Bohr-Mottelson Model Research Articles

Shapes and structure for the lowest states of the 42,44Ca isotopes

The shape and the associated dynamics of the 42,44Ca isotopes are investigated within the Bohr-Mottelson Model and the Covariant Density Functional Theory for the presence of the shape coexistence and mixing phenomena. The corresponding experimental energy spectrum and most of the electromagnetic transitions are well reproduced only by taking into account such phenomena. New possible developments of the models are indicated where improvements in agreement with the experimental data are needed.

Proton-neutron symplectic model description of 106Cd

In this study, a microscopic shell-model description of the low-lying collective states in the weakly deformed nucleus Cd within the recently proposed microscopic version of the Bohr-Mottelson model is provided. A good description of the excitation energies of the lowest ground, γ, and β quasibands is obtained without the adjustable kinetic energy term. Furthermore, γ degrees of freedom are shown to play a crucial role in the description of spectroscopy of this nucleus. A modified preserving high-order interaction is used to produce a γ-unstable type of odd-even staggering, observed experimentally between the states of the quasi-γ band. The current approach enables the characterization of observed intraband and interband quadrupole collectivity. The findings of this study propose an alternative interpretation of the fundamental question regarding the nature of low-energy vibrations, as well as the emergence of deformation and collectivity in weakly deformed atomic nuclei.

Microscopic shell-model description of irrotational-flow dynamics in 102Pd

The structure of the low-lying collective excitations in Pd is examined within the recently proposed microscopic shell-model version of the Bohr-Mottelson (BM) collective model. A good description of the excitation energies of the lowest ground, γ, and β bands, as well as the staggering function between the collective states of the γ band and some other energy-dependent quantities, is obtained. The low-energy intraband and interband quadrupole dynamics is reasonably well described within the present proton-neutron sympletic based shell-model approach without the use of an effective charge and compared with the predictions of nuclear structure models. The obtained results of the present study shed light on the question of the existence of irrotational-flow type quadrupole dynamics, which lies on the ground of the original BM model of quantized vibrations and surface-wave rotations in atomic nuclei.

On the microscopic shell-model version of the Bohr-Mottelson collective model

The recently proposed microscopic shell-model version of the Bohr-Mottelson (BM) collective model is considered in more detail in the coordinate representation. The latter possesses a clear and transparent physical meaning, which reveals several features of the new version of the collective model missed in the previous formulation. The relationship to the original BM model is considered, along with the relationships between the different limiting submodels of the microscopic version of the BM model, which closely resemble the relationships of the original Wilets-Jean and rotor models. The kinematically correct many-particle wave functions of the microscopic version of the BM model, conserving the experimentally observed integrals of motion, are shown to consist of collective irrotational-flow and intrinsic components in accordance with the original BM unified model. The general BM Hamiltonian is obtained as a contraction limit of the microscopic many-particle nuclear Hamiltonian, or, alternatively, by restricting the latter to the scalar irreducible collective space.

Longitudinal and transverse form factors from 65Cu and 71Ga nuclei

In the present work, the inelastic electron scattering for longitudinal and transverse form factors of 65Cu and 71Ga nuclei lies in the fp-shell region are studied in the framework of the shell model. The calculation is performed in the (1f5/2, 2p3/2, 2p1/2, 1g9/2) model space using jun45 effective interaction. The wavefunctions employed to conduct the shell model calculations are extracted from the jun45 effective interaction for these nuclei with the jj44 shell model space and (Sk35−Skzs ∗) residual interaction to evaluate the interactions matrix element between initial and final states. The effective charges used to account for the core-polarization (CP) effect are created using calculations of microscopic perturbations that include intermediate one-particle, one-hole excitation from the core and the model space (MS) orbits into all upper orbits with \(n\hslash \omega\) excitations following the same approach done in [Radhi et al. in Euro. Phys. J. A 50:1–9, 2014]. To account for the (CP) effects contribution, the inelastic form factor is obtained by employing the shape of Tassie and Bohr–Mottelson models with appropriate proton and neutron effective charges. The calculated form factors were compared with available experimental data.

