L. Fortunato

Solutions of the Bohr Hamiltonian, a compendium

Abstract.The Bohr Hamiltonian, also called collective Hamiltonian, is one of the cornerstones 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 discussion. It is the aim of the present monograph to present in detail all the known solutions in γ-unstable and γ-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. The quasi-dynamical SO(2) symmetry is proposed in connection with the labeling of bands in triaxial nuclei.

Analytically solvable potentials for γ-unstable nuclei

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) × SO(5) algebraic structure is discussed.

New analytic solutions of the collective Bohr Hamiltonian for a β-soft, γ-soft axial rotor

New analytic solutions of the quadrupole collective Bohr Hamiltonian are proposed, exploiting an approximate separation of the β and γ variables to describe γ-soft prolate axial rotors. The model potential is a sum of two terms: a β-dependent term taken either with a Coulomb-like or a Kratzer-like form, and a γ-dependent term taken as a harmonic oscillator. In particular, it is possible to give a one-parameter paradigm for a β-soft, γ-soft axial rotor that can be applied, with considerable agreement, to the spectrum of 234U.

Signatures of the Giant Pairing Vibration in the 14C and 15C atomic nuclei

Giant resonances are collective excitation modes for many-body systems of fermions governed by a mean field, such as the atomic nuclei. The microscopic origin of such modes is the coherence among elementary particle-hole excitations, where a particle is promoted from an occupied state below the Fermi level (hole) to an empty one above the Fermi level (particle). The same coherence is also predicted for the particle–particle and the hole–hole excitations, because of the basic quantum symmetry between particles and holes. In nuclear physics, the giant modes have been widely reported for the particle–hole sector but, despite several attempts, there is no precedent in the particle–particle and hole–hole ones, thus making questionable the aforementioned symmetry assumption. Here we provide experimental indications of the Giant Pairing Vibration, which is the leading particle–particle giant mode. An immediate implication of it is the validation of the particle–hole symmetry.

Solution of the Bohr Hamiltonian for soft triaxial nuclei

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 \ensuremath{\gamma}, which is solved with an approximated algebraic technique; and Coulomb/Kratzer, harmonic/Davidson, and infinite square-well potentials in \ensuremath{\beta}, 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 compare our results with experimental information and previous calculations.

The electron screening puzzle and nuclear clustering

Abstract Accurate measurements of nuclear reactions of astrophysical interest within, or close to, the Gamow peak show evidence of an unexpected effect attributed to the presence of atomic electrons in the target. The experiments need to include an effective “screening” potential to explain the enhancement of the cross sections at the lowest measurable energies. Despite various theoretical studies conducted over the past 20 years and numerous experimental measurements, a theory has not yet been found that can explain the cause of the exceedingly high values of the screening potential needed to explain the data. In this letter we show that instead of an atomic physics solution of the “electron screening puzzle”, the reason for the large screening potential values is in fact due to clusterization effects in nuclear reactions, in particular for reaction involving light nuclei.

Phase transitions in the interacting boson fermion model: The γ -unstable case

The phase transition around the critical point in the evolution from spherical to deformed {gamma}-unstable shapes is investigated in odd nuclei within the interacting boson fermion model. We consider the particular case of an odd j=3/2 particle coupled to an even-even boson core that undergoes a transition from spherical U(5) to {gamma}-unstable O(6) situation. The particular choice of the j=3/2 orbital preserves in the odd case the condition of {gamma}-instability of the system. As a consequence, energy spectrum and electromagnetic transitions, in correspondence of the critical point, display behaviors qualitatively similar to those of the even core. The results are also in qualitative agreement with the recently proposed E(5/4) model, although few differences are present, due to the different nature of the two schemes.

Role of higher multipole excitations in the electromagnetic dissociation of one-neutron halo nuclei

Abstract.We investigate the role of higher multipole excitations in the electromagnetic dissociation of one-neutron halo nuclei within two different theoretical models --a finite-range distorted-wave Born approximation and another in a more analytical method with a finite-range potential. We also show, within a simple picture, how the presence of a weakly bound state affects the breakup cross-section.

Electromagnetic response and breakup of light weakly bound nuclei in a dicluster model

Abstract.The light weakly bound nucleus 7Li is studied within a dicluster α + t picture. Different observables obtained within our simple model are compared with previous calculations and experiments showing good agreement. In particular, we calculate dipole and quadrupole electromagnetic response to the continuum. The energy distribution of B(Eλ) values are consistent with the energy-weighted molecular sum rule and display a sizable contribution of non-resonant character arising from the weak binding property. The corresponding form factors for excitations to the continuum are used in a semiclassical coupled-channel scheme to get estimates for the breakup cross-section in a heavy-ion reaction. The nuclear contribution is found to play an important role in the process for bombarding energies around the Coulomb barrier. The masses and charges ratios of the two clusters are shown to lead to features of the cluster halo that may significantly differ from the one usually associated with one-nucleon haloes.

Soft triaxial rotovibrational motion in the vicinity of γ= π/6

A solution of the Bohr collective hamiltonian for the