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Algebraic collective model and its applications

G. Thiamová

Published Oct 8, 2015• 0 citations
OpenAlex

Abstract

I will review developments and applications of an algebraic version of Bohr’scollective model (BM), known as the algebraic collective model (ACM) withinwhich fully converged calculations can be performed for a range of Hamiltonians.Examining the algebraic structure underlying the BM has clarified itsrelationship with the interacting boson model (IBM), with which it has relatedsolvable limits. I will review results of comparison of the two models for thecase of both axially-symmetric and triaxial rotor. Predictions for single andmulti-phonon excitations of deformed nuclei will be discussed and comparedwith data.

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FRANCESCO IACHELLO, P. Van Isacker

Introduction In many cases in physics, one has to deal simultaneously with collective and single-particle excitations of the system. The collective excitations are usually bosonic in nature while the single-particle excitations are often fermionic. One is therefore led to consider a system which includes bosons and fermions. In this book we discuss applications of a general algebraic theory of mixed Bose- Fermi systems to atomic nuclei. The collective degrees of freedom here can be described in terms of a system of interacting bosons as discussed in a previous book (Iachello and Arima, 1987), henceforth referred to as Volume 1. The single-particle degrees of freedom represent the motion of individual nucleons in the average nuclear field. They are described in terms of a system of interacting fermions. The coupling of fermions and bosons leads to the interacting boson-fermion model which has been used extensively in recent years to discuss the properties of nuclei with an odd number of nucleons. The interacting boson-fermion model was introduced by Arima and one of us in 1975 (Arima and Iachello, 1975). It was subsequently expanded by Iachello and Scholten (1979) and cast into a form more readily amenable to calculations. As in the corresponding case of even-mass systems, the algebra of creation and annihilation operators can be realized in several ways. One of these is the Hoistein-Primakoff realization which leads to a slightly different version of the interacting boson-fermion model called the truncated quadrupole phonon-fermion model (Paar, 1980; Paar and Brant, 1981), based on the boson realization introduced by Janssen, Jolos and Donau in 1974 and discussed in Sect. 1.4.6 of Volume 1.

Cambridge University Press eBooks•Aug 30, 1991•0 citations

Microscopic calculation for deformed nuclei

Takaharu Otsuka

The microscopic basis of the Interacting Boson Model for deformed nuclei is discussed. The IBM Hamiltonian is constructed microscopically in the following two steps. In the first step, the collective nucleon pairs of J = 0/sup +/ (S), 2/sup +/ (D), etc. are mapped onto the corresponding bosons. Nucleon-nucleon interactions are also mapped onto boson-boson interactions. This mapping method for deformed nuclei was proposed recently, and it turned out that this method is consistent with the Hartree-Fock-Bogoliubov + angular momentum projection calculation. Low-lying collective states primarily consist of S and D pairs. Consequently, the corresponding boson states mainly consist of s and d bosons, while there are some admixture of g-bosons. In the second step, effects of these g-bosons are included within the s-d boson space by a unitary transformation which transforms a combination of d and g bosons into a new d-boson. By minimizing the coupling between new d and g bosons with an appropriate mixing angle, one can neglect the coupling and obtain the IBM Hamiltonian with s and d bosons. It is demonstrated that the s-d Hamiltonian thus derived indeed reproduces spectra of the original s-d-g Hamiltonian.

•Sep 24, 1984•0 citations

The geometrical Bohr-Mottelson model:analytic solutions and an algebraic Cartan-Weyl perspective

