A symmetry for heavy nuclei: Proxy-SU(3)

TL;DR

This paper introduces the proxy-SU(3) approximation, a new fermionic method that restores SU(3) symmetry in heavy nuclei, complementing existing bosonic and pseudo-SU(3) approaches.

Contribution

The paper presents the novel proxy-SU(3) approximation for heavy nuclei, expanding the tools for symmetry-based nuclear structure modeling.

Findings

Proxy-SU(3) restores SU(3) symmetry in heavy nuclei.

Comparison with pseudo-SU(3) highlights similarities and differences.

Potential applications in nuclear structure calculations.

Abstract

The SU(3) symmetry realized by J. P. Elliott in the sd nuclear shell is destroyed in heavier shells by the strong spin-orbit interaction. However, the SU(3) symmetry has been used for the description of heavy nuclei in terms of bosons in the framework of the Interacting Boson Approximation, as well as in terms of fermions using the pseudo-SU(3) approximation. We introduce a new fermionic approximation, called the proxy-SU(3), and we comment on its similarities and differences with the other approaches.

Full text

1 Introduction

The SU(3) symmetry has been introduced in nuclear structure by J. P. Elliott Elliott1 ; Elliott2 , who considered the sd shell nuclei and showed the microscopic origins of the connection between the nuclear quadrupole deformation and SU(3). A generalization of the Elliott SU(3) scheme to more than one nuclear shell has been obtained in the framework of the microscopic symplectic model Rosensteel . Since then the SU(3) symmetry has been used in the framework of various algebraic models, especially for the study of medium-mass and heavy deformed nuclei, where the LS coupling scheme of the Elliott model breaks down Talmi , while microscopic calculations are still out of reach. Descriptions in terms of bosons have been given in the frameowork of the Interacting Boson Model (IBM) IA and of the Interacting Vector Boson Model (IVBM) Georgieva , while fermionic descriptions have been provided by the Fermion Dynamical Symmetry Model (FDSM) FDSM . It underlies also the pseudo-SU(3) scheme pseudo1 ; pseudo2 ; DW1 ; DW2 ; Bahri68 ; Blokhin74 ; Ginocchio , which we will discuss below, as well as the quasi-SU(3) symmetry Zuker1 ; Zuker2 , in which an approximate restoration of LS coupling in heavy nuclei is obtained, based on the smallness of certain $\Delta j=1$ matrix elements.

On the other hand, many properties of heavy deformed nuclei have been successfully described in detail in terms of the Nilsson model Nilsson1 ; Nilsson2 ; RN . Nilsson states are labelled by $K[Nn_{z}\Lambda]$, where $N$ is the number of oscillator quanta, $n_{z}$ is the number of quanta along the cylindrical symmetry axis, $\Lambda$ is the projection of the orbital angular momentum along the symmetry axis, and $K$ is the the projection of the total angular momentum along the symmetry axis, connected to $\Lambda$ by $K=\Lambda+\Sigma$, where $\Sigma$ is the the projection of the spin along the symmetry axis. For large deformations, the Nilsson wave functions reach the asymptotic limit, in which these quantum numbers become good quantum numbers, and they remain rather good even at intermediate deformation values Nilsson2 .

Ben Mottelson has remarked Mottelson that the asymptotic quantum numbers of the Nilsson model can be seen as a generalization of Elliott’s SU(3), applicable to heavy deformed nuclei. Working in this direction, we have shown PRC1 ; PRC2 that a proxy-SU(3) symmetry of the Elliott type can be developed in heavy deformed nuclei. The development of the proxy-SU(3) scheme is based on the so-called 0[110] pairs of Nilsson orbits related by $\Delta K[\Delta N\Delta n_{z}\Delta\Lambda]=0[110]$ Cakirli . These pairs, which are characterized by high overlaps Karampagia , have been shown to play a key role in the onset and development of nuclear deformation in the rare earth region Cakirli ; Karampagia .

In the proxy-SU(3) scheme we also focus attention on Nilsson 0[110] pairs, but in a different way. Instead of taking advantage of proton-neutron pairs, we use proton-proton and neutron-neutron pairs. In this way we reveal an approximate SU(3) symmetry in heavy deformed nuclei, which can be used for predicting nuclear properties within the SU(3) symmetry using algebraic methods, as we shall see in Refs. IoMartinou ; IoSarant .

2 The proxy-SU(3) model

We are going to explain the basic idea behind the proxy-SU(3) scheme by considering as an example the 50-82 major nuclear shell, shown in Fig. 1.

In the 50-82 major shell one finds the 3s1/2, 2d3/2, 2d5/2, and 1g7/2 orbitals (shown in Fig. 1 by solid lines). These are the pieces of the full sdg shell remaining after the desertion (because of the spin-orbit force) of the 1g9/2 orbitals (indicated by dashed lines) into the next shell below, i.e. into the 28-50 nuclear shell. In addition, the 50-82 major shell contains the 1h11/2 orbitals (shown by dashed lines plus one dotted line), which have invaded this shell from above, forced down and out of the pfh shell also by the spin-orbit force.

