1. Introduction
Different models of nuclear structure exist for describing a particular set of experimental data or aspect of nuclear excitations. These models can be roughly divided into two groups—phenomenological and microscopic models. It is well known that a characteristic feature that distinguishes between the two groups is provided by the Pauli principle. The models are referred to as microscopic if they fulfil the Pauli principle, which originates from the fermionic nature of the atomic nucleus. For the phenomenological models the situation is opposite—they do not respect the Pauli principle, i.e. the composite fermion structure of the nucleus is not taken into account. A well known example of a microscopic model in nuclear physics is provided, for instance, by the algebraic Elliott SU(3) model of nuclear rotations [1]. A widely exploited phenomenological model of the nuclear collective motion, which has conceptually influenced the development of the other collective models of nuclear structure, is presented by the Bohr–Mottelson (BM) collective model [2]. In its standard formulation [3, 4], the latter cannot be naturally related to the microscopic many-fermion nuclear theory. In particular, it is not clear how the state vectors in the Bohr–Mottelson model, which characterize the quantized surface vibrations and rotations of atomic nuclei, can be identified with the wave functions in the Hilbert space of A nucleon antisymmetric states. This is a common property of all phenomenological collective models, which usually describe the nuclear collective motion in terms of shape parameters or bosons of certain type.
It turns out that many phenomenological models of nuclear structure can be given a microscopic foundation. For example, this is achieved by considering the fermion composite substructure of the bosons within the boson-type models (see, for example, [5]). A more powerful and elegant method to do this is provided by the algebraic approach. The idea is to embed the desired phenomenological model into the many-particle microscopic shell-model theory [6, 7] by using spectrum generating algebras (SGA) and dynamical groups [8]. It is well known that the nuclear shell model (see, for example, [9]) provides such a basic formal framework for understanding nuclei in terms of interacting protons and neutrons. In this way, the algebraic approach appears as an unifying concept in nuclear structure physics, relating different models (irrespective of whether they are of phenomenological or microscopic nature) by means of their algebraic structures.
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. In this regard, the problem of the embedding of a certain collective model is largely solved when once it is recognized that both the collective model under consideration and the shell model can be formulated as algebraic models with dynamical groups. Thus, a certain collective model becomes a submodel of the shell model if its dynamical group is expressed as a subgroup of a dynamical group of the shell model (see, for example, [6, 7]). The full Lie algebra of observables of the shell model is huge (strictly speaking, infinite), which is the reason for making the shell model (with major-shell mixing) an unsolvable problem and for seeking its tractable approximations. Fortunately, it has a subalgebra which is easier to manage; i.e. the Lie algebra of all one-body operators. The corresponding dynamical group is then the group of one-body unitary transformations. An example of a complete algebraic model that is a submodel of the shell model is provided by the Elliott SU(3) model [1] already mentioned above.
Then, in general, to give a certain collective model a microscopic shell-model interpretation, the following three steps are required within the framework of the algebraic approach: (a) algebraic formulation of the collective model in terms of a Lie algebra of observables; (b) seeking of a microscopic, many-particle realization of this algebra in terms of all position and momentum coordinates of the particles of the system; and (c) construction of its shell-model representations. Sometimes, it is necessary to adjust the considered phenomenological model, so that its algebraic structure becomes compatible with the microscopic shell-model structure of the nucleus. Such an example is provided by the embedding of the BM model in the one-component shell-model theory. To make the BM model microscopically realizable, one first needs to replace the shape variables, which do not have a microscopic expression, by the microscopic quadrupole moment operators. The latter, together with their time derivatives, yield a new set of commutation relations, defining the Lie algebra of the so-called CM(3) model of Weaver, Biedenharn and Cusson [10, 11]. In this way, one obtains a microscopic many-particle realization of the BM model augmented by the intrinsic vortex spin degrees of freedom, but which is not compatible with the shell-model structure of the nucleus (i.e. only the first two steps are performed). In order to make the CM(3) model compatible with the fermionic nature of the nucleus, its dynamical group was extended to the non-compact symplectic group Sp(6, R) by including the many-particle kinetic energy operator to its set of collective observables. In this way, the Sp(6, R) model [12], sometimes called a microscopic collective model, is obtained as a result of embedding the BM model into the one-component shell-model theory. It has well-defined shell-model representations, which are constructed by means of the three-dimensional creation and annihilation operators of harmonic oscillator quanta. The Sp(6, R) model is a submodel of the nuclear shell model, as should be according to the prescription described above. Another example of embedding into the nuclear shell model is presented by the phenomenological interacting vector boson model [13], in which the nuclear collective motion is described by means of two types of vector bosons. This model possesses only one shell-model representation for even–even nuclei—namely, the trivial scalar representation of its Sp(12, R) dynamical group. A microscopic foundation of this model was obtained by augmenting it with an intrinsic microscopic many-particle U(6) structure, which already admits many nonscalar Sp(12, R) representations compatible with the fermion structure of the nucleus. This was achieved first by expressing its SGA observables in terms of many-particle proton and neutron position and momentum observables and then by construction of the Pauli-allowed shell-model representations (i.e. performing the second and third steps). As a result, a completely new microscopic model of nuclear collective motion has appeared, which is referred to as a proton–neutron symplectic model (PNSM) [14, 15]. At the same time, the PNSM generalizes the microscopic one-component Sp(6, R) model [12] for the case of the two-component proton–neutron nuclear systems, which becomes evident by the embedding Sp(6, R) ⊂ Sp(12, R). In this way, the PNSM has appeared as a simultaneous generalization of the phenomenological interacting vector boson model and the microscopic Sp(6, R) symplectic model.
Recently, the BM model was embedded [16, 17] into the two-component shell-model theory within the framework of the PNSM. It was demonstrated that a microscopic shell-model version of the BM model is defined by one of dynamical symmetry chains of the PNSM. It is the purpose of the present paper to consider in detail the shell-model irreducible representations of this new version and to show that for SO(6) irreps with $\upsilon$ ≥ $\upsilon$0 they represent Pauli-allowed many-particle subspaces of the Hilbert space of the nucleus.
