D.J. Rowe

Nuclear collective motion : models and theory

Phenomenological Models: General Trends and Coupling Schemes The Collective Vibrational Model The Unified Model for Vibrations The Vibrating Potential Model The E1 Photoresonance The Collective Rotational Model The Unified Model for Rotations The Moment of Inertia Microscopic Theories: Hartree?Fock Self-Consistent Field Theory: Spherical Nuclei Hartree?Fock Self-Consistent Field Theory: Deformed Nuclei Pairing-Force Theory The Tamm?Dancoff Approximation or Simple Particle?Hole Theory An Equations-of-Motion Method The Random Phase Approximation or Sophisticated Particle-Hole Theory Time-Dependent Hartree?Fock (TDHF) Theory The Microscopic Foundations of the Unified Vibrational Model Concluding Remarks.

On the algebraic formulation of collective models III. The symplectic shell model of collective motion

Abstract The symplectic model is a microscopic theory which provides a practical technique for identifying the shell configurations necessary for the description of quadrupole and monopole vibrations as well as collective rotations of the nucleus. The model is based on the non-compact symplectic algebra sp (3, R ) and is a natural generalization of Elliotts su (3) model to include many major oscillator shells in addition to core excitations. It is also simultaneously the shell model adaptation of the collective rotational [ R 5 ] so (3), the Bohr-Mottelson cm (3) = [ R 6 ] sl (3) and the mass quadrupole collective MQC = [ R 6 ] gl (3) models. In contrast to the su (3) algebra, the sp (3, R ) algebra makes no 0ħω approximations and treats all observables in the algebra exactly, thereby achieving a microscopic theory of large amplitude collective motion. The observables in the algebra include the quadrupole and monopole moments, the kinetic energy, the harmonic oscillator Hamiltonian and the angular and vibrational momenta. Numerical results are reported for 20 Ne using an 8 ħω truncation and a phenomenological potential V ( β , γ ). Satisfactory agreement with experiment is obtained for the absolute B ( E 2) rates without resorting to an effective charge.

Microscopic theory of the nuclear collective model

This articles reviews the development of a microscopic theory of nuclear collective structure as a submodel of the nuclear-shell model. It starts by showing how the so-called geometrical (Bohr-Mottelson-Frankfurt) collective model must be augmented by the addition of vortex spin degrees of freedom to make it compatible with the shell model. A unified symplectic model emerges that can be applied both with phenomenological and microscopic interactions. Examples are given of both kinds of calculation. It is shown how the full shell model space can be expressed in an Sp(3R) contains/implies U(3) basis in which form it naturally factors into collective and intrinsic subspaces. In this way, the collective content of a shell model state becomes immediately apparent. Thus a shell model interpretation is given of collective states, including the low-lying rotational bands, the so-called beta and gamma vibrations and the giant monopole and quadrupole resonances.

The splitting of giant multipole states of deformed nuclei

Abstract A vibrating potential model is applied to deformed nuclei with a deformed harmonic oscillator potential in order to discuss the splitting of isoscalar giant quadrupole states. Eigenfrequencies of the collective states are estimated to be √2ω(1 − 1 3 δ) , √2ω(1 − 1 6 δ) and √2ω(1 + 1 3 δ) fo and 2 + modes, respectively. The splitting of isovector dipole and isovector quadrupole states is also studied according to a schematic model as proposed by Bohr and Mottelson. It is shown that. isovector dipole states are split, as in a hydrodynamic model, while isovector quadrupole states with the same scaling factors as those of isoscalar quadrupole modes.

Coherent state theory of the noncompact symplectic group

An extended coherent state theory is presented for the noncompact Sp(3,R) group which reveals a simple relationship between the Sp(3,R) algebra and its contracted u(3)‐boson limit. The relationship is used to derive a remarkably accurate analytic expression for Sp(3,R) matrix elements for the generic lowest‐weight representations. The expression is shown to be exact whenever the states involved are multiplicity free with respect to the u(3) subalgebra. It is further shown how exact matrix elements are easily calculated in general. Dyson and Holstein–Primakoff type u(3)‐boson expansions are given.

Vector coherent state representation theory

A vector coherent state theory is formulated as a natural extension of standard coherent state theory. It is shown that the Godement representations and the coherent state representations of the Sp(N,R) groups of Rowe and of Deenen and Quesne are special cases of this more general theory.

The algebraic CM(3) model

Abstract Rotational bands are shown to exist under the assumption that the states of a nucleus carry an irreducible representation of CM(3). A generalization of both the SL(3, R) model of Weaver and Biedenharn and the rotational [R5]SU(2) model of Weaver, Biedenharn, and Cusson, the CM(3) model has a more general multiband structure than either of these models. The CM(3) model predicts E2 transition rates and energy levels with four adjustable parameters. Extensions of the CM(3) model and the rotational model that predict Eλ transition rates for all λ are found.

A computationally tractable version of the collective model

Abstract A computationally tractable version of the Bohr–Mottelson collective model is presented which makes it possible to diagonalize realistic collective models and obtain convergent results in relatively small appropriately chosen subspaces of the collective model Hilbert space. Special features of the proposed model are that it makes use of the beta wave functions given analytically by the softened-beta version of the Wilets–Jean model, proposed by Elliott et al., and a simple algorithm for computing SO(5)⊃SO(3) spherical harmonics. The latter has much in common with the methods of Chacon, Moshinsky, and Sharp but is conceptually and computationally simpler. Results are presented for collective models ranging from the spherical vibrator to the Wilets–Jean and axially symmetric rotor–vibrator models.

Dynamical symmetries of nuclear collective models

Abstract The construction of algebraic models has become popular over the past twenty years. However, the fact that almost every dynamical model has an underlying algebraic structure is often overlooked. Indeed, a model is often described as geometric to distinguish it from models that are manifestly algebraic. In fact, the algebraic and geometric perspectives of a model are complementary and emerge naturally when the model is expressed in terms of its dynamical symmetry group. There is much to be gained by examining the dynamical symmetry of a model. For example, dynamical group transformations map out phase spaces and reveal the dynamical content of a model in classical terms. On the other hand, the unitary representations of a models Lie algebra provide a quantization of its dynamics and the means to classify basis states and do calculations. Even more important, perhaps, is the discovery that identifying the dynamical symmetry of a phenomenological model provides a means to determine if the model is compatible with the microscopic many-nucleon structure of the nucleus and, when it is. give it a microscopic interpretation. Finally, as discussed in the last section of this review, dynamical symmetry concepts can be invoked to discover if the dynamics associated with two different models are compatible and can be combined. The models considered in this review are restricted (because of page limitations) to those which concern quadrupole vibrations and rotational motions.

Vector coherent state theory and its application to the orthogonal groups

Vector coherent state theory is developed and presented in a form that explicitly exhibits its general applicability to the ladder representations of all semisimple Lie groups and their Lie algebras. It is shown that, in a suitable basis, the vector coherent state inner product can be inferred algebraically, by K‐matrix theory, and changed to a simpler Bargmann inner product thereby facilitating the explicit calculation of the matrix representaions of Lie algebras. Applications are made to the even and odd orthogonal Lie algebras.