) can be written in an U(3)-coupled form as ${\rm{\Psi }}={[{{\rm{\Psi }}}_{R}\times {{\rm{\Psi }}}_{C}]}_{kLM}^{\rho E}$ with the identifications: $\Psi$R = f(qR), $\Psi$C = φ(A − 2) and φ(A − 2) = φ1(A1 − 1)φ2(A2 − 1).
$\begin{eqnarray}\begin{array}{rcl}{G}_{ij}^{\alpha }| {\sigma }^{\alpha }\rangle & = & 0,\\ {A}_{ij}^{\alpha }| {\sigma }^{\alpha }\rangle & = & 0,\qquad i\lt j\\ {A}_{ii}^{\alpha }| {\sigma }^{\alpha }\rangle & = & \left(\right.{\sigma }_{i}^{\alpha }+\frac{{m}_{\alpha }}{2}\left)\right.| {\sigma }^{\alpha }\rangle ,\end{array}\end{eqnarray}$
and are classified by the following reduction chain $\begin{eqnarray}\begin{array}{lcl}Sp{(6,R)}_{\alpha } & \quad \supset \quad & {U}_{\alpha }(3).\,\\ \,\langle {\sigma }^{\alpha }\rangle \quad & {n}_{\alpha }{\rho }_{\alpha } & {[{E}_{1}^{\alpha },{E}_{2}^{\alpha },{E}_{3}^{\alpha }]}_{3}\end{array}\end{eqnarray}$
For the R-subsystem nR = ρR = mR = 1 and the allowed UR(3) irreps are only fully symmetric, i.e. of the type ${[{E}^{R},0,0]}_{3}$. Similarly, for the C-subsystem mC = A − 2 and the UC(3) irreps are of general type ${[{E}_{1}^{C},{E}_{2}^{C},{E}_{3}^{C}]}_{3}$. The Sp(6, R)α basis states can then be represented in the following coupled form $\begin{eqnarray}| {\rm{\Psi }}({\sigma }^{\alpha }{n}_{\alpha }{\rho }_{\alpha }{E}^{\alpha }{\eta }_{\alpha })\rangle ={[{P}^{({n}_{\alpha })}({F}^{\alpha })\times | {\sigma }^{\alpha }\rangle ]}_{{\eta }_{\alpha }}^{{\rho }_{\alpha }{E}^{\alpha }},\end{eqnarray}$
where ${E}^{\alpha }={[{E}_{1}^{\alpha },{E}_{2}^{\alpha },{E}_{3}^{\alpha }]}_{3}$ denotes the coupled Uα(3) irrep and ρα is a multiplicity label of its appearance in the product n ⨂ σα with ${n}^{\alpha }={[{n}_{1}^{\alpha },{n}_{2}^{\alpha },{n}_{3}^{\alpha }]}_{3}$ and ${\sigma }^{\alpha }={[{\sigma }_{1}^{\alpha },{\sigma }_{2}^{\alpha },{\sigma }_{3}^{\alpha }]}_{3}$. Finally, the symbol ηα labels basis states of the group Uα(3). Hence, the total wave functions of the whole two-cluster system with respect to the whole chain (Generally, the physical operators of interest (e.g., Hamiltonian and transition operators) within the fully algebraic approach should be expressed through the generators of different subgroups in (4 ). For instance, the quadrupole generators of the groups Sp(6, R)R and Sp(6, R)C can be written in the form
$\begin{eqnarray}{Q}_{ij}^{R}={q}_{i}^{R}{q}_{j}^{R}={A}_{ij}^{R}+\frac{1}{2}\left(\right.{F}_{ij}^{R}+{G}_{ij}^{R}\left)\right.,\end{eqnarray}$
$\begin{eqnarray}{Q}_{ij}^{C}=\displaystyle \sum _{s=1}^{A-2}{q}_{is}{q}_{js}={A}_{ij}^{C}+\frac{1}{2}\left(\right.{F}_{ij}^{C}+{G}_{ij}^{C}\left)\right..\end{eqnarray}$
Additionally, to obtain the charge quadrupole operators, the cluster subsystem quadrupole operators ${Q}_{ij}^{C}$ must be multiplied by the factor eeff(Z − 1)/(A − 2). The total quadrupole operator of the whole system, according to the group part Sp(6, R)R ⨂ Sp(6, R)C of (4 ), can be represented as 4 ) with obvious identifications α1 = eeff and α2 = eeff(Z − 1)/(A − 2) (i.e. α1 ≠ α2). Alternatively, according to the group part UR(3) ⨂ UC(3) of (4 ), one could consider as a transition operator only the SU(3) components of the charge quadrupole operator
$\begin{eqnarray}{Q}_{ij}={\alpha }_{1}{Q}_{ij}^{R}+{\alpha }_{2}{Q}_{ij}^{C}.\end{eqnarray}$
The equal strengths α1 = α2 = α would correspond to the chain Sp(6, R)R ⨂ Sp(6, R)C ⊃ Sp(6, R), which differs from the chain ( $\begin{eqnarray}{\tilde{Q}}_{ij}={e}_{\mathrm{eff}}{A}_{ij}^{R}+{e}_{\mathrm{eff}}\left(\displaystyle \frac{Z-1}{A-2}\right){A}_{ij}^{C}.\end{eqnarray}$
In spherical coordinates the latter operators become $\begin{eqnarray}{\tilde{Q}}_{2m}=\sqrt{3}\left({e}_{\mathrm{eff}}{A}_{2m}^{R}+{e}_{\mathrm{eff}}\left(\displaystyle \frac{Z-1}{A-2}\right){A}_{2m}^{C}\right).\end{eqnarray}$
At this point we want to make a comment, concerning the other approaches to clustering in atomic nuclei that are more or less related to the symplectic symmetry. The relation of the Sp(6, R) and α-cluster model states has been done in [33–35], using complicated overlap integrals between the symplectic and cluster bases. In [33, 34] 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 2nℏω. Sometimes, subsets of the full set of the SU(3) basis states, contained in the Sp(6, R) irreducible spaces, are used in practical applications. In this respect, the Sp(2, R) ⊂ Sp(6, R) [51] and Sp(4, R) ⊂ Sp(6, R) submodels [52] represent the giant resonance collective excitations along the z-direction or the z and x directions, respectively. The connection of the α-cluster and Sp(2, R) states has been investigated in [53, 54] for the case of 8Be, again confirming their complementary nature. The Sp(2, R) and Sp(4, R) symplectic models of nuclear structure, however, don't contain an SO(3) subgroup, which requires the usage of sophisticated angular-momentum projection techniques (often combined with the complicated GCM [55] calculation of the Hamiltonian and norm overlap kernels). Recently, the no-core symplectic model (NCSpM) [56, 57] 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 12C [56, 58, 59]. Actually, the approach of [56, 58, 59] exploits the symplectic symmetry, related to clustering, indirectly by using the complicated technique of Y. Suzuki [34] for computing, just as in [33–35], the overlaps between the complementary symplectic shell model and cluster model bases to project the cluster wave functions out of the NCSpM [56, 57] (or ab initio symmetry-adapted no-core–shell model (SA-NCSM) [58, 60]) microscopic wave functions. The NCSpM differs from the standard Sp(6, R) symplectic model of Rosensteel and Rowe [36], proposed for description of the quadrupole collectivity of atomic nuclei, only by the type of nuclear interactions exploited in the calculations. Thus, using the overlap technique [34], the cluster wave functions can be projected out of the standard Sp(6, R) model instead of those of the NCSpM. In this way the symplectic symmetry of [56, 58, 59] is actually related to the standard Sp(6, R) model [36] of quadrupole collectivity and the cluster effects in atomic nuclei are implicitly included by considering sufficiently large model symplectic spaces able to describe spatially expanded nuclear configurations (high-lying np-nh excitations). In contrast, the microscopic cluster-model type wave functions within the present SSAC are computed directly, using the symplectic symmetry (even for the 0p-0h valence subspace). As can be seen from equation (4 ), the symplectic symmetry appears at many levels within the present approach and is associated with both the intercluster and intracluster (internal) nuclear excitations.
