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CRC Press eBooks

Quantum algebraic symmetries in atomic nuclei and metal clusters

Dennis Bonatsos A, B. A. Kotso, P. Raychev, P. A. Terziev

Published Nov 30, 2003• 0 citations
Publisher

Abstract

Using irreducible tensor operators under su q (2) a rotationally invariant Hamiltonian which provides a good description of nuclear rotational spectra is constructed and its relation to existing nuclear models is considered. Using the same techniques a 3dimensional g-deformed harmonic oscillator with u q (3)⊃so q (3) symmetry is constructed, compared to the modified oscillator of Nilsson, and used for the successful description of magic numbers and supershells in metal clusters.

DOI
10.1201/9781482269079-68

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Using irreducible tensor operators under su q (2) a rotationally invariant Hamiltonian which provides a good description of nuclear rotational spectra is constructed and its relation to existing nuclear models is considered. Using the same techniques a 3dimensional g-deformed harmonic oscillator with u q (3)⊃so q (3) symmetry is constructed, compared to the modified oscillator of Nilsson, and used for the successful description of magic numbers and supershells in metal clusters.
Quantum algebras are a mathematical tool which provides us with a class of symmetries wider than that of Lie algebras, which are contained in the former as a special case. After a self-contained introduction to the necessary mathematical tools (qqq-numbers, qqq-analysis, qqq-oscillators, qqq-algebras), the suq_qq​(2) rotator model and its extensions, the construction of deformed exactly soluble models (Interacting Boson Model, Moszkowski model), the use of deformed bosons in the description of pairing correlations, and the symmetries of the anisotropic quantum harmonic oscillator with rational ratios of frequencies, which underly the structure of superdeformed and hyperdeformed nuclei, are discussed in some detail. A brief description of similar applications to molecular structure and an outlook are also given.
The rotational invariance under the usual physical angular momentum of the SUq(2) Hamiltonian for the description of rotational molecular spectra is explicitly proved and a connection of this Hamiltonian to the formalism of Amal'sky is provided. In addition, a new Hamiltonian for rotational spectra is introduced, based on the construction of irreducible tensor operators (ITOs) under SUq(2) and use of q-deformed tensor products and q-deformed Clebsch-Gordan coefficients. The rotational invariance of this SUq(2) ITO Hamiltonian under the usual physical angular momentum is explicitly proved and a simple closed expression for its energy spectrum (the ``hyperbolic tangent formula'') is introduced. Numerical tests against an experimental rotational band of HF are provided.
Abstract The rotational invariance under the usual physical angular momentum of the suq(2) Hamiltonian for the description of rotational molecular spectra is explicitly proved and a connection of this Hamiltonian to the formalism of Amal'sky is provided. In addition, a new Hamiltonian for rotational spectra is introduced, based on the construction of irreducible tensor operators (ITOs) under suq(2) and use of q‐deformed tensor products and q‐deformed Clebsch–Gordan coefficients. The rotational invariance of this suq(2) ITO Hamiltonian under the usual physical angular momentum is explicitly proved and a simple closed expression for its energy spectrum (the “hyperbolic tangent formula”) is introduced. Numerical tests against an experimental rotational band of the HF molecule are provided. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 95: 1–20, 2003
A deformed version of the vibron model for diatomic molecules is constructed. Both the O(4) and U(3) dynamical symmetries of the model are rewritten, using the concept of complementary subalgebras, in a more convenient form, which is subsequently deformed. The present model unifies the so far independent successful quantum-algebraic approaches to rotational and to vibrational spectra of diatomic molecules. In addition, the method can be used for the construction of deformed versions of the U(5) and O(6) limits of the interacting boson model of nuclear structure.
As a direct generalization of our previous work, we construct a manifestly rotation invariant Hamiltonian describing a system composed of a tensor boson with spin I = 2 — such as, for example, phonon of nuclear quadrupole surface vibration — in which special cubic and quartic self-interactions are taken into account. All the ground state solutions are obtained. It is shown that these solutions can be classified into three classes: The first is the normal vacuum of boson, which is manifestly rotation symmetry. The second class of solutions can be viewed as generated from a single chosen state by rotating it into an arbitrary direction. The third one is Goldstone modes built on this chosen ground state. This chosen state — an exact ground state solution of our rotation invariant Hamiltonian — is not an eigenstate of angular momentum, but contains infinitely many different angular momentum states in it; thus, typifying “spontaneous breakdown” of “rotation symmetry”. It is further shown that this state can be naturally identified to an “intrinsic” deformed state in nuclear collective model of Bohr and Mottelson. The results are examined from the standpoint of Goldstone theorem and detailed discussions are given to explore how Goldstone theorem is modified in our finite dimensional system.