Analytical solutions of the Bohr Hamiltonian are obtained in the $γ$-unstable case, as well as in an exactly separable rotational case with $γ\approx 0$, called the exactly separable Morse (ES-M) solution. Closed expressions for the energy eigenvalues are obtained through the Asymptotic Iteration Method (AIM), the effectiveness of which is demonstrated by solving the relevant Bohr equations for the Davidson and Kratzer potentials. All medium mass and heavy nuclei with known $β_1$ and $γ_1$ bandheads have been fitted by using the two-parameter $γ$-unstable solution for transitional nuclei and the three-parameter ES-M for rotational ones. It is shown that bandheads and energy spacings within the bands are well reproduced for more than 50 nuclei in each case.
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.
In the present work, we have obtained closed analytical expressions for eigenvalues and eigenfunctions of the Bohr Hamiltonian with the Manning-Rosen potential for γ-unstable nuclei as well as exactly separable rotational ones with γ=0. Some heavy nuclei with known \b{eta} and γ bandheads have been fitted by using two parameters in the ν-unstable case and three parameters in the axially symmetric prolate deformed one. A good agreement with experimental data has been achieved.
The following special solutions of the Bohr Hamiltonian are briefly described: 1) Z(5) (approximately separable solution in five dimensions with gamma close to 30 degrees), 2) Z(4) (exactly separable gamma-rigid solution in four dimensions with gamma = 30 degrees), 3) X(3) (exactly separable gamma-rigid solution in three dimensions with gamma =0). The analytical solutions obtained using Davidson potentials in the E(5), X(5), Z(5), and Z(4) frameworks are also mentioned.
Approximate analytical solutions in closed form are obtained for the 5-dimensional Bohr Hamiltonian with the Woods-Saxon potential, taking advantage of the Pekeris approximation and the exactly soluble one-dimensional extended Woods-Saxon potential with a dip near its surface. Comparison to the data for several gamma-unstable and prolate deformed nuclei indicates that the potential can describe well the ground state and gamma-1 bands of many prolate deformed nuclei corresponding to large enough "well size" and diffuseness, while it fails in describing the beta-1 bands, due to its lack of a har…
The Bohr Hamiltonian describing the collective motion of atomic nuclei is modified by allowing the mass to depend on the nuclear deformation. Exact analytical expressions are derived for spectra and wave functions in the case of a gamma-unstable Davidson potential, using techniques of supersymmetric quantum mechanics. Numerical results in the Xe-Ba region are discussed.
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…
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 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.
Analytical expressions of the wave functions are derived for a Bohr Hamiltonian with the Manning{Rosen potential in the cases of γ-unstable nuclei and axially symmetric prolate deformed ones with γ=0. By exploiting the results we have obtained in a recent work on the same theme Ref. [1], we have calculated the B(E2) transition rates for 34 γ-unstable and 38 rotational nuclei and compared to experimental data, revealing a qualitative agreement with the experiment and phase transitions within the ground state band and showing also that the Manning-Rosen potential is more appropriate for such cal…
In this paper, we present new analytical solutions of the Bohr Hamiltonian problem that we derived with the Tietz-Hua potential, here used for describing the β-part of the nuclear collective potential plus harmonic oscillator one for the γ-part. Also, we proceed to a systematic comparison of the numerical results obtained with this kind of β-potential with others which are widely used in such a framework as well as with the experiment. The calculations are carried out for energy spectra and electromagnetic transition probabilities for γ-unstable and axially symmetric deformed nuclei. In the sa…
In this paper, we present an analytical solution for the Bohr Hamiltonian with the trigonometric Pöschl Teller (P.T) potential in the cases of γ unstable nuclei and γ stable axially symmetric prolate deformed ones with γ = 0. The energy spectra and corresponding wave functions are derived by means of the asymptotic iteration method. In addition, B(E2) transition rates are calculated and compared with experimental data. Overall good agreement is obtained for inter and intra band transitions within ground state and \b{eta} bands. Our numerical results, particularly for transition rates are much …
An exact analytical solution for the Bohr Hamiltonian with an energy dependent Coulomb-like $γ$-unstable potential is presented. Due to the linear energy dependence of the potential's coupling constant, the corresponding spectrum in the asymptotic limit of the slope parameter resembles the spectral structure of the spherical vibrator, however with a different state degeneracy. The parameter free energy spectrum as well as the transition rates for this case are given in closed form and duly compared with those of the harmonic $U(5)$ dynamical symmetry. The model wave functions are found to exhi…
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…
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.
Bohr-Mottelson Hamiltonian with octic potential applied to the $^{106-116}$Cd isotopes
Department of Theoretical Physics, "Horia Hulubei" − - National Institute for R&D in Physics and Nuclear Engineering（希里亚·胡卢贝伊理论物理系，核物理与核工程研发国家研究所）；Academy of Romanian Scientists（罗马尼亚科学家学院）
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…
An analytical solution of the collective Bohr equation with a Coulomb-like and a Kratzer-like $γ-$unstable potential in quadrupole deformation space is presented. Eigenvalues and eigenfunctions are given in closed form and transition rates are calculated for the two cases. The corresponding SO(2,1)$\times$SO(5) algebraic structure is discussed.