The Sp(6, R)R ⨂ Sp(6, R)C group structure exploited in a full account in the present paper, to our knowledge, has never been used in the literature. In this way, in contrast to [33–35, 56, 58, 59] in which the overlaps between the cluster model and the Sp(6, R) model wave functions have been calculated in the attempt to “unify" these two models of nuclear excitations (and revealing their complementary character), the present work introduces a fully algebraic cluster model that is directly based on the symplectic symmetry. Through the Sp(6, R)R ⨂ Sp(6, R)C group substructure, the conventional Sp(6, R) (associated now with the group Sp(6, R)C) symplectic excitations, related to the quadrupole collectivity in atomic nuclei, are also naturally incorporated in a purely algebraic and self-consistent way within the present SSAC. In the limiting case of no clustering, the group of the whole system Sp(6(A − 1), R) reduces to Sp(6, R) ⨂ O(A − 1), i.e. we obtain the Sp(6(A − 1), R) ⊃ Sp(6, R) ⨂ O(A − 1) group structure, by means of which the standard Sp(6, R) symplectic model of quadrupole collectivity is recovered.
The UR(3) ⨂ UC(3) ⊃ U(3) subgroup structure of the Sp(6, R)R ⨂ Sp(6, R)C group further allows us to establish close relationships to the other algebraic cluster models (both microscopic or phenomenological). We note that the SUR(3) ⨂ SUC(3) ⊃ SU(3) underlying substructure of various cluster models, i.e. the SU(3)-based RGM, has first been considered in [61]. We note also that the proper permutational symmetry and the related spin or spin–isospin content within the present approach, in contrast to the SACM, for instance, is ensured by the complementary orthogonal group structure $O({A}_{\alpha }-1)\supset {S}_{{A}_{\alpha }}$, which together with the other relevant groups is contained in the Sp(6(A − 1), R) dynamical symmetry group of the whole nuclear system. This is another characteristic feature of the SSAC that distinguishes it from the SACM, which is known to be a hybrid model of Elliott's SU(3) shell model [10] and the U(4) vibron model of Iachello [62], in which the spin–isospin symmetry is determined by the complementary Wigner UST,α(4) (α = 1, 2) symmetry group [63].
3. Physical operators and basis states
3.1. Hamiltonian
A quite general shell-model Hamiltonian within the fully algebraic symplectic symmetry approach can be represented in the form 17 ) by means of the elements of the corresponding symplectic dynamical groups.
$\begin{eqnarray}H={H}_{0}+V(F,G,A)+{H}_{\mathrm{res}},\end{eqnarray}$
where H0 is the three-dimensional harmonic oscillator Hamiltonian and V(F, G, A) is a potential, which can be expressed by mean of the symplectic generators F, G, A of the Sp(6, R)R and Sp(6, R)C groups. In particular, the interaction, related to the intercluster relative motion, can be represented by means of the Sp(6, R)R generators only as $V=V({F}_{ij}^{R},{G}_{ij}^{R},{A}_{ij}^{R})$. H0 corresponds to the mean field and determines the shell structure of the nuclear system. It orders the shell-model states with respect to the number of oscillator quanta. Within a given oscillator shell the oscillator states are degenerated, but the introduction of the potential V(F, G, A) splits them in energy. H${}_{\mathrm{res}}$ is a residual part, not included in V(F, G, A). It can include, e.g., symplectic symmetry breaking interactions, like the spin-orbit or/and pairing interaction. But, often, like in the present work, H${}_{\mathrm{res}}$ consists of a residual-rotor part, representing the shell-model image of the rotor-model Hamiltonian. We note also that a large class of microscopic cluster-model Hamiltonians of the type $\begin{eqnarray}H={H}_{{C}_{1}}+{H}_{{C}_{2}}+T({q}_{R})+V({q}_{{C}_{1}},{q}_{{C}_{2}},{q}_{R}),\end{eqnarray}$
with ${q}_{{C}_{1}}=({q}_{1},\ldots ,{q}_{{A}_{1}-1})$ and ${q}_{{C}_{2}}=({q}_{1},\ldots ,{q}_{{A}_{2}-1})$, can be expressed in the algebraic form (3.2. E2 transition operator
As E2 transition operators we chose 19 )), were used as E2 transition operators in [64, 65] in contrast to the widely exploited in-shell or SU(3) components [14, 30, 66, 67]. We note also that in [68] the in-shell SUR(3) quadrupole operators ${\widetilde{Q}}_{2M}^{R}$ are multiplied by the factor ($\left.\tfrac{{e}_{\mathrm{eff}}Z}{A}\right)$, which in our opinion is redundant since the intercluster motion is associated with the single relative Jacobi vector qR (hence the related charge factor of the R-subsystem is simply eeff since AR = ZR = 1).