The deserter 1g9/2 orbital consists of the Nilsson orbitals 1/2[440], 3/2[431], 5/2[422], 7/2[413], 9/2[404]. These happen to be 0[110] partners of the 1h11/2 Nilsson orbitals 1/2[550], 3/2[541], 5/2[532], 7/2[523], 9/2[514], in the same order. Two orbitals being 0[110] partners possess exactly the same values of the projections of orbital angular momentum, spin, and total angular momentum, thus they are expected to exhibit identical behavior as far as properties related to angular momentum projections are concerned. 0[110] partners have been first used in relation to proton-neutron pairs Cakirli , found to correspond to increased strength of the proton-neutron interaction, because of their large overlaps Karampagia .

One can thus think of replacing all of the invading 1h11/2 orbitals (the upper group of dashed lines in Fig. 1), except the 11/2[505] orbital (the dotted line in Fig. 1) in the 50-82 shell by their deserting 1g9/2 counterparts (the lower group of dashed lines in Fig. 1), expecting nuclear properties related to angular momentum to be little affected, since angular momentum projections remain intact. However, one should take carefully into account that during this replacement the $N$ and $n_{z}$ quantum numbers have been changed by one unit each, thus changing the sign of the parity. These changes will obviously affect the selection rules of various relevant matrix elements, as well as the avoided crossings Cejnar in the Nilsson diagrams. Detailed calculations to be shown in Ref. IoAssim will demonstrate that the changes inflicted in the Nilsson diagrams by these modifications are indeed minimal.

The 1h11/2 11/2[505] orbit has no 0[110] partner in the 1g9/2 shell, thus it has been excluded from this replacement. However, this orbit lies at the top of the 50-82 shell in the Nilsson diagrams Nilsson1 ; Nilsson2 , where it is unlikely to find nuclei with large deformations. The same remark applies to similar orbits in other shells such as the 13/2[606] orbit in the 82-126 shell.

After these two approximations have been performed, one is left with a collection of orbitals which form exactly the full sdg shell, which is known to possess a U(15) symmetry, having an SU(3) subalgebra BK . However, in axially symmetric deformed nuclei the relevant symmetry is not spherical, but cylindrical Takahashi . As a consequence, the relevant algebras are not U(N) Lie algebras, but more complicated versions of deformed algebras RD ; ND ; PVI ; Lenis ; Sugawara ; Arima . Nevertheless, one can expect that some of the SU(3) features would appear within the approximate scheme.

Since the present approximation scheme is based on the replacement of the invading from above abnornal parity orbitals (except the one with highest angular momentum) by their 0[110] counterparts deserting to the lower shell, with the latter being used as proxies of the former in subsequent considerations, we are going to call this approximation the proxy-SU(3) model.

The same approximation can be made in the 28-50, 82-126, 126-184 shells, which thus become approximate pf, pfh, sdgi shells, respectivel. These shells are known to correspond to U(10), U(21), U(28) algebras having SU(3) subalgebras (see BK and references therein).

3 The pseudo-SU(3) scheme

The present approach exhibits several similarities with and differences from the pseudo-SU(3) scheme, which has been extremely useful in the study of many properties of medium-mass and heavy nuclei away from closed shells. We list here some of these studies.

1. Yrast DW1 ; DW2 ; Naqvi516 and non-Yrast Popa62 ; Popa69 ; Vargas70 ; Popa403 ; Vargas49 ; Vargas53 bands of even-even deformed nuclei.

2. Normal parity bands and excited bands in odd-mass nuclei Naqvi536 ; Vargas61 ; Vargas673 ; Vargas64 ; Vargas66 .

3. The scissors mode and magnetic dipole excitations Castanos180 ; Beuschel57 ; Rompf57 ; Draayer25 ; Beuschel61 ; Vargas551 .

4. Superdeformed bands Dudek59 ; Nazarewicz64 .

5. Double-beta decay Castanos571 ; Hirsch51 ; Hirsch589 , neutrinoless double-beta decay Hirsch582 , and double-electron capture Ceron471 .

6. It should be pointed out that in the case of pseudo-SU(3) a unitary transformation connecting the normal parity orbitals to the pseudo-SU(3) space is known unitary1 ; unitary2 .

It is expected that using a large number of results regarding the study of fermionic systems by algebraic techniques, already developed and used in the pseudo-SU(3) framework, one would be able to perform further complementary studies using the proxy-SU(3) model.

4 Conclusions

A new approximate SU(3) symmetry applicable in heavy deformed nuclei has been suggested PRC1 ; PRC2 , called the proxy-SU(3) scheme. In Ref. IoAssim a detailed numerical study will demonstrate the validity of the approximation, while in Refs. IoMartinou ; IoSarant some first applications of the method in predicting nuclear properties will be described.