Bohr-Mottelson Hamiltonian Research Articles

Study of Bohr Mottelson Hamiltonian with minimal length effect for Woods-Saxon potential and its thermodynamic properties

The Bohr Mottelson Hamiltonian with the variable of β collective shape for the Woods-Saxon potential in the rigid deformed nucleus for γ=0 and the X(3) model was investigated in the presence of the minimal length formalism. The Bohr Mottelson Hamiltonian was solved approximately by proposing a new wave function. The q-deformed hyperbolic potential concept such that the rigid deformed nucleus of the Bohr Mottelson equation in the minimal length formalism for Woods-Saxon potential was used, so that the equation was reduced to the form of Schrodinger-like equation with cotangent hyperbolic potential. The hypergeometric method was used to obtain the energy spectra equation and the unnormalized wave function of the system. The results showed that the energy spectra were affected by the quantum number, the minimal length parameter, and the atomic mass. The larger mass of the atom affected the energy spectra to decrease, the increase of the values of the minimal length affected the increase of the energy spectra of all atoms. The energy spectra were used to determine the thermodynamic properties including the partition function, mean energy, specific heat, free energy, and entropy of the quantum system with the help of the imaginary error function.

On the conformable fractional E(5) critical point symmetry

A new category of critical point symmetries is introduced. It is suggested that an element of this category, associated with zeros of conformable fractional Bessel functions, be utilized to describe spectra of nuclei around the E(5) critical point. The exact eigenvalue and eigenfunction solutions of local fractional Bohr-Mottelson Hamiltonian (with infinite square well potential) are obtained. The evolution of the spectra of the fractional E(5) critical point in correspondence with the fractional derivative order is investigated. Using the fractional E(5) critical point, a satisfactory description of the energy levels is obtained in the 102Pd and 104Ru nuclei.

The Deformation Dependent Mass Davidson Model

The Deformation Dependent Mass Davidson Model is an extension of the well known Bohr-Mottelson Hamiltonian for the atomic nuclei. It primarily refers to the mass dependence on the deformation and secondary to the Davidson behavior for the potential of the Ø-vibration. This article will be devoted solely in the solution of the radial equation. Fitting results for the 162Dy and 238U ground state, β1 and γ1 bands are also presented.

Study of Bohr–Mottelson with minimal length effect for Q-deformed modified Eckart potential and Bohr–Mottelson with Q-deformed quantum for three-dimensional harmonic oscillator potential

Bohr–Mottelson Hamiltonian on the γ-rigid regime for Q-deformed modified Eckart and three-dimensional harmonic oscillator potentials in the β-collective shape variable was investigated in the presence of minimal length formalism and Q-deformed of the radial momentum part. By introducing new wave function and using the Q-deformed potential concept in Bohr–Mottelson Hamiltonian in the minimal length formalism, the un-normalized wave function and energy spectra equation were obtained by using the hypergeometric method. Meanwhile, the Bohr–Mottelson Hamiltonian in the presence of the quadratic spatial deformation to the momentum in collective shape variable was investigated using transformation of a new variable such as the Schrodinger-like equation with shape invariant potential. The energy equation and un-normalized wave function were obtained using the hypergeometric method. The results showed that the Bohr–Mottelson equations with different energy potentials and different deformation forms in the radial momentum reduced to the similar Schrodinger-like equation with the modified Poschl–Teller potential.

Is 198Hg a soft triaxial nucleus with $\gamma = 30^{\circ}$?

Our purpose with this paper is mainly to raise the question about the shape of 198Hg and to suggest it as a testing ground for different models. The spectroscopic properties of this nucleus have been investigated by applying a computational code based on the diagonalization the Bohr-Mottelson Hamiltonian with the generalized Gneuss-Greiner potential. The coefficients of the kinetic energy operator and potential energy for this nucleus have been obtained by including the experimental data of some levels in the fitting code. These calculations indicate that the $0^{+}_{2}$ level of nucleus 198Hg is found somewhere between $ 1.23 \lesssim 0^{+}_{2} \lesssim 1.65$ that is precisely the region where a few excited 0+ states are found. Also, they predict a rather rigid triaxial shape around $ \gamma = 30^{\circ}$, that is soft in the $ \beta$ variable.

Collective motion in prolate [formula omitted]-rigid nuclei within minimal length concept via a quantum perturbation method

Based on the minimal length concept, inspired by Heisenberg algebra, a closed analytical formula is derived for the energy spectrum of the prolate γ-rigid Bohr–Mottelson Hamiltonian of nuclei, within a quantum perturbation method (QPM), by considering a scaled Davidson potential in β shape variable. In the resulting solution, called X(3)-D-ML, the ground state and the first β-band are all studied as a function of the free parameters. The fact of introducing the minimal length concept with a QPM makes the model very flexible and a powerful approach to describe nuclear collective excitations of a variety of vibrational-like nuclei. The introduction of scaling parameters in the Davidson potential enables us to get a physical minimum of this latter in comparison with previous works. The analysis of the corrected wave function, as well as the probability density distribution, shows that the minimal length parameter has a physical upper bound limit.