$\begin{eqnarray}{T}_{2M}^{E2}=\sqrt{\displaystyle \frac{5}{16\pi }}\left[{e}_{\mathrm{eff}}{Q}_{2M}^{R}+{e}_{\mathrm{eff}}\left(\displaystyle \frac{Z-1}{A-2}\right){Q}_{2M}^{C}\right],\end{eqnarray}$
which generally have both (internal) cluster and intercluster excitation components. ${Q}_{2M}^{R}=\sqrt{3}\left[\right.{A}_{2M}^{R}+\frac{1}{2}\left(\right.{F}_{2M}^{R}+{G}_{2M}^{R}\left)\right.\left]\right.$ and ${Q}_{2M}^{C}=\sqrt{3}\left[\right.{A}_{2M}^{C}+\frac{1}{2}\left(\right.{F}_{2M}^{C}+{G}_{2M}^{C}\left)\right.\left]\right.$ are the quadrupole generators of the groups Sp(6, R)R and Sp(6, R)C, respectively. We recall that Sp(6, R)R represents the intercluster excitations, whereas Sp(6, R)C is associated with the internal cluster excitations. In the case of unchanged internal cluster structure, only the in-shell SU(3) components ${\widetilde{Q}}_{2M}^{C}=\sqrt{3}{A}_{2M}^{C}$ of ${Q}_{2M}^{C}$, which does not affect the internal structure, will contribute to the B(E2) transition probabilities. In present approach, we use the bare electric charge, i.e. eeff = e. We point out that within the cluster models the full quadrupole operators associated with the intercluster excitations, i.e. the symplectic Sp(6, R)R generators ${Q}_{2M}^{R}$ only (without the ${Q}_{2M}^{C}$ components; cf equation (We stress that for light nuclei the quadrupole moment Q2M usually is multiplied by the numerical factor $\sqrt{\frac{5}{16\pi }}$ [10, 37–39, 68], i.e. one uses ${T}_{2M}^{E2}=\sqrt{\tfrac{5}{16\pi }}\left(\tfrac{{e}_{\mathrm{eff}}Z}{A}\right){Q}_{2M}$. This factor distinguishes the form of the quadrupole operator from that which is commonly used in the shell model, in which ${T}_{2M}^{E2}=\left.\tfrac{{e}_{\mathrm{eff}}Z}{A}\right){Q}_{2M}$ [2, 8, 69, 70]. We recall that the units of Q2M are in ${b}_{0}^{2}$, where ${b}_{0}=\sqrt{\frac{\hslash }{M\omega }}$ is the standard oscillator length parameter. Then, using the bare electric charge eeff = e, one obtains the charge quadrupole moments $\left(\right.\frac{eZ}{A}\left)\right.{Q}_{2M}$ in units of $e{b}_{0}^{2}$. Making use of the expression for the harmonic oscillator length b0 = 1.010A1/6 fm [2], the units of charge quadrupole moments become efm2. To obtain the B(E2) transition strengths in Weisskopf units, one thus must multiply the corresponding expression by the numerical factor $WU={(1.010{A}^{1/6})}^{4}\frac{1}{(5.940\times 1{0}^{-2}){A}^{4/3}}$. This factor as a function of even A values in the range 2 − 200 is shown in figure 1, from which it can be seen that for light nuclei it produces too huge values. In this respect, the factor $\frac{5}{16\pi }$ entering in the expression for the B(E2) values will reduce the latter approximately 10 times. When an effective charge is used, the numerical factor $\sqrt{\frac{5}{16\pi }}$ is irrelevant. But it is crucial in obtaining the proper experimentally observed B(E2) values when no effective charge is used. Otherwise, too huge values are obtained for the light nuclei due to the large values of the WU factor given above. For instance, for A = 20 − 24, WU[A] ≃ 2.2, while for A ≈ 100, WU[A] ≃ 0.8. This means that the B(E2) values for light sd-shell nuclei with A = 20 − 24 will be about three times larger compared to the B(E2) values for heavy nuclei with A ≈ 100. For example, assuming a pure SU(3) structure (8, 4), the B(E2) transition probability produced by the transition operator (19 ) without the factor $\sqrt{\frac{5}{16\pi }}$ is 124.7 W.u., which is too huge compared to the experimental value 20.3 W.u. Using a vertical mixing of different SU(3) states, corresponding to the relative motion cluster excitations, will increase the B(E2) values even more dramatically. The same kind of dramatic difference will be obtained for lighter nuclei from the s and p major shells without the factor $\sqrt{\frac{5}{16\pi }}$ in the definition of the E2 transition operator (19 ).
Figure 1. The factor $WU[A]={(1.010{A}^{1/6})}^{4}\frac{1}{(5.940\times 1{0}^{-2}){A}^{4/3}}$, representing the Weisskopf units, as a function of the even values of the mass number A in the range 2 − 200. |
3.3. Basis states
The basis along the reduction chain (4 ) can be written in the form 4 ), and κ is a multiplicity label in the reduction U(3) ⊃ SO(3). Using the standard Elliott notations (λR, μR) = (ER, 0), $({\lambda }_{C}={E}_{1}^{C}-{E}_{2}^{C},{\mu }_{C}={E}_{2}^{C}-{E}_{3}^{C})$, and (λ = E1 − E2, μ = E2 − E3) for the various SU(3) subgroups and the relation N = (E1 + E2 + E3) + 3(A − 1)/2 for the number of oscillator quanta (including the zero-point motion), the basis (20 ) can alternatively be presented as Appendix .
$\begin{eqnarray}| {\rm{\Gamma }};{[{E}_{R},0,0]}_{3},{[{E}_{1}^{C},{E}_{2}^{C},{E}_{3}^{C}]}_{3};{[{E}_{1},{E}_{2},{E}_{3}]}_{3};\kappa L\rangle ,\end{eqnarray}$
where ${[{E}_{R},0,0]}_{3}$, $[{E}_{1}^{C},{E}_{2}^{C}$, ${E}_{3}^{C}{]}_{3}$, ${[{E}_{1},{E}_{2},{E}_{3}]}_{3}$, and L denote the irreducible representations of the UR(3), UC(3), U(3), and SO(3) groups, respectively. The symbol Γ labels the set of remaining quantum numbers of other subgroups in ( $\begin{eqnarray}| {\rm{\Gamma }}N;({E}_{R},0),({\lambda }_{C},{\mu }_{C});(\lambda ,\mu );\kappa L\rangle ,\end{eqnarray}$
which turns to be more convenient and will be used in what follows. The matrix elements of different physical operators then can be represented in this basis in terms of the SU(3) coupling and recoupling coefficients. The required computational technique for performing a realistic microscopic calculations is shortly presented in the 4. Application
In the present application, we consider only the 20Ne + α → 24Mg channel and the ground state configurations of the two clusters, the latter being unchanged as it is assumed in the RGM [3]. The α particle represents a closed-shell nucleus and hence is characterized by the scalar SU(3) irrep (0, 0). The 20Ne cluster has two protons and two neutrons in the valence sd shell. Using the supermultiplet spin–isospin coupling scheme and the codes [71, 72] one easily obtains the following Pauli allowed SU(3) states (8, 0), (4, 2), (0, 4), (2, 0). A common practice is to choose the leading, i.e. maximally deformed SU(3) state, especially for the strongly deformed nuclei. Hence, for the ground-state cluster SUC(3) irrep (λC, μC) we obtain $({\lambda }_{{C}_{1}},{\mu }_{{C}_{1}})\otimes ({\lambda }_{{C}_{2}},{\mu }_{{C}_{2}})=(8,0)\otimes (0,0)=(8,0)$. The Wildermuth condition [3] requires for the minimum Pauli allowed number of oscillator quanta ER = 8 of the intercluster excitations. We note that nonzero values for the minimal value of ER, in contrast to the case of phenomenological cluster models, in the present microscopic approach means that we have deformed two-cluster nuclear system that can rotate, which crucially changes the traditional interpretation of the U(3) (or, equivalently, SU(3)) excitations as being of a pure vibrational nature. Then the 20Ne + α model space is obtained by the outer product (ER, 0) ⨂ (λC, μC), which produces the following set of SU(3) cluster states: 22 ) defines the Pauli allowed 20Ne + α cluster model space for the lowest value ER = 8: (8, 4), (4, 6), (0, 8). The relevant 20Ne + α cluster model space for the lowest values of ER is given in table 1. The members of each SU(3) multiplet form a rotational-like sequence of levels, which can be interpreted as a ‘cluster band'. Similar considerations, for instance, produce the following Pauli allowed sets of SU(3) states: a) 3(8, 4), 3(7, 3), (8, 1), 2(6, 2), (5, 1), (4, 0) and b) (8, 4) for the 12C + 12C → 24Mg and 16O + 8Be → 24Mg channels, respectively.
