Bohr-Mottelson Model Research Articles

Isoscalar giant monopole resonance within the Bohr-Mottelson model

Background: With the possibility to measure the isoscalar giant monopole resonance (ISGMR) in the short-lived nuclei, the compressibility of the open-shell and exotic nuclei has attracted much attention.Purpose: The present work is an attempt to develop a semiclassical model to describe the ISGMR and the compressibility of the finite nuclei.Method: The fluid dynamical reduction of the Boltzmann-Langevin equation was carried out to describe the collective motion of the nucleus [Z. Phys. A 337, 413 (1990); Z. Phys. A 349, 119 (1994)]. In this work, by including the Skyrme energy density functional, we develop the model and apply it to the ISGMR. Both the self-consistent and linearized methods are used to solve the corresponding Langevin equation.Results: It is shown that the calculations of the ISGMR energies in this work agree with those by the relativistic mean-field theory and the random phase approximation, and reproduce the general trend of the data. The model only includes the two-body dissipation, and underestimates the width of the ISGMR. Twelve sets of the Skyrme interactions are applied to perform the calculations and study the incompressibility parameters in the leptodermous expansion. The calculated surface parameter ${K}_{S}$ within the model is smaller than the extracted value, and decreases linearly over mass number of the nucleus. The calculated values of the symmetry parameter ${K}_{\ensuremath{\tau}}$ and the Coulomb parameter ${K}_{C}$ agree with the extracted values by fitting the available data for the nuclei from $^{12}\mathrm{C}$ to $^{238}\mathrm{U}$.Conclusions: The Bohr-Mottelson model provides a reasonable method to understand the ISGMR in a macroscopic approach.

Theoretical Study of Density Distributions and Size Radii of 8B and 17Ne

The proton, neutron and matter density distributions, the corresponding size radii and elastic electron scattering form factors of one-proton 8 B and two-proton 17 Ne halo nuclei are calculated. The theoretical technique used to fulfill calculations is by assuming that both nuclei under study are composed of two main parts; the first is the compact core and the second is the unstable halo part. The single-particle radial wavefunctions of harmonic-oscillator (HO) and Woods-Saxon (WS) potentials are used to study core and halo parts, respectively. And other approach is studied by using HO potential for both core and halo parts, but using two HO size parameters for both supposed parts. The long tail behavior which is the main characteristic of halo nuclei are well produced for both 8 B and 17 Ne. The calculated size radii are in general in good agreement with the available experimental data. The electron scattering form factors of the C0+C2 and C0 components are also calculated for 8 B and 17 Ne, respectively and compared with corresponding stable 10 B and 20 Ne nuclei. For 8 B calculations, the core-polarization (CP) effects are taken into account by using Tassie and Bohr-Mottelson models. The contribution from model-space (MS) part C2 component is taken through pwt interaction. The results of the calculated charge form factors are left for the planned electron-radioactive ion beam colliders where the study of skin or halo on the charge form factors are going to be studied.

Theoretical description of the deformation properties for 154–164Gd isotopes

The 154–164Gd isotopes in SU(3) region were investigated. For these nuclei, the positive ground-state band of 154–164Gd has been calculated using Bohr–Mottelson Model, Interacting Vector Boson Model and Interacting Boson Model, while the negative-parity band of 154–164Gd has been calculated using Bohr–Mottelson Model and Interacting Vector Boson Model only. To determine the property of the ground-state band, the ratio (r(I+2)/I) and E-GOS curve (Eγ/I) as a function of the spin (I) have been drawn. The reduced transition probabilities B(E2) of these nuclei were calculated. The results were compared with the experimental data and the previous calculations. It is shown that there is a good agreement between the results found and especially with the experimental ones. The potential energy surface (PES) has been calculated within the framework using interacting boson model (IBM-1). The contour plot of the potential energy surfaces shows that the interested nuclei are deformed and have rotational characters.

Investigation of Energy Levels and Electromagnetic Transitions for Yb–Pt Nuclei with N=108 Using IBM, IVBM, and BMM

The interacting boson and vector boson models, as well as the Bohr–Mottelson one, are employed to describe the energy levels and electromagnetic transitions of the 178Yb–186Pt (N = 108) nuclei. For the purpose of determining the evolution of the ground state, both r((I + 2)/I) and E-GOS ratios have been calculated as functions of the spin I. Based on the interacting vector boson model and Bohr–Mottelson model, the negative-parity and GSB bands have been calculated, while the interacting boson model is only employed to calculate GSB, y, and b. The interacting boson model is also used to calculate the reduced transition probabilities B(E2). The obtained findings show a very well agreement with experimentally obtained results elsewhere. We also used the intrinsic coherent state to obtain the potential energy surfaces. These results indicate that these nuclei have a rotational property SU(3), while 186Pt has property O(6).