Bohr Hamiltonian with a potential having spherical and deformed minima at the same depth

A solution for the Bohr-Mottelson Hamiltonian with an anharmonic oscillator potential of sixth order, obtained through a diagonalization in a basis of Bessel functions, is presented. The potential is consid- ered to have simultaneously spherical and deformed minima of the same depth separated by a barrier (a local maximum). This particular choice is appropriate to describe the critical point of the nuclear phase transition from a spherical vibrator to an axial rotor. Up to a scale factor, which can be cancelled by a corresponding normalization, the energy spectra and the electromagnetic E2 transition probabilities depend only on a single free parameter related to the height of the barrier. Investigations of the numerical data revealed that the model represents a good tool to describe this critical point.

Anharmonic oscillator potentials in the prolate γ-rigid regime of the collective geometrical model

Based on higher order JWKB approximation, an analytical formula is derived for the energy spectrum of the prolate γ-rigid Bohr-Mottelson Hamiltonian with an oscillator potential and a quartic or sextic anharmonicity in β shape variable. The model is found to provide good description for near vibrational nuclei situated in vicinity of a closed neutron shell.

Investigation of the information entropy for the X(3) model

In this work, we have studied the information entropy for the Bohr-Mottelson Hamiltonian relating to the X(3) model. For this purpose, first the eigenvalues as well as eigenstates are determined through solving the relevant wave equation. Then, the information entropy density and its quantity for the obtained wave functions are determined. Moreover, uncertainty for both the $\beta$ shape variable and momentum are specified. Furthermore, the uncertainty principle is investigated for the nuclei studied by the X(3) model.

Investigation of Bohr–Mottelson Hamiltonian in γ-rigid version with position dependent mass

In this paper, we consider the Bohr–Mottelson Hamiltonian in γ-rigid version with position dependent mass. The separation of variables has been done for the related wave equation. The obtained radial wave equation is solved for Kratzer potential. Then, the corresponding wave function, energy spectra and transition rates have been obtained for some nuclei. In addition, our results have been compared with experimental data.

Alternative solution of the gamma-rigid Bohr Hamiltonian in minimal length formalism

The Bohr–Mottelson Hamiltonian on γ-rigid regime has been extended to the minimal length formalism for the infinite square well potential and the corresponding wave functions as well as the spectra are obtained. The effect of minimal length on energy spectra is studied via various figures and tables and numerical calculations are included for some nuclei and the results are compared with other results and existing experimental data.

Gamma-rigid regime of the Bohr–Mottelson Hamiltonian in energy-dependent approach

We determine the energy spectrum and wave function for the Bohr–Mottelson Hamiltonian on [Formula: see text]-rigid regime separately with the harmonic and Coulomb energy-dependent potentials. We study the effect of potential parameters on the energy levels and probability density distribution. The transition rates are determined in each case.

On γ -rigid regime of the Bohr–Mottelson Hamiltonian in the presence of a minimal length

A prolate {\gamma}-rigid regime of the Bohr-Mottelson Hamiltonian within the minimal length formalism, involving an infinite square well like potential in {\beta} collective shape variable, is developed and used to describe the spectra of a variety of vibrational-like nuclei. The effect of the minimal length on the energy spectrum and the wave function is duly investigated. Numerical calculations are performed for some nuclei revealing a qualitative agreement with the available experimental data.

Sextic potential for -rigid prolate nuclei

The equation of the Bohr–Mottelson Hamiltonian with a sextic oscillator potential is solved for -rigid prolate nuclei. The associated shape phase space is reduced to three variables which are exactly separated. The angular equation has the spherical harmonic functions as solutions, while the equation is converted to the quasi-exactly solvable case of the sextic oscillator potential with a centrifugal barrier. The energies and the corresponding wave functions are given in closed form and depend, up to a scaling factor, on a single parameter. The and states are exactly determined, having an important role in the assignment of some ambiguous states for the experimental bands. Due to the special properties of the sextic potential, the model can simulate, by varying the free parameter, a shape phase transition from a harmonic to an anharmonic prolate -soft rotor crossing through a critical point. Numerical applications are performed for 39 nuclei: Ru, Mo, Xe, Ce, Nd, Sm, Gd, Dy, 172Os, Pt, 190Hg and 222Ra. The best candidates for the critical point are found to be 104Ru and Xe, followed closely by 128Xe, 172Os, 196Pt and 148Nd.

Harmonic oscillator potential with a sextic anharmonicity in the prolate γ-rigid collective geometrical model

An analytical expression for the energy spectrum of the ground and β bands was obtained through the JWKB approximation in the axially symmetric γ-rigid regime of the Bohr–Mottelson Hamiltonian with an oscillator potential and a sextic anharmonicity in the β shape variable. Due to the scaling property of the problem, the resulting energy depends up to an overall multiplicative constant on a single parameter. Studying the behavior of the energy spectrum as a function of the free parameter, one establishes the present model's place among other prolate γ-rigid models and in the more general extent of collective solutions. The agreement with experiment is achieved through model fits for few nuclei.