$\begin{eqnarray}\begin{array}{rcl}(8,0)\displaystyle \otimes (8,0) & = & (16,0),(14,1),(12,2),(10,3),\\ & & \times (8,4),(6,5),(4,6),(2,7),(0,8).\end{array}\end{eqnarray}$
To obtain the Pauli allowed cluster states, however, one needs to compare them to the standard shell-model SU(3) states of the combined two-cluster system 24Mg. Using again the supermultiplet spin–isospin scheme, the codes [71, 72] for eight valence nucleons in the sd shell produce the following set: (8, 4), (7, 3), (8, 1), (4, 6), (5, 4), (6, 2), (3, 5), (4, 3), (5, 1), (0, 8), (2, 4), (3, 2), (4, 0), (1, 3), (0, 2). The matching SU(3) condition between the latter and the set (Table 1. SU(3) basis states (λ, μ) of the 20Ne + α cluster model space for 24Mg, classified according to the chain ( |
| ER | ℏω | $\qquad SU(3)\,I{R}^{{\prime} }s\,\,\varrho (\lambda ,\mu )$ |
|---|---|---|
| ⋮ | ⋮ | … |
| 10 | 2 | (10, 4), (8, 5), 2(6, 6), (9, 3), (7, 4), (8, 2), (4, 7), 2(2, 8), (5, 5), (3, 6), (4, 4), (1, 7), (0, 6), |
| 9 | 1 | (9, 4), (7, 5), (8, 3), (5, 6), (3, 7), (4, 5), (1, 8), (0, 7) |
| 8 | 0 | (8, 4), (4, 6), (0, 8) |
In the present application we use the following model Hamiltonian 4 ). The Hamiltonian (23 ) is a particular case of the algebraic form, given by equation (17 ). The SU(3) Casimir operators in the Hamiltonian (24 ) will arrange the 0ℏω SU(3) irreps in energy with the (8, 4) multiplet becoming the lowest. In order to account for the experimentally observed bandhead energies of the lowest ${K}^{\pi }={0}_{1}^{+}$ and ${K}^{\pi }={2}_{1}^{+}$ bands contained in the (8, 4) multiplet, we introduce also a K2 term, i.e. Hres = bK2, which is a common practice. In addition, to take into account the different experimental moments of inertia of various bands we take them energy dependent, i.e. ${ \mathcal J }={{ \mathcal J }}_{0}(1+{\alpha }_{i}{E}_{i})$, where Ei is the excitation energy of the corresponding bandhead. Such energy- and/or spin-dependent moments of inertia of the type ${ \mathcal J }\,={{ \mathcal J }}_{0}(1+{\alpha }_{i}{E}_{i}+\beta L)$ are often used in the literature, e.g. [73–75]. Related to 24Mg, e.g., energy-dependent moments of inertia were use in [40], whereas in [39] the third- and fourth-order SU(3) symmetry-braking interactions X3 and X4 were used to split the degeneracy of the ${K}^{\pi }={0}_{1}^{+}$ and ${K}^{\pi }={2}_{1}^{+}$ bands, instead of the simpler K2 operator (which is a special linear combination of the operators X3, X4 and L2) used in the present work and in [40].
$\begin{eqnarray}H={H}_{\mathrm{DS}}+{H}_{\mathrm{res}}+{H}_{\mathrm{vmix}},\end{eqnarray}$
where the dynamical symmetry Hamiltonian $\begin{eqnarray}\begin{array}{rcl}{H}_{\mathrm{DS}} & = & {H}_{0}+\xi {C}_{2}[Sp{(6,R)}_{R}]+B{C}_{2}[SU(3)]\\ & & +C{({C}_{2}[SU(3)])}^{2}+\displaystyle \frac{1}{2{ \mathcal J }}{C}_{2}[SO(3)]\end{array}\end{eqnarray}$
is expressed by means of the Casimir operators of different subgroups in the chain (The second-order Sp(6, R)R Casimir operator splits in energy different Sp(6, R)R irreducible representations. In particular it distinguishes between the even and odd Sp(6, R)R irreps $\langle {\sigma }^{R}\rangle =\langle {\sigma }_{1}^{R}+\frac{1}{2},{\sigma }_{2}^{R}+\frac{1}{2},{\sigma }_{3}^{R}+\frac{1}{2}\rangle $, determined by the Sp(6, R)R lowest-weight state labels ${\sigma }^{R}\equiv ({\sigma }_{1}^{R}={E}_{R},0,0)$ or $({\sigma }_{1}^{R}={E}_{R}+1,0,0)$, which in the present approach correspond to the positive- and negative-parity cluster-model states. The lowest even and odd Sp(6, R)R irreps can conveniently be denoted simply as 0p-0h ${[{E}_{R}]}_{3}$ and 1p-1h ${[E{{\prime} }_{R}]}_{3}$, respectively. They consist, respectively, of the following SU(3) shell-model states: ER, ER + 2, ER + 4, … and $E{{\prime} }_{R},E{{\prime} }_{R}+2,E{{\prime} }_{R}+4,\ldots \,$ with $E{{\prime} }_{R}={E}_{R}+1$. These two Sp(6, R)R irreps can be considered as comprising a single irreducible representation of the semi-direct product group WSp(6, R)R = [HW(3)R]Sp(6, R)R with the set of SU(3) states: ER, ER + 1, ER + 2, ER + 3, … . The Sp(6, R)R Casimir operator within the symplectic representation $\langle {\sigma }^{R}\rangle \,=\langle {\sigma }_{1}^{R}+1/2,{\sigma }_{2}^{R}+1/2,{\sigma }_{3}^{R}+1/2\rangle $ takes the following eigenvalue [77]:
$\begin{eqnarray}\langle {C}_{2}[Sp{(6,R)}_{R}]\rangle =\displaystyle \sum _{i=1}^{3}{\sigma }_{i}({\sigma }_{i}+8-2i).\end{eqnarray}$
The diagonal part of the Hamiltonian (23 ), i.e. neglecting the mixing term, thus has the following eigenvalues
$\begin{eqnarray}\begin{array}{rcl}E & = & N\hslash \omega +\xi \langle {C}_{2}[Sp{(6,R)}_{R}]\rangle +B\langle {C}_{2}[SU(3)]\rangle \\ & & +C{(\langle {C}_{2}[SU(3)]\rangle )}^{2}+\frac{1}{2{ \mathcal J }}L(L+1)+b{K}^{2},\end{array}\end{eqnarray}$
where ⟨C2[SU(3)]⟩ = 2(λ2 + μ2 + λμ + 3λ + 3μ)/3 is the eigenvalue of the second-order Casimir operator of SU(3) group.Finally, we introduce a simple vertical mixing term of the algebraic form 27 ) corresponds directly to the intercluster excitations. In this way, we diagonalize the model Hamiltonian (23 ) in the space of stretched SU(3) cluster states only, built on the (8, 4)/(9, 4) multiplet (see table 1), up to energy 20ℏω. The stretched states are the SU(3) cluster states of the type (λ0 + 2n, μ0) [78] with n = 0, 1, 2, …. The results of the diagonalization for the excitation energies of the lowest ${K}^{\pi }={0}_{1}^{+}$, ${K}^{\pi }={2}_{1}^{+}$ and ${K}^{\pi }={0}_{1}^{-}$ bands in 24Mg are compared with experiment [76] in figure 2. The values of the model parameters, obtained by fitting to the excitation energies and the $B(E2;{2}_{1}^{+}\to {0}_{1}^{+})$ transition strength, are: ξ = 0.314, B = −1.203, C = 0.0017, b = 0.841, vmix = −0.0022 (in MeV), and ${{ \mathcal J }}_{0}=2.6$, ${\alpha }_{{2}_{1}^{+}}=0.076$, and ${\alpha }_{{0}_{1}^{-}}=0.145$ (in MeV−1). From the figure one sees a good description of the excitation energies for the three bands under consideration in 24Mg. The description of energy levels of the ground ${K}^{\pi }={0}_{1}^{+}$ band can be, for instance, improved further by introducing a spin-dependence of the ground-state moment of inertia at the price of introducing one more extra parameter. For instance, in [38, 39] an L4-term has been used, which is known to mimics the effect of a spin-dependent moment of inertia (stretching effect). At first sight, it may seems that too many parameters are used in the present microscopic calculations. In this respect, we point out that the two ${K}^{\pi }={0}_{1}^{+}$ (ground) and ${K}^{\pi }={2}_{1}^{+}$ (gamma) bands are described in [39] and [38] by using five and six parameters, respectively, which is comparable with the six parameters used in the present approach. The extra two parameters in our calculations are used for the description of the additional ${K}^{\pi }={0}_{1}^{-}$ band of negative parity, determining its experimentally observed excitation bandhead energy and moment of inertia, respectively.