Stijn De Baerdemacker

In this introductory chapter, a brief overview is given on nuclear structure physics and how symmetry principles can enhance insight in the solutions of the Schrodinger equation. A pairing Hamiltonian is discussed as an example of the power of Lie algebras, and extended towards the Interacting Boson Model (IBM). Via a mean field approach, the connection between the IBM and geometric collective models is shown and the topic of quantum shape phase transitions is introduced. Section 1.1 Different levels of nuclear structure ’The atomic nucleus is like a huge soccer team: although the actions are set by the individual players, in the end, it is the team that wins the game’. The physical meaning of this metaphor is that all the interactions between the players in the field (protons and neutrons) contribute to the final structure of the team. Nevertheless, no matter how tempting, one cannot extend the metaphor over the whole line. In contrast to the popular sport discipline, the interactions between the particles are not fully known nor understood. This is partly due to the fact that the protons and neutrons are not elementary particles, rather composite particles in the sense that they are also built from other constituent parts, whichmeans that one is forced to start from the underlying quark degrees of freedom if a fundamental nucleon-nucleon interaction is desired. Another problem is that the atomic nucleus is a typical example of a many-body system. One could

•Jan 1, 2007•2 citations

Quantum Algebras in Nuclear Structure

Dennis Bonatsos A, C. Daskaloyannis, P. Kolokotronis +1 more

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 ($q$-numbers, $q$-analysis, $q$-oscillators, $q$-algebras), the su$_q$(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.

arXiv:nucl-th/9512017•Romanian Journal of Physics•Dec 12, 1995•5 citations

Algebraic Approach to the Theory of Collective Motion: Spherical to Deformed Transition in the Single- j Shell

Mathilde Vallières, Abraham Klein, R. M. Dreizler

The program of the algebraic approach to nuclear collective motion is summarized. In essence, it should permit an accurate solution of the many-body problem within a subspace of the many-particle space if the conditions defining collective motion are present. The formulation, containing both dynamical and kinematical aspects, leads to coupled nonlinear algebraic equations for the matrix elements between collective states of fermion pair operators. An explicit set of equations is given for the problem of $N$ (even) particles confined to a single-$j$ shell and described by the conventional pairing plus quadrupole-quadrupole interaction Hamiltonian. This formulation is designed to be nearly exact in the pairing-dominant (seniority) limit and to be accurate enough (based on plausibility arguments) to describe the transition from spherical to deformed shape. That this is so is verified for configurations, the largest being ${j}^{N}={(\frac{21}{2})}^{6}$, for which exact solutions are available. The present method can be viewed as an angular momentum conserving alternative to the description of the transition based on the Hartree-Bogoliubov-BCS method. Qualitative and detailed comparisons of the two methods are carried out. In particular, it is shown from approximate analytic results that the two theories predict the transition to occur for almost identical values of the parameters. Over-all agreement with exact results is much better for the algebraic method, however, confirming its status as an approximate method of variation after projection.

Physical Review C•Jun 1, 1973•12 citations

Use of a Boson Mapping to Elucidate the Relationship Between the IBM and the Bohr Collective Hamiltonian and Between a Simplified Shell Model and the Bohr Collective Hamiltonian

