The interacting boson model.

Signatures of criticality in the evolution of the nuclear ground-state shapes across the NxZ plane are discussed. Attention is paid to specific data indicating sudden structural changes in various isotopic and isotonic chains of medium-mass and heavy even-even nuclei, as well as to diverse theoretical aspects of the models used to describe these changes. The interacting boson model and the geometric collective model, in particular, are discussed in detail, the former providing global predictions for the evolution of collective observables in nuclei between closed shells and the latter yielding a parameter-efficient description of nuclei at the critical points of shape transitions. Some issues related to the mechanism of first- and second-order quantum phase transitions in general many-body systems are also outlined.

Quantum phase transitions in the shapes of atomic nuclei

Quantum phase transitions in the interacting boson model

Recently, the idea of quantum phase transitions has been applied to equilibrium shape changes in finite atomic nuclei. An introduction to and discussion of this concept, both from the macroscopic perspective and from a microscopic approach, in terms of shell structure and residual interactions is presented. Following this, the quantitative predictions of the critical point symmetries (CPS) X(5) and E(5) are discussed in some detail and compared to the data in a number of nuclei and mass regions. Any successful new model paradigm soon generates modifications to improve the predictions and, likewise, often spawns related models inspired by the original ansatz. This is eminently true of the CPS and we make an effort to briefly discuss this rapidly advancing area in terms of a number of recent modifications to the CPS as well as related models that are couched in a geometrical perspective.

Quantum phase transitions and structural evolution in nuclei

This review is focused on various properties of quantum phase transitions (QPTs) in the Interacting Boson Model (IBM) of nuclear structure. The model in its infinite-size limit exhibits shape-phase transitions between spherical, deformed prolate, and deformed oblate forms of the ground state. Finite-size precursors of such behavior are verified by robust variations of nuclear properties (nuclear masses, excitation energies, transition probabilities for low lying levels) across the chart of nuclides. Simultaneously, the model serves as a theoretical laboratory for studying diverse general features of QPTs in interacting many-body systems, which differ in many respects from lattice models of solid-state physics. We outline the most important fields of the present interest: (a) The coexistence of first- and second-order phase transitions supports studies related to the microscopic origin of the QPT phenomena. (b) The competing quantum phases are characterized by specific dynamical symmetries and novel symmetry related approaches are developed to describe also the transitional dynamical domains. (c) In some parameter regions, the QPT-like behavior can be ascribed also to individual excited states, which is linked to the thermodynamic and classical descriptions of the system. (d) The model and its phase structure can be extended in many directions: by separating proton and neutron excitations, considering odd-fermion degrees of freedom or different particle-hole configurations, by including other types of bosons, higher order interactions, and by imposing external rotation. All these aspects of IBM phase transitions are relevant in the interpretation of experimental data and important for a fundamental understanding of the QPT phenomenon.

The relation of the recently proposed $E(5)$ critical point symmetry with the interacting boson model is investigated. The large-$N$ limit of the interacting boson model at the critical point in the transition from U(5) to O(6) is obtained by solving the Richardson equations. It is shown explicitly that this algebraic calculation leads to the same results as the solution of the Bohr differential equation with a ${\ensuremath{\beta}}^{4}$ potential.

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U(5)-O(6) transition in the interacting boson model and the E(5) critical point symmetry

We investigate the phase transition in odd nuclei within the Interacting Boson Fermion Model in correspondence with the transition from spherical to stable axially deformed shape. The odd particle is assumed to be moving in the single-particle orbitals with angular momenta j=1/2,3/2,5/2 with a boson-fermion Hamiltonian that leads to the occurrence of the SU{sup BF}(3) boson-fermion symmetry when the boson part approaches the SU(3) condition. Both energy spectra and electromagnetic transitions show characteristic patterns similar to those displayed by the even nuclei at the corresponding critical point. The role of the additional particle in characterizing the properties of the critical points in finite quantal systems is investigated by resorting to the formalism based on the intrinsic frame.

C. E. Alonso

Papers

U(5)-O(6) transition in the interacting boson model and the E(5) critical point symmetry

UBF(5) to SUBF(3) shape phase transition in odd nuclei for j=1/2, 3/2, and 5/2 orbits: The role of the odd particle at the critical point

Backbending of Dy isotopes described with the neutron-photon IBA plus two-quasiparticle model

Odd-even nuclei in the A ≈ 100 nuclear region

Intrinsic structure of two-phonon states in the interacting boson model