The 3-D Bohr-Mottelson Hamiltonian for $γ$-rigid prolate isotopes, known as $X(3)$, is solved via inverse square potential having only one free parameter, $β_{0}$. The exact form of the wave functions and the energy spectra are obtained as a function of the free parameter of the potential that determines the changes in the spectra ratios and the $B(E2)$. Since $X(3)$ is an exactly separable $γ$-rigid version of $X(5)$, the solutions are compared with the $X(5)$ model and some new set of equations that show the relationships between the two models are stated. In other to show the dynamical symmetry nature of the solutions, the entire solutions from $β_{0}=0$ to $β_{0}=\infty$ are compared with $U(5)$, $X(5)$ and $SU(3)$. The solutions spread from the region around $U(5)$ over $X(5)$ and approach $SU(3)$ at $β_{0}=\infty$. The exact solutions obtained via variational procedure are compared favourably with some existing $X(3)$ models found in the literature. The strong agreement between the present model and $X(3)$ via infinite square well potential is discussed. Twelve best critical point isotopes, $^{102}$Mo, $^{104-108}$Ru, $^{120-126}$Xe, $^{148}$Nd, $^{184-188}$Pt are chosen for experimental realization of the model and moderate agreements are recorded. An excellent agreement which appears in the first $β$-excited state in the comparison of the present model with three $N=90$ isotones: $^{150}$Nd, $^{154}$Gd, and $^{156}$Dy, known to be $X(5)$ candidates, suggests that the present model compensates the $X(5)$ models whose predictions are excellent in the ground states but moderately bad in the first $β$-excited states.
The article reviews the general version of the Bohr collective model for the description of quadrupole collective states, including a detailed study the model's kinematics. The general form of the classical and quantum Bohr Hamiltonian is presented together with a discussion of the tensor structure of the collective wave functions and with a short review of various methods of solving the Bohr Hamiltonian eigenvalue equation.The methods of derivation of the classical and quantum Bohr Hamiltonian from the microscopic many-body theory are recalled and the microscopic approach to the Bohr Hamilton…
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.
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…
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.
We calculate the eigenvalues and their corresponding eigenfunctions of the Bohrs collective Hamiltonian with the help of the modified Poschl-Teller potential model within -unstable structure. Our numerical results for the ground state beta and gamma band heads together with the electric quadrupole B(E2) transition rates are displayed and compared with those available experimental data.
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.
A gamma-rigid solution of the Bohr Hamiltonian is derived for gamma=0 utilizing the Davidson potential in the beta variable. This solution is going to be called X(3)-D. The energy eigenvalues and wave functions are obtained by using an analytic method which has been developed by Nikiforov and Uvarov. BE(2) transition rates are calculated. A variational procedure is applied to energy ratios to determine whether or not the X(3) model is located at the critical point between spherical and deformed 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.
Nuclei exhibit quantum phase transitions (earlier called ground state phase transitions) between different shapes as the number of nucleons is modified, resulting in changes in the ground and low lying nuclear states. Special solutions of the Bohr Hamiltonian appropriate for the critical point of such shape phase transitions, as well as other special solutions applicable to relevant nuclear regions are described.
An analytic, parameter-free (up to overall scale factors) solution of the Bohr Hamiltonian involving axially symmetric quadrupole and octupole deformations, as well as an infinite well potential, is obtained, after separating variables in a way reminiscent of the Variable Moment of Inertia (VMI) concept. Normalized spectra and B(EL) ratios are found to agree with experimental data for 226-Ra and 226-Th, the nuclei known to lie closest to the border between octupole deformation and octupole vibrations in the light actinide region.
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…
A gamma-rigid solution of the Bohr Hamiltonian for gamma = 30 degrees is derived, its ground state band being related to the second order Casimir operator of the Euclidean algebra E(4). Parameter-free (up to overall scale factors) predictions for spectra and B(E2) transition rates are in close agreement to the E(5) critical point symmetry, as well as to experimental data in the Xe region around A=130.
Embedding of the 5-dimensional (5D) space of the Bohr Hamiltonian with a deformation-dependent mass (DDM) into a 6-dimensional (6D) space shows that the free parameter in the dependence of the mass on the deformation is connected to the curvature of the 5D space, with the special case of constant mass corresponding to a flat 5D space. Comparison of the DDM Bohr Hamiltonian to the 5D classical limit of Hamiltonians of the 6D interacting boson model (IBM), shows that the DDM parameter is proportional to the strength of the pairing interaction in the U(5) (vibrational) symmetry limit, while it is…
A prolate γ-rigid regime of the Bohr-Mottelson Hamiltonian within the minimal length formalism, involving an infinite square well like potential in β 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.