$\begin{eqnarray}{H}_{\mathrm{vmix}}={v}_{\mathrm{mix}}\left({A}_{2}^{R}\cdot {F}_{2}^{R}+h.c.\right),\end{eqnarray}$
which mixes the SUR(3) and, hence, the SU(3) irreducible representations only vertically. The vertical mixing interaction (
Figure 2. (Color online) Comparison of the excitation energies of the lowest ${K}^{\pi }={0}_{1}^{+}$, ${K}^{\pi }={2}_{1}^{+}$ and ${K}^{\pi }={0}_{1}^{-}$ bands in 24Mg with experiment [76]. The values of the model parameters are: ξ = 0.314, B = −1.203, C = 0.0017, b = 0.841, νmix = −0.0022 (in MeV), and ${{ \mathcal J }}_{0}=2.6$, ${\alpha }_{{2}_{1}^{+}}=0.076$, and ${\alpha }_{{0}_{1}^{-}}=0.145$ (in MeV−1). |
In figure 3 we show the intraband B(E2) transition probabilities in Weisskopf units between the states of the ground band, compared with the experimental data [76] and the predictions of Sp(6, R) collective model calculations [38] and the microscopic cluster model [41] using the GCM. We note that the Sp(6, R) collective-model calculations of [38] and [39] actually coincide and we provide only the results of the first reference. From the figure we see practically identical results for the two symplectic approaches, in which no effective charge is used (i.e., eeff = e). In addition, in table 2 we compare the known experimental B(E2) values [39, 40, 76] with the theory for the nonyrast states of the ${K}^{\pi }={2}_{1}^{+}$ and ${K}^{\pi }={0}_{1}^{-}$ bands in 24Mg and the predictions of the Sp(6, R) collective model calculation [38], and those of the GCM [41] and the SACM [40]. For the quadrupole moment of the excited ${2}_{1}^{+}$ state we obtain $Q({2}_{1}^{+})=-0.56$ eb, to be compared with the experimental value −0.29(3) eb [79], respectively. From figure 3 and table 2 one sees a practically equal description of the B(E2) transition strengths from the states of the ${K}^{\pi }={2}_{1}^{+}$ and ${K}^{\pi }={0}_{1}^{-}$ bands in 24Mg within the different theoretical approaches. In the present and the Sp(6, R) collective-model results, no effective charge has been used, in contrast to those of the SACM approach.
Table 2. Comparison of the theoretical interband or intraband B(E2) transition probabilities (in Weisskopf units) for the lowest states of the ${K}^{\pi }={2}_{1}^{+}$and ${K}^{\pi }={0}_{1}^{-}$ bands in 24Mg with the known experimental data [39, 40, 76] and the predictions of [38] (Sp(6,R)), [41] (GCM), and those of SACM [40]. No effective charge is used in the symplectic based and GCM calculations, where for SACM the adopted values qR = 1.272 and qC = 1.066 are used for the transition operator ${T}_{2M}^{E2}={q}_{R}{\widetilde{Q}}_{2M}^{R}+{q}_{C}{\widetilde{Q}}_{2M}^{C}$. ${\widetilde{Q}}_{2M}^{R}$ and ${\widetilde{Q}}_{2M}^{C}$ are the quadrupole operators of the SUR(3) and SUC(3) groups, respectively. |
| i | f | Exp | Th | Sp(6, R) | GCM | SACM |
|---|---|---|---|---|---|---|
| ${2}_{2}^{+}$ | ${0}_{1}^{+}$ | 1.4(0.3) | 2.0 | 1.3 | 0.7 | 1.6 |
| ${2}_{2}^{+}$ | ${2}_{1}^{+}$ | 2.7(0.4) | 4.3 | 1.9 | 6.4 | 3.2 |
| ${3}_{1}^{+}$ | ${2}_{1}^{+}$ | 2.1(0.3) | 3.6 | 2.3 | 1.2 | 2.9 |
| ${4}_{2}^{+}$ | ${2}_{1}^{+}$ | 1.0(0.2) | 0.6 | 0.9 | 7.9 | 0.6 |
| ${4}_{2}^{+}$ | ${4}_{1}^{+}$ | 1.0(1.0) | 5.1 | 2.3 | 4.10−3 | 3.9 |
| ${5}_{1}^{+}$ | ${4}_{1}^{+}$ | 3.9(0.8) | 2.3 | 2.0 | 8.6 | 2.0 |
| ${6}_{2}^{+}$ | ${4}_{1}^{+}$ | $0.8{(}_{-0.3}^{+0.8})$ | 0.2 | 1.0 | 0.4 | 0.3 |
| ${3}_{1}^{+}$ | ${2}_{2}^{+}$ | 34(6) | 34.8 | 35.3 | 19.4 | 37.5 |
| ${4}_{2}^{+}$ | ${2}_{2}^{+}$ | 16(3) | 11.4 | 11.0 | 5.4 | 11.4 |
| ${5}_{1}^{+}$ | ${3}_{1}^{+}$ | 28(5) | 17.3 | 16.6 | 9.7 | 17.5 |
| ${5}_{1}^{+}$ | ${4}_{2}^{+}$ | 14(6) | 17.4 | 17.7 | 4.1 | 19.5 |
| ${6}_{2}^{+}$ | ${4}_{2}^{+}$ | $23{(}_{-8}^{+23})$ | 17.6 | 18.3 | 1.3 | 18.1 |
| ${7}_{1}^{+}$ | ${5}_{1}^{+}$ | 18.8 | 8.2 | 19.7 | ||
| ${8}_{2}^{+}$ | ${6}_{2}^{+}$ | ≥3 | 17.9 | 15.9 | 5.4 | 13.7 |
| ${3}_{1}^{-}$ | ${1}_{1}^{-}$ | <200 | 23.4 | 17.6 | 32.0 | |
| ${5}_{1}^{-}$ | ${3}_{1}^{-}$ | $20{(}_{-5}^{+8})$ | 25.8 | 22.5 | 34.7 | |
| ${7}_{1}^{-}$ | ${5}_{1}^{-}$ | 51(10) | 24.3 | 10.5 | 32.3 |
In figure 4 we give the SU(3) decomposition of the wave functions for the cluster-model states of the ${K}^{\pi }={0}_{1}^{+}$, ${K}^{\pi }={2}_{1}^{+}$ and ${K}^{\pi }={0}_{1}^{-}$ bands in 24Mg for different angular momentum values. From the figure, we see a similar structure for the cluster states of the three bands with predominant contribution of the 0ℏω/1ℏω SU(3) multiplet (8, 4)/(9, 4) and some admixtures due to the mixing to the excited cluster SU(3) configurations. A similar microscopic structure for the wave functions for the states of the ${K}^{\pi }={0}_{1}^{+}$ and ${K}^{\pi }={2}_{1}^{+}$ bands is obtained in [38, 39] within the Sp(6, R) collective model. From the figure, one can see also that the SU(3) decomposition coefficients are approximately angular-momentum independent (especially for the states of the ${K}^{\pi }={2}_{1}^{+}$, and ${K}^{\pi }={0}_{1}^{-}$ bands). The latter indicates the presence of a new type of symmetry, referred to as a quasi-dynamical symmetry in the sense of [80, 81]. Hence, despite the mixing of the SU(3) cluster-model basis states, the microscopic structure of the experimentally observed cluster states of the three bands under consideration shows the presence of an approximate SU(3) quasi-dynamical symmetry.
