Quasi-dynamical Symmetry Research Articles

PHASE-TRANSITIONS AND NUCLEAR CLUSTERIZATION

After reviewing some basic features of the temperature-governed phase-transitions in macroscopic systems and in atomic nuclei we consider non-thermal phase-transitions of nuclear structure in the example of cluster states. Phenomenological and semimicroscopical algebraic cluster models with identical interactions are applied to binary cluster systems of closed and non-closed shell clusters. Phase-transitions are observed in each case between the rotational (rigid molecule-like) and vibrational (shell-like) cluster states. The phase of this finite quantum system shows a quasi-dynamical symmetry.

Clusters and the quasi-dynamical symmetry

The possible role of the quasi-dynamical symmetry in nuclear clusterization is discussed. Two particular examples are considered: i) the phases and phase-transitions of some algebraic cluster models, and ii) the clusterization in heavy nuclei. The interrelation of exotic (superdeformed, hyperdeformed) nuclear shapes and cluster-configurations are also investigated both for light, and for heavy nuclei, based on the dynamical and quasi-dynamical SU(3) symmetries, respectively.

Symmetry, quasisymmetry, and critical phenomena

This presentation is an analysis of the role of symmetry in second-order quantum phase transitions. It seeks to explain why transitions between phases of systems, associated with different symmetries, exhibit critical phenomena. It transpires that a system in one phase, tends to hold onto the symmetry associated with that phase until a breaking point is reached at which a rapid transition occurs to a new phase associated with a different symmetry. Understanding what is happening presents the challenge of explaining how a system "holds on" to a symmetry in the face of strong symmetry-breaking interactions. The explanation is of fundamental interest in physics, for understanding why models with built in symmetries often work extremely well even when the models ignore large symmetry-breaking interactions that are known to be present. When this phenomenon occurs, we say that the system has a quasidynamical symmetry. This concept is of interest in mathematics because it turns out that quasidyndamical symmetries are the physical realizations of a new concept in group theory, which we refer to as an embedded representation. PACS Nos.: 21.60.Fw, 21.60.Ev, 64.6–.Ht, 68.18.Jk

PERSISTENCE OF ROTATIONAL STRUCTURE IN NUCLEI AND QUASI-DYNAMICAL SYMMETRY

The existence of quasi-dynamical symmetry (QDS) in physical systems and its significance for understanding the persistence of rotational structure in nuclei is explained in terms of the mathematical concept of an embedded representation. We consider the spectra obtained by coupling two SU (3) irreps by means of a quadrupole-quadrupole interaction. For a particular large value of this interaction, the two irreps combine to form a single (strongly-coupled) irrep while for zero interaction the weakly-coupled results are mixtures of many irreps. A notable result is the persistence of the rotor character of the low-energy states for a wide range of the interaction strength which can be explained by coherent mixing of SU (3) irreps. Such a coherent mixing of representations is an indication that the model has a QDS. Also notable is the fact that, for very weak interaction strengths, the rotational states of the yrast band approach those of a vibrational sequence while the B( E 2) transition strengths remain close to those of an axially symmetric rotor.

Phase transitions and quasi-dynamical symmetry in nuclear collective models, III: The U(5) to SU(3) phase transition in the IBM

This paper studies the dependence of the U(5) to SU(3) phase transition in the interacting boson model (IBM) on the finite number N of bosons. Such investigations illuminate the relationship between a first order transition in a macroscopic system, which exhibits a discontinuous change of some of its properties at a highly singular critical point, and a transition in a corresponding finite particle system, which shows a critical point smeared by fluctuations. The IBM model Hamiltonian H ( α ) = ( 1 − α ) H 1 + α H 2 of this paper is an interpolation between a U(5)-invariant Hamiltonian H 1 , given by the d-boson number operator, and an SU(3) Hamiltonian H 2 , given by a quadrupole–quadrupole operator. For N sufficiently large, the low energy eigenstates of H ( α ) fall into either a U(5) phase or an SU(3) phase depending on the value of α. An exception is a narrow zone around a critical value of α. The width of this critical zone decreases as N increases. In this zone the spectra for the N values considered show an approximate X(5) dynamical symmetry to within an N-dependent scale factor. The states in the SU(3) phase show an SU(3) quasi-dynamical symmetry that becomes more well-defined as N increases. The states in the U(5) phase show a U(5) quasi-dynamical symmetry albeit one that closely approaches a pure U(5) dynamical symmetry for smaller values of α and which also becomes better defined the larger the particle number. The concluding section reviews studies of phase transitions in nuclear physics.