Investigation of the Nuclear Structure of Some Ni and Zn Isotopes with Skyrme-Hartree-Fock Interaction

The inelastic C2 form factors and the charge density distribution (CDD) for 58,60,62Ni and 64,66,68Zn nuclei has been investigated by employing the Skyrme-Hartree-Fock method with (Sk35-Skzs*) parametrization. The inelastic C2 form factor is calculated by using the shape of Tassie and Bohr-Mottelson models with appropriate proton and neutron effective charges to account for the core-polarization effects contribution. The comparison of the predicted theoretical values was conducted with the available measured data for C2 and CDD form factors and showed very good agreement.

Form Factor and Charge Density Distribution Calculations for 58Ni, 70,72Ge Isotopes by Using Skyrme Interaction in NushellX@MSU Code

The inelastic electron scattering form factors for several chosen states of the 58Ni and 70,72Ge nuclei that reside in the fp-shell region were examined in this research. NushellX@MSU, a recent version of the shell model code, was used to do the computations. Skyrme effective interactions were utilized to calculate the energy Eigen values and eigen vectors that were used to generate the one body transition densities (OBTD) that were used in the inelastic electron scattering form factor calculations. To determine the inelastic electron scattering form factors, Tassie and Bohr-Mottelson models are used. The experimental data was compared to the estimated inelastic electron scattering form factors and charge density distribution for the examined nucleus.

Longitudinal form Factors of 25Mg with Bohr-Mottelson Model Using Large-scale Shell Model

The longitudinal electron scattering form factors in large scale SDPF-shell for 25Mg with SDPF now as effective interaction have been discussed using Bohr-Mottelson model to computation core polarization (CP). The calculation are done with the help of NuShell code. To determine the single-particle wave function, the SKyrme-Hartree Fock potential was used to compute the matrix elements' radial wave function containing a single particle. Bohr-Mottelson model is used in these calculation. The result show good agreement with experimental data for 25Mg nucleus.

Microscopic Version of the Bohr-Mottelson Model and Its Application

Published from 1956, the Bulgarian Journal of Physics accepts articles in the fields of: General and Mathematical Physics; Particle Physics and Field Theory; Atomic and Molecular Physics; Nuclear Physics; Fundamental Areas of Phenomenology; Theory of Relativity and Astrophysics; Condensed Matter, Statistical Physics; Cross-Disciplinary Physics; and Related Areas of Science and Technology.

Mixing of the coexisting shapes in the ground states of 74Ge and 74Kr

Ground state shape coexistence with mixing is suggested in nuclei, based on empirical evidences. The claim is demonstrated by the investigation of deformation properties for 74Ge and 74Kr nuclei within the phenomenological Bohr-Mottelson model, having as input the experimental collective energy states, as well with Covariant Density Functional Theory based on microscopic structural information. The results of the two approaches are shown to be compatible in what concerns the presence of coexisting shapes in the ground state of the considered nuclei, while the mixing characteristics are deduced from the phenomenologically calculated collective states.

Collective states of even–even nuclei in γ-rigid quadrupole Hamiltonian with minimal length under the sextic potential

In the present paper, we study the collective states of even–even nuclei in γ-rigid mode within the sextic potential and the minimal length (ML) formalism in Bohr–Mottelson model. The eigenvalues problem for the latter is solved by combined means of quasi-exact solvability and a quantum perturbation method. Numerical calculations are performed for 35 nuclei: 98−108Ru, 100−102Mo, 116−130Xe, 180−196Pt, 172Os, 146−150Nd, 132−134Ce, 154Gd, 156Dy and 150−152Sm. Through this study, it appears that our elaborated model leads to an improved agreement of the theoretical results with the corresponding experimental data by reducing the rms with a rate going up to 63% for some nuclei. This comes from the fact that we have combined the sextic potential, which is a very useful phenomenological potential, with the formalism of the ML, which is based on the generalized uncertainty principle and which is in turn a quantum concept widely used in quantum physics. In addition, we investigate the effect of ML on energy ratios, transition rates, moments of inertia and a shape phase transition for the most numerous isotopic chains, namely Ru, Xe, Nd and Pt.