2. The proton–neutron symplectic model
The Sp(12, R) SGA of the PNSM has many subalgebra chains, which can be divided in two types–the collective-model and shell-model chains, respectively. For the shell-model purposes, the Sp(12, R) SGA of the PNSM can be represented by its complexification Sp(12, R) = {Fij(α, β), Gij(α, β), Aij(α, β)}. In this realization, the Sp(12, R) generators [15]
$\begin{eqnarray}{F}_{{ij}}(\alpha ,\beta )=\displaystyle \sum _{s=1}^{m}{b}_{i\alpha ,s}^{\dagger }{b}_{j\beta ,s}^{\dagger },\end{eqnarray}$
$\begin{eqnarray}{G}_{{ij}}(\alpha ,\beta )=\displaystyle \sum _{s=1}^{m}{b}_{i\alpha ,s}{b}_{j\beta ,s},\end{eqnarray}$
$\begin{eqnarray}{A}_{{ij}}(\alpha ,\beta )=\displaystyle \frac{1}{2}\displaystyle \sum _{s=1}^{m}({b}_{i\alpha ,s}^{\dagger }{b}_{j\beta ,s}+{b}_{j\beta ,s}{b}_{i\alpha ,s}^{\dagger }),\end{eqnarray}$
are expressed as bilinear combinations of the standard creation and annihilation operators of harmonic oscillator quanta $\begin{eqnarray}\begin{array}{rcl}{b}_{i\alpha ,s}^{\dagger } & = & \sqrt{\displaystyle \frac{{M}_{\alpha }\omega }{2{\hslash }}}\left({x}_{{is}}(\alpha )-\displaystyle \frac{i}{{M}_{\alpha }\omega }{p}_{{is}}(\alpha )\right),\\ {b}_{i\alpha ,s} & = & \sqrt{\displaystyle \frac{{M}_{\alpha }\omega }{2{\hslash }}}\left({x}_{{is}}(\alpha )+\displaystyle \frac{i}{{M}_{\alpha }\omega }{p}_{{is}}(\alpha )\right).\end{array}\end{eqnarray}$
In the last expressions, xis(α) and pis(α) denote the coordinates and corresponding momenta of the translationally invariant relative Jacobi vectors of the m-quasiparticle two-component nuclear system and A is the number of protons and neutrons. The range of indices is as follows: i, j = 1, 2, 3; α, β = p, n and s = 1,…,m = A − 1.The microscopic shell-model version of the BM model is defined by the following dynamical symmetry chain [16, 17]:
$\begin{eqnarray}\begin{array}{l}{Sp}(12,R)\supset {SU}(1,1)\otimes {SO}(6)\\ \qquad \langle \sigma \rangle \quad \qquad \quad {\lambda }_{\upsilon }\qquad \quad \upsilon \\ \qquad \supset U(1)\otimes {{SU}}_{{pn}}(3)\otimes {SO}(2)\supset {SO}(3),\\ \,p\qquad \ (\lambda ,\mu )\qquad \quad \nu \quad \ q\quad \ L\end{array}\end{eqnarray}$
which represents a PNSM shell-model coupling scheme. The labels under the different groups stand for their irreducible representations. $\langle \sigma \rangle \equiv \langle {\sigma }_{1}+\tfrac{m}{2},\ldots ,{\sigma }_{6}+\tfrac{m}{2}\rangle $ labels the Sp(12, R) irreducible representation or the symplectic bandhead intrinsic structure. The SU(1, 1) ⊃ U(1) quantum numbers λ$\upsilon$ = $\upsilon$ + 6/2 and p are related to the U(6) ⊃ SO(6) quantum numbers E = [E, 0,…,0]6 and $\upsilon \equiv {\left(\upsilon \right)}_{6}=(\upsilon ,0,0)$ via the expression p = (E − $\upsilon$)/2, and, hence, define the oscillator shell structure. The quantum numbers (λ, μ), ν, and L stand for the irreducible representations of the groups SUpn(3), SO(2) and SO(3), respectively. Finally, q is a multiplicity index for the reduction SUpn(3) ⊃ SO(3).According to the chain (5 ), the combined monopole–quadrupole nuclear dynamics splits into radial and orbital motions and the wave functions can be represented in the form [16]:1 )−(3 ) by contraction with respect to both indices i and α. The orbital motion group SO(6) can be expressed through the U(6) generators ALM(α, β), equation (3 ), in a standard way by taking their antisymmetric combination [16]:5 ) are given by the following operators:11 )−(12 ) and equation (13 ), the two sets of operators are irreducible tensors of different rank with respect to the group SO(3). The two groups SUpn(3) and SO(2), therefore, are mutually complementary [18] within the fully symmetric SO(6) irreps $\upsilon \equiv {\left(\upsilon ,\mathrm{0,0}\right)}_{6}$. The SUpn(3) irrep labels (λ, μ) in this case are in one-to-one correspondence with the SO(6) and SO(2) quantum numbers $\upsilon$ and ν, given by the following expression [16]:
$\begin{eqnarray}{{\rm{\Psi }}}_{{\lambda }_{\upsilon }p;\upsilon \nu {qLM}}(r,{{\rm{\Omega }}}_{5})={R}_{p}^{{\lambda }_{\upsilon }}(r){Y}_{\nu {qLM}}^{\upsilon }({{\rm{\Omega }}}_{5}).\end{eqnarray}$
For more details concerning the structure of these function we refer the readers to [17]. The radial SU(1, 1) Lie algebra is generated by the shell-model operators [16]: $\begin{eqnarray}{S}_{+}^{({\lambda }_{\upsilon })}=\displaystyle \frac{1}{2}\displaystyle \sum _{\alpha }{F}^{0}(\alpha ,\alpha ),\end{eqnarray}$
$\begin{eqnarray}{S}_{-}^{({\lambda }_{\upsilon })}=\displaystyle \frac{1}{2}\displaystyle \sum _{\alpha }{G}^{0}(\alpha ,\alpha ),\end{eqnarray}$
$\begin{eqnarray}{S}_{0}^{({\lambda }_{\upsilon })}=\displaystyle \frac{1}{2}\displaystyle \sum _{\alpha }{A}^{0}(\alpha ,\alpha ),\end{eqnarray}$
which are obtained from equations ( $\begin{eqnarray}{{\rm{\Lambda }}}^{{LM}}(\alpha ,\beta )={A}^{{LM}}(\alpha ,\beta )-{\left(-1\right)}^{L}{A}^{{LM}}(\beta ,\alpha ).\end{eqnarray}$
The generators of different SO(6) subgroups along the chain ( $\begin{eqnarray}{\widetilde{q}}^{2M}=\sqrt{3}i[{A}^{2M}(p,n)-{A}^{2M}(n,p)],\end{eqnarray}$
$\begin{eqnarray}{L}^{1M}=\sqrt{2}[{A}^{1M}(p,p)+{A}^{1M}(n,n)],\end{eqnarray}$
and $\begin{eqnarray}M=-\sqrt{3}{{\rm{\Lambda }}}^{0}(\alpha ,\beta )=-i\sqrt{3}[{A}^{0}(\alpha ,\beta )-{A}^{0}(\beta ,\alpha )],\end{eqnarray}$
which generate the SUpn(3) and SO(2) groups, respectively. As can be seen from equations ( $\begin{eqnarray}{\left(\upsilon \right)}_{6}=\mathop{\displaystyle \bigoplus }\limits_{\nu =\pm \upsilon ,\pm (\upsilon -2),\ldots ,0(\pm 1)}\left(\lambda =\displaystyle \frac{\upsilon +\nu }{2},\mu =\displaystyle \frac{\upsilon -\nu }{2}\right)\otimes {\left(\nu \right)}_{2}.\end{eqnarray}$
The reduction rules for SUpn(3) ⊃ SO(3) are given in terms of a multiplicity index q, which distinguishes the same L values in the SUpn(3) multiplet (λ, μ) [1]: $\begin{eqnarray}\begin{array}{rcl}q & = & \min (\lambda ,\mu ),\min (\lambda ,\mu )-2,\ldots ,0\,(1)\\ L & = & \max (\lambda ,\mu ),\max (\lambda ,\mu )-2,\ldots ,0\,(1);q=0\\ L & = & q,q+1,\ldots ,q+\max (\lambda ,\mu );q\ne 0.\end{array}\end{eqnarray}$
For our present purposes, however, it is more convenient to use the equivalent [16, 19] dynamical chain
$\begin{eqnarray}{Sp}(12,R)\supset U(6)\supset {SO}(6)\supset {{SU}}_{{pn}}(3)\otimes {SO}(2)\supset {SO}(3)\end{eqnarray}$
to classify the many-particle shell-model states of the nucleus. The branching rules for the reduction U(6) ⊃ SO(6) in the case of fully symmetric representations [E]6 of U(6) are given by [20]: $\begin{eqnarray}{[E]}_{6}=\mathop{\displaystyle \bigoplus }\limits_{\upsilon =E,E-2,\ldots ,0(1)}{\left(\upsilon ,\mathrm{0,0}\right)}_{6}=\underset{i=0}{\overset{\left\langle \tfrac{E}{2}\right\rangle }{\displaystyle \bigoplus }}{\left(E-2i\right)}_{6},\end{eqnarray}$
where ⟨E/2⟩ = E/2 if E is even and (E − 1)/2 if E is odd. From the latter expression, we see that only fully symmetric ${\left(\upsilon ,\mathrm{0,0}\right)}_{6}\equiv {\left(\upsilon \right)}_{6}$ irreps of SO(6) appear. For non-symmetric U(6) irreducible representations one can use, for example, the SCHUR computer program [21] to obtain the corresponding SO(6) subrepresentations.A simple model Hamiltonian that can be used in practical applications can be written in the form:5 ), including the harmonic oscillator Hamiltonian H0 = p ℏω. The mixing Hamiltonians Hhmix and Hvmix within the fully algebraic approach can be expressed by means of the Sp(12, R) generators and mix different SUpn(3) multiplets belonging to the same (horizontal mixing) or different (vertical mixing) SO(6) irreducible representations, the latter being from different U(6) harmonic oscillator shells, respectively.