In this work, the Davydov-Chaban Hamiltonian, describing the collective motion of $γ$-rigid atomic nuclei, is amended by allowing the mass parameter to depend on the nuclear deformation. Further, Z(4)-DDM (Deformation-Dependent Mass) model is proposed by considering the Kratzer potential for the $β$ variable, and solving the problem by techniques of asymptotic iteration method (AIM). The results of the calculated spectra and $B(E2)$ transition rates for series of $^{192-196}$Pt isotopes are compared with the corresponding experimental data as well as with other theoretical models. Exact analyt…
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…
Quantum scattering in the presence of a potential valley followed by a barrier is examined for the case of a Morse potential, for which exact analytic solutions to the Schr\UNICODE{0xf6}dinger equation are known in terms of confluent hypergeometric functions. For our application the potential is characterized by three parameters: the height of the barrier, the distance of the barrier from the origin of the radial variable $r,$ and a diffuseness parameter. The wave function, defined in the interval $0\leq r<\infty ,$ is required to vanish at $r=0,$ and hence represents a radial partial wave for…
For non-zero $\ell$ values, we present an analytical solution of the radial Schrödinger equation for the rotating Morse potential using the Pekeris approximation within the framework of the Asymptotic Iteration Method. The bound state energy eigenvalues and corresponding wave functions are obtained for a number of diatomic molecules and the results are compared with the findings of the super-symmetry, the hypervirial perturbation, the Nikiforov-Uvarov, the variational, the shifted 1/N and the modified shifted 1/N expansion methods.
We investigate phase transitions in boson-fermion systems. We propose an analytically solvable model (E(5/12)) to describe odd nuclei at the critical point in the transition from the spherical to $γ$-unstable behaviour. In the model, a boson core described within the Bohr Hamiltonian interacts with an unpaired particle assumed to be moving in the three single particle orbitals j=1/2,3/2,5/2. Energy spectra and electromagnetic transitions at the critical point compare well with the results obtained within the Interacting Boson Fermion Model, with a boson-fermion Hamiltonian that describes the s…
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 \b{eta} shape variable. In the resulting solution, called X(3)-D-ML, the ground state and the first \b{eta}-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 exc…
The quantitative formulation of Bohr's complementarity proposed by Greenberger and Yasin is applied to some physical situations for which analytical expressions are available. This includes a variety of conventional double-slit experiments, but also particle oscillations, as in the case of the neutral-kaon system, and Mott scattering of identical nuclei. For all these cases, a unified description can be achieved including a new parameter, $ν$, which quantifies the effective number of fringes one can observe in each specific interferometric set-up.
A sixth order quadrupole boson Hamiltonian is treated through a time dependent variational principle approach choosing as trial function a coherent state with respect to zeroth $b^{\dagger}_0$ and second $b^{\dagger}_2+b^{\dagger}_{-2}$ components of the quadrupole bosons. The coefficients involved in the model Hamiltonian are chosen so that the classical effective potential energy term has two distinct minima. The equation of motion for the radial coordinate is analytically solved and the resulting trajectories are extensively studied. One distinguishes three energy regions exhibiting differe…
The real Scarf II potential is discussed as a radial problem. This potential has been studied extensively as a one-dimensional problem, and now these results are used to construct its bound and resonance solutions for $l=0$ by setting the origin at some arbitrary value of the coordinate. The solutions with appropriate boundary conditions are composed as the linear combination of the two independent solutions of the Schrödinger equation. The asymptotic expression of these solutions is used to construct the $S_0(k)$ s-wave $S$-matrix, the poles of which supply the $k$ values corresponding to the…
More recently, comprehensive application results of approximate analytical solutions of the Woods-Saxon potential in closed form for the 5-dimensional Bohr Hamiltonian have been appeared [14] and its comparison to the data for many different nuclei has clearly revealed the domains for the sucsess and failure in case of using such potential forms to analyse the data concerning with the nuclear structure of deformed nuclei within the frame of the collective model. Gaining confidence from this work, exact solvability of the Woods-Saxon type potentials in lower dimensions for the bound states havi…
In this paper, we determine deuteron's static properties, low energy scattering parameters, total cross-section and form factors from inverse S-wave potentials constructed using Morse function. The scattering phase shifts (SPS) at different lab energies are determined using phase function method. The model parameters are optimised using both machine learning algorithm and traditional data analysis by choosing mean squared error as cost function. The mean absolute error between experimental and obtained SPS for states 3S1 and 1S0 are found to be 0.35 and 0.70 respectively. The low energy scatte…