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.
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 t…
The generalized Bohr Hamiltonian is applied to a description of low-lying collective excitations in even-even isotopes of Te, Xe, Ba, Ce, Nd and Sm. The collective potential and inertial functions are determined by means of the Strutinsky method and the cranking model, respectively. A shell-dependent parametrization of the Nilsson potential is used. An approximate particle-number projection is performed in treatment of pairing correlations. The effect of coupling with the pairing vibrations is taken into account approximately when determining the inertial functions. The calculation does not co…
New analytic solutions of the quadrupole collective Bohr hamiltonian are proposed, exploiting an approximate separation of the beta and gamma variables to describe gamma-soft prolate axial rotors. The model potential is a sum of two terms: a beta-dependent term taken either with a Coulomb-like or a Kratzer-like form, and a gamma-dependent term taken as an harmonic oscillator. In particular it is possible to give a one parameter paradigm for a beta-soft, gamma-soft axial rotor that can be applied, with a considerable agreement, to the spectrum of 234U.
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…
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…
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…
We investigate the properties of the excited spectra of the even-even isotopes of krypton using a Generalised Bohr Hamiltonian with three different Skyrme functionals. In particular, we investigate the evolution of the low-lying $2^{+}_{1}$ and $4^{+}_{1}$ states and their associated electromagnetic transitions. The model reproduces quite nicely the energy trends apart from ${}^{88}$Kr, where none of the interactions used here are able to grasp a sudden change in the energy spectrum. Additionally, we explore the neutron deficient region ${}^{72-76}$Kr which is a proposed region for shape coexi…
An exactly separable version of the Bohr Hamiltonian is developed using a potential of the form u(beta)+u(gamma)/beta^2, with the Davidson potential u(beta)= beta^2 + beta_0^4/beta^2 (where beta_0 is the position of the minimum) and a stiff harmonic oscillator for u(gamma) centered at gamma=0. In the resulting solution, called exactly separable Davidson (ES-D), the ground state band, gamma band and 0_2^+ band are all treated on an equal footing. The bandheads, energy spacings within bands, and a number of interband and intraband B(E2) transition rates are well reproduced for almost all well-de…
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.
In a recent work (Phys.Rev.C84, 044321, 2011) M.J. Ermamatov and P.R. Fraser have studied rotational and vibrational excited states of axially symmetric nuclei within the Bohr Hamiltonian with different mass parameters. However, the energy formula that the authors have used contains some inaccuracies. So, the numerical results they obtained seem to be controversial. In this paper, we revisit all calculations related to this problem and determine the appropriate formula for the energy spectrum. Moreover, in order to improve such calculations, we reconsider this problem within the framework of t…
We discuss the nature of the low-frequency quadrupole vibrations from small-amplitude to large-amplitude regimes. We consider full five-dimensional quadrupole dynamics including three-dimensional rotations restoring the broken symmetries as well as axially symmetric and asymmetric shape fluctuations. Assuming that the time-evolution of the self-consistent mean field is determined by five pairs of collective coordinates and collective momenta, we microscopically derive the collective Hamiltonian of Bohr and Mottelson, which describes low-frequency quadrupole dynamics. We show that the five-dime…
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…
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 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…
Experimental data on $^{96}$Zr indicate coexisting spherical and deformed structures with small mixing amplitudes. We investigate the properties of the low-lying collective states of $^{96}$Zr based on the collective quadrupole Bohr Hamiltonian. The $β$-dependent collective potential having two minima -- spherical and deformed, is fixed so to describe experimental data in the best way.Good agreement with the experimental data on the excitation energies, $B(E2)$ and $B(M1)$ reduced transition probabilities is obtained. It is shown that the low-energy structure of $^{96}$Zr can be reproduced in …
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 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…
Shape/phase transitions in atomic nuclei have first been discovered in the framework of the Interacting Boson Approximation (IBA) model. Critical point symmetries appropriate for nuclei at the transition points have been introduced as special solutions of the Bohr Hamiltonian, stirring the introduction of additional new solutions describing wide ranges of nuclei. The complementarity of the IBA and geometrical approaches will be demonstrated by three examples. First, it will be shown that specific special solutions of the Bohr Hamiltonian correspond to the borders of the critical region of the …
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…
Experimental data on $^{96}$Zr indicate coexisting spherical and deformed structures with small mixing amplitudes. Several collective low-lying states and E2 and M1 transitions are observed for this nucleus. The quadrupole-collective Bohr Hamiltonian depending on both $β$ and $γ$ shape variables with a potential having spherical and deformed minima, is applied consideration of these data. The relative depth of two minima, height and width of the barrier, rigidity of the potential near both minima are determined so as to achieve a satisfactory description of the observed properties of the low-l…
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 …
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 $β$ shape variable, while the $γ$ potential is taken to be equal to the…
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.