Abraham Klein, Mathilde Vallières

Springer eBooks•Jan 1, 1982•0 citations
I will review developments and applications of an algebraic version of Bohr’scollective model (BM), known as the algebraic collective model (ACM) withinwhich fully converged calculations can be performed for a range of Hamiltonians.Examining the algebraic structure underlying the BM has clarified itsrelationship with the interacting boson model (IBM), with which it has relatedsolvable limits. I will review results of comparison of the two models for thecase of both axially-symmetric and triaxial rotor. Predictions for single andmulti-phonon excitations of deformed nuclei will be discussed and comparedwith data.
Introduction In many cases in physics, one has to deal simultaneously with collective and single-particle excitations of the system. The collective excitations are usually bosonic in nature while the single-particle excitations are often fermionic. One is therefore led to consider a system which includes bosons and fermions. In this book we discuss applications of a general algebraic theory of mixed Bose- Fermi systems to atomic nuclei. The collective degrees of freedom here can be described in terms of a system of interacting bosons as discussed in a previous book (Iachello and Arima, 1987), henceforth referred to as Volume 1. The single-particle degrees of freedom represent the motion of individual nucleons in the average nuclear field. They are described in terms of a system of interacting fermions. The coupling of fermions and bosons leads to the interacting boson-fermion model which has been used extensively in recent years to discuss the properties of nuclei with an odd number of nucleons. The interacting boson-fermion model was introduced by Arima and one of us in 1975 (Arima and Iachello, 1975). It was subsequently expanded by Iachello and Scholten (1979) and cast into a form more readily amenable to calculations. As in the corresponding case of even-mass systems, the algebra of creation and annihilation operators can be realized in several ways. One of these is the Hoistein-Primakoff realization which leads to a slightly different version of the interacting boson-fermion model called the truncated quadrupole phonon-fermion model (Paar, 1980; Paar and Brant, 1981), based on the boson realization introduced by Janssen, Jolos and Donau in 1974 and discussed in Sect. 1.4.6 of Volume 1.
The microscopic basis of the Interacting Boson Model for deformed nuclei is discussed. The IBM Hamiltonian is constructed microscopically in the following two steps. In the first step, the collective nucleon pairs of J = 0/sup +/ (S), 2/sup +/ (D), etc. are mapped onto the corresponding bosons. Nucleon-nucleon interactions are also mapped onto boson-boson interactions. This mapping method for deformed nuclei was proposed recently, and it turned out that this method is consistent with the Hartree-Fock-Bogoliubov + angular momentum projection calculation. Low-lying collective states primarily consist of S and D pairs. Consequently, the corresponding boson states mainly consist of s and d bosons, while there are some admixture of g-bosons. In the second step, effects of these g-bosons are included within the s-d boson space by a unitary transformation which transforms a combination of d and g bosons into a new d-boson. By minimizing the coupling between new d and g bosons with an appropriate mixing angle, one can neglect the coupling and obtain the IBM Hamiltonian with s and d bosons. It is demonstrated that the s-d Hamiltonian thus derived indeed reproduces spectra of the original s-d-g Hamiltonian.
In this introductory chapter, a brief overview is given on nuclear structure physics and how symmetry principles can enhance insight in the solutions of the Schrodinger equation. A pairing Hamiltonian is discussed as an example of the power of Lie algebras, and extended towards the Interacting Boson Model (IBM). Via a mean field approach, the connection between the IBM and geometric collective models is shown and the topic of quantum shape phase transitions is introduced. Section 1.1 Different levels of nuclear structure ’The atomic nucleus is like a huge soccer team: although the actions are set by the individual players, in the end, it is the team that wins the game’. The physical meaning of this metaphor is that all the interactions between the players in the field (protons and neutrons) contribute to the final structure of the team. Nevertheless, no matter how tempting, one cannot extend the metaphor over the whole line. In contrast to the popular sport discipline, the interactions between the particles are not fully known nor understood. This is partly due to the fact that the protons and neutrons are not elementary particles, rather composite particles in the sense that they are also built from other constituent parts, whichmeans that one is forced to start from the underlying quark degrees of freedom if a fundamental nucleon-nucleon interaction is desired. Another problem is that the atomic nucleus is a typical example of a many-body system. One could
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 program of the algebraic approach to nuclear collective motion is summarized. In essence, it should permit an accurate solution of the many-body problem within a subspace of the many-particle space if the conditions defining collective motion are present. The formulation, containing both dynamical and kinematical aspects, leads to coupled nonlinear algebraic equations for the matrix elements between collective states of fermion pair operators. An explicit set of equations is given for the problem of NNN (even) particles confined to a single-jjj shell and described by the conventional pairing plus quadrupole-quadrupole interaction Hamiltonian. This formulation is designed to be nearly exact in the pairing-dominant (seniority) limit and to be accurate enough (based on plausibility arguments) to describe the transition from spherical to deformed shape. That this is so is verified for configurations, the largest being jN=(212)6{j}^{N}={(\frac{21}{2})}^{6}jN=(221​)6, for which exact solutions are available. The present method can be viewed as an angular momentum conserving alternative to the description of the transition based on the Hartree-Bogoliubov-BCS method. Qualitative and detailed comparisons of the two methods are carried out. In particular, it is shown from approximate analytic results that the two theories predict the transition to occur for almost identical values of the parameters. Over-all agreement with exact results is much better for the algebraic method, however, confirming its status as an approximate method of variation after projection.