SU(3) gauge theory of nuclear rotations

The legacy Bohr-Mottelson model of collective rotational modes has a hidden differential geometric structure that enables its natural generalization to a nuclear model that has the mathematical structure of Yang-Mills theory. The essential differential geometry ingredients for Yang-Mills are a base manifold, a gauge group, and a connection or covariant derivative. In this letter, the base manifold is the space of nuclear orientations and quadrupole-monopole deformations, the gauge group is either SO(3) or SU(3), and the covariant derivative determines a new gauge-invariant “magnetic-type” interaction. The high-lying energy states of the legacy irrotational flow model enter, as a direct result of gauge coupling, the domain of low-energy yrast rotational bands, as observed by experiment. Although the relevant SU(3) representation for a deformed nucleus is the same as the Elliott model, the non-Abelian SU(3) gauge group's physical interpretation is very different and concerns the Kelvin circulation.

A possible quantum fluid-dynamical approach to vortex motion in nuclei

The essential point of Bohr–Mottelson theory is to assume an irrotational flow. As was already suggested by Marumori and Watanabe, the internal rotational motion, i.e., the vortex motion, however, may exist also in nuclei. So, we must take the vortex motion into consideration. In classical fluid dynamics, there are various ways to treat the internal rotational velocity. The Clebsch representation, [Formula: see text] is very powerful and allows for the derivation of the equations of fluid motion from a Lagrangian. Making the best use of this advantage, Kronig–Thellung, Ziman and Ito obtained a Hamiltonian including the internal rotational motion, the vortex motion, through the term [Formula: see text]. Going to quantum fluid dynamics, Ziman and Thellung finally derived the roton spectrum of liquid Helium II postulated by Landau. Is it possible to follow a similar procedure in the description of the collective vortex motion in nuclei? The description of such a collective motion has not been considered in the context of the Bohr–Mottelson model (BMM) for a long time. In this paper, we will investigate the possibility of describing the vortex motion in nuclei on the basis of the theories of Ziman and Ito together with Marumori’s work.

Microscopic description of the even-even 140-148Ba isotopes using BM, IBM and IVBM

A description of the even-even Ba isotopes for A = 140 to 148 in framework of Bohr-Mottelson model, interacting boson model and interacting vector boson model are carried out. The E-GOS curve (E $\gamma$ /I and the ratio between the energies of the (I + 2) and (I) states (r(I + 2)/I) as a function of the spin (I have been drawn to determine the property of the ground-state band. The positive ground-state band of 140-148Ba has been calculated using Bohr-Mottelson model, interacting boson model and interacting vector boson model, while the negative-parity band of 140-148Ba has been calculated using Bohr-Mottelson model and interacting vector boson model only. The reduced transition probabilities B(E2) of these nuclei were calculated. The parameters of the best fit to the measured data are determined. The potential energy surfaces (PESs) to the IBM Hamiltonian have been obtained using the intrinsic coherent state.

The Evolution of 146Nd– 156Nd Isotope States under the Framework of BM, IBM-1, IVBM and D-G

The properties of the ground, beta and negative parity state bands of 146Nd–156Nd isotopes have been studied. The ratio of gamma energy over spin has been calculated and drawn as a function of spin (I) to show the ground-state evolution for each isotope. The ratio between the excitation energies of successive (I+2) and (I) states as a function of spin (I) has been calculated and drawn to determine the property of the ground-state bands of 146Nd–156Nd isotopes. The odd-even staggering has been drawn to determine the energy difference of the states of the ground-state bands and the negative-parity bands. The energy levels for the ground-state band of 146Nd–156Nd have been calculated using the Bohr-Mottelson Model, the Interacting Boson Model, the Interacting Vector Boson Model and the Doma–El-Gendy Relation. The energy levels of the beta band for the measured experimentally levels of 148Nd, 150Nd and 152Nd have been calculated using the Interacting Boson Model. The energy levels of the negative parity band of the 146Nd, 148Nd, 150Nd, 152Nd and 156Nd isotopes have also been calculated using the Bohr-Mottelson Model and the Interacting Vector Boson Model. The calculated energy levels compared with the experimental data indicate the quality of the fit presented in this paper.

The rotational–vibrational properties of 178−188Os isotopes

The properties of the ground and excited-state bands of 178−188Os nuclei have been studied. The ratio Eγ/I has been calculated as a function of the spin (I) to determine the ground-state evolution. The ratio between the energies of the (I + 2) and (I) states as a function of the spin (I) has been drawn to determine the property of the ground-state band. The energy levels for the ground-state band of 178−188Os have been calculated using Bohr–Mottelson model, Interacting Boson Model, interacting vector boson model and Doma–El-Gendy relation. The negative parity band of 178−188Os has been also calculated using above models. The γ-band of 178−188Os isotopes and β-band states of 178,182,184Os isotopes have been calculated using Interacting Boson Model. The parameters of the best fit to the measured data are determined.