Recently, a variant of the Bohr Hamiltonian was proposed where the mass term is allowed to depend on the beta variable of nuclear deformation. Analytic solutions of this modified Hamiltonian have been obtained using the Davidson and the Kratzer potentials, by employing techniques from supersymmetric quantum mechanics. Apart from the new set of analytic solutions, the newly introduced Deformation-Dependent Mass (DDM) model offered a remedy to the problematic behaviour of the moment of inertia in the Bohr Hamiltonian, where it appears to increase proportionally to the square of beta. In the DDM …
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 \betha and γvibrations and taking into account the Cor…
Basing on the Bohr collective quadrupole Hamiltonian the $A$-dependence of the Grodzins product is derived and the proportionality coefficient for the Grodzins relation is evaluated. The result obtained is in a correspondence with the experimental data.
Using in the Bohr Hamiltonian the approximations leading to the Bohr and Mottelson description of wobbling motion in even nuclei, a W(5) model for wobbling bands, coexisting with a X(5) ground state band, is obtained. Separation of variables is achieved by assuming that the relevant potential has a sharp minimum at gamma_0, which is the only parameter entering in the spectra and B(E2) transition rates (up to overall scale factors). B(E2) transition rates exhibit the features expected in the wobbling case, while the spectrum for gamma_0=20 degrees is in good agreement with experimental data for…
It is proved that the potentials of the form $β^{2n}$ (with $n$ being integer) provide a ``bridge'' between the U(5) symmetry of the Bohr Hamiltonian with a harmonic oscillator potential (occuring for $n=1$) and the E(5) model of Iachello (Bohr Hamiltonian with an infinite well potential, materialized for infinite $n$). Parameter-free (up to overall scale factors) predictions for spectra and B(E2) transition rates are given for the potentials $β^4$, $β^6$, $β^8$, corresponding to $R_4=E(4)/E(2)$ ratios of 2.093, 2.135, 2.157 respectively, compared to the $R_4$ ratios 2.000 of U(5) and 2.199 of…
Exact numerical results of the interacting boson model Hamiltonian along the integrable line from U(5) to O(6) are obtained by diagonalization within boson seniority subspaces. The matrix Hamiltonian reduces to a block tridiagonal form which can be diagonalized for large number of bosons. We present results for the low energy spectrum and the transition probabilities for systems up to 10000 bosons, which confirm that at the critical point the system is equally well described by the Bohr Hamiltonian with a $β^4$ potential.
Davidson potentials of the form $β^2 +β_0^4/β^2$, when used in the original Bohr Hamiltonian for $γ$-independent potentials bridge the U(5) and O(6) symmetries. Using a variational procedure, we determine for each value of angular momentum $L$ the value of $β_0$ at which the derivative of the energy ratio $R_L=E(L)/E(2)$ with respect to $β_0$ has a sharp maximum, the collection of $R_L$ values at these points forming a band which practically coincides with the ground state band of the E(5) model, corresponding to the critical point in the shape phase transition from U(5) to O(6). The same pote…
We study the spherical to prolate-deformed shape transition in $^{144-158}$Sm and $^{146-160}$Gd isotopes with modern calculations beyond the mean field with the Gogny D1S force. We compare the results with the shape-phase transition predicted by the collective hamiltonian model and with the experimental data. Our calculations do not support the existence of a first order phase transition in these isotopic chains in the viewpoint of the Bohr hamiltonian neither the interpretation of the nuclei N=90 as critical points.
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.
The generalized Bohr Hamiltonian was used to describe the low-lying collective excitations in even-even isotopes of Ru, Pd, Te, Ba and Nd. The Strutinsky collective potential and cranking inertial functions were obtained using the Nilsson potential. The effect of coupling with the pairing vibrations is taken into account approximately when determining the inertial functions. The calculation does not contain any free parameter.
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…
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…
