D. Petrellis

Z(5): critical point symmetry for the prolate to oblate nuclear shape phase transition

Abstract 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°. 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 Pt).

X(3): an exactly separable γ-rigid version of the X(5) critical point symmetry

Abstract A γ-rigid version (with γ = 0 ) of the X(5) critical point symmetry is constructed. The model, to be called X(3) since it is proved to contain three degrees of freedom, utilizes an infinite well potential, is based on exact separation of variables, and leads to parameter free (up to overall scale factors) predictions for spectra and B ( E 2 ) transition rates, which are in good agreement with existing experimental data for 172Os and 186Pt. An unexpected similarity of the β 1 -bands of the X(5) nuclei 150Nd, 152Sm, 154Gd, and 156Dy to the X(3) predictions is observed.

Ground state bands of the E(5) and X(5) critical symmetries obtained from Davidson potentials through a variational procedure

Abstract Davidson potentials of the form β 2 + β 0 4 / β 2 , when used in the original Bohr Hamiltonian for γ -independent potentials bridge the U(5) and O(6) symmetries. Using a variational procedure, we determine for each value of angular momentum L the value of β 0 at which the derivative of the energy ratio R L = E ( L )/ E (2) with respect to β 0 has a sharp maximum, the collection of R L values at these points forming a band which practically coincides with the ground state band of the E(5) model, corresponding to the critical point in the shape phase transition from U(5) to O(6). The same potentials, when used in the Bohr Hamiltonian after separating variables as in the X(5) model, bridge the U(5) and SU(3) symmetries, the same variational procedure leading to a band which practically coincides with the ground state band of the X(5) model, corresponding to the critical point of the U(5) to SU(3) shape phase transition. A new derivation of the Holmberg–Lipas formula for nuclear energy spectra is obtained as a by-product.

γ-rigid solution of the Bohr Hamiltonian for γ=30° compared to the E(5) critical point symmetry

A gamma-rigid solution of the Bohr Hamiltonian for gamma = 30 degrees is derived, its ground state band being related to the second order Casimir operator of the Euclidean algebra E(4). Parameter-free (up to overall scale factors) predictions for spectra and B(E2) transition rates are in close agreement to the E(5) critical point symmetry, as well as to experimental data in the Xe region around A=130.

Exactly separable version of the Bohr Hamiltonian with the Davidson potential

An exactly separable version of the Bohr Hamiltonian is developed using a potential of the form u({beta})+u({gamma})/{beta}{sup 2}, with the Davidson potential u({beta})={beta}{sup 2}+{beta}{sub 0}{sup 4}/{beta}{sup 2} (where {beta}{sub 0} is the position of the minimum) and a stiff harmonic oscillator for u({gamma}) centered at {gamma}=0 deg. In the resulting solution, called the exactly separable Davidson (ES-D) solution, the ground-state, {gamma}, and 0{sub 2}{sup +} bands 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 ({beta}{sub 0},{gamma} stiffness). Insights are also obtained regarding the recently found correlation between {gamma} stiffness and the {gamma}-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.

Nuclear collective motion with a coherent coupling interaction between quadrupole and octupole modes

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 {beta}{sub 2} and {beta}{sub 3}. The system is considered to oscillate between positive and negative {beta}{sub 3} values by rounding an infinite potential core in the ({beta}{sub 2},{beta}{sub 3}) plane with {beta}{sub 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 {sup 150}Nd, {sup 152}Sm, {sup 154}Gd, and {sup 156}Dy is presented.

Analytic description of critical-point actinides in a transition from octupole deformation to octupole vibrations

An analytic collective model in which the relative presence of the quadrupole and octupole deformations is determined by a parameter ({phi}{sub 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 {phi}{sub 0}, draws the border between the regions of octupole deformation and octupole vibrations in an essentially parameter-independent way, and describes well {sup 226}Th and {sup 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 {phi}{sub 0}=45 deg. 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 {beta}{sup 2}+{beta}{sub 0}{sup 4}/{beta}{sup 2}, leading to the AQOA spectrum through a variational procedure.

Exactly separable version of X(5) and related models

Abstract One-parameter exactly separable versions of the X(5) and X(5)- β 2 models, labelled as ES-X(5) and ES-X(5)- β 2 respectively, are derived by using in the Bohr Hamiltonian potentials of the form u ( β ) + u ( γ ) / β 2 . Unlike X(5), in these models the β 1 and γ 1 bands are treated on equal footing. Spacings within the γ 1 band are well reproduced by both models, while spacings within the β 1 band are well reproduced only by ES-X(5)- β 2 , for which several nuclei with R 4 / 2 = E ( 4 1 + ) / E ( 2 1 + ) ratios and [normalized to E ( 2 1 + ) ] β 1 and γ 1 bandheads corresponding to the model predictions have been found.

Octupole deformation in light actinides within an analytic quadrupole octupole axially symmetric model with a Davidson potential

The analytic quadrupole octupole axially symmetric model, which had successfully predicted 226Ra and 226Th as lying at the border between the regions of octupole deformation and octupole vibrations in the light actinides using an infinite well potential (AQOA-IW), is made applicable to a wider region of nuclei exhibiting octupole deformation, through the use of a Davidson potential (AQOA-D). Analytic expressions for energy spectra and B(E1), B(E2), B(E3) transition rates are derived. The spectra of 222-226Ra and 224,226Th are described in terms of the two parameters phi_0 (expressing the relative amount of octupole vs. quadrupole deformation) and beta_0 (the position of the minimum of the Davidson potential), while the recently determined B(EL) transition rates of 224Ra, presenting stable octupole deformation, are successfully reproduced. A procedure for gradually determining the parameters appearing in the B(EL) transitions from a minimum set of data, thus increasing the predictive power of the model, is outlined.

Z(4): γ‐rigid Solution of the Bohr Hamiltonian for γ = 30° Compared to the E(5) Critical Point Symmetry

A γ‐rigid solution of the Bohr Hamiltonian for γ = 30° is derived, its β‐part being related to the second order Casimir operator of the Euclidean algebra E(4). The solution is called Z(4), since it corresponds to the Z(5) model with the γ variable “frozen”. Parameter‐free (up to overall scale factors) predictions for spectra and B(E2) transition rates are in close agreement to the E(5) critical point symmetry, as well as to experimental data in the Xe region around A = 130.