Figure 4. SU(3) decomposition of the wave functions for the cluster states of the lowest ${K}^{\pi }={0}_{1}^{+}$, ${K}^{\pi }={2}_{1}^{+}$ and ${K}^{\pi }={0}_{1}^{-}$ bands in 24Mg for different angular momentum values. |
5. Conclusions
In the present paper, the recently proposed SSAC has been modified and applied for the first time to a specific two-cluster nuclear system, namely to 24Mg. We have considered the single channel 20Ne + α → 24Mg for the microscopic description of the low-lying cluster states of the lowest ${K}^{\pi }={0}_{1}^{+}$, ${K}^{\pi }={2}_{1}^{+}$ and ${K}^{\pi }={0}_{1}^{-}$ bands in 24Mg. According to the RGM [3], the internal structure of the two clusters is considered unchanged. If needed in the specific practical applications of the SSAC, one may relax the requirement of unchanged internal cluster structure as a next approximation to the nuclear many-body problem.
The purpose of this work was not so to obtain a full and detailed microscopic description of the low-lying excitation spectrum in 24Mg, but to illustrate and test the newly proposed microscopic SSAC to a real nuclear system, consisting of clusters with non-scalar internal structure that involves more general representations (in contrast, e.g., to the case of 20Ne proposed initially in [32]). This allows us to illustrate the more general nontrivial mathematical structures that appear in the construction of the microscopic Pauli allowed model space of the SU(3) cluster states within the present approach.
A good description of the low-lying energy levels for the lowest ${K}^{\pi }={0}_{1}^{+}$, ${K}^{\pi }={2}_{1}^{+}$ and ${K}^{\pi }={0}_{1}^{-}$ bands in 24Mg, as well as for the low-energy intra- and interband B(E2) transition probabilities is obtained within the SSAC by using a simple dynamical symmetry Hamiltonian to which residual-rotor part and vertical-mixing terms are also added. This Hamiltonian is diagonalized within the subspace of stretched SU(3) cluster states, built on the leading (8, 4)/(9, 4) multiplet for the positive/negative-parity states up to energy 20ℏω. The microscopic structure of the cluster states of the three bands under consideration shows a similar picture (cf. figure 4) with a predominant contribution of the 0ℏω (for ${K}^{\pi }={0}_{1}^{+}$ and ${K}^{\pi }={2}_{1}^{+}$ bands) and 1ℏω (for ${K}^{\pi }={0}_{1}^{-}$ band) SU(3) irreps, respectively, with the presence of some admixtures due to the higher cluster SU(3) oscillator configurations. Similarly to the other symplectic based approaches to nuclear structure, no effective charge is used in the calculation of the B(E2) transition strengths.
The results obtained in the present paper are shown to be of the same quality as those obtained in the microscopic Sp(6, R) collective model calculations performed in [38, 39], but the physics behind the SSAC and that of Sp(6, R) model is different. The latter results in a different, microscopic cluster interpretation of the observed low-lying states of the ${K}^{\pi }={0}_{1}^{+}$, ${K}^{\pi }={2}_{1}^{+}$ and ${K}^{\pi }={0}_{1}^{-}$ bands in 24Mg.
The present work opens the path for the further applications of the SSAC to other two-cluster nuclear systems from different mass regions. Generally, a horizontal mixing of different SU(3) cluster states can also be added to the model Hamiltonian, together with the vertical mixing, in order to perform more sophisticated microscopic cluster-model calculations within the SSAC. The SSAC can also be extended to three- and four-cluster nuclear systems. Note that in the limiting case, the group Sp(6(A − 1), R) can be decomposed into Sp(6, R)1 ⨂ Sp(6, R)2 ⨂ …Sp(6, R)A−1, in which the Sp(6, R)s symplectic excitations are associated with each relative Jacobi vector qs with s = 1, 2, …, A − 1. An interesting example of a three-cluster nuclear system is provided by the 3α cluster system of 12C, and, in particular, by the structure of its ${0}_{2}^{+}$ (Hoyle) excited state which is well known to be represented by a 4p − 4h excitation. The extension of the present SSAC to the case of three-cluster nuclear systems will produce the SUR(3) leading irrep (0, 4) for the ground state. The α particle, as a closed-shell nucleus, has a scalar SUC(3) irrep (0, 0). Since the internal structure of the α particles, in first approximation, is considered unexcited, then the complete cluster dynamics of the α + α + α → 12C system, governed by the Sp(6, R)R ⨂ Sp(6, R)C group structure, will be determined by the dynamics of the R-subsystem only. (The contribution of the C-subsystem to the energies and the B(E2) transition probabilities identically vanishes.) In this case, the α + α + α → 12C model space will be spanned purely by the irreps of the group Sp(6, R)R. Then, the relevant cluster-model space for the Hoyle state will be spanned by the excited irrep 4p − 4h (12, 0) of Sp(6, R)R.
