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HNPS Advances in Nuclear Physics

Quantum algebraic symmetries in nuclear physics

Dennis Bonatsos A, C. Dascaloyannis

Published Dec 5, 2019• 0 citations
Publisher

Abstract

The pairing correlations in a single-j nuclear shell are considered. It is proven that a simple boson mapping in terms of q-defonned bosons exists, which reproduces correctly both the commutation relations and the energy up to first order corrections, the parameter q being connected to the size of the shell. An exact solution in terms of a generalized deformed oscillator is also found

DOI
10.12681/hnps.2384

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The pairing correlations in a single-j nuclear shell are considered. It is proven that a simple boson mapping in terms of q-defonned bosons exists, which reproduces correctly both the commutation relations and the energy up to first order corrections, the parameter q being connected to the size of the shell. An exact solution in terms of a generalized deformed oscillator is also found
In a single-j shell the author considers fermion pair and multipole operators coupled to zero angular momentum. The commutation relations of these operators can be satisfied up to first-order corrections by suitably defined q-bosons, onto which the fermion pair operators are mapped. After performing the same mapping to a simple pairing Hamiltonian, it is proved that the pairing energies are also correctly reproduced up to the same order. The small parameter used (T=In Q) is found to be inversely proportional to the size of the shell.
The quon algebra, which interpolates between the Bose and Fermi algebras and depends on a free paramenter qqq, is used to generate a deformed Dyson boson expansion of the quadrupole operator. Then we obtain a quadrupole-quadrupole hamiltonian, for a single j-shell, in terms of this deformed bosonic operator. The hamiltonian is diagonalized and its eigenvalues are compared with the ones obtained from the fermionic quadrupole-quadrupole hamiltonian. The deformation parameter helps in achieving the correct energy levels, what cannot be encountered in practice with the usual non-deformed Dyson expansion.
The quon algebra, which interpolates between the Bose and Fermi algebras and depends on a free parameter q , is used to generate deformed Marumori boson expansions for the pair and quadrupole operators. Then we obtain a pairing plus quadrupole-quadrupole Hamiltonian, for a single j -shell, in terms of these deformed bosonic operators. The Hamiltonian is diagonalized and its eigenvalues are compared with the ones obtained from the counterpart fermionic Hamiltonian. The deformation parameter helps in achieving the correct energy levels, which cannot be encountered in practice with the usual non-deformed boson expansion.
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 formalism derived for the boson–fermion and the baryon mapping in Paper I of this series is applied to the single-orbit quark shell model developed by Petry and co-workers for the description of nuclear properties. The multiquark space is reduced to the collective subspace of multinucleon states. A collective representation of states and operators in terms of bosons and ideal fermions or in terms of ideal baryons, respectively, is constructed. The results of the original model in the multinucleon space are exactly reproduced. Finally, the two transformations are compared with a related mapping technique recently published by Pittel, Engel, Dukelsky, and Ring.
The validity of the essential ingredients of the microscopic formulation of the interacting boson model: (i) truncation of the shell-model or fermion space and (ii) the procedures for mapping the truncated fermion space on to the boson space, is tested over various (spherical, deformed and transitional) regions both in the simplistic single j-shell, identical boson systems and the more realistic multi j-shell, proton-neutron systems depending on the feasibility of such calculations. In the situations where the truncation and mapping procedures fail to reproduce the exact shell-model results, it is demonstrated that renormalization and higher order mapping procedures cure the associated problems, respectively and provide excellent agreements. In addition, a Hamiltonian of the interacting boson-fermion model which is relevant for a study of odd mass nuclei is derived for a single j-shell case in the shell-model framework, which yields results identical to those in the shell-model for the monopole pairing interaction.