$\begin{eqnarray}H={H}_{{DS}}+{H}_{{h}{\rm{mix}}}+{H}_{{v}{\rm{mix}}},\end{eqnarray}$
where the dynamical symmetry Hamiltonian HDS consists of the Casimir operators of various groups in the chain (3. Shell-model representations
The symplectic basis for an irreducible representation $\langle \sigma \rangle \equiv \langle {\sigma }_{1}+\tfrac{m}{2},\ldots ,{\sigma }_{6}+\tfrac{m}{2}\rangle $ of the group Sp(12, R) is constructed by the symplectic raising operators, equation (1 ), acting on the Sp(12, R) lowest-weight state ∣σ⟩. This can be symbolically represented in the following form [15]:
$\begin{eqnarray}| {\rm{\Psi }}(\sigma n\rho E\eta )\rangle ={\left[{P}^{(n)}(F)\times | \sigma \rangle \right]}_{\eta }^{\rho E},\end{eqnarray}$
where ${P}^{(n)}(F)={\left[F\times \ldots \times F\right]}^{(n)}$ and n = [n1,…,n6] is a partition with even integer parts. E = [E1,…,E6] indicates the U(6) quantum numbers of the coupled state, η labels a basis of states for the coupled U(6) irrep E and ρ is a multiplicity index. In this way, we obtain a basis of Sp(12, R) states that reduces the subgroup chain Sp(12, R) ⊃ U(6). In addition, in our practical applications we usually restrict the model space only to the fully symmetric U(6) irreps $E={[{E}_{1}\equiv E,0,\ldots ,0]\equiv [E]}_{6}$.The symplectic basis states are further classified by the remaining groups in the chain (16 ). This means that the symplectic states are characterized by their irreducible representations, i.e. we fix the basis index η = $\upsilon$νqLM in equation (19 ). But using the relation (14 ), one alternatively obtains for the basis index η = $\upsilon$(λ, μ)qLM. The latter choice is more convenient for the analysis of the SU(3) content of the shell-model representations of Sp(12, R). We note also that the lowest-weight state of Sp(12, R) is simultaneously a highest-weight state for the U(6) irreducible representation σ ≡ [σ1,…,σ6]. Such a Sp(12, R) lowest-weight but U(6) highest-weight state is sometimes referred to as a lowest-grade U(6) state. For this intrinsic U(6) structure we will simply use the term symplectic bandhead or Sp(12, R) bandhead.
To understand better the type and the structure of shell-model representations of the Sp(12, R) basis states that are classified either by the dynamical chains (5 ) or (16 ), we will consider the relevant representations for three nuclei with varying collective properties and belonging to different mass regions. First, we consider the relevant Sp(12, R) shell-model irreducible representation for the light nucleus 20Ne.
3.1. Shell-model representations of 20Ne
It is well known that possible SU(3) states in the nuclear shell model are obtained by taking all possible distributions of protons and neutrons within the considered valence shells. The set of Pauli-allowed states within a given three-dimensional oscillator shell ${ \mathcal N }$ can be obtained by the so-called plethysm operation, according to which the set of the SU(3) shell-model states are defined by the reduction chain U(d) ⊃ SU(3), where $d=\tfrac{1}{2}({ \mathcal N }+1)({ \mathcal N }+2)$ for each nuclear shell ${ \mathcal N }$. Computer codes [22, 23] exist for the evaluation of the SU(3) irreps contained in U(d). For the case of a two-component nuclear system, one should first consider Up(d) ⊃ SUp(3) and Un(d) ⊃ SUn(3) with the consequent coupling of the proton and neutron SUα(3) (α = p, n) multiplets, i.e. (λp, μp) ⨂ (λn, μn), to the SU(3) irreducible representation (λ, μ) of the combined proton–neutron nuclear system. Generally, we have many possible proton–neutron SU(3) multiplets, i.e. (λp, μp) ⨂ (λn, μn) = ∑(λ, μ).
Filling pairwise the levels of the three-dimensional harmonic oscillator by protons and neutrons separately at the experimentally observed quadrupole deformation, starting from bottom, we obtain completely filled s and p shells, plus two protons and two neutrons in the sd shell. That is, we obtain the same many-particle configuration ${\left(0\right)}^{2}{\left(1\right)}^{6}{\left(2\right)}^{2}$ for the proton and neutron subsystem. Then, using the codes [22, 23], for 20Ne one readily obtains the following SU(3) irreducible representations for the proton (neutron) subsystem: (4, 0) and (0, 2). The Pauli-allowed SU(3) multiplets for the combined proton–neutron nuclear system are obtained by the direct products of these two irreps, i.e.: (a) (4, 0) ⨂ (4, 0) = (8, 0), (6, 1), (4, 2), (2, 3), (0, 4); (b) (4, 0) ⨂ (0, 2) = (4, 2), (3, 1), (2, 0); and (c) (0, 2) ⨂ (0, 2) = (0, 4), (1, 2), (2, 0). Each one of these SU(3) multiplets, for example, can serve as an intrinsic SU(3) structure for the construction of an Sp(6, R) shell-model representation. The many-particle Hilbert space for 20Ne, therefore, can be represented as a direct sum of different Sp(6, R) shell-model irreducible representations, including—beyond the 0p-0h representations built on the sd valence shell SU(3) multiplets just obtained—also the excited Sp(6, R) representations by taking all possible distributions of the protons and neutrons over the higher major shells. Usually, the leading SU(3) representation is used, which is obtained by coupling the leading, i.e. most deformed, proton and neutron representations. Thus, for 20Ne, one obtains the leading proton–neutron (8, 0) multiplet. The irreducible collective space within the Sp(6, R) model, built upon this (8, 0) multiplet, is given in table 1 as an example.
Table 1. Irreducible collective space 0p-0h (8, 0) of Sp(6, R), relevant to 20Ne. |
| ⋯ | ⋯ |
|---|---|
| N0 + 4 | (12, 0), (10, 1), 2(8, 2), (6, 3), (7, 1), (4, 4), (6, 0) |
| N0 + 2 | (10, 0), (8, 1), (6, 2) |
| N0 | (8, 0) |
Alternatively, one can use the supermultiplet spin–isospin scheme to obtain the Pauli-allowed SU(3) states. Thus, filling each level by four nucleons, one obtains the many-particle configuration: ${\left(0\right)}^{4}{\left(1\right)}^{12}{\left(2\right)}^{4}$. For four nucleons in the ${ \mathcal N }=2$ sd shell, the codes [22, 23] produce: (8, 0), (4, 2), (0, 4), (2, 0). We see that the odd SU(3) irreps obtained in the proton–neutron scheme are now missing. The even SU(3) irreps are the same.