Appendix: The SSAC matrix elements
Generally, if we have a tensor operator ${T}^{({l}_{R},{m}_{R})\,{L}_{R}{M}_{M}}$ acting on the R-subsystem and a tensor operator ${S}^{({l}_{C},{m}_{C})\,{L}_{C}{M}_{C}}$ acting on the C-subsystem, then the SO(3)-reduced matrix elements of the coupled tensor operator ${[{T}^{({l}_{R},{m}_{R})\,{L}_{R}}\times {S}^{({l}_{C},{m}_{C})\,{L}_{C}}]}^{\sigma (l,m)\,L^{\prime\prime} M^{\prime\prime} }$ can be expressed in the form [82, 83]: A.1 ), the equivalence of the U(3) and SU(3) recoupling coefficients is used. If the tensor operator acts only in the one subsystem subspace (R or C), then the SU(3) 9 − (λ, μ) recoupling coefficients reduce to the 6 − (λ, μ) U-coefficients.
$\begin{eqnarray}\begin{array}{l}\langle {\rm{\Gamma }};N^{\prime} ;(E{{\prime} }_{R},0),(\lambda {{\prime} }_{C},\mu {{\prime} }_{C});(\lambda ^{\prime} ,\mu ^{\prime} );\kappa ^{\prime} L^{\prime} | | [{T}^{({l}_{R},{m}_{R})\,{L}_{R}}\\ \times {S}^{({l}_{C},{m}_{C})\,{L}_{C}}{]}^{(l,m)\,L^{\prime\prime} M^{\prime\prime} }\\ \times | {\rm{\Gamma }};N;({E}_{R},0),({\lambda }_{C},{\mu }_{C})| ;(\lambda ,\mu );\kappa L\rangle \\ =\displaystyle \sum _{{\rho }_{R},{\rho }_{C},{\rho }_{f}}\left\{\begin{array}{cccc}({E}_{R},0) & ({l}_{R},{m}_{R}) & (E{{\prime} }_{R},0) & {\rho }_{R}\\ ({\lambda }_{C},{\mu }_{C}) & ({l}_{C},{m}_{C}) & (\lambda {{\prime} }_{C},\mu {{\prime} }_{C}) & {\rho }_{C}\\ (\lambda ,\mu ) & (l,m) & (\lambda ^{\prime} ,\mu ^{\prime} ) & {\rho }_{f}\\ 1 & \sigma & 1 & \end{array}\right\}\\ \times \langle (E{{\prime} }_{R},0)| | | {T}^{({l}_{R},{m}_{R}})| | | ({E}_{R},0){\rangle }_{{\rho }_{R}}\\ {\times \langle (\lambda {{\prime} }_{C},\mu {{\prime} }_{C})| | | {S}^{({l}_{C},{m}_{C})}| | | ({\lambda }_{C},{\mu }_{C})\rangle }_{{\rho }_{C}}\\ \times {\langle (\lambda ,\mu )\kappa L,(l,m)kL^{\prime\prime} | | (\lambda ^{\prime} ,\mu ^{\prime} )\kappa ^{\prime} L^{\prime} \rangle }_{{\rho }_{f}},\end{array}\end{eqnarray}$
where {…} and $\langle (\lambda ,\mu )\kappa L,(l,m)kL^{\prime\prime} | | (\lambda ^{\prime} ,\mu ^{\prime} )\kappa ^{\prime} L^{\prime} \rangle $ are the SU(3) recoupling and coupling coefficients, respectively. In equation (As an example consider first the matrix elements with respect to the whole chain (4 ) of the tensors ${A}_{2M}^{R}\,\equiv {A}_{(1,1)\,(0,0)\,(1,1)}^{R\qquad \quad 2M}$ and ${F}_{2M}^{R}\equiv {F}_{(2,0)\,(0,0)\,(2,0)}^{R\qquad \quad 2M}$, entering in ${Q}_{2M}^{R}$. For the SO(3)-reduced matrix elements of ${A}_{2M}^{R}$ we obtain: 4 ) are embedded in Sp(6, R)R and Sp(6, R)C, respectively. In obtaining the expression (A.3 ) the standard symplectic computational technique is used. The SO(3)-reduced matrix elements of the lowering symplectic generators ${G}_{2M}^{R}\equiv {G}_{(0,2)\,(0,0)\,(0,2)}^{R\qquad \quad 2M}$ can be obtained from those of the ${F}_{2M}^{R}$ by conjugate operation. Similarly, one can obtain the matrix elements of the quadrupole operators ${Q}_{2m}^{C}$, acting in the C-space.
$\begin{eqnarray}\begin{array}{l}\langle {\rm{\Gamma }};N;({E}_{R},0),({\lambda }_{C},{\mu }_{C});(\lambda ,\mu );\kappa ^{\prime} L^{\prime} | | {A}_{2M}^{R}| \\ \times | {\rm{\Gamma }};N;({E}_{R},0),({\lambda }_{C},{\mu }_{C});(\lambda ,\mu );\kappa L\rangle \\ =\,\left\{\begin{array}{cccc}({E}_{R},0) & (1,1) & \,({E}_{R},0) & 1\\ ({\lambda }_{C},{\mu }_{C}) & (0,0) & ({\lambda }_{C},{\mu }_{C}) & 1\\ (\lambda ,\mu ) & (1,1) & (\lambda ,\mu ) & 1\\ 1 & 1 & 1 & \end{array}\right\}\langle ({E}_{R},0)| | | {A}_{2M}^{R}| | | ({E}_{R},0)\rangle \\ \times \langle (\lambda ,\mu )\kappa L,(1,1)12| | (\lambda ,\mu )\kappa ^{\prime} L^{\prime} \rangle \\ =\,\sqrt{\langle 2{C}_{2}[S{U}_{R}(3)]({E}_{R},0)\rangle }\\ \times \left\{\begin{array}{cccc}({E}_{R},0) & (1,1) & ({E}_{R},0) & 1\\ ({\lambda }_{C},{\mu }_{C}) & (0,0) & ({\lambda }_{C},{\mu }_{C}) & 1\\ (\lambda ,\mu ) & (1,1) & (\lambda ,\mu ) & 1\\ 1 & 1 & 1 & \end{array}\right\}\\ \times \langle (\lambda ,\mu )\kappa L,(1,1)12| | (\lambda ,\mu )\kappa ^{\prime} L^{\prime} \rangle ,\end{array}\end{eqnarray}$