Phase transitions and quasidynamical symmetry in nuclear collective models. II. The spherical vibrator to gamma-soft rotor transition in an SO(5)-invariant Bohr model

A model of a second-order shape phase transition is investigated in the Bohr collective model. The model contains two variable parameters, a mass parameter M and a control parameter α, and is such that when α = 0 the Hamiltonian is that of a harmonic spherical vibrator and when α is large it approaches that of an adiabatically decoupled rotor-vibrator. The results obtained by diagonalization of this Hamiltonian show that the range of α, in which the low-energy states of the model are in a transition region between that of a harmonic spherical vibrator phase (for small α) and that of an adiabatic rotor-vibrator phase (for large α), shrinks as M increases and as M → ∞ a critical point develops at α = 0.5 . The dynamical symmetries associated with the limiting phases of this model, which appear to persist in the small and large α domains, are interpreted as quasidynamical symmetries. For finite values of M, the results closely parallel those of the corresponding phase transition of an interacting boson model, studied in paper I of this series, when the mass M of the collective model is set equal to twice the boson number N of the IBM. The various solvable submodels of the Bohr model are related to corresponding limits of the IBM by contraction maps. Such contraction maps imply a correspondence between subsets of states in the domains of the two models for which a given contraction map applies. Thus, by considering the contraction limit of an IBM Hamiltonian in the Bohr model, one can interpret and even anticipate what the results of an IBM calculation would be in its macroscopic N → ∞ limit. The asymptotic scaling of the spectrum at the critical point and Iachello's critical point symmetry in the Bohr model and IBM are considered from this perspective.

Phase transitions and quasidynamical symmetry in nuclear collective models: I. The [formula omitted] to [formula omitted] phase transition in the IBM

The transition from a spherical-vibrator to a deformed-rotor phase is examined in this and succeeding papers in the interacting boson and Bohr–Mottelson collective models. This paper explores the transition from a U ( 5 ) to an O ( 6 ) phase in the interacting boson model with variation of a control parameter α. As expected from other studies, it is observed that a low-lying state of the model can be assigned to one of three phases: a U ( 5 ) phase, an O ( 6 ) phase, or a transition phase. It is also shown that the range of values of the control parameter, for which the low-lying states are in the transition phase, shrinks as the dimension of the IBM Hilbert space increases and that a sharp phase transition develops as the infinite-dimensional limit is approached. A detailed examination of the states of the two phases shows that the U ( 5 ) symmetry of the α = 0 limit is mixed throughout much of the U ( 5 ) phase and the O ( 6 ) symmetry of the α = 1 limit is even more strongly mixed throughout much of the O ( 6 ) phase. Nevertheless, it is found that quasidynamical U ( 5 ) and O ( 6 ) symmetries continue to characterize the phases and provide an explanation of the apparent persistence of these symmetries in the face of strong symmetry-breaking interactions. Thus, the analysis suggests a useful definition of a phase associated with a dynamical symmetry. It also adds a perspective to Landau's theory of second-order phase transitions and to Iachello's concept of a critical point symmetry with analytical solutions in the transition region.

Quasidynamical symmetry in an interacting boson model phase transition.

The apparent persistence of symmetry in the face of strong symmetry-breaking interactions is examined in a many-boson model. The model exhibits a transition between two phases associated with U(5) and O(6) symmetries, respectively, as the value of a control parameter progresses from 0 to 1. The remarkable fact is that, in spite of strong mixing of the symmetries for intermediate values of the control parameter, the model continues to exhibit the characteristics of its closest symmetry limit for all but a relatively narrow transition region that becomes progressively narrower as the boson number increases. This phenomenon is explained in terms of quasidynamical symmetry.

SU(3) quasi-dynamical symmetry as an organizational mechanism for generating nuclear rotational motions

The phenomenological symplectic model with a Davidson potential is used to construct rotational states for a rare-earth nucleus with microscopic wave functions. The energy levels and E2 transitions obtained are in remarkably close agreement (to within a few percent) with those of the rotor model with vibrational shape fluctations that are adiabatically decoupled from the rotational degrees of freedom. An analysis of the states in terms of their SU(3) content shows that SU(3) is a very poor dynamical symmetry but an excellent quasi-dynamical symmetry for the model. It is argued that such quasi-dynamical symmetry can be expected for any Hamiltonian that reproduces the observed low-energy properties of a well-deformed nucleus, whenever the latter are well-described by the nuclear rotor model.

Angular-momentum projection of rotational model wave functions

It is known that angular momentum projection can be carried out analytically from highest weight states of ( λ,0) SU(3) irreps. It is shown here that they can also be projected analytically from deformed harmonic oscillator (asymptotic Nilsson model) states and that these states become intrinsic states for rotational bands in the limit of large deformation. It is also shown that SU(3) is a good quasi-dynamical symmetry for a soft rotor with a large deformation.