The relevant irreducible collective space for 20Ne, spanned by the Sp(12, R) irreducible representation 0p-0h [8]6 (or using an equivalent notation, ⟨σ⟩ = ⟨10 + 19/2, 2 + 19/2,…,2 + 19/2⟩) that is restricted only to the fully symmetric U(6) irreps and which basis states are classified by the chain (16 ), is given in table 2. This Sp(12, R) representation is defined by the intrinsic U(6) structure [10, 2, 2, 2, 2, 2]6≡[8]6, which in turn is fixed by the leading SU(3) irrep (8, 0). From the table, the structure of the symplectic basis become evident. Some points are of importance here. First, the collective potential that can be expressed along the chain (16 ) as a function of the second- and third-order Casimir operators of SUpn(3) will organize the space of SU(3) irreps according to their deformation. That is, the lowest in energy will be the SUpn(3) multiplet (8, 0) from the maximal seniority SO(6) irrep $\upsilon$0 = 8 of the symplectic bandhead. We note that, in the absence of the third-order SUpn(3) Casimir operator that distinguishes between the prolate and oblate shapes, the same energy will be obtained for the conjugate multiplet (0, 8) from the SO(6) irrep $\upsilon$0 = 8. The other SUpn(3) multiplets belonging to the SO(6) irrep $\upsilon$0 = 8 will be higher in energy, followed by the SUpn(3) multiplets for other SO(6) irreps with $\upsilon$ < $\upsilon$0 belonging to the lowest-grade U(6) irrep [8]6 characterized also by N0. For the other U(6) shells the situation will be similar. Note that the different U(6) shells are separated by the harmonic oscillator energy ℏω = 41A−1/3 and the n-th excited shell will have an energy nℏω. Second, the horizontal set of SU(3) irreducible representations of, for example, the row defined by N0, at first sight looks different compared to that obtained by the plethysm operation via the reduction U(d) ⊃ SU(3) [22, 23] and given above. That is, the overlap of the two sets of SU(3) irreps looks partial. This is because the many-particle configurations in the PNSM are classified by the basis states of the six-dimensional harmonic oscillator rather than the standard three-dimensional one. But the SU(3) states contained in the U(6) group structure can be organized in different ways since different choices for the group G in the reduction U(6) ⊃ G ⊃ SU(3) are possible. Then each shell in the present approach is determined by the corresponding U(6) representation, which in turn contains different seniority SO(6) irreducible representations $\upsilon$ or subshells (see table 2). Consider first the SU(3) irreps belonging to the maximal seniority SO(6) irrep $\upsilon$0 = 8 of the Sp(12, R) bandhead structure N0, which is of particular interest in the practical application of the microscopic shell-model version of the BM model. We will show now that the horizontal set of the remaining SU(3) irreps, which are placed to the right from the axially symmetric multiplet (λ = 8, 0), the latter being in the most left position, actually represent many-particle-many-hole (mp-mh) excitations of the nuclear system.
Table 2. Irreducible collective space 0p-0h [8]6 of Sp(12, R), relevant to 20Ne, which SUpn(3) basis states are classified according to chain ( |
| N | $\upsilon$\ν | ⋯ | 10 | 8 | 6 | 4 | 2 | 0 | −2 | − 4 | − 6 | − 8 | − 10 | ⋯ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ⋮ | ⋮ | ⋱ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋰ |
| | 10 | (10,0) | (9, 1) | (8, 2) | (7, 3) | (6, 4) | (5, 5) | (4, 6) | (3, 7) | (2, 8) | (1, 9) | (0, 10) | ||
| 8 | (8, 0) | (7, 1) | (6, 2) | (5, 3) | (4, 4) | (3, 5) | (2, 6) | (1, 7) | (0, 8) | |||||
| N0 + 2 | 6 | (6, 0) | (5, 1) | (4, 2) | (3, 3) | (2, 4) | (1, 5) | (0, 6) | ||||||
| 4 | (4,0) | (3, 1) | (2, 2) | (1, 3) | (0, 4) | |||||||||
| 2 | (2, 0) | (2, 0) | (1, 1) | (0, 2) | ||||||||||
| 0 | . | . | . | . | . | . | (0, 0) | . | . | . | . | . | ||
| 8 | (8, 0) | (7, 1) | (6, 2) | (5, 3) | (4, 4) | (3, 5) | (2, 6) | (1, 7) | (0, 8) | |||||
| 6 | (6, 0) | (5, 1) | (4, 2) | (3, 3) | (2, 4) | (1, 5) | (0, 6) | |||||||
| N0 | 4 | (4, 0) | (3, 1) | (2, 2) | (1, 3) | (0, 4) | ||||||||
| 2 | (2, 0) | (1, 1) | (0, 2) | |||||||||||
| 0 | . | . | . | . | (0, 0) | . | . | . | . | |||||
We recall that the raising symplectic generators Flm(α, β) transform according to the U(6) irreducible representation [2]6. According to equation (17 ), [2]6 decomposes to the SO(6) irreps ${\left(2\right)}_{6}$ and ${\left(0\right)}_{6}$, respectively. Further, using equation (14 ), we find the SUpn(3) content for each of these two SO(6) irreps: (a) ${\left(0\right)}_{6}\downarrow (0,0);$ (b) ${\left(2\right)}_{6}\downarrow (2,0),(1,1),(0,2)$. These results actually coincide with the last two subrows with $\upsilon$ = 2 and 0 of table 2. The symplectic lowering generators Glm(α, β) transform according to the conjugate U(6) representation ${\left[2\right]}_{6}^{* }={[-2]}_{6}\,\equiv {[222220]}_{6}$, which decomposes to the same SO(6) irreps ${\left(2\right)}_{6}$ and ${\left(0\right)}_{6}$. In turn, we obtain the same SUpn(3) content for the lowering symplectic generators, as given above for the raising operators. The product of the lowering and the raising symplectic generators will transform according to the direct product [−2]6⨂[2]6 of the corresponding U(6) representations, producing the set: ${\left[2,-2\right]}_{6}^{* }$, ${\left[1,-1\right]}_{6}^{* }$ and ${\left[0,-0\right]}_{6}^{* }$. Then it is easy to show that acting on the SUpn(3) multiplet (8, 0) by the tensor operator ${G}^{2}(a,a)\cdot {F}^{2}(b,b)=\tfrac{2}{3}\sqrt{5}{\left[{G}^{2}(a,a)\times {F}^{2}(b,b)\right]}_{-4100}^{4}$, classified by the whole chain (16 ), we obtain the SUpn(3) multiplet (6, 2). The operators a's and b's are defined as a linear combination of the proton and neutron creation or annihilations operators. In particular, ${a}_{j}^{\dagger }=\tfrac{1}{\sqrt{2}}\left(-{{iB}}_{j}^{\dagger }(p)+{B}_{j}^{\dagger }(n)\right)$ and ${b}_{j}^{\dagger }=\tfrac{1}{\sqrt{2}}\left({{iB}}_{j}^{\dagger }(p)+{B}_{j}^{\dagger }(n)\right)$ [19], which transform as (1, 0) and (0, 1) SU(3) tensors, respectively, where the following notation is also used ${B}_{i}^{\dagger }(\alpha )={\sum }_{s}{b}_{i\alpha ,s}^{\dagger }$. Their conjugate operators aj and bj transform as (0, 1) and (1, 0) SU(3) tensors, respectively. By repeated action with the same operator ${G}^{2}(a,a)\cdot {F}^{2}(b,b)\,=\tfrac{2}{3}\sqrt{5}{\left[{G}^{2}(a,a)\times {F}^{2}(b,b)\right]}_{-4100}^{4}$, one can obtain the remaining even SUpn(3) multiplets belonging to the SO(6) irrep $\upsilon$0 = 8. In this way, the operator ${G}^{2}(a,a)\cdot {F}^{2}(b,b)\,\,=\tfrac{2}{3}\sqrt{5}{\left[{G}^{2}(a,a)\times {F}^{2}(b,b)\right]}_{-4100}^{4}$ can be interpreted as a 2p-2h-like operator of the core excitations that creates two oscillator quanta in the shell above and annihilates two oscillator quanta in the shell below, i.e. it promotes two oscillator quanta up. For instance, the SUpn(3) multiplet (6, 2) within the SO(6) irrep $\upsilon$0 = 8 of the symplectic Sp(12, R) bandhead, defined by N0 oscillator quanta, is obtained by promoting two oscillator quanta from the shell ${ \mathcal N }=1$ to ${ \mathcal N }=2$, i.e. changing the many-particle shell-model configuration ${\left(0\right)}^{4}{\left(1\right)}^{12}{\left(2\right)}^{4}$ to ${\left(0\right)}^{4}{\left(1\right)}^{10}{\left(2\right)}^{6}$, the latter producing the excited SUpn(3) irrep (6, 2) from (8, 0) of the former configuration. The odd SUpn(3) multiplet (7, 1) can be obtained in a similar manner from (8, 0) by acting with an 1p-1h-like operator, i.e. by the U(6) operator ${\left[{A}^{0}(b,a)\right]}_{-2100}^{2}$. Note that the latter operator together with ${G}^{2}(a,a)\cdot {F}^{2}(b,b)\,=\tfrac{2}{3}\sqrt{5}{\left[{G}^{2}(a,a)\times {F}^{2}(b,b)\right]}_{-4100}^{4}$ preserve the number of U(6) harmonic oscillator quanta N of each shell.