whereas for the matrix elements of the raising symplectic generators ${F}_{2M}^{R}$ we get $\begin{eqnarray}\begin{array}{l}\langle {\rm{\Gamma }};N+2;({E}_{R}+2,0),({\lambda }_{C},{\mu }_{C});(\lambda ^{\prime} ,\mu ^{\prime} );\kappa ^{\prime} L^{\prime} | \\ | {F}_{2M}^{R}| | {\rm{\Gamma }};N;({E}_{R},0),({\lambda }_{C},{\mu }_{C});(\lambda ,\mu );\kappa L\rangle \\ =\,\left\{\begin{array}{cccc}({E}_{R},0) & (2,0) & ({E}_{R}+2,0) & 1\\ ({\lambda }_{C},{\mu }_{C}) & (0,0) & ({\lambda }_{C},{\mu }_{C}) & 1\\ (\lambda ,\mu ) & (2,0) & (\lambda ^{\prime} ,\mu ^{\prime} ) & 1\\ 1 & 1 & 1 & \end{array}\right\}\\ \times \langle ({E}_{R}+2),0)| | | {F}_{2M}^{R}| | | ({E}_{R},0)\rangle \\ \times \,\langle (\lambda ,\mu )\kappa L,(2,0)2| | (\lambda ^{\prime} ,\mu ^{\prime} )\kappa ^{\prime} L^{\prime} \rangle \\ =\,\sqrt{({n}_{R}+1)({n}_{R}+2)}\sqrt{{\rm{\Delta }}{\rm{\Omega }}({\sigma }_{R}n^{\prime} E^{\prime} ;{n}_{R}{E}_{R})}\\ \times \left\{\begin{array}{cccc}({E}_{R},0) & (2,0) & ({E}_{R}+2,0) & 1\\ ({\lambda }_{C},{\mu }_{C}) & (0,0) & ({\lambda }_{C},{\mu }_{C}) & 1\\ (\lambda ,\mu ) & (2,0) & (\lambda ^{\prime} ,\mu ^{\prime} ) & 1\\ 1 & 1 & 1 & \end{array}\right\}\\ \times \langle (\lambda ,\mu )\kappa L,(2,0)2| | (\lambda ^{\prime} ,\mu ^{\prime} )\kappa ^{\prime} L^{\prime} \rangle ,\end{array}\end{eqnarray}$
where $\sqrt{{\rm{\Delta }}{\rm{\Omega }}({\sigma }_{R}n^{\prime} E^{\prime} ;{n}_{R}{E}_{R})}=\sqrt{{\rm{\Omega }}({\sigma }_{R}n^{\prime} E^{\prime} )-{\rm{\Omega }}({\sigma }_{R}{n}_{R}{E}_{R})}$ and ${\rm{\Omega }}({\sigma }_{R}{n}_{R}{E}_{R})=\frac{1}{4}\displaystyle \sum _{i=1}^{3}[2{({E}_{i}^{R})}^{2}-{({n}_{i}^{R})}^{2}+8({E}_{i}^{R}-{n}_{i}^{R})\,-2i(2{E}_{i}^{R}-{n}_{i}^{R})]$ [84–86]. We note that the matrix elements of the raising and lowering symplectic generators are modified, compared to those with respect to the standard UR(3) ⨂ UC(3) ⊃ U(3) chain, because the UR(3) and UC(3) groups according to (Finally, we give the SO(3)-reduced matrix elements of the tensor operator ${[{A}_{2}^{R}\times {F}_{2}^{R}]}_{(2,0)\,(0,0)\,(2,0)}^{l=0,m=0}$: 17 ), containing a simple vertical mixing term that couples cluster states of different oscillator quanta. The other matrix elements of interest can be obtained in a similar way. We recall that computer codes [87–90] exist for the numerical calculation of the required SU(3) coupling and recoupling coefficients.
$\begin{eqnarray*}\begin{array}{l}\langle {\rm{\Gamma }};N+2;({E}_{R}+2,0),({\lambda }_{C},{\mu }_{C});(\lambda ^{\prime} ,\mu ^{\prime} );\kappa ^{\prime} L\,| \\ | {[{A}_{2}^{R}\times {F}_{2}^{R}]}_{(2,0)\,(0,0)\,(2,0)\,| }^{l=0,m=0}\\ \times | {\rm{\Gamma }};N;({E}_{R},0),({\lambda }_{C},{\mu }_{C});(\lambda ,\mu );\kappa L\rangle \\ =\,\left\{\begin{array}{cccc}({E}_{R},0) & (2,0) & ({E}_{R}+2,0) & 1\\ ({\lambda }_{C},{\mu }_{C}) & (0,0) & ({\lambda }_{C},{\mu }_{C}) & 1\\ (\lambda ,\mu ) & (2,0) & (\lambda ^{\prime} ,\mu ^{\prime} ) & 1\\ 1 & 1 & 1 & \end{array}\right\}\\ \times \langle ({E}_{R}+2,0)| | | {[{A}_{2}^{R}\times {F}_{2}^{R}]}_{(2,0)\,(0,0)\,(2,0)}^{l=0,m=0}| | | ({E}_{R},0)\rangle \\ \times \langle (\lambda ,\mu )\kappa L,(2,0)0| | (\lambda ^{\prime} ,\mu ^{\prime} )\kappa ^{\prime} L\rangle ,\end{array}\end{eqnarray*}$
where the triple-bared SUR(3) matrix elements can be obtained by means of the 6 − (λ, μ) Racah recoupling coefficients: $\langle ({E}_{R}+2,0)| | | {[{A}_{2}^{R}\times {F}_{2}^{R}]}_{(2,0)\,(0,0)\,(2,0)}^{l=0,m=0}| | | ({E}_{R},0)\rangle =$ $U\left(\right.({E}_{R},0);(2,0);({E}_{R}+2,0);(1,1)\left|\right.\left|\right.({E}_{R}+2,0);(2,0)\left)\right.$ $\times \langle ({E}_{R}+2,0)| | | {A}_{2M}^{R}| | | ({E}_{R}+2,0)\rangle \langle ({E}_{R}+2,0)| | | {F}_{2M}^{R}| | | ({E}_{R},0)\rangle $. Using the latter, one finally obtains for the SO(3)-reduced matrix elements of the tensor operator ${[{A}_{2}^{R}\times {F}_{2}^{R}]}_{(2,0)\,(0,0)\,(2,0)}^{l=0,m=0}$: $\begin{eqnarray}\begin{array}{l}\langle {\rm{\Gamma }};N+2;({E}_{R}+2,0),({\lambda }_{C},{\mu }_{C});(\lambda ^{\prime} ,\mu ^{\prime} );\kappa ^{\prime} L\\ \times | {[{A}_{2}^{R}\times {F}_{2}^{R}]}_{(2,0)\,(0,0)\,(2,0)}^{l=0,m=0}| \\ \times | {\rm{\Gamma }};N;({E}_{R},0),({\lambda }_{C},{\mu }_{C});(\lambda ,\mu );\kappa L\rangle \\ =\,\sqrt{\langle 2{C}_{2}[S{U}_{R}(3)]({n}_{R}+2,0)\rangle }\\ \times \sqrt{({n}_{R}+1)({n}_{R}+2)}\sqrt{{\rm{\Delta }}{\rm{\Omega }}({\sigma }_{R}n^{\prime} E^{\prime} ;{n}_{R}{E}_{R})}\\ \times \left\{\begin{array}{cccc}({E}_{R},0) & (2,0) & ({E}_{R}+2,0) & 1\\ ({\lambda }_{C},{\mu }_{C}) & (0,0) & ({\lambda }_{C},{\mu }_{C}) & 1\\ (\lambda ,\mu ) & (2,0) & (\lambda ^{\prime} ,\mu ^{\prime} ) & 1\\ 1 & 1 & 1 & \end{array}\right\}\\ \times \,U\left(\right.({E}_{R},0);(2,0);({E}_{R}+2,0);(1,1)\left|\right.\left|\right.\\ \times | ({E}_{R}+2,0);(2,0)\left)\right.\langle (\lambda ,\mu )\kappa L,(2,0)0| | (\lambda ^{\prime} ,\mu ^{\prime} )\kappa ^{\prime} L\rangle .\end{array}\end{eqnarray}$
The latter matrix elements allows us to perform a numerical diagonalization of the Hamiltonian (