Nuclear Gamma-soft Character in ¹²⁸Ba

We report the properties of gamma soft O(6) of 128Ba isotones with neutron N = 72 using Interacting Vector Boson Model (IVBM), interacting Boson Model (IBM-1), Bohr-Mottelson Model (BM), and Doma-El-Gendy (D-G) relation. The first energy level ( ) and ratio have been investigated which show that 128Ba has gamma-soft character. The curves Eγ/Vs.J of E-GOS of even 128Ba nucleus were compared with the standard curves of vibrational, gamma soft and rotational limits. The staggering factors were studied of available measured data of 128Ba nucleus. The yrast levels of this isotope are calculated by the model of VBMI, IBM-1, BM and D-G and they were compared by measured data. The negative parity band of 128Ba was calculated by IVBM and BM model and compared with experimental values.

Microscopic shell-model counterpart of the Bohr–Mottelson model

In the present paper we demonstrate that there exists a fully microscopic shell-model counterpart of the Bohr-Mottelson model by embedding the latter in the microscopic shell-model theory of atomic nucleus within the framework of the recently proposed fully microscopic proton-neutron symplectic model (PNSM). For this purpose, another shell-model coupling scheme of the PNSM is considered in which the basis states are classified by the algebraic structure $SU(1,1) \otimes SO(6)$. It is shown that the configuration space of the PNSM contains a six-dimensional subspace that is closely related to the configuration space of the generalized quadrupole-monopole Bohr-Mottelson model and its dynamics splits into radial and orbital motions. The group $SO(6)$ acting in this space, in contrast, e.g., to popular IBM, contains an $SU(3)$ subgroup which allows to introduce microscopic shell-model counterparts of the exactly solvable limits of the Bohr-Mottelson model that closely parallel the relationship of the original Wilets-Jean and rotor models. The Wilets-Jean-type dynamics in the present approach, in contrast to the original collective model formulation, is governed by the microscopic shell-model intrinsic structure of the symplectic bandhead which defines the relevant Pauli allowed $SO(6)$, and hence $SU(3)$, subrepresentations. The original Wilets-Jean dynamics of the generalized Bohr-Mottelson model is recovered for the case of closed-shell nuclei, for which the symplectic bandhead structure is trivially reduced to the scalar or equivalent to it irreducible representation.

The Effect of Core Polarization By Means of Tassie and Bohr-mottelson Models for Some FP-Shell Nuclei

The Effect of Core Polarization By Means of Tassie and Bohr-mottelson Models for Some FP-Shell Nuclei

Nuclear shape coexistence from the perspective of an algebraic many-nucleon version of the Bohr-Mottelson unified model

A fully quantal algebraic version of the Bohr-Mottelson unified model is presented with the important property that its quantisation is defined by its irreducible unitary representations which span the many-nucleon Hilbert space of every nucleus. The model is uniquely defined by the requirement that its Lie algebra of observables includes the nuclear quadrupole moments and kinetic energy. It then follows that there can be no non-zero isoscalar E2 transitions between any states belonging to its different irreducible representations and, as a result, the states of the model are uniquely defined with the property that observed transitions between rotational states of nuclei are to be expressed in terms of mixtures of the model irreps. The algebraic version of the unified model parallels the Bohr-Mottelson model in most respects, including the possibility of including the effects of Coriolis and centrifugal forces as subsequent perturbations. However, it corrects its treatment of angular momentum quantisation and no longer uses an over-complete set of coordinates. These changes have significant implications for the dynamics of nuclear rotations which are hidden when its moments of inertia are considered as inertial masses in the standard expression of rotational kinetic energies. Thus, the developments put a new perspective on the phenomenon of shape coexistence.