Thus, we have seen that the SU(3) many-particle shell-model configurations are organized in a different way by means of the group SO(6) through the reduction U(6) ⊃ SO(6) ⊃ SU(3) (more precisely, Sp(12, R) ⊃ U(6) ⊃ SO(6) ⊃ SUpn(3) ⨂ SO(2) for different U(6) shells), compared to the standard shell-model plethysm reduction U(d) ⊃ SU(3). Each horizontal subset of the SU(3) multiplets is now characterized by the same value of the SO(6) irrep $\upsilon$ = λ + μ. In this regard, we want to point out that the SU(3) content of the U(6) shells defined by the PNSM dynamical chain Sp(12, R) ⊃ U(6) ⊃ SUp(3) ⨂ SUn(3) ⊃ SU(3) considered, e.g. in [14, 15], will coincide precisely with that generated first by the separate reductions Uα(d) ⊃ SUα(3) (α = p, n) with the subsequent coupling of the proton (λp, μp) and neutron (λn, μn) subsystem representations to the combined proton–neutron SU(3) irreducible representation (λ, μ), since in this case the PNSM many-particle SU(3) configurations are organized by means of the group structure SUp(3) ⨂ SUn(3) ⊃ SU(3) within the U(6) harmonic oscillator shell. For 20Ne, the U(6) irrep [8]6 according to the underlying algebraic structure SUp(3) ⨂ SUn(3) ⊃ SU(3) produces the three sets: (a) (4, 0) ⨂ (4, 0) = (8, 0), (6, 1), (4, 2), (2, 3), (0, 4); (b) (4, 0) ⨂ (0, 2) = (4, 2), (3, 1), (2, 0); and (c) (0, 2) ⨂ (0, 2) = (0, 4), (1, 2), (2, 0). The latter are exactly those obtained by means of the plethysm operation given earlier.
The SUpn(3) states belonging to the SO(6) representations with $\upsilon$ > $\upsilon$0 within the irreducible collective space of Sp(12, R) can be obtained from those belonging to the SO(6) irrep $\upsilon$0 by using the raising symplectic generators Flm(δ, τ) (δ, τ = a, b). In particular, the SUpn(3) states with $\upsilon$ = $\upsilon$0 + 2 belonging to the next U(6) shell are readily obtained by acting on the SUpn(3) states contained in the SO(6) representation $\upsilon$0 with the raising generators F2m(a, a), F2m(a, b), and F2m(b, b), which transform as (2, 0), (1, 1) and (0, 2) SU(3) tensors, respectively. The construction of the remaining SU(3) basis states of the irreducible collective space of Sp(12, R) then becomes straightforward.
In this way, we have shown that the SUpn(3) states within the SO(6) irrep $\upsilon$0 are Pauli allowed. Then the states generated from them and belonging to $\upsilon$ > $\upsilon$0 are also Pauli allowed. The states with $\upsilon$ < $\upsilon$0 are not particularly interesting, but we will make a comment concerning them. First, note that each of the SU(3) multiplets in the U(d) ⊃ SU(3) set (8, 0), (4, 2), (0, 4), (2, 0) is contained, respectively, in the SO(6) representation with $\upsilon$ = 8, 6, 4 and 2 of the U(6) irrep [8]6, given in table 2 by the row with N0. The state (0, 0), for this specific example of 20Ne, is missing in the U(d) ⊃ SU(3) set. Thus it is Pauli forbidden. The SUpn(3) states, which are placed to the right of the Pauli-allowed multiplets (8, 0), (4, 2), (2, 0) in each row that is defined by the corresponding SO(6) irrep with $\upsilon$ < $\upsilon$0, are Pauli allowed, since they also can be represented as multi-particle-multi-hole excitations built upon the (8, 0), (4, 2) or (2, 0), correspondingly. The SUpn(3) states on the left from these multiplets are Pauli forbidden, since they correspond to promoting two oscillator quanta down to filled shells.
Up to now, nothing has been said about the spin content. In this respect, we recall that the proper permutational symmetry in the PNSM is ensured by the reduction
$\begin{eqnarray}O(m)\supset {S}_{A.}\end{eqnarray}$
of the complementary group O(m) in the reduction of the many-particle dynamical group Sp(12m, R) of the whole system, i.e. Sp(12m, R) ⊃ Sp(12, R) ⨂ O(m) [14, 15]. The irrep ω of O(m) is determined by the irrep ⟨σ⟩ of Sp(12, R) and vice versa. But, since the antisymmetry should be satisfied separately for protons and neutrons, in order to ensure the proper permutational symmetry, we consider further the reduction [14, 15]: $\begin{eqnarray}\begin{array}{l}O(m)\quad \supset \quad {S}_{A}\quad \supset \quad {S}_{Z}\quad \otimes \quad {S}_{N},\\ \quad \omega \qquad \delta \quad \ [f]\quad {\delta }_{0}\quad [{f}_{p}]\qquad \ \ [{f}_{n}]\end{array}\end{eqnarray}$
where the quantum numbers below different groups stand for their irreducible representations, and δ and δ0 are multiplicity indices. Due to the overall antisymmetry, the spin wave functions for the proton and neutron subsystems are characterized by the conjugate representations $[{\widetilde{f}}_{p}]$ and $[{\widetilde{f}}_{n}]$ [14, 15], respectively. In the standard shell model, using the proton–neutron formalism, the spin content is obtained by considering the reduction ${U}_{\alpha }(2d)\supset {U}_{\alpha }(d)\otimes {U}_{{S}_{\alpha }}(2)$ (α = p, n). For example, in the case of 20Ne, the maximal spatial symmetry of two protons in the sd shell is provided by the Up(6) irrep [fp] = [2], which in addition must be compatible with the permutational symmetry of the whole proton subsystem, i.e. [fZ] = [f0][f1][f2] ≡ [25] with fi related to a single-shell configuration ${\left(i\right)}^{{p}_{i}}$. Then, the conjugate proton spin symmetry is $[{\widetilde{f}}_{p}]=[11]$, from which one obtains ${S}_{p}=\displaystyle \frac{1}{2}({\widetilde{f}}_{1p}-{\widetilde{f}}_{2p})=0$. Similar considerations are valid for the neutron subsystem, i.e. Sn = 0. Hence the total spin is also zero. In the PNSM, the O(19) irrep of 20Ne is determined by the Sp(12, R) bandhead σ = [10, 2, 2, 2, 2, 2]6, i.e. ω = (10, 2, 2, 2, 2, 2). Using the computer program [21] and the Pauli-allowed spatial symmetries of the type [444…422…2] for even–even nuclei, one sees that the O(19) irrep (10, 2, 2, 2, 2, 2) reduces to the maximal space symmetry S20 irrep [444422], which in turn reduces to the S10 ⨂ S10 irrep [25][25]. Then, from the proton spatial symmetry [fp] = [25], one obtains the conjugate spin symmetry $[{\widetilde{f}}_{p}]=[55]$, from which it follows that Sp = 0. Similarly, one obtains Sn = 0, and hence, S = 0. In this way, similarly to the standard shell model, within the PNSM the spatial symmetry is also accompanied by the corresponding conjugate spin symmetry. We will not consider further the spin content, which can be recovered when this is required. We note too that because the Sp(12, R) generators are O(m)-scalar operators, they are also SA-scalar operators, and, therefore, they preserve the permutational symmetry.We recall that to account for the four Pauli-allowed SU(3) multiplets (8, 0), (4, 2), (0, 2), (2, 0) of the shell ${ \mathcal N }=2$, one needs to consider the direct sum of these four irreducible collective many-particle subspaces in the Elliott SU(3) or Sp(6, R) shell models. In the microscopic shell-model version of the BM model, they result in a single Sp(12, R) irrep. The Pauli-allowed SUpn(3) states with $\upsilon$ ≥ $\upsilon$0 of the Sp(12, R) irreducible collective space 0p-0h [8]6 were used in [24] for the description of the ground and first two beta bands in 20Ne by using both vertical and horizontal mixing interactions.