Phonon coupling effects in magnetic moments of magic and semimagic nuclei

Phonon coupling (PC) corrections to magnetic moments of odd neighbors of magic and semi-magic nuclei are analyzed within the self-consistent Theory of Finite Fermi Systems (TFFS) based on the Energy Density Functional by Fayans et al. The perturbation theory in g_L^2 is used where g_L is the phonon-particle coupling vertex. A model is developed with separating non-regular PC contributions, the rest is supposed to be regular and included into the standard TFFS parameters. An ansatz is proposed to take into account the so-called tadpole term which ensures the total angular momentum conservation with g_L^2 accuracy. An approximate method is suggested to take into account higher order terms in g_L^2. Calculations are carried out for four odd-proton chains, the odd Tl, Bi, In and Sb ones. Different PC corrections strongly cancel each other. In the result, the total PC correction to the magnetic moment in magic nuclei is, as a rule, negligible. In non-magic nuclei considered it is noticeable and, with only one exception, negative. On average it is of the order of -(0.1 - 0.5) \mu_N and improves the agreement of the theory with the data. Simultaneously we calculated the gyromagnetic ratio g_L^{ph} of all low-lying phonons in 208Pb. For the 3^-_1 state it is rather close to the Bohr-Mottelson model prediction whereas for other L-phonons, two 5^- and six positive parity states, the difference from the Bohr-Mottelson values is significant.

Self-consistent calculation of phonon gyromagnetic ratios in 208Pb

Within the Theory of Finite Fermi Systems the gyromagnetic ratios gLph of all low-lying phonons in 208Pb are calculated. The input data, i.e., single-particle energies, single-particle wavefunctions, and the ph interaction are derived from the Energy Density Functional by Fayans et al. For the 31− phonon which is the most collective state, the gLph value is close to the prediction of the collective Bohr-Mottelson (BM) model. Gyromagnetic ratios of other phonons that are included in our calculations, two 5− states and six positive parity phonons, differ significantly from the BM model prediction.

COMPARISONS OF THE ODD–EVEN STAGGERING PATTERNS BETWEEN THE Ba–Dy REGION WITH 88 NEUTRONS AND Ra ISOTOPES WITH 88 PROTONS

Useful information about the shape transitions of even–even isotones 144 Ba , 146 Ce , 148 Nd , 150 Sm , 152 Gd , 154 Dy with 88 neutrons and 218-226 Ra isotopes with 88 protons are obtained from the ratios of [Formula: see text] as a function of neutron number N or proton number Z and the relations [Formula: see text] and r((I+2)/I) as a function of the angular momentum I. Interacting Boson Model (IBM-1), Bohr–Mottelson Model (BM) and Interacting Vector Boson Model (IVBM) have been employed to study the energy levels for the ground and octupole bands of the above nuclei. The best input parameters for the above approaches which lead to the best fit to experimental data are determined. The ΔI = 1 staggering (odd–even staggering) in octupole bands of the studied nuclei is found to exhibit a "beat" behavior as a function of angular momentum I. Comparisons of the displacement energies δE(I) and the staggering factor ΔE1, γ(I) are presented between the Ba–Dy region with 88 neutrons and Ra isotopes with 88 protons.

Fixing the moment of inertia in the Bohr Hamiltonian through Supersymmetric Quantum Mechanics

A well known problem of the Bohr-Mottelson model is the behavior of nuclear moments of inertia. The coefficient of the angular part (γ, θi) of the kinetic energy is proportional to β2. It is shown how a SUSYQM treatment of the β vibrations, emerges moments of inertia compatible with the experimental picture. This emergence is the result of the SUSYQM solution of the radial equation. Fitting results for the 162Dy and 238U in ground state, β1 and γ1 bands are also presented.

Search for E(5) behavior: IBM and Bohr–Mottelson model with Davidson potential calculations of some even–even Xe isotopes

The interacting boson model (IBM) has been widely used for describing the quadrupole collective states of the medium-heavy nuclei and no distinction is made between proton and neutron variables when the first version of the model is applied. However, the neutrons' and protons' degrees of freedom are described explicitly in the second version of the model (IBM-2). Moreover, the microscopic foundations certainly state that it is very important to describe the proton and neutron variables explicitly and this is also the generalized definition of the second version of the IBA model (IBM-2 model). So, triaxiality can be described explicitly through the introduction of cubic terms in the boson operators. Using the best-fitted values of parameters in the Hamiltonian of the IBM-2, we have calculated energy levels and B(E2) values for a number of transitions in 122,124,126,128,130,132,134Xe. The results were compared with the previous experimental and theoretical data and it has been observed that they are in good agreement. Many B(E2) values that are still not known so far are stated and the set of parameters used in these calculations is the best approximation that has been carried out so far. It has also turned out that the IBA and Bohr–Mottelson Hamiltonian with Davidson potential are fairly reliable models for the calculation of spectra in the entire set of 122,124,126,128,130,132,134Xe isotopes.

