E(5) and X(5) critical point symmetries obtained from Davidson potentials through a variational procedure

Abstract

Davidson potentials of the form β 2 +β 4 0 /β 2 , when used in the E(5) framework (i.e., in the original Bohr Hamiltonian after separating variables as in the E(5) model, describing the critical point of the U(5) to O(6) shape phase transition) bridge the U(5) and O(6) symmetries, while they bridge the U(5) and SU(3) symmetries when used in the X(5) framework (i.e., in the original Bohr Hamiltonian after separating variables as in the X(5) model, corresponding to the critical point of the U(5) to SU(3) transition). Using a variational procedure, we determine for each value of angular momentum L the value of β 0 at which the rate of change of various physical quantities (energy ratios, intraband B(E2) ratios, quadrupole moment ratios) has a maximum, the collection of the values of the physical quantity formed in this way being a candidate for describing its behavior at the relevant critical point. Energy ratios lead to the E(5) and X(5) results (whice correspond to an infinite well potential in β), while intraband B(E2) ratios and quadrupole moments lead to the E(5)-β 4 and X(5)-β 4 results, which correspond to the use of a β 4 potential in the relevant framework. A new derivation of the Holmberg-Lipas formula for nuclear energy spectra is obtained as a by-product. (5) [2] models are supposed to describe shape phase transitions in atomic nuclei, the former being related to the transition from U(5) (vibrational) to O(6) (γ-unstable) nuclei, and the latter corresponding to the transition from U(5) to SU(3) (rotational) nuclei. In both cases the original Bohr collective Hamiltonian [3] is used, with an infinite well potential in the collective β-variable. Separation of variables is achieved in the E(5) case by assuming that the potential is independent of the collective γ-variable, while in the X(5) case the potential is assumed to be of the form u(β) + u(γ). We are going to refer to these two cases as "the E(5) framework" and "the X(5) framework" respectively. The selection of an infinite well potential in the β-variable in both cases is justified by the fact that the potential is expected to be flat around the point at which a shape phase transition occurs. Experimental evidence for the occurence of the E(5) and X(5) symmetries in some appropriate nuclei is growing ([4, 5] and [6,7] respectively).In the present work we examine if the choice of the infinite well potential is the optimum one for the description of shape phase transitions. For this purpose, we need one-parameter potentials which can span the U(5)-O(6) region in the E(5) framework, as well as the U(5)-SU(3) region in the X(5) framework. It turns out that the exactly soluble [8,9] Davidson potentials [10]

E(5), X(5), and prolate to oblate shape phase transitions in relativistic Hartree-Bogoliubov theory

“…The presence of a bump might be related to the success of the confined β-soft (CBS) rotor model [43,44], the relevance of Davidson potentials in the description of X (5) propetries [47,48], as well as to the existence of a significant five-dimensional centrifugal effect found recently [49].…”

“…However, the existence of a bump in the PES corresponding to good experimental examples of X(5) might be related to the success of the confined β-soft (CBS) rotor model [43,44], employing an infinite square well potential displaced from zero, as well as to the relevance of Davidson potentials [45,46] is the minimum of the potential) in the description of X(5) properties [47,48]. It can also be related to the significant five-dimensional centrifugal effect [49] The prediction that the nucleus 186 Pt is critical is supported by several pieces of evidence.…”

Description of nuclear octupole and quadrupole deformation close to the axial symmetry and phase transitions in the octupole mode

“…Other candidates for the X(5) symmetry in different nuclear regions have been reported later [6,7,8,9,10,11,12,13]. From the theoretical point of view, slightly different potentials have been explored by Caprio [14] and by Bonatsos et al [15], while the evolution of the s = 2 band when the lower border of the square well potential is displaced from zero has been investigated by Pietralla and Gorbachenko [16].…”

Bohr Hamiltonian with Davidson potential for triaxial nuclei

“…Using a variational procedure applied earlier in the E(5) and X(5) frameworks [26,27], one can see that the Z(5) solution can be interpreted as corresponding to the critical point of a shape phase transition between a triaxial vibrator and a rigid triaxial rotator.…”