3.2. Shell-model representations of 106Cd
As a second example, we consider the weakly deformed nucleus 106Cd. For heavy nuclei we use the pseudo-SU(3) scheme [25–27]. Filling pairwise the pseudo-Nilsson levels with protons at observed quadrupole deformation β ≈ 0.17 [28], one obtains a completely filled $\widetilde{{ \mathcal N }}=2$ pseudo-shell $({\widetilde{d}}_{5/2},{\widetilde{d}}_{3/2},{\widetilde{s}}_{1/2})$ plus eight protons in the unique-parity level g9/2. We notice that the filling at the experimentally observed quadrupole deformation determines the number of particles on the normal- and abnormal-parity levels, respectively. Recall also that the Nilsson levels of the real-SU(3) scheme map into the corresponding pseudo-SU(3) scheme levels that have one less quanta, i.e. $\widetilde{{ \mathcal N }}={ \mathcal N }-1$. The relevant Nilsson levels for different oscillator shells with ${ \mathcal N }\geqslant 4$ (or $\widetilde{{ \mathcal N }}\geqslant 3$) are given, for example, in [27]. The leading proton SUp(3) irrep is the scalar irrep (0, 0), since the particles in the unique-parity levels in the pseudo-SU(3) scheme are considered in a seniority zero configuration. Similarly, for neutrons one obtains a completely filled $\widetilde{{ \mathcal N }}=2$ pseudo-shell plus six (or eight) neutrons occupying the ${\widetilde{f}}_{7/2}$ and ${\widetilde{f}}_{5/2}$ levels of the $\widetilde{{ \mathcal N }}=3$ pseudo-shell and two (or zero) neutrons in the unique-parity level h11/2. The two alternative choices for neutron occupancies appear because at β ≈ 0.17 the two relevant pseudo-Nilsson levels cross. Using the codes [22, 23] and allowing the distribution of the normal-parity neutrons over the pseudo $\widetilde{{ \mathcal N }}=3$ shell levels (${\widetilde{f}}_{7/2},{\widetilde{f}}_{5/2},{\widetilde{p}}_{3/2},{\widetilde{p}}_{1/2}$), one obtains the set of Pauli-allowed SU(3) states: (12, 0), (9, 3), (6, 6), (7, 4), (8, 2), … or (10, 4), (12, 0), (8, 5), (9, 3), (10, 1), (5, 8), (6, 6), (7, 4), (8, 2), …, considering six or eight active neutrons, respectively. Further, the proton and neutron irreps should be coupled to obtain the combined proton–neutron SUpn(3) representation of the whole nuclear system. Since for the proton subsystem, only the scalar representation (0, 0) is admitted, the set of combined proton–neutron multiplets coincides with that of the neutron subsystem since (λp, μp) ⨂ (λn, μn) = (0, 0) ⨂ (λn, μn) ≡ (λ, μ).
Alternatively, using the supermultiplet scheme, one readily obtains that six nucleons fill the last valence $\widetilde{{ \mathcal N }}=3$ pseudo-shell. Again, using the codes [22, 23], one gets the set of SU(3) states:16 ), is given in table 3. By comparing the set (22 ) with the SUpn(3) multiplets of the symplectic bandhead structure for the SO(6) irreps $\upsilon$ < $\upsilon$0 = 12, given in table 3 at the row N0, one now sees, in contrast to the case of 20Ne, that there are many more Pauli-allowed states, including the scalar SUpn(3) irrep (0, 0). Recall that all SUpn(3) multiplets that are on the right from a giving $(\lambda ^{\prime} ,\mu ^{\prime} )$ multiplet at each $\upsilon$ < $\upsilon$0, which is matched with a certain SU(3) irrep of the set (22 ), are Pauli allowed. In practical applications, however, only the SUpn(3) states with $\upsilon$ ≥ $\upsilon$0 are used in the diagonalization of the model Hamiltonian. Hence, only the Pauli-allowed states are retained in the many-particle irreducible collective space of a given Sp(12, R) shell-model representation.
$\begin{eqnarray}\begin{array}{l}(14,2),(12,3),(13,1),(10,4),(11,2),(12,0),\ldots ,\\ (9,3),(6,6),\ldots ,(8,2),(5,5),(2,8),\ldots ,\\ (7,1),(4,4),(1,7),\ldots ,(6,0),(3,3),(0,6),\ldots ,\\ (2,2),(1,1),(0,0),\ldots \end{array}\end{eqnarray}$
Using the Nilsson model ideas [29–32], we choose the SU(3) irrep (12, 0), contained in the two alternatively obtained sets of Pauli-allowed many-particle SU(3) states, by means of which we fix the appropriate Sp(12, R) irreducible representation 0p-0h [12]6 (or ⟨σ⟩ = ⟨27 + m/2, 15 + m/2,…,15 + m/2⟩ using an equivalent notation), which turns out to be useful for the description of the low-energy quadrupole collectivity observed in 106Cd. The relevant irreducible collective space for 106Cd, spanned by the Sp(12, R) irreducible representation 0p-0h [12]6, restricted to the fully symmetric U(6) irreps only, and which SUpn(3) basis states are classified according to the chain (Table 3. Irreducible collective space 0p-0h [12]6 of Sp(12, R), relevant to 106Cd, which SUpn(3) basis states are classified according to chain ( |
| N | $\upsilon$\ν | ⋯ | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 0 | −2 | −4 | −6 | −8 | − 10 | −12 | −14 | ⋯ | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ⋮ | ⋮ | ⋱ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋰ | |
| 14 | (14, 0) | (13, 1) | (12, 2) | (11, 3) | (10, 4) | (9, 5) | (8, 6) | (7, 7) | (6, 8) | (5, 9) | (4, 10) | (3, 11) | (2, 12) | (1, 13) | (0, 14) | ||||
| 12 | (12, 0) | (11, 1) | (10, 2) | (9, 3) | (8, 4) | (7, 5) | (6, 6) | (5, 7) | (4, 8) | (3, 9) | (2, 10) | (1, 11) | (0, 12) | ||||||
| 10 | (10, 0) | (9, 1) | (8, 2) | (7, 3) | (6, 4) | (5, 5) | (4, 6) | (3, 7) | (2, 8) | (1, 9) | (0, 10) | ||||||||
| N0 + 2 | ⋮ | ⋱ | ⋮ | ⋮ | ⋮ | ⋰ | |||||||||||||
| 2 | (2, 0) | (1, 1) | (0, 2) | ||||||||||||||||
| 0 | (0, 0) | ||||||||||||||||||
| | |||||||||||||||||||
| 12 | (12, 0) | (11, 1) | (10, 2) | (9, 3) | (8, 4) | (7, 5) | (6, 6) | (5, 7) | (4, 8) | (3, 9) | (2, 10) | (1, 11) | (0, 12) | ||||||
| 10 | (10, 0) | (9, 1) | (8, 2) | (7, 3) | (6, 4) | (5, 5) | (4, 6) | (3, 7) | (2, 8) | (1, 9) | (0, 10) | ||||||||
| N0 | ⋮ | ⋱ | ⋮ | ⋮ | ⋮ | ⋰ | |||||||||||||
| 2 | (2, 0) | (1, 1) | (1, 1) | ||||||||||||||||
| 0 | (0, 0) | ||||||||||||||||||
3.3. Shell-model representations of 158Gd