Solution of the Bohr Hamiltonian for soft triaxial nuclei

The Bohr-Mottelson model is solved for a generic soft triaxial nucleus, separating the Bohr Hamiltonian exactly and using a number of different model potentials: a displaced harmonic oscillator in \ensuremath{\gamma}, which is solved with an approximated algebraic technique; and Coulomb/Kratzer, harmonic/Davidson, and infinite square-well potentials in \ensuremath{\beta}, which are solved exactly. In each case we derive analytic expressions for the eigenenergies, which are then used to calculate energy spectra. Here we study the chain of osmium isotopes and compare our results with experimental information and previous calculations.

Semi-classical and anharmonic quantum models of nuclear wobbling motion

A semi-classical model for wobbling motion is presented as an extension to the Bohr–Mottelson model of wobbling motion. Using the resultant wobbling potential, a quantum mechanical equation is derived for anharmonic wobbling motion. We then attempt to explain the anharmonicity observed in the excited bands of two wobbling phonons in the A≃160 region.

Prolate–oblate shape-phase transition in the O(6) description of nuclear rotation

Recently it has been shown that the O(6)-based Hamiltonian of the interacting boson model (IBM) with three-body interactions parallels the traditional SU(3) description of nuclear rotation in a way which is closely related to the Bohr–Mottelson model. Here we focus on properties of quantum phase transitions between prolate and oblate ground-state shapes induced by the sign inversion of the cubic interaction. The classical-limit analysis shows that the phase structure of our model is partly analogous to that of the standard IBM with only two-body interactions. A triple point of spherical, prolate and oblate phases is identified within the O(6) dynamical symmetry. Quadrupole shape invariants and wave-function overlaps are applied to measure the phase-transitional rate at a realistic number of bosons.

Quantum Systems on Linear Groups

Discussed are quantized dynamical systems on orthogonal and affine groups. The special stress is laid on geodetic systems with affinely-invariant kinetic energy operators. The resulting formulas show that such models may be useful in nuclear and hadronic dynamics. They differ from traditional Bohr–Mottelson models where SL(n,ℝ) is used as a so-called non-invariance group. There is an interesting relationship between classical and quantized integrable lattices.

Problems in nuclear wobbling motion

Experimental results in Lu isotopes in the A ≈ 160 mass region have been interpreted as indicating the existence of the wobbling mode proposed by Bohr and Mottelson. Despite the success analysing the obtained data by use of the phenomenological particle-plus-rotor model (PRM) by Hamamoto, questions still remain. These questions relate to the lack of evidence of even–even wobbling systems, the anharmonicity of the wobbling spectrum and the nonlinear spin dependence of the excitation energy. An extension of the original Bohr–Mottelson model is suggested that incorporates spin dependence into the moment of inertia which provides a possible mechanism for explaining the latter two questions, whilst phenomenological arguments are adopted for the first.

The many relationships between the IBM and the Bohr model

Relationships between the IBM-1 and the Bohr collective model are explored in which the states of the IBM in its various dynamical symmetry limits are identified with subsets of Bohr model states of corresponding dynamical symmetries. The maps of interest are ones which give the contractions of the IBM in the limit of large boson number. The known map from the IBM into the Bohr model gives a contraction appropriate for the U(5) dynamical symmetry limit. A new map is given consistent with a contraction of the O(6) dynamical symmetry of the IBM. This map gives an explicit identification of IBM states in an O(6) ⊃ O(5) basis with states of the Wilets–Jean model and makes it possible to benefit from the different but complementary perspectives of the two models. For example, it leads to explicit expressions for the matrix elements of an IBM O(6) irrep in terms of the known matrix elements of the corresponding Wilets–Jean model. The relationship also shows how to obtain the familiar rotor plus beta- and gamma-vibrational bands of the Bohr–Mottelson model, in the IBM, by the addition to an O(6) Hamiltonian of a scalar cubic in the quadrupole moment operators of the type considered recently by Van Isacker. The establishment of close relationships between the two models enables one to benefit from the different and complementary perspectives they afford. For example, recent developments of an algebraic version of the collective model has shown that the Bohr model has an SU ( 1 , 1 ) × O ( 5 ) dynamical group with representations ranging from those of a spherical vibrator to a beta-vibrational soft-gamma rotor of the Wilets–Jean limit and, with the addition of an interaction, to a rotor of the standard Bohr–Mottelson type with beta- and gamma-vibrational bands. A parallel range of results can now be obtained in the IBM, albeit with some differences, within the framework of an SU ( 1 , 1 ) × O ( 5 ) dynamical group.