Geometric Collective Model Research Articles

A novel four parameter formula for the rotational spectra of even-even nuclei

In this article, the authors presents a novel four parameter formula based on developing an expression for the moment of inertia in terms of Bohr-Mottelson Geometric Collective Model (GCM). The formula provides an excellent agreement with the experimental data for all nuclei under consideration. Some key quantities will also be derived such as the rotational frequency, the kinematic and dynamical moment of inertia, and their variation with spin will be investigated. A comparison between our proposed four parameter model with both the variable moment of inertia model, and the recently proposed β-stretching model is also presented. Various ΔI=1 and ΔI=2 energy staggering indices will be presented and compared to the experimental counterpart in the paper.

Influence of Triaxiality on the Description of Low-Energy Excitation Spectrum of 96Zr

The observed properties of the low-lying collective states of $^{96}$Zr are investigated within the geometrical collective model. The quadrupole-collective Bohr Hamiltonian with the potential having spherical and axially-symmetric deformed minima is applied. The role of triaxiality is investigated by rotating the potential in $\gamma$-degree of freedom so that the deformed minimum occurs at various axially asymmetric shapes. The change of excitation energies and reduced matrix elements of quadrupole transitions with increase of triaxiliaty is analyzed.

Shell-model study of octupole collectivity near Pb208

We show that the collectivity of the particle-hole wave function of low-lying octupole 3^- states in doubly magic nuclei is mainly due to the neutron-proton interaction. Both the enhanced reduced transition probability to the ground state, B(E3;3-_1 -> 0+_1), and the coupling of the octupole excitation to a nucleon result from the coherent action of all the components of the collective state. The results obtained with a realistic shell-model interaction both for 208Pb and 209Pb agree with the geometric collective model of Bohr and Mottelson, where octupole excitations are associated with phonons corresponding to collective shape oscillations of the surface of the nucleus.

Evolution of the phenomenologically determined collective potential along the chain of Zr isotopes

The properties of the collective low-lying states of Zr isotopes which include excitation energies and $E2$ reduced transition probabilities indicate that some of these states are mainly spherical and the other are mainly deformed ones. We investigate the properties of the low-lying collective states of $^{92-102}$Zr and their evolution with increase of the number of neutrons based on the five-dimensional Geometrical Quadrupole Collective Model. The quadrupole-collective Bohr Hamiltonian with a potential having spherical and deformed minima, is applied. The relative depth of two minima, height and width of the barrier, rigidity of the potential near both minima are determined so as to achieve the best possible description of the observed properties of the low-lying collective quadrupole states of $^{92-102}$Zr. Satisfactory agreement with the experimental data on the excitation energies and the $E2$ reduced transition probabilities is obtained. The evolution of the collective potential with increase of $A$ is described and the distributions of the wave functions of the collective states in $\beta-\gamma$ plane are found. The resulting potential evolves with $A$ increase from having only one spherical minimum in $^{92}$Zr, through the potentials having both spherical and deformed minima, to the potential with one deformed minimum in $^{102}$Zr.

Prediction of the excitation energies of the 21+ states for superheavy nuclei based on the microscopically derived Grodzins relation

As the result of synthesis of nuclei with large proton numbers a new region of investigations of the structure of nuclei has been discovered. Due to the recent significant increase in the yield of superheavy nuclei their gamma-spectroscopic studies became possible. The purpose of paper is to predict the excitation energies of the $2^+_1$ states of nuclei with Z$\ge 100$ using the microscopic variant of the Grodzins relation derived based on the geometrical collective model. The excitation energies of the $2^+_1$ states of the even-even nuclei from $^{256}$Fm to $^{296}_{120}$X which differ from each other in the number of $\alpha$-particles are predicted. It is shown that at the beginning of the chain of the studied nuclei the excitation energies of the $2^+_1$ states don't exceed 100 keV. Then $E(2^+_1)$ sharply increases with $A$ and reaches maximum value of $400-500$ keV in $^{284}$Fl or $^{292}$Og.

