In this work, we solve the eigenvalues problem with the Bohr collective Hamiltonian for triaxial nuclei within Deformation-Dependent Mass formalism (DDM) using the Hulthén potential. We shall call the solution developed here Z(5)-HDDM. Analytical expressions for energy spectra are derived by means of a recent version of the Asymptotic Iteration Method. The calculated numerical results are compared with the experimental data, and the model Z(5)-H using the Hulthén potential without DDM formalism as well as theoretical predictions of Z(5)-DDDM model with Davidson potential.
The approximate solution of Bohr-Mottelson Hamiltonian in rigid deformed nucleus case for Hulthen potential with minimal length effect was investigated using Asymptotic Iteration Method. Asymptotic Iteration Method was used to solve approximately the Bohr-Mottelson Hamiltonian to obtain energy spectrum and un-normalized wave function. The energy spectrum was calculated numerically using the Matlab software. The un-normalized wave function was expressed in the hypergeometric term. The results showed that the energy spectrum increased due to the increasing minimal length parameter. The energy sp…
In this paper, we solve the eigenvalues and eigenvectors problem with Bohr collective Hamil- tonian for triaxial nuclei. The ? beta part of the collective potential is taken to be equal to Hulth?en potential while the gamma part is defined by a new generalized potential obtained from a ring shaped one. Analytical expressions for spectra and wave functions are derived by means of a recent version of the asymptotic iteration method and the usual approximations. The calculated energies and B(E2) transition rates are compared with experimental data and the available theoretical results in the lite…
In this work, we present a new version of the Bohr collective Hamiltonian for triaxial nuclei within Deformation-Dependent Mass formalism (DDM) using the Hulthén 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 Hulthén potential without DDM formalism as well as theoretical predictions…
In this work we solve the Schrödinger equation for Bohr Hamiltonian with Coulomb and Hulthén potentials within the formalism of minimal length in order to obtain analytical expressions for the energy eigenvalues and eigenfunctions by means of asymptotic iteration method. The obtained formulas of the energy spectrum and wave functions, are used to calculate excitation energies and transition rates of $γ$-rigid nuclei and compared with the experimental data at the shape phase critical point X(3) in nuclei.
A solution of the Bohr Hamiltonian appropriate for triaxial shapes, involving a Davidson potential in beta and a steep harmonic oscillator in gamma, centered around gamma=30 degrees, is developed. Analytical expressions for spectra and B(E2) transition rates ranging from a triaxial vibrator to the rigid triaxial rotator are obtained and compared to experiment. Using a variational procedure it is pointed out that the Z(5) solution, in which an infinite square well potential in beta is used, corresponds to the critical point of the shape phase transition from a triaxial vibrator to the rigid tri…
Detailed quantitative predictions are obtained for phonon and multiphonon excitations in well-deformed rotor nuclei within the geometric framework, by exact numerical diagonalization of the Bohr Hamiltonian in an SO(5) basis. Dynamical gamma deformation is found to significantly influence the predictions through its coupling to the rotational motion. Basic signatures for the onset of rigid triaxial deformation are also obtained.
The triaxial nature of low-lying rotational bands of $^{166}$Er is presented from the viewpoint of the Bohr Hamiltonian and from that of many-fermion calculations by the Monte Carlo shell model and the constrained Hartree-Fock method with projections. A recently proposed novel picture of those bands suggests definite triaxial shapes of those bands, in contrast to the traditional view with the prolate ground-state band and the $γ$-vibrational excited band. Excitation level energies and E2 transitions can be described well by the Bohr Hamiltonian and by the many-fermion approaches, where rather …
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 $γ$, which is solved with an approximated algebraic technique, and Coulomb/Kratzer, harmonic/Davidson and infinite square well potentials in $β$, 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 we compare our results with experimental information and previous cal…
The collective Bohr hamiltonian is solved for the soft triaxial rotor around $γ_0=π/6$ 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.
Exact numerical diagonalization of the Bohr Hamiltonian by SU(1,1)xSO(5) methods is used to obtain detailed quantitative predictions for single-phonon and multi-phonon excitations in well-deformed rotor nuclei. Dynamical gamma deformation is found to significantly influence the predictions through its coupling to the rotational motion. Basic signatures for the onset of rigid triaxial deformation are obtained.