As a final example, we consider the strongly deformed nucleus 158Gd. Similarly, using the pseudo-SU(3) scheme [25–27] one obtains the following many-particle configurations: (a) ${\left(\widetilde{2}\right)}^{20}{\left(\widetilde{3}\right)}^{8}$ plus six protons occupying the unique-parity level h11/2 and (b) ${\left(\widetilde{2}\right)}^{20}{\left(\widetilde{3}\right)}^{20}{\left(\widetilde{4}\right)}^{6}$ plus six neutrons occupying the unique-parity level i13/2. They are obtained by filling the pseudo-Nilsson levels at the experimentally observed quadrupole deformation β ≈ 0.35. The relevant normal-parity levels are (${\widetilde{f}}_{7/2},{\widetilde{f}}_{5/2},{\widetilde{p}}_{3/2},{\widetilde{p}}_{1/2}$) for protons, and (${\widetilde{g}}_{9/2},{\widetilde{g}}_{7/2},{\widetilde{d}}_{5/2},{\widetilde{d}}_{3/2},{\widetilde{s}}_{1/2}$) for neutrons, respectively. The codes [22, 23] produce the following two sets: (a) (10, 4), (12, 0), (8, 5), (9, 3),...; and (b) (18, 0), (15, 3), (12, 6), (13, 4), (14, 2), …. Coupling the leading SUp(3) and SUn(3) irreps, i.e. (10, 4) ⨂ (18, 0), one obtains the leading (most deformed) combined proton–neutron multiplet (28, 4). Alternatively, using the many-particle configuration ${\left(\widetilde{2}\right)}^{40}{\left(\widetilde{3}\right)}^{34}$ (plus 24 nucleons occupying unique-parity level h11/2) based on the pseudo-SU(3) and supermultiplet schemes, one obtains the following set of SU(3) states: (2, 14), (3, 12), (4, 10), (5, 8), (6, 6), (7, 4), (8, 2), (1, 13), (2, 11), (3, 9), …, which consists of predominantly oblate-like SU(3) multiplets with λ < μ. The most deformed prolate-like SU(3) irreducible representations are (7, 4) and (8, 2). The leading SU(3) multiplet is (2, 14), corresponding to an oblate-like shape of the combined proton–neutron nuclear system. Hence, the supermultiplet scheme of filling the pseudo-Nilsson levels at the observed quadrupole deformation is not appropriate for this nucleus. This is a well known result for nuclei, in which the valence protons and neutrons occupy different shells. Using again the Nilsson model ideas [29–32], we choose the axially symmetric prolate SU(3) representation (36, 0), which is now a linear combination of Slater determinants. The latter is obtained by coupling axially symmetric proton and neutron multiplets, i.e.16 ), is given in table 4. Similar considerations are also valid concerning the SUpn(3) multiplets belonging to the SO(6) irreps with $\upsilon$ < $\upsilon$0 for 158Gd, not all of which are Pauli permitted. For instance, those which are on the left from the SUpn(3) multiplets $(\lambda ^{\prime} ,\mu ^{\prime} )$, the latter matching the corresponding irreps from set (23 ), are not allowed for 36 < $\upsilon$ ≤ 18, as well as all SUpn(3) multiplets belonging to $\upsilon$ ≤ 16. The Pauli-allowed SUpn(3) multiplets contained in the maximal seniority SO(6) irrep $\upsilon$0 = 36 of the symplectic bandhead were used for studying the low-energy quadrupole dynamics in 158Gd [33] using a horizontal mixing interaction in the model Hamiltonian.
$\begin{eqnarray}\begin{array}{l}(18,0)\otimes (18,0)=(36,0),(34,1),(32,2),(30,3),\\ \quad \ \ldots ,(2,17),(0,18).\end{array}\end{eqnarray}$
The relevant irreducible collective space for 158Gd, spanned by the Sp(12, R) irreducible representation 0p-0h [36]6 restricted to the fully symmetric U(6) irreps only and which SUpn(3) basis states are classified according to the chain (Table 4. Irreducible collective space 0p-0h [36]6 of Sp(12, R), relevant to 158Gd, which SUpn(3) basis states are classified according to chain ( |
| N | $\upsilon$\ν | ⋯ | 38 | 36 | 34 | 32 | ⋯ | 4 | 2 | 0 | −2 | −4 | ⋯ | −32 | −34 | −36 | −38 | ⋯ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ⋮ | ⋮ | ⋱ | ⋮ | ⋮ | ⋮ | ⋮ | ⋯ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋯ | ⋮ | ⋮ | ⋮ | ⋮ | ⋰ |
| 38 | (38, 0) | (37, 1) | (36, 2) | (35, 3) | ⋯ | (21, 17) | (20, 18) | (19, 19) | (18, 20) | (17, 21) | ⋯ | (3, 35) | (2, 36) | (1, 37) | (0, 38) | |||
| 36 | (36, 0) | (35, 1) | (34, 2) | ⋯ | (20, 16) | (19, 17) | (18, 18) | (17, 19) | (16, 20) | ⋯ | (2, 34) | (1, 35) | (0, 36) | |||||
| N0 + 2 | 34 | (34, 0) | (33, 1) | ⋯ | (19, 15) | (18, 16) | (17, 17) | (16, 18) | (15, 19) | ⋯ | (1, 33) | (0, 34) | ||||||
| ⋮ | ⋱ | ⋮ | ⋮ | ⋮ | ⋰ | |||||||||||||
| 2 | (2, 0) | (1, 1) | (0, 2) | |||||||||||||||
| 0 | (0, 0) | |||||||||||||||||
| 36 | (36, 0) | (35, 1) | (34, 2) | ⋯ | (20, 16) | (19, 17) | (18, 18) | (17, 19) | (16, 20) | ⋯ | (2, 34) | (1, 35) | (0, 36) | |||||
| 34 | (34, 0) | (33, 1) | ⋯ | (19, 15) | (18, 16) | (17, 17) | (16, 18) | (15, 19) | ⋯ | (1, 33) | (0, 34) | |||||||
| N0 | ⋮ | ⋱ | ⋮ | ⋮ | ⋮ | ⋰ | ||||||||||||
| 2 | (2, 0) | (1, 1) | (0, 2) | |||||||||||||||
| 0 | (0, 0) |
In this way, we have demonstrated how the irreducible collective spaces of Sp(12, R), defined by the symplectic bandhead structure ⟨σ⟩ or equivalent to it O(A − 1) irrep ω, can be constructed according to the chain (5 ) (or (16 )). We note that the irreducible collective subspaces of the many-particle Hilbert space, in which the model Hamiltonians act, distinguish between the phenomenological and microscopic models of nuclear structure. In this respect, we recall that the original BM collective model for even–even nuclei has only one irreducible collective space—the scalar one, corresponding to the scalar Sp(12, R) irrep ⟨σ⟩ = ⟨0⟩. The latter is trivially obtained when the intrinsic lowest-weight state ∣σ⟩ of Sp(12, R) is simply reduced to the vacuum state ∣0⟩. This, in the microscopic version of the BM collective model, corresponds to the case of doubly closed shell nuclei, for which there are no low-lying vibrational and rotational degrees of freedom; only the irrotational-flow high-energy collective dynamics of the original BM-type survives. In contrast to the original BM model, its microscopic shell-model version contains many nonscalar ⟨σ⟩ ≠ ⟨0⟩ irreducible representations of Sp(12, R), each of which represents a Pauli-allowed irreducible collective subspace ${{\mathbb{H}}}^{({{\rm{\Lambda }}}_{0}\omega \ne (0){\rm{\Gamma }})}$ with $\omega \geqslant {\omega }_{{\min }}$ of the many-particle nuclear Hilbert space, containing both low- and high-energy vibrational plus rigid-flow and irrotational-flow degrees of freedom. The full collective Hilbert space of the nucleus can then be represented as a direct sum of O(A − 1) (or, equivalently Sp(12, R)) irreducible collective subspaces: ${{\mathbb{H}}}^{({{\rm{\Lambda }}}_{0})}={{\mathbb{H}}}^{({{\rm{\Lambda }}}_{0}\omega {\rm{\Gamma }})}\oplus {{\mathbb{H}}}^{({{\rm{\Lambda }}}_{0}\omega ^{\prime} {\rm{\Gamma }}^{\prime} )}\oplus \ldots $, where Λ0 denotes the set of experimentally observed exact integrals of motion and Γ stands for the set of remaining quantum numbers required to classify the many-particle shell-model states. It has been shown in [17] that the original BM collective model is not kinematically correct, since for A − 1 > 4 the scalar O(A − 1) irreps are not permitted and, strictly speaking, the standard many-nucleon quantum mechanics is not well-defined in the phenomenological Hilbert space ${{\mathbb{H}}}_{{phen}}={{\mathbb{H}}}^{({{\rm{\Lambda }}}_{0}\omega =(0){\rm{\Gamma }})}$ of the original BM model. In this way, in the phenomenological version of the BM model, the collective dynamics, governed by the intrinsic fermion structure and related to the basic low-energy vibrational and rotational degrees of freedom, is suppressed (identically vanishes).
