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Register a new userConnection between Elliott SU(3) and spherical shell model bases
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In the Elliott SU(3) symmetry scheme the single particle basis is derived from the isotropic harmonic oscillator Hamiltonian in the Cartesian coordinate system. These states are transformed into the solutions of the same Hamiltonian within the spherical coordinate system. Then the spin-orbit coupling can be added in a straightforward way. The outcome is a transformation between the Elliott single particle basis and the spherical shell model space.
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Rotational $SU(3)$ algebraic symmetry continues to generate new results in the shell model (SM). Interestingly, it is possible to have multiple $SU(3)$ algebras for nucleons occupying an oscillator shell $eta$. Several different aspects of the multiple $SU(3)$ algebras are investigated using shell model and also deformed shell model based on Hartree-Fock single particle states with nucleons in $sdg$ orbits giving four $SU(3)$ algebras. Results show that one of the $SU(3)$ algebra generates prolate shapes, one oblate shape and the other two also generate prolate shape but one of them gives quiet small quadrupole moments for low-lying levels. These are inferred by using the standard form for the electric quadrupole transition operator and using quadrupole moments and $B(E2)$ values in the ground $K=0^+$ band in three different examples. Multiple $SU(3)$ algebras extend to interacting boson model and using $sdg$IBM, the structure of the four $SU(3)$ algebras in this model are studied by coherent state analysis and asymptotic formulas for $E2$ matrix elements. The results from $sdg$IBM further support the conclusions from the $sdg$ shell model examples.
The proxy-SU(3) symmetry has been proposed for spin-orbit like nuclear shells using the asymptotic deformed oscillator basis for the single particle orbitals, in which the restoration of the symmetry of the harmonic oscillator shells is achieved by a change of the number of quanta in the z-direction by one unit for the intruder parity orbitals. The same definition suffices within the cartesian basis of the Elliott SU(3) model. Through a mapping of the cartesian Elliott basis onto the spherical shell model basis, we translate the proxy-SU(3) approximation into spherical coordinates, proving, that in the spherical shell model basis the proxy-SU(3) approximation corresponds to the replacement of the intruder parity orbitals by their de Shalit--Goldhaber partners. Furthermore it is shown, that the proxy-SU(3) approximation in the cartesian Elliott basis is equivalent to a unitary transformation in the z-coordinate, leaving the x-y plane intact, a result which in the asymptotic deformed oscillator coordinates implies, that the z-projections of angular momenta and spin remain unchanged. The present work offers a microscopic justification of the proxy-SU(3) approximation and in addition paves the way, for taking advantage of the proxy-SU(3) symmetry in shell model calculations.
A novel dual-shell mechanism for the phenomenon of shape coexistence in nuclei within the Elliott SU(3) and the proxy-SU(3) symmetry is proposed for all mass regions. It is supposed, that shape coexistence is activated by large quadrupole-quadrupole interaction and involves the interchange among the spin-orbit (SO) like shells within nucleon numbers 6-14, 14-28, 28-50, 50-82, 82-126, 126-184, which are being described by the proxy-SU(3) symmetry, and the harmonic oscillator (HO) shells within nucleon numbers 2-8, 8-20, 20-40, 40-70, 70-112, 112-168 of the Elliott SU(3) symmetry. The outcome is, that shape coexistence may occur in certain islands on the nuclear map. The dual-shell mechanism predicts without any free parameters, that nuclei with proton number (Z) or neutron number (N) between 7-8, 17-20, 34-40, 59-70, 96-112, 146-168 are possible candidates for shape coexistence. In the light nuclei the nucleons flip from the HO shell to the neighboring SO-like shell, which means, that particle excitations occur. For this mass region, the predicted islands of shape coexistence, coincide with the islands of inversion. But in medium mass and heavy nuclei, in which the nucleons inhabit the SO-like shells, shape coexistence is accompanied by a merging of the SO-like shell with the open HO shell. The shell merging can be accomplished by the outer product of the SU(3) irreps of the two shells and represents the unification of the HO shell with the SO-like shell.
We present a review of the pseudo-SU(3) shell model and its application to heavy deformed nuclei. The model have been applied to describe the low energy spectra, B(E2) and B(M1) values. A systematic study of each part of the interaction within the Hamiltonian was carried out. The study leads us to a consistent method of choosing the parameters in the model. A systematic application of the model for a sequence of rare earth nuclei demonstrates that an overarching symmetry can be used to predict the onset of deformation as manifested through low-lying collective bands.The scheme utilizes an overarching sp(4,R) algebraic framework.
A pseudo shell SU(3) model description of normal parity bands in 159-Tb is presented. The Hamiltonian includes spherical Nilsson single-particle energies, the quadrupole-quadrupole and pairing interactions, as well as three rotor terms. A systematic parametrization is introduced, accompained by a detailed discussion of the effect each term in the Hamiltonian has on the energy spectrum. Yrast and excited band wavefunctions are analyzed together with their B(E2) values.
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