Solutions of the Bohr hamiltonian, a compendium

Abstract

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 $V(\beta ,\gamma )$ 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 discussion. It is the aim of the present monograph to present in detail all the known solutions in $\gamma -$unstable and $\gamma -$stable cases, in a taxonomic and didactical way. In pursuing this task we especially stressed the mathematical side leaving the discussion of the physics to already published comprehensive material. The paper contains also a new approximate solution for the linear potential, and a new solution for prolate and oblate soft axial rotors, as well as some new formulae and comments, and an appendix on the analysis of a few interesting numerical sequences appearing in this context. The quasi-dynamical SO(2) symmetry is proposed in connection with the labeling of bands in triaxial nuclei.

48 pages, 28 figures, 6 tables

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Cited by in corpus (38)

• Quadrupole collective states within the Bohr collective Hamiltonian
• Bohr Hamiltonian with deformation-dependent mass term for the Kratzer potential
• Bohr Hamiltonian with deformation-dependent mass term for the Davidson potential
• Exactly separable version of the Bohr Hamiltonian with the Davidson potential
• Analytical solution for the Davydov-Chaban Hamiltonian with sextic potential for $\gamma =30^{\circ }$
• Sextic potential for $\gamma$-rigid prolate nuclei
• Bohr Hamiltonian with Davidson potential for triaxial nuclei
• Bohr Hamiltonian with Hulth?en plus Ring shaped potential for triaxial nuclei
• Application of the sextic oscillator with centrifugal barrier and the spheroidal equation for some X(5) candidate nuclei
• Solution of the Bohr hamiltonian for soft triaxial nuclei
• Phase diagram for a Cubic Consistent-Q Interacting Boson Model Hamiltonian: signs of triaxiality
• The fractional symmetric rigid rotor
• Vibrational and rotational excited states within Bohr Hamiltonian with deformation-dependent mass formalism
• Exactly separable version of X(5) and related models
• Quantum chaos in the nuclear collective model: I. Classical-quantum correspondence
• Octupole deformation in light actinides within an analytic quadrupole octupole axially symmetric model with Davidson potential
• Description of the isotope chain ${}^{180-196}$Pt within some solvable approaches
• Electric quadrupole transitions of the Bohr Hamiltonian with Manning-Rosen potential
• Triaxial rigidity of ${}^{166}$Er and its Bohr-model realization
• Bohr Hamiltonian with an energy dependent $\gamma$-unstable Coulomb-like potential
• Gamma-soft Analog of the Confined Beta-soft Rotor Model
• A solvable model which has X(5) as a limiting symmetry and removes some inherent drawbacks
• Solvable models for the gamma deformation having X(5) as limiting symmetry. Removing some drawbacks of the existent descriptions
• Davydov-Chaban Hamiltonian with deformation-dependent mass term for γ = 30°
• Bohr Hamiltonian with a deformation-dependent mass term: physical meaning of the free parameter
• Conjunction of $\gamma$-rigid and $\gamma$-stable collective motion in the critical point of the phase transition from spherical to deformed nuclear shapes
• Quadrupole collective variables in the natural Cartan-Weyl basis
• Shape phase mixing in critical point nuclei
• Prolate-oblate asymmetric shape phase transition in the interacting boson model with SU (3) higher-order interactions
• Competing $\gamma$-rigid and $\gamma$-stable vibrations in neutron rich Gd and Dy isotopes
• Excited collective states of nuclei within Bohr Hamiltonian with Tietz-Hua potential
• The quadrupole collective model from a Cartan-Weyl perspective
• Spectral properties of a tractable collective Hamiltonian
• Effective field theories for collective excitations of atomic nuclei
• Observation of Ultra Fine Structures in Energy Levels of Prolate Nuclei
• Bohr-Mottelson Hamiltonian with octic potential applied to the ${}^{106-116}$Cd isotopes
• Coriolis contribution to excited states of odd-mass nuclei with different deformation-dependent mass coeffcients
• An overview of the scientific contribution of Andrea Vitturi to Nuclear Physics (being an account of the recent TNP19 meeting held in Padova)