Bohr Collective Hamiltonian Research Articles

Bohr Hamiltonian with sextic potential for γ -rigid prolate nuclei with deformation-dependent mass term

The main aim of the present paper is to solve the eigenvalues problem with the Bohr collective Hamiltonian for $\ensuremath{\gamma}$-rigid nuclei within a model we have elaborated by combining two model approaches: the quantum mechanical formalism, namely, deformation-dependent mass formalism (DDM), and the anharmonic sextic oscillator potential for the variable $\ensuremath{\beta}$ and $\ensuremath{\gamma}=0$. The model developed in this way is conventionally called the sextic and DDM approach. Analytical expressions for energy spectra are conjointly derived by means of quasiexact solvability and a quantum perturbation method. Due to the scaling property of the problem, the energy and $B(E2)$ transition ratios depend on two free parameters apart from an integer number which limits the number of allowed states. Numerical results are given for 35 nuclei---$^{98--108}\mathrm{Ru}, ^{100--102}\mathrm{Mo}, ^{116--130}\mathrm{Xe}, ^{180--196}\mathrm{Pt}, ^{172}\mathrm{Os}, ^{146--150}\mathrm{Nd}, ^{132--134}\mathrm{Ce}, ^{154}\mathrm{Gd}, ^{156}\mathrm{Dy}$, and $^{150--152}\mathrm{Sm}$---revealing a good agreement with experiment. Moreover, as proved for the first time by Bonatsos et al. [D. Bonatsos, P. Georgoudis, D. Lenis, N. Minkov, and C. Quesne, Phys. Lett. B 683, 264 (2010)], the dependence of the mass on the deformation with the sextic potential moderates the increase of the moment of inertia with the deformation, removing an important drawback that has been revealed in the constant mass case [Buganu and Budaca, J. Phys. G: Nucl. Part. Phys. 42, 105106 (2015)]. Additionally, the correlation between the DDM and the minimal length formalism persists for the sextic potential. Finally, the DDM effects on the shape phase transition for the most numerous isotopic chains, namely, Ru, Xe, Nd, and Pt, have been duly investigated.

Deformation-dependent mass Bohr Hamiltonian with Anharmonic potential for triaxial nuclei

In this work, we study the collective Bohr Hamiltonian of triaxial nuclei within deformation-dependent mass formalism (DDM) using the new Anharmonic potential. The Schrödinger equation with Bohr Hamiltonian is separated into two parts and the [Formula: see text]-part is solved by mean of Nikiforov–Uvarov method to obtain the energy spectra. The [Formula: see text]-part is defined by a ring shape potential and solved with the same method. The energies and the [Formula: see text] transition rates for nuclide [Formula: see text]Pt are calculated and compared with the experimental data and with some others models in the same formalism.

Triaxial projected shell model study of $$\gamma $$-bands in atomic nuclei

A systematic study of $$\gamma $$ -bands observed in atomic nuclei is performed using the triaxial projected shell model (TPSM) approach. The staggering phase between the even and odd spin members of the $$\gamma $$ -band for most of the nuclei investigated in the present work is found to have even-I-below-odd-I, which in the framework of the collective model is considered as a signature of $$\gamma $$ -softness. It is observed that out of twenty-three systems studied, only four nuclei, namely, $$^{76}$$ Ge, $$^{112}$$ Ru, $$^{170}$$ Er and $$^{232}$$ Th depict staggering phase with odd-I-below-even-I, which is regarded as an indication of the static $$\gamma $$ -deformation in the collective model picture. The inclusion of the quasiparticle excitations in the framework of configuration mixing is shown to reverse the staggering phase from odd-I-down to the even-I-down for all the studied nuclei, except for the aforementioned four nuclei. Furthermore, by fitting a collective Bohr Hamiltonian to the TPSM energies, the differences between the two models are delineated through a comparison of the transition probabilities.

