Shape coexistence in nuclei appears to be unique in the realm of finite many-body quantum systems It differs from the various geometrical arrangements that sometimes occur in a molecule in that in a molecule the various arrangements are of the widely separated atomic nuclei In nuclei the various ''arrangements'' of nucleons involve (sets of) energy eigenstates with different electric quadrupole properties such as moments and transition rates, and different distributions of proton pairs and neutron pairs with respect to their Fermi energies Sometimes two such structures will ''invert'' as a function of the nucleon number, resulting in a sudden and dramatic change in ground-state properties in neighboring isotopes and isotones In the first part of this review the theoretical status of coexistence in nuclei is summarized Two approaches, namely, microscopic shell-model descriptions and mean-field descriptions, are emphasized The second part of this review presents systematic data, for both even- and odd-mass nuclei, selected to illustrate the various ways in which coexistence is observed in nuclei The last part of this review looks to future developments and the issue of the universality of coexistence in nuclei Surprises continue to be discovered With the major advances in reaching to extremes of proton-neutronmore » number, and the anticipated new ''rare isotope beam'' facilities, guidelines for search and discovery are discussed« less

Shape coexistence in atomic nuclei

https://people.nscl.msu.edu/~brown/brown-all-papers/058-1984-npa.425.205.pdf

Folding-model analysis of elastic and inelastic α-particle scattering using a density-dependent force

The IBA-1 is reviewed with particular emphasis on the symmetry structure that arises naturally from its inherently algebraic approach. The formulation of the model, in both its algebraic and its numerical aspects, is presented and the basic character of its predictions is discussed and compared with nuclear data. The limitations of the model, and efforts to ameliorate these by appropriate extensions, are also reviewed in some detail. An effort is made to provide a simple, transparent understanding of the basic physics by clarifying the relation of the mathematical structure to underlying physical ideas, and to provide guidance in understanding and carrying out practical calculations.

The interacting boson approximation

A new method of determining nuclear shapes is proposed which avoids the assumption of a specific nuclear model. The concept of an equivalent ellipsoid, whose charge and moments equal those of the nucleus, is employed. But the method is equally valid for spherical, deformed, and intermediate nuclei. It can be employed for any nucleus (evev-evevn, odd-$\mathbf{A}$, or odd-odd) provided enough $E2$ matrix elements are available. The example of $^{152}\mathrm{Sm}$ is discussed.

Intrinsic Quadrupole Moments and Shapes of Nuclear Ground States and Excited States

We present a method based on mean-field states generated by triaxial quadrupole constraints that are projected on particle number and angular momentum and mixed by the generator coordinate method on the quadrupole moment. This method is equivalent to a seven-dimensional GCM calculation, mixing all five degrees of freedom of the quadrupole operator and the gauge angles for protons and neutrons. A first application to $^{24}\mathrm{Mg}$ permits a detailed analysis of the effects of triaxial deformations and of $K$ mixing.

Configuration mixing of angular-momentum and particle-number projected triaxial Hartree-Fock-Bogoliubov states using the Skyrme energy density functional

The question of validity of the pairing-plus-quadrupole model to well-deformed nuclei is examined. The calculated energy levels, B (E2) values, E2/M1 mixing ratios, and electromagnetic moments are compared with the experimental values of /sup 156/Gd which is used as a test case. In particular, the structure of the second excited K = 0 band and the K = 4 band are examined in terms of model wave functions. For the sake of comparison and discussion, some results are also given for /sup 154/Gd.

Pairing-plus-quadrupole model calculations for /sup 154/ /sup 156/Gd

The effects of triaxial deformations in the structure of the 2(1)+ transition charge density in 58Ni

The coupled channel method has been extended so as to include the possibilities of spherical-transitional-deformed target nuclei, different shapes for target and final nuclei, and different shapes for different reaction channels. Deformation-dependent nuclear structure wave functions, solutions of a Bohr Hamiltonian deduced from a microscopic Hamiltonian, are employed to weigh the scattering potentials for different shapes and channels. The deformation and K-dependent form factors are integrated over deformation variables and summed over K in a general theory applicable to a wide variety of targets and projectiles. Illustrative examples, including comparisons with experimental cross sections, are presented for /sup 4/He scattering from /sup 24/Mg and /sup 28/Si, and for total neutron cross sections of /sup 148,150,152,154/Sm.

Extended coupled channel method for baryon scattering based on the dynamics of the Bohr Hamiltonian deduced from a microscopic nucleon-nucleon Hamiltonian.

A certain two-channel s-wave scattering model containing a resonance at low energy is studied, and it is shown that both the multichannel effective range approximation and the K matrix pole test for compositeness of the resonance fail for this model. When the model is applied to the low-energy I = 0 KN interaction, it leads to a K matrix which is very different from those used in the phenomenological K matrix parametrizations hitherto carried out. Implications of this are discussed.

Alternate form for the low-energy KN K matrix

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