Description of the low-lying collective states of Zr96 based on the collective Bohr Hamiltonian including the triaxiality degree of freedom

Experimental data on $^{96}$Zr indicate coexisting spherical and deformed structures with small mixing amplitudes. Several collective low-lying states and E2 and M1 transitions are observed for this nucleus. The quadrupole-collective Bohr Hamiltonian depending on both $\beta$ and $\gamma$ shape variables with a potential having spherical and deformed minima, is applied consideration of these data. The relative depth of two minima, height and width of the barrier, rigidity of the potential near both minima are determined so as to achieve a satisfactory description of the observed properties of the low-lying collective quadrupole states of $^{96}$Zr. Good agreement with the experimental data on the excitation energies, $B(E2)$ and $B(M1; 2^+_2\rightarrow 2^+_1)$ reduced transition probabilities is obtained. It is shown that the low-energy structure of $^{96}$Zr can be described in a satisfactory way within the Geometrical Collective Model with a potential function supporting shape coexistence without other restrictions of its shape. However, the excitation energy of the $2^+_2$ state can be reproduced only if the rotation inertia coefficient is taken by four times smaller than the vibrational one in the region of the deformed well. It is shown also that shell effects are important for the description of the $B(M1; 2^+_2 \rightarrow 2^+_1)$ and $B(M1; 3^+_1 \rightarrow 2^+_1)$ transition probabilities. An indication for the influence of the pairing vibrational mode on the $0^+_2 \rightarrow 0^+_1$ transition is confirmed in agreement with the previous result.

Connecting the X(5)-β2, X(5)-β4, and X(3) models to the shape/phase transition region of the Interacting Boson Model

The parameter independent (up to overall scale factors) predictions of the X(5)-β2, X(5)-β4, and X(3) models, which are variants of the X(5) critical point symmetry developed within the framework of the geometric collective model, are compared to two- parameter calculations in the framework of the interacting boson approximation (IBA) model. The results show that these geometric models coincide with IBA parameters consistent with the phase/shape transition region of the IBA for boson numbers of physical interest (close to 10). 186Pt and 172Os are identified as good examples of X(3), while 146Ce, 174Os and 158Er, 176Os are identified as good examples of X(5)-β2 and X(5)-β4 behavior respectively.

GCM Solver (Ver. 3.0): A Mathematica Notebook for Diagonalization of the Geometric Collective Model (Bohr Hamiltonian) with Generalized Gneuss–Greiner Potential

The program diagonalizes the Geometric Collective Model (Bohr Hamiltonian) with generalized Gneuss–Greiner potential with terms up to the sixth power in β . In nuclear physics, the Bohr–Mottelson model with later extensions into the rotovibrational Collective model is an important theoretical tool with predictive power and it represents a fundamental step in the education of a nuclear physicist. Nuclear spectroscopists might find it useful for fitting experimental data, reproducing spectra, EM transitions and moments and trying theoretical predictions, while students might find it useful for learning about connections between the nuclear shape and its quantum origin. Matrix elements for the kinetic energy operator and for scalar invariants as β 2 and β 3 cos ( 3 γ ) have been calculated in a truncated five-dimensional harmonic oscillator basis with a different program, checked with three different methods and stored in a matrix library for the lowest values of angular momentum. These matrices are called by the program that uses them to write generalized Hamiltonians as linear combinations of certain simple operators. Energy levels and eigenfunctions are obtained as outputs of the diagonalization of these Hamiltonian operators.

Lifetime Measurements of Excited States in ^{172}Pt and the Variation of Quadrupole Transition Strength with Angular Momentum.