Summarizing, the microscopic shell-model version, in contrast to the original version of the BM model, admits many (0p-0h, 1p-1h, 2p-2h, etc.) shell-model irreducible representations determined by the intrinsic symplectic Sp(12, R) bandhead structure ⟨σ⟩ ≠ ⟨0⟩, which strongly affects the proton–neutron rigid-flow quadrupole collectivity.
4. Conclusions
The structure of the irreducible collective space of the Sp(12, R) shell-model representations with respect to the SU(3) symmetry of the many-particle nuclear states, which are classified by the dynamical chain Sp(12, R) ⊃ U(6) ⊃SO(6) ⊃ SUpn(3) ⨂ SO(2) ⊃ SO(3) within the framework of the PNSM, is considered in detail. This chain has been shown to corresponds to a microscopic shell-model version of the BM model, obtained recently by embedding the original BM model into the two-component proton–neutron shell-model theory [16, 17]. The construction of the relevant shell-model representations of the Sp(12, R) dynamical group has been considered for three nuclei with varying collective properties belonging to different mass regions. A comparison with the standard consideration of the Pauli-allowed SU(3) states within the valence shells using the proton–neutron formalism, or within a single valence shell using the supermultiplet scheme, both exploiting the plethysm operation defined by the reduction U(d) ⊃ SU(3) with $d=\tfrac{1}{2}({ \mathcal N }+1)({ \mathcal N }+2)$ for any major shell ${ \mathcal N }$, is given. It was shown that, in the present proton–neutron shell-model approach, the SUpn(3) many-particle states are organized in a different way into different SO(6) subshells of the six-dimensional harmonic oscillator with a given SO(6) irreducible representation $\upsilon$ = λ + μ. This is in contrast to the case of filling the levels of the standard three-dimensional harmonic oscillator and using the plethysm operation. In this way, in contrast to the standard proton–neutron shell-model reduction ${U}_{\alpha }(2d)\,\supset [{U}_{\alpha }(d)\supset {{SU}}_{\alpha }(3)]$ $\otimes [{U}_{{S}_{\alpha }}(2)\supset {{SU}}_{{S}_{\alpha }}(2)]$, new structures and organization (coupling schemes) of the many-particle nuclear states appear within the framework of the PNSM with the direct-product dynamical group Sp(12, R) ⨂ O(m) ⊂ Sp(12m, R). In particular, the structure SU(1, 1) ⨂ SO(6) ⊂ Sp(12, R), missing in the other microscopic shell-model approaches, allows a microscopic counterpart [16, 17] of the BM collective model, with its exactly solvable limits, to be established that closely parallels the original construction. For example, the Sp(6, R) or the proton–neutron (real or pseudo) SUp(3) ⨂ SUn(3) shell models contain the SU(3) submodel, which obviously can be associated only with the rotational limit of the BM model. The Sp(6, R) (or its Sp(6, R) ⨂ SUT(2) and Sp(6, R) ⨂ SUST(4) extensions) and SUp(3) ⨂ SUn(3) shell models do not contain an O(5) or O(6) group, which can be associated with the important class of γ-unstable Wilets-Jean type submodels of the BM collective model [16]. In [17], it was shown that any generalized BM Hamiltonian of the type ${H}_{{BM}}=-\tfrac{{{\hslash }}^{2}}{2{\mathfrak{B}}}{{\rm{\nabla }}}_{{Bohr}}^{2}+V(r,\beta ,\gamma )$ immediately defines a microscopic shell-model Hamiltonian along the chain (5 ). The structure SUp(3) ⨂ SUn(3) ⊃ SU(3), which is also contained in the another limit of the PNSM, for instance corresponds to the microscopic shell-model counterpart of the coupled two-rotor model and is appropriate for studying the isovector, out-of-phase proton–neutron excitations in nuclear systems. In this way, different subgroups of Sp(12, R) enrich the dynamical content of possible collective motions in the two-component proton–neutron nuclear systems, whereas the O(m) group ensures the proper permutational symmetry.
Further, it was shown that the SUpn(3) states belonging to the SO(6) irreps $\upsilon$ ≥ $\upsilon$0 are always Pauli permitted, whereas for $\upsilon$ < $\upsilon$0 not all of the SUpn(3) multiplets are Pauli allowed. $\upsilon$0 denotes the maximal seniority SO(6) irreducible representation contained in the symplectic Sp(12, R) bandhead. The situation, in some respect, resembles that encountered in the algebraic cluster models (see, e.g. [34, 35]) based on the SU(3) dynamical group, for which the Pauli-allowed SU(3) states are restricted from below by the minimal number of oscillator quanta n0. All states with n < n0 are Pauli forbidden, and hence they are excluded from the many-particle subspaces of the Hilbert space for the cluster configuration under consideration. For the PNSM which states are classified by the chain (16 ), we also have a restriction of the many-particle states by the minimal Pauli-allowed number of harmonic oscillator quanta N0. But due to the repeating substructure for each subsequent U(6) representation (cf tables 2−4), resulting from the properties of the group SO(6) in the reduction U(6) ⊃ SO(6) ⊃ SU(3), the restriction with respect to N0 is not enough. One needs a similar restriction with respect to the SO(6) quantum number, although not all of the corresponding SUpn(3) states belonging to the SO(6) irreps $\upsilon$ < $\upsilon$0 are Pauli forbidden. For any case, it is safe to discard them and to consider only the Pauli-allowed SUpn(3) multiplets belonging to the SO(6) representations $\upsilon$ ≥ $\upsilon$0, spanning the many-particle irreducible collective space of the relevant Sp(12, R) shell-model representation of the concrete nuclear system. These sets are exactly those SUpn(3) many-particle states that are exploited in the practical applications of the microscopic shell-model version of the BM model. The results obtained in the present work are important, because they allow construction of the Pauli-allowed shell-model representations of the microscopic version of the BM collective model, defined by the chain Sp(12, R) ⊃ U(6) ⊃ SO(6) ⊃ SUpn(3) ⨂ SO(2) ⊃ SO(3), for its wide applications over the nuclear chart. The microscopic counterparts of the BM model in its Wilets-Jean and rotor model limits, in contrast to the original versions, are endowed with microscopic many-particle wave functions with the proper permutational symmetry. The shell-model representations, determined by the nonscalar O(A − 1) irreps ω ≠ (0) (or, equivalently, by the intrinsic Sp(12, R) structure ⟨σ⟩ ≠ ⟨0⟩), define the proper irreducible collective subspaces of the nuclear Hilbert space in which the standard many-nucleon quantum mechanics can be fully exploited for studying the quadrupole–monopole collective dynamics of BM-type. In addition, the octupole degrees of freedom can easily be included in the present proton–neutron shell-model approach by considering the trilinear combinations of the many-particle position Jacobi coordinates xis(α), i.e. by considering the octupole Oijk(α, β, δ) = ∑sxis(α)xjs(β)xks(δ) shell-model operator.
Finally, it should be pointed out that for the medium- and heavy-mass nuclei, one may alternatively use the proxy-SU(3) scheme [36] (see also, e.g. [37] for a recent review) instead of the pseudo-SU(3) one, as was done, for instance, for the description of the irrotational-flow quadrupole dynamics in 102Pd in [38].