A critical point symmetry for the prolate to oblate shape phase transition is introduced, starting from the Bohr Hamiltonian and approximately separating variables for γ = 30 o . Parameter-free (up to overall scale factors) predictions for spectra and B(E2) transition rates are found to be in good agreement with experimental data for 194 Pt, which is supposed to be located very close to the prolate to oblate critical point, as well as for its neighbours ( 192 Pt, 196 (5) [3], related to the transition from axial to triaxial shapes. All these critical point symmetries have been constructed by considering the original Bohr equation [8], separating the collective β and γ variables, and making different assumpions about the u(β) and u(γ) potentials involved.Furthermore, it has been demonstrated [9] that experimental data in the Hf-Hg mass region indicate the presence of a prolate to oblate shape phase transition, the nucleus 194 Pt being the closest one to the critical point. No critical point symmetry for the prolate to oblate shape phase transition originating from the Bohr equation has been given so far, although it has been suggested [10,11] that the (parameter-dependent) O(6) limit of the Interacting Boson Model (IBM) [12] can serve as the critical point of this transition, since various physical quantities exhibit a drastic change of behaviour at O(6), as they should [13].In the present work a parameter-free (up to overall scale factors) critical point symmetry, to be called Z(5), is introduced for the prolate to oblate shape phase transition, leading to parameter-free predictions which compare very well with the experimental data for 194 Pt. The path followed for constructing the Z(5) critical point symmetry is described here: 1) Separation of variables in the Bohr equation [8] is achieved by assuming γ = 30 o . When considering the transition from γ = 0 o (prolate) to γ = 60 o (oblate), it is reasonable to expect that the triaxial region (0 o < γ < 60 o ) will be crossed, γ = 30 o lying in its middle. Indeed, there is experimental evidence supporting this assumption [14].2) For γ = 30 o the K quantum number (angular momentum projection on the bodyfixedẑ ′ -axis) is not a good quantum number any more, but α, the angular momentum projection on the body-fixedx ′ -axis is, as found [15] in the study of the triaxial rotator [16,17].3) Assuming an infinite well potential in the β-variable and a harmonic oscillator potential having a minimum at γ = 30 o in the γ-variable, the Z(5) model is obtained.On these choices, the following comments apply: 1) Taking γ = 30 o does not mean that rigid triaxial shapes are prefered. In fact, it has been pointed out [18] that a nucleus in a γ-flat potential [19] (as it should be expected for a prolate to oblate shape phase transition) oscillates uniformly over γ from γ = 0 o to γ = 60 o , having an average value of γ av = 30 o , and, therefore, the triaxial case to which it should be compared is the one with γ = 30 o . Furthermore, it is known [20] that many prediction...

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An exactly separable version of the Bohr Hamiltonian is developed using a potential of the form u(β) + u(γ)/β 2 , with the Davidson potential u(β) = β 2 + β 4 0 /β 2 (where β 0 is the position of the minimum) and a stiff harmonic oscillator for u(γ) centered at γ = 0 • . In the resulting solution, called exactly separable Davidson (ES-D), the ground state band, γ band and 0 + 2 band are all treated on an equal footing. The bandheads, energy spacings within bands, and a number of interband and intraband B(E2) transition rates are well reproduced for almost all well-deformed rare earth and actinide nuclei using two parameters (β 0 , γ stiffness). Insights regarding the recently found correlation between γ stiffness and the γ-bandhead energy, as well as the long standing problem of producing a level scheme with Interacting Boson Approximation SU(3) degeneracies from the Bohr Hamiltonian, are also obtained.

The effect of a fermion with angular momentum j on quantum phase transitions of a (s,d) bosonic system is investigated. It is shown that the presence of a fermion strongly modifies the critical value at which the transition occurs, and its nature, even for small and moderate values of the coupling constant. The analogy with a bosonic system in an external field is mentioned. Experimental evidence for precursors of quantum phase transitions in bosonic systems plus a fermion (odd-even nuclei) is presented.Comment: 4 pages, 3 figure

The Deformation Dependent Mass (DDM) Kratzer model is constructed by considering the Kratzer potential in a Bohr Hamiltonian, in which the mass is allowed to depend on the nuclear deformation, and solving it by using techniques of supersymmetric quantum mechanics (SUSYQM), involving a deformed shape invariance condition. Analytical expressions for spectra and wave functions are derived for separable potentials in the cases of γ-unstable nuclei, axially symmetric prolate deformed nuclei, and triaxial nuclei, implementing the usual approximations in each case. Spectra and B(E2) transition rates are compared to experimental data. The dependence of the mass on the deformation, dictated by SUSYQM for the potential used, moderates the increase of the moment of inertia with deformation, removing a main drawback of the model.

A collective Hamiltonian for the rotation-vibration motion of nuclei is considered, in which the axial quadrupole and octupole degrees of freedom are coupled through the centrifugal interaction. The potential of the system depends on the two deformation variables β 2 and β 3 . The system is considered to oscillate between positive and negative β 3 -values, by rounding an infinite potential core in the (β 2 , β 3 )-plane with β 2 > 0. By assuming a coherent contribution of the quadrupole and octupole oscillation modes in the collective motion, the energy spectrum is derived in an explicit analytic form, providing specific parity shift effects. On this basis several possible ways in the evolution of quadrupole-octupole collectivity are outlined. A particular application of the model to the energy levels and electric transition probabilities in alternating parity spectra of the nuclei 150 Nd, 152 Sm, 154 Gd and 156 Dy is presented.

An analytic collective model in which the relative presence of the quadrupole and octupole deformations is determined by a parameter (φ 0 ), while axial symmetry is obeyed, is developed. The model [to be called the analytic quadrupole octupole axially symmetric model (AQOA)] involves an infinite well potential, provides predictions for energy and B(EL) ratios, which depend only on φ 0 , draws the border between the regions of octupole deformation and octupole vibrations in an essentially parameter-independent way, and describes well 226 Th and 226 Ra, for which experimental energy data are shown to suggest that they lie close to this border. The similarity of the AQOA results with φ 0 = 45 • for ground-state band spectra and B(E2) transition rates to the predictions of the X(5) model is pointed out. Analytic solutions are also obtained for Davidson potentials of the form β 2 + β 4 0 /β 2 , leading to the AQOA spectrum through a variational procedure.

One-parameter exactly separable versions of the X(5) and X(5)-beta^2 models, labelled as ES-X(5) and ES-X(5)-beta^2 respectively, are derived by using in the Bohr Hamiltonian potentials of the form u(beta)+u(gamma)/beta^2. Unlike X(5), in these models the beta_1 and gamma_1 bands are treated on equal footing. Spacings within the gamma_1 band are well reproduced by both models, while spacings within the beta_1 band are well reproduced only by ES-X(5)-beta^2, for which several nuclei with R_{4/2}=E(4_1^+)/E(2_1^+) ratios and [normalized to E(2_1^+)] beta_1 and gamma_1 bandheads corresponding to the model predictions have been found.Comment: 12 pages, LaTeX, including three .eps figure