Coriolis contribution to excited states of odd-mass nuclei with different deformation-dependent mass coefficients

Within the collective Bohr Hamiltonian, the adoption of a mass tensor as a function of collective coordinates has demonstrated its importance for describing the structure of nuclei. On the other hand, for odd-mass nuclei, the Coriolis interaction between the rotational and single-particle motions affects significantly the structure of nuclear excited states. In the framework of a recently developed extended Bohr Hamiltonian, by considering the deformation-dependent mass formalism whith different mass parameters for the rotation and the two β and γ vibrations and taking into account the Coriolis contribution, we investigate the bands structure of the 173Yb, 163Dy, 155Eu and 153Eu nuclei. Excited-state energies and B(E2) transition probabilities are calculated and compared with the available experimental data. Besides, we investigate the effect of DDMF and the Coriolis force on nuclear observables.

Bohr Hamiltonian with Hulthén plus ring-shaped potential for triaxial nuclei with Deformation-Dependent Mass term

In this work, we present a new version of the Bohr collective Hamiltonian for triaxial nuclei within Deformation-Dependent Mass formalism (DDM) using the Hulth\'en potential. We shall call the developed model Z(5)-HD. Analytical expressions for energy spectra are derived by means of the recent version of the Asymptotic Iteration Method. The calculated numerical results of energies and B(E2) transition rates are compared with the experimental data, and several theoretical results from Z(5) model, the model Z(5)-H using the Hulth\'en potential without DDM formalism as well as theoretical predictions of Z(5)-DD model with Davidson potential using DDM formalism. The obtained results show an overall agreement with experimental data and an important improvement in respect to the other models.

Excited states of odd-mass nuclei with different deformation-dependent mass coefficients

Experimental data indicate that the mass tensor of collective Bohr Hamiltonian cannot be considered as a constant but should be considered as a function of the collective coordinates. In this work our purpose is to investigate the properties of low-lying collective states of the odd nuclei $^{173}$Yb and $^{163}$Dy by using a new generalized version of the collective quadrupole Bohr Hamiltonian with deformation-dependent mass coefficients. The proposed new version of the Bohr Hamiltonian is solved for Davidson potential in $\beta$ shape variable, while the $\gamma$ potential is taken to be equal to the harmonic oscillator. The obtained results of the excitation energies and B(E2) reduced transition probabilities show an overall agreement with the experimental data. Moreover, we investigate the effect of the deformation dependent mass parameter on energy spectra and transition rates in both cases, namely: when the mass coefficients are different and when they are equal. Besides, we will show the positive effect of the present formalism on the moment of inertia.

Nuclear energy excitation by Bohr Hamiltonian for triaxial nuclei using Killingbeck plus Morse potential

This paper proposes an improved potential for the [Formula: see text]-part of the collective Bohr Hamiltonian, namely, a Killingbeck plus Morse potential, while the [Formula: see text]-part is solved for a triaxial deformation close to [Formula: see text]. The Asymptotic Iteration Method is used, involving the Pekeris approximation, to calculate the energy eigenvalues and the eigenfunctions after an exact separation of the Bohr Hamiltonian into its variables is achieved. The results of these calculations are applied for energy spectra of the low-lying states and for corresponding [Formula: see text] quadrupole transition probabilities of the [Formula: see text] isotopes. Moreover, the results of the present solution are compared with those of the well-known [Formula: see text] and [Formula: see text] models.

Davydov–Chaban Hamiltonian with deformation-dependent mass term for γ = 30∘

MotivationSeveral theoretical comparisons with experimental data have recently pointed out that the mass tensor of the collective Bohr Hamiltonian cannot be considered as a constant and should be taken as a function of the collective coordinates. MethodThe Davydov–Chaban Hamiltonian, describing the collective motion of γ-rigid atomic nuclei, is modified by allowing the mass to depend on the nuclear deformation. Moreover, the eigenvalue problem for this Hamiltonian is solved for Davidson potential and γ=30∘ involving an Asymptotic Iteration Method (AIM). The present model is conventionally called Z(4)-DDM-D (Deformation Dependent Mass with Davidson potential), in respect to the so called Z(4) model. ResultsExact analytical expressions are derived for energy spectra and normalized wave functions, for the present model. The obtained results show an overall agreement with the experimental data for 108–116Pd, 128–132Xe, 136,138Ce and 190–198Pt and an important improvement in respect to other models. Prediction of a new candidate nucleus for triaxial symmetry is made. ConclusionThe dependence of the mass on the deformation reduces the increase rate of the moment of inertia with deformation, removing a main drawback of the model and leading to an improved agreement with the corresponding experimental data.