Lifetimes of the first excited 2^{+} and 4^{+} states in the extremely neutron-deficient nuclide ^{172}Pt have been measured for the first time using the recoil-distance Doppler shift and recoil-decay tagging techniques. An unusually low value of the ratio B(E2:4_{1}^{+}→2_{1}^{+})/B(E2:2_{1}^{+}→0_{gs}^{+})=0.55(19) was found, similar to a handful of other such anomalous cases observed in the entire Segré chart. The observation adds to a cluster of a few extremely neutron-deficient nuclides of the heavy transition metals with neutron numbers N≈90-94 featuring the effect. No theoretical model calculations reported to date have been able to explain the anomalously low B(E2:4_{1}^{+}→2_{1}^{+})/B(E2:2_{1}^{+}→0_{gs}^{+}) ratios observed in these cases. Such low values cannot, e.g., be explained within the framework of the geometrical collective model or by algebraic approaches within the interacting boson model framework. It is proposed that the group of B(E2:4_{1}^{+}→2_{1}^{+})/B(E2:2_{1}^{+}→0_{gs}^{+}) ratios in the extremely neutron-deficient even-even W, Os, and Pt nuclei around neutron numbers N≈90-94 reveal a quantum phase transition from aseniority-conserving structure to a collective regime as a function of neutron number. Although a system governed by seniority symmetry is the only theoretical framework for which such an effect may naturally occur, the phenomenon is highly unexpected for these nuclei that are not situated near closed shells.

Aspects of Shape Coexistence in the Geometric Collective Model of Nuclei

We examine the coexistence of spherical and γ-unstable deformed nuclear shapes, described by an SO(5)-invariant Bohr Hamiltonian, along the critical-line. Calculations are performed in the Algebraic Collective Model by introducing two separate bases, optimized to accommodate simultaneously different forms of dynamics. We demonstrate the need to modify the β-dependence of the moments of inertia, in order to obtain an adequate description of such shape-coexistence.

Yang–Mills equation for the nuclear geometrical collective model connexion

The Bohr–Mottelson collective model of rotations and quadrupole vibrations is a foundational model in nuclear structure physics. A modern formulation using differential geometry of bundles builds on this legacy collective model to allow a deformation-dependent interaction between rotational and vortical degrees of freedom. The interaction is described by the bundle connexion. This article reports the Yang–Mills equation for the connexion. For a class of solutions to the Yang–Mills equation, the differential geometric collective model attains agreement between experiment and theory for the moments of inertia of deformed isotopes. More generally, the differential geometric framework applies to models of emergent phenomena in which two interacting sets of degrees of freedom must be unified.

Geometric criterion for chaos in collective dynamics of nuclei

A local geometrical criterion for chaos, based on an embedding of a Hamiltonian system into an appropriately chosen curved space, is applied in the classical version of the Geometric Collective Model of nuclei. The criterion is shown to be equivalent with a simpler indicator of chaos based on the convex–concave change of equipotential curves. It is tested by comparing its predictions with a detailed numerical analysis of global dynamics. The results indicate a capacity of the method to estimate the energy of the onset of chaos, but also demonstrate convincing counterexamples that disprove its general applicability.

Connexions for the nuclear geometrical collective model

The Bohr–Mottelson–Frankfurt model of nuclear rotations and quadrupole vibrations is a foundational model in nuclear structure physics. The model, also called the geometrical collective model or simply GCM(3), has two hidden mathematical structures, one group theoretic and the other differential geometric. Although the group structure has been understood for some time, the geometric structure is a new feature that this paper investigates in some detail. Using the de Rham Laplacian for the kinetic energy extends significantly the physical scope of the GCM(3) model. This Laplacian contains a ‘magnetic’ term due to the connexion between base manifold rotational and fibre vortex degrees of freedom. When the connexion specializes to irrotational flow, the Laplacian reduces to the Bohr–Mottelson kinetic energy operator.