Quartic oscillator potential in the γ-rigid regime of the collective geometrical model

A prolate γ-rigid version of the Bohr-Mottelson Hamiltonian with a quartic anharmonic oscillator potential in β collective shape variable is used to describe the spectra for a variety of vibrational-like nuclei. Speculating the exact separation between the two Euler angles and the β variable, one arrives at a differential Schrödinger equation with a quartic anharmonic oscillator potential and a centrifugal-like barrier. The corresponding eigenvalue is approximated by an analytical formula depending only on a single parameter up to an overall scaling factor. The applicability of the model is discussed in connection to the existence interval of the free parameter, which is limited by the accuracy of the approximation, and by comparison with the predictions of the related X(3) and X(3)-β 2 models. The model is applied to qualitatively describe the spectra for nine nuclei which exhibit near-vibrational features.

Application of the sextic oscillator potential together with Mathieu and spheroidal functions for triaxial and X(5) type nuclei

The Bohr-Mottelson Hamiltonian is amended with a potential which depends on both β and γ deformation variables and which allows us to separate the β variable from the other variables. The equation for the β variable is quasi-exactly solved for a sextic oscillator with centrifugal barrier potential. Concerning the γ equation, its solutions are the angular spheroidal and Mathieu functions for X(5) type and triaxial nuclei, respectively. The models developed in this way are conventionally called the Sextic and Spheroidal Approach (SSA) and the Sextic and Mathieu Approach (SMA). SSA and SMA was successfully applied for several nuclei, details being presented below.

Description of the isotope chain180−196Pt within several solvable approaches

Energies of the ground, $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ bands as well as the associated $B(E2)$ values are determined for each even-even isotope of the ${}^{180\ensuremath{-}196}$Pt chain by the exact solutions of some differential equations which approximate the generalized Bohr-Mottelson Hamiltonian. The emerging approaches are called the sextic and spheroidal approach (SSA), the sextic and Mathieu approach (SMA), the infinite square well and spheroidal approach (ISWSA), and the infinite square well and Mathieu approach (ISWMA). While the first three methods were formulated in some earlier papers, ISWMA is an unedited approach of this work. Numerical results are compared with those obtained with the so-called X(5) and Z(5) models. A contour plot for the probability density as function of the intrinsic dynamic deformations is given for a few states of the three considered bands with the aim of evidencing the shape evolution along the isotope chain and pointing out possible shape coexistence.

Towards a new solvable model for the even-even triaxial nuclei

The even-even triaxial nuclei are described by amending the Bohr-Mottelson Hamiltonian with an energy potential consisting of two terms: a sextic oscillator with centrifugal barrier in the β variable and a periodic function in the γ variable. After the variable separation is performed, the β equation is quasi-exactly solved, while the γ equation is satisfied by the Mathieu function. The reduced E2 transition probabilities are determined using an anharmonic transition operator. The formalism is conventionally called the Sextic and Mathieu Approach (SMA). Numerical applications concerned seven non-axial nuclei: 188Os, 190Os, 192Os, 228Th, 230Th, 182W and 180Hf. SMA results are compared with the experimental data as well as with those yielded by the Coherent State Model (CSM).

Application of the sextic oscillator with a centrifugal barrier and the spheroidal equation for some X(5) candidate nuclei

The eigenvalue equation associated with the Bohr–Mottelson Hamiltonian is considered in the intrinsic reference frame and amended by replacing the harmonic oscillator potential in the β variable with a sextic oscillator potential with centrifugal barrier plus a periodic potential for the γ variable. After the separation of variables, the β equation is quasi-exactly solved, while the solutions for the γ equation are just the angular spheroidal functions. An anharmonic transition operator is used to determine the reduced E2 transition probabilities. The formalism is conventionally called the sextic and spheroidal approach (SSA) and applied for several X(5) candidate nuclei: 176, 178, 180, 188, 190Os, 150Nd, 170W, 156Dy and 166, 168Hf. The SSA predictions are in good agreement with the experimental data of the mentioned nuclei. The comparison of the SSA results with those yielded by other models, such as X(5) (Iachello 2001 Phys. Rev. Lett.87052502), infinite square well (Raduta et al 2009 Nucl. Phys.A 81946) and Davidson like potential (Raduta et al) for the β, otherwise keeping the spheroidal functions for the γ , and the coherent state model (Raduta et al 1981 Phys. Lett. B 99 444, Raduta et al 1982 Nucl. Phys.A 381 253, Raduta et al 1987 Phys. Rev. C 36 2111, Raduta et al 1983 Z. Phys. A 313 69, Raduta et al 1997 Phys. Rev. C 55 1747, Raduta et al 2002 Phys. Rev. C 65 064322, Raduta and Sabac 1983 Ann. Phys., NY 148 1, Raduta 2004 Recent Research Developments in Nuclear Physics vol 1) respectively, suggests that SSA represents a good approach to describe nuclei achieving the critical point of the U(5)→SU(3) shape phase transition.