Bohr model description of the critical point for the first order shape phase transition

The critical point of the shape phase transition between spherical and axially deformed nuclei is described by a collective Bohr Hamiltonian with a sextic potential having simultaneous spherical and deformed minima of the same depth. The particular choice of the potential as well as the scaled and decoupled nature of the total Hamiltonian leads to a model with a single free parameter connected to the height of the barrier which separates the two minima. The solutions are found through the diagonalization in a basis of Bessel functions. The basis is optimized for each value of the free parameter by means of a boundary deformation which assures the convergence of the solutions for a fixed basis dimension. Analyzing the spectral properties of the model, as a function of the barrier height, revealed instances with shape coexisting features which are considered for detailed numerical applications.

Exactly separable Bohr Hamiltonian with the Morse potential for triaxial nuclei

In this paper, the Morse potential is used in the β-part of the collective Bohr Hamiltonian for triaxial nuclei. Energy eigenvalues and eigenfunctions are obtained in a closed form through exactly separating the Hamiltonian into its variables by using an appropriate form of the potential. The results are applied to generate the nuclear spectrum of 192 Pt , 194 Pt and 196 Pt isotopes which are known to be the best candidate exhibiting triaxiality. Electric quadrupole transition ratios are calculated and then compared with the experimental data and the Z(5) model results.

Quadrupole shape dynamics from the viewpoint of a theory of large-amplitude collective motion

Low-lying quadrupole shape dynamics is a typical manifestation of large-amplitude collective motion in finite nuclei. To describe the dynamics on a microscopic foundation, we have formulated a consistent scheme in which the Bohr collective Hamiltonian for the five-dimensional quadrupole shape variables is derived on the basis of the time-dependent Hartree–Fock–Bogoliubov theory. It enables us to incorporate the Thouless–Valatin effect on the shape inertial functions, which has been neglected in previous microscopic Bohr Hamiltonian approaches. Quantitative successes are illustrated for the low-lying spectra in Se, Mg and Cr, which display shape-coexistence, shape-mixing and shape-transitional behavior.

Solution of the Bohr Hamiltonian with the Davidson potential for a soft triaxial shape around γ = π/6 in nuclei

The triaxial shape in the vicinity of γ = π/6 in nuclei has been studied making use of a potential with the exactly separable form u(β) + v(γ)/β 2 in the Bohr collective Hamiltonian. The Davidson potential for u(β) and a harmonic potential centered at γ = π/6 for v(γ) are assumed to solve the Bohr Hamiltonian. We present analytic expressions for the energy spectra and the B(E2) transition rates, and compare theoretical results with available experimental data for 192Pt.

Electric quadrupole transitions of the Bohr Hamiltonian with the Morse potential

Eigenfunctions of the collective Bohr Hamiltonian with the Morse potential have been obtained by using the Asymptotic Iteration Method (AIM) for both gamma-unstable and rotational structures. B(E2) transition rates have been calculated and compared to experimental data. Overall good agreement is obtained for transitions within the ground state band, while some interband transitions appear to be systematically underpredicted in gamma-unstable nuclei and overpredicted in rotational nuclei.

Experimental and theoretical investigations of quadrupole collective degrees of freedom in 104Ru

The magnitudes and signs of twenty eight E2 and three M1 matrix elements involving 17 low-lying excited states in 104Ru have been measured by Coulomb excitation using 208Pb, 136Xe and 58Ni ions. The completeness of the set of E2 matrix elements is sufficient to extract, directly from the data, the expectation values of the intrinsic-frame E2 moments that provide considerable insight into the underlying collectivity. The measured E2 properties strongly correlate with macroscopic quadrupole collective degrees of freedom. Detailed comparison of the experimental results and theoretical microscopic calculations within the general quadrupole collective Bohr Hamiltonian are presented. Very good agreement of calculations and experiment is achieved without any parameters fitted to the 104Ru experimental data. Results of phenomenological γ-unstable and γ-rigid models are also given.