Quantum phase transitions in the shapes of atomic nuclei

Signatures of criticality in the evolution of the nuclear ground-state shapes across the $N\ifmmode\times\else\texttimes\fi{}Z$ plane are discussed. Attention is paid to specific data indicating sudden structural changes in various isotopic and isotonic chains of medium-mass and heavy even-even nuclei, as well as to diverse theoretical aspects of the models used to describe these changes. The interacting boson model and the geometric collective model, in particular, are discussed in detail, the former providing global predictions for the evolution of collective observables in nuclei between closed shells and the latter yielding a parameter-efficient description of nuclei at the critical points of shape transitions. Some issues related to the mechanism of first- and second-order quantum phase transitions in general many-body systems are also outlined.

Chaotic dynamics in collective models of nuclei

We present results of an extensive analysis of classical and quantum signatures of chaos in the geometric collective model (GCM) and the interacting boson model (IBM) of nuclei. Apart from comparing the regular fraction of the classical phase space and the Brody parameter for the nearest neighbor spacing distribution in the quantum case, we also adopt (i) the Peres lattices allowing one to distinguish ordered and disordered parts of spectra and to reveal main ordering principles of quantum states, (ii) the geometrical method to determine the position where the transition from order to chaos occurs, and (iii) we look for the 1/fα power law in the power spectrum of energy level fluctuations. The Peres method demonstrates the adiabatic separation of collective rotations in the IBM.

Regularities and symmetries of subsets of collective0+states

The energies of subsets of excited ${0}^{+}$ states in geometric collective models are investigated and found to exhibit intriguing regularities. In models with an infinite square well potential, it is found that a single formula, dependent on only the number of dimensions, describes a subset of ${0}^{+}$ states. The same behavior of a subset of ${0}^{+}$ states is seen in the large boson number limit of the interacting boson approximation (IBA) model near the critical point of a first-order phase transition, in contrast to the fact that these ${0}^{+}$ state energies exhibit a harmonic behavior in all three limiting symmetries of the IBA. Finally, the observed regularities in ${0}^{+}$ energies are analyzed in terms of the underlying group theoretical framework of the different models.

Quantum chaos in the nuclear collective model. II. Peres lattices

This is a continuation of our paper [Phys. Rev. E 79, 046202 (2009)] devoted to signatures of quantum chaos in the geometric collective model of atomic nuclei. We apply the method by Peres to study ordered and disordered patterns in quantum spectra drawn as lattices in the plane of energy vs average of a chosen observable. Good qualitative agreement with standard measures of chaos is manifested. The method provides an efficient tool for studying structural changes in eigenstates across quantum spectra of general systems.

Quantum chaos in the nuclear collective model: Classical-quantum correspondence

Spectra of the geometric collective model of atomic nuclei are analyzed to identify chaotic correlations among nonrotational states. The model has been previously shown to exhibit a high degree of variability of regular and chaotic classical features with energy and control parameters. Corresponding signatures are now verified also on the quantum level for different schemes of quantization and with a variable classicality constant.

Order and chaos in the geometric collective model

We investigate collective vibrations and rotations of atomic nuclei from the classical viewpoint within the geometric collective model. We quantify the proportion of regular and chaotic orbits in the phase space, observing very complex behavior of its dependence on the model control parameters, energy, and angular momentum.

Connecting the X(5)−β 2, X(5)−β 4, and X(3) models to the shape/phase transition region of the interacting boson model

The parameter-independent (up to overall scale factors) predictions of the X(5)−β 2,X(5)−β 4, and X(3) models, which are variants of the X(5) critical point symmetry developed within the framework of the geometric collective model, are compared to two-parameter calculations in the framework of the interacting boson approximation (IBA) model. The results show that these geometricmodels coincide with IBA parameters consistentwith the phase/shape transition region of the IBA for boson numbers of physical interest (close to 10). 186Pt and 172Os are identified as good examples of X(3), while 146Ce, 174Os, and 158Er, 176Os are identified as good examples of X(5)−β 2 and X(5)−β 4 behavior, respectively