In this work, we derive a closed solution of the Shr$ \ddot{o} $dinger equation for Bohr Hamiltonien within the minimal length formalism. This formalism is inspired by Heisenberg algebra and a generlized uncertainty principle (GUP), applied to the geometrical collective Bohr- Mottelson model (BMM) of nuclei by means of deformed canonical commutation relation and the Pauli-Podolsky prescription. The problem is solved by means conjointly of asymptotic iteration method (AIM) and a quantum perturbation method (QPM) for transitional nuclei near the critical point symmetry Z(4) corresponding to phas…
A solution of the Bohr collective hamiltonian for the $β-$soft, $γ-$soft triaxial rotor with $γ\sim π/6$ is presented making use of a harmonic potential in $γ$ and Coulomb-like and Kratzer-like potentials in $β$. It is shown that, while the $γ-$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 $β$ 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 agr…
The Deformation Dependent Mass (DDM) Kratzer model is constructed by considering the Kratzer potential in a Bohr Hamiltonian, in which the mass is allowed to depend on the nuclear deformation, and solving it by using techniques of supersymmetric quantum mechanics (SUSYQM), involving a deformed shape invariance condition. Analytical expressions for spectra and wave functions are derived for separable potentials in the cases of gamma-unstable nuclei, axially symmetric prolate deformed nuclei, and triaxial nuclei, implementing the usual approximations in each case. Spectra and B(E2) transition ra…
We present a computer program which solves the Schrodinger equation of the stationary states for an average nuclear potential of Woods-Saxon type. In this work, we take specifically into account triaxial (i.e. ellipsoidal) nuclear surfaces. The deformation is specified by the usual Bohr parameters. The calculations are carried out in two stages. In the first, one calculates the representative matrix of the Hamiltonian in the cartesian oscillator basis. In the second stage one diagonalizes this matrix with the help of subroutines of the EISPACK library. If it is wished, one can calculate all ei…
The phenomenological classification of collective quadrupole excitations by means of the Bohr Hamiltonian is reviewed with focus on signatures for triaxility. The variants of the microscopic Bohr Hamiltonian derived by means of the Adiabatic Time Dependent Mean Field theory from the Pairing plus Quadrupole-Quadrupole interaction, the Shell Correction Method, the Skyrme Energy Density Functional, the Relativistic Mean Field Theory, and the Gogny interaction are discussed and applications to concrete nuclides reviewed. The Generator Coordinate Method for the five dimensional quadrupole deformati…
The Bohr hamiltonian, also called collective hamiltonian, is one of the cornerstone of nuclear physics and a wealth of solutions (analytic or approximated) of the associated eigenvalue equation have been proposed over more than half a century (confining ourselves to the quadrupole degree of freedom). Each particular solution is associated with a peculiar form for the $V(β,γ)$ potential. The large number and the different details of the mathematical derivation of these solutions, as well as their increased and renewed importance for nuclear structure and spectroscopy, demand a thorough discussi…
The concept of minimal length, inspired by Heisenberg algebra, is applied to the geometrical collective Bohr- Mottelson model (BMM) of nuclei. With the deformed canonical commutation relation and the Pauli-Podolsky prescription, we have derived the quantized Hamiltonian operator for triaxial nuclei as we have previously done for axial prolate $γ$-rigid ones (M. Chabab et al., Phys. Lett. B 758 (2016) 212-218). By considering an infinite square well like potential in $β$ collective shape variable, the eigenvalues of the Hamiltonian are obtained in terms of zeros of Bessel functions of irrationa…
Analytical expressions for spectra and wave functions are derived for a Bohr Hamiltonian, describing the collective motion of deformed nuclei, in which the mass is allowed to depend on the nuclear deformation. Solutions are obtained for separable potentials consisting of a Davidson potential in the beta variable, in the cases of gamma-unstable nuclei, axially symmetric prolate deformed nuclei, and triaxial nuclei, implementing the usual approximations in each case. The solution, called the Deformation Dependent Mass (DDM) Davidson model, is achieved by using techniques of supersymmetric quantu…
A systematic study of $γ$-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 $γ$-band for most 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 $γ$-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 indicati…
A microscopic quantum ideal rotor-model intrinsic Hamiltonian for triaxial rotation is derived from the nuclear Schrodinger equation by applying a rotation operator to a deformed nuclear ground state. This Hamiltonian is obtained only when a rigid-flow prescription is used for the three rotation angles in the rotation operator. Using Hartree-Fock variational and second quantization methods, the rotor Hamiltonian is transformed into that of a self-consistent triaxial cranking model (MSCRM-3) with a self-consistent angular-velocity vector, plus residual terms associated with the square of the an…
Triaxial shapes and the angular structure of nuclear three-body correlations
Institute of Modern Physics, Fudan University（复旦大学现代物理研究所）；School of Physics and Center for High Energy Physics, Peking University（北京大学物理学院高能物理中心）；Theoretical Physics Department, CERN（欧洲核子研究中心理论物理部）；Instituto de Física, Universidade de São Paulo（圣保罗大学物理研究所）
Motivation : Several 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. Method : The 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 conventiona…