Soft triaxial rotor in the vicinity of γ = π/6 and its extensions

The collective Bohr Hamiltonian is solved for the soft triaxial rotor around γo = π/6 with a displaced harmonic oscillator potential in γ and a Kratzer-like potential in β. The properties of the spectrum are outlined and a generalization for the more general triaxial case with 0 < γ < π/6 is proposed.

Soft triaxial rotovibrational motion in the vicinity ofγ=π∕6

A solution of the Bohr collective hamiltonian for the $\beta-$soft, $\gamma-$soft triaxial rotor with $\gamma \sim \pi/6$ is presented making use of a harmonic potential in $\gamma$ and Coulomb-like and Kratzer-like potentials in $\beta$. It is shown that, while the $\gamma-$angular part in the present case gives rise to a straightforward extension of the rigid triaxial rotor energy in which an additive harmonic term appears, the inclusion of the $\beta$ part results instead in a non-trivial expression for the spectrum. The negative anharmonicities of the energy levels with respect to a simple rigid model are in qualitative agreement with general trends in the experimental data.

New analytic solutions of the collective Bohr Hamiltonian for a β-soft, γ-soft axial rotor

New analytic solutions of the quadrupole collective Bohr Hamiltonian are proposed, exploiting an approximate separation of the β and γ variables to describe γ-soft prolate axial rotors. The model potential is a sum of two terms: a β-dependent term taken either with a Coulomb-like or a Kratzer-like form, and a γ-dependent term taken as a harmonic oscillator. In particular, it is possible to give a one-parameter paradigm for a β-soft, γ-soft axial rotor that can be applied, with considerable agreement, to the spectrum of 234U.

THE NUCLEAR BORN–OPPENHEIMER METHOD APPLIED TO NUCLEAR COLLECTIVE MOTION

We deal with the application of the nuclear Born–Oppenheimer (NBO) method to the study of nuclear collective motion. In particular, we look at the description of nuclear rotations and vibrations. The collective operators are specified within the NBO method only to the extent of identifying the type of collective degrees of freedom we intend to describe; the operators are then determined from the dynamics of the system. To separate the collective degrees of freedom into rotational and vibrational terms, we transform the collective tensor operator from the lab fixed frame of reference to the frame defined by the principal axes of the system; this transformation diagonalizes the tensor operator. We derive a general expression for the NBO mean energy and show that it contains internal, collective and coupling terms. Then, we specify the approximations that need to be made in order to establish a connection between Bohr's collective model and the NBO method. We show that Bohr's collective Hamiltonian can be recovered from the NBO Hamiltonian only after adopting some rather crude approximations. In addition, we try to understand, in light of the NBO approach, why Bohr's collective model gives the wrong inertial parameters. We show that this is due to two major reasons: the ad hoc selection of the collective degrees of freedom within the context of Bohr's collective model and the unwarranted neglect of several important terms from the Hamiltonian.

Proton scattering from the unstable nuclei 30S and 34Ar: structural evolution along the sulfur and argon isotopic chains

Proton elastic and inelastic scattering angular distributions to the 2 1 + and 3 1 − collective states of the proton-rich nuclei 30S and 34Ar were measured at 53 MeV/ A and 47 MeV/ A, respectively, using secondary beams from the GANIL facility and the MUST silicon strip detector array. Data for the stable 32S nucleus were also obtained at 53 MeV/ A for comparison. A phenomenological analysis was used to deduce the deformation parameters β p,p′ for the low-lying collective excitations. A microscopic analysis was performed by generating matter and transition densities from self-consistent QRPA calculations. Configuration mixing calculations based on a collective Bohr Hamiltonian were also performed. DWBA and coupled-channel calculations using microscopic optical potentials built from these densities and the JLM interaction are compared to the data. There is no indication for the presence of proton skins in these nuclei. The microscopic calculations are extended to the even–even sulfur and argon isotopes from A=30 to A=40, and A=34 to A=44, respectively, and compared to available experimental results. On the basis of this analysis predictions are made for the 42,44S and 46Ar nuclei concerning ground state and transition densities.

BRST treatment of the Bohr collective hamiltonian at high spins

The BRST treatment of triaxial systems rotating at high spins is used to solve perturbatively the γ-independent Bohr collective hamiltonian.