Properties of some Nuclei with Mass Number A=140

The positive ground-state band (GSB) of isobars was calculated using the Bohr – Mottelson (BM) model, interacting vector boson model (IVBM), and interacting boson model (IBM-1), while the negative-parity band (NPB) of was calculated using the BM model and IVBM. To determine the properties of the ground-state band, the back-bending curve, the E-GOS curve (Eγ/I), the ratio (r(I +2)/I) and the staggering have been plotted as a function of the spin (I). The contour plots show that all nuclei under consideration have complicate properties.

Critical Point of the 152Sm, 154Gd, and 156Dy Isotones

The even—even rare-earth nuclei in U(5)—SU(3) region at neutron number (N) = 90, have been systematically studied using the Bohr—Mottelson Model (BM), Interacting Vector Boson Model (IVBM), and Interacting Boson Model (IBM). The positive ground-state band (GSB) of 152Sm, 154Gd, and 156Dy nuclei has been calculated by using BM, IVBM, and IBM, while the negative-parity band (NPB) of those nuclei are calculated by BM and IVBM only. To determine the intermediate structure, the ratio (r(I + 2)/I) and E-GOS curve (E-Gamma Over Spin) as a function of the spin (I) have been drawn. In the IBM, the calculated reduced B(E2) transition probabilities of the GSB in the 152Sm, 154Gd, and 156Dy nuclei are analyzed and compared to the prediction of vibrational U(5) and rotational SU(3) limits. In the Sm—Dy nuclei with N = 90, the U(5)—SU(3) IBM potential energy surfaces (PES’s) are analyzed and evolve from spherical to deformed shapes with increasing the boson number. The critical phase transition points are identified in the space of model parameters and 152Sm,154Gd, and 156Dy nuclei have been found to be close to critical points.

Differential geometry of collective models

The classical astrophysical theory of Riemann ellipsoids and the quantum nuclear theory of Bohr and Mottelson share a common mathematical foundation in terms of the differential geometry of a principal bundle ${\cal P}$ and its associated vector bundle E, respectively. The bundle ${\cal P}=GL_+(3,R)$ is the connected component of the general linear group, the structure group G=SO(3) is the vorticity group, and the base manifold is the space of positive-definite real $3\times 3$ symmetricmatrices, identified geometrically with the space of inertia ellipsoids. The bundle is a Riemannian manifold whose metric is inherited from three-dimensional Euclidean space. A nonholonomic constraint force, like irrotational flow, determines a connection on the bundle.Wave functions of the Bohr-Mottelson model are sections of the associated vector bundle E = ${\cal P}\times_\rho$V, where $\rho$ denotes an irreducible representation of the vorticity group on the vector space V. Using the de Rham Laplacian $\triangle = \star d_\nabla \star d_\nabla$ for the kinetic energy introduces a magnetic term due to the connection between base manifold rotational and fiber vortex degrees of freedom. A class of Ehresmann connections creates a new model of nuclear rotation that predicts moments of inertia in agreement with experiment.

Description of Energy Levels for 84,86,96,98Sr Isotopes

The positive ground-state band (GSB) of 84,86,96,98Sr has been calculated using the Bohr–Mottelson (BM) model, interacting vector boson model (IVBM), and interacting boson model (IBM-1), while the negative-parity band (NPB) of 84,86,96,98Sr has been calculated using the BM model and IVBM. IBM-1 has been used to calculate the energy of the gamma and beta bands. To determine the properties of the ground-state band, the ratio (r(I +2)/I) and the E-GOScurve (Eγ/I) have been plotted as a function of the spin (I). The potential energy surface (PES) has been calculated using IBM-1. The contour plots of the PESs show that 84,86,96Sr nuclei have vibrational properties and 98Sr has rotational characteristics.

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.