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Physical Review C

Collective motion of nuclear mixed states: Thermal boson expansions

A. Kuriyama, M. Yamamura, João da Providência, Constança Providência

Published May 1, 1992• 4 citations
Publisher

Abstract

A classical microscopic theory of mixed states has recently been formulated in order to deal with the theoretical description of nuclear phenomena observed in highly excited states. In this theory, which is here briefly reviewed, the degrees of freedom of real fermions are doubled by fictitious fermions, in the context of the formalism of thermofield dynamics. The requirement of static stability of the thermal state of equilibrium plays, in the theory, a relevant role. The possibility of constraining the dynamics of the fictitious degrees of freedom, in such a way as to eliminate spurious thermal excitations, is discussed. The concept of canonical collective variables for mixed states is analyzed. Boson expansions appropriate to describe collective excitations of hot quantal systems are introduced and their properties investigated. It is shown that thermal boson expansions based on the formalism of thermofield dynamics are free from formal inconsistencies which may be present in boson expansions derived by an ad hoc quantization of conventional thermal mean field theories. The formalism is applied to the Lipkin model, and the correlation energy of a hot system is discussed.

DOI
10.1103/physrevc.45.2196

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A classical microscopic theory of mixed states has recently been formulated in order to deal with the theoretical description of nuclear phenomena observed in highly excited states. In this theory, which is here briefly reviewed, the degrees of freedom of real fermions are doubled by fictitious fermions, in the context of the formalism of thermofield dynamics. The requirement of static stability of the thermal state of equilibrium plays, in the theory, a relevant role. The possibility of constraining the dynamics of the fictitious degrees of freedom, in such a way as to eliminate spurious thermal excitations, is discussed. The concept of canonical collective variables for mixed states is analyzed. Boson expansions appropriate to describe collective excitations of hot quantal systems are introduced and their properties investigated. It is shown that thermal boson expansions based on the formalism of thermofield dynamics are free from formal inconsistencies which may be present in boson expansions derived by an ad hoc quantization of conventional thermal mean field theories. The formalism is applied to the Lipkin model, and the correlation energy of a hot system is discussed.
With the aim of giving a microscopic description of various nuclear phenomena observed in highly excited states, the equation of the collective submanifold for mixed states is presented. The basic idea is a possible TDHF-like variation in the enlarged space which is adopted in the thermo field dynamics formalism. A set of equations, which determines the collective submanifold, is obtained. The form is analogous to that given in the conventional TDHF theory. At the small amplitude limit, an equation, the form of which is similar to the conventional RPA equation, can be derived in a natural way.
A variational derivation of the Liouville-von Neumann equation of quantum-statistical mechanics is presented, in order to formulate mean~field approximations appropriate to mixed states. The Hartree-Fock and the RPA at finite temperatures are particular cases of the general formalism. A thennal boson expansion is defined, which allows us to describe anhannonic motion around a thennal excited state. In a numerical application on the basis of the Lipkin model, temperature-dependent phase transitions are observed. I. Introduction Studies of the decay of the compound nuclei formed in heavy-ion reactions have added a new dimension to the field of nuclear structure. The temperature of the compound nucleus at a given excitation energy rises from zero at the yrast state to a few MeV at angular momentum zero. Recently the decay of high-excited states above the yrast line, like the giant dipole resonance, has been observed 1,2). These measurements were carried out making use of sum-spectrometer multiplicity techniques, thus allowing to determine the average angular momentum of the system and from this knowledge the temperature of the nucleus. One can then study experimentally the nuclear structure at finite values of both the angular momentum and temperature, and the corresponding theoretical analysis of giant resonances above thermal excited states is now the object of intensive research 3-7). On the other hand, the observation of deep-inelastic reactions has been an important sourCe of information on the behaviour of nuclei under strong external fields. In these reactions the two interacting ions emerge after the collision with approximatley the same charge and mass, but having lost a large fraction of the energy and angular momentum of relative motion they had in the entrance channel (cf. e.g. ref. 8) and references therein). Different mechanisms play a role in these reactions, like the transfer of particles and the excitation of giant resonances (cf. ref. 9) and references therein). Because these vibrations have a damping width the energy and angular momentum absorbed through them into the nuclei lead to a finite temperature and to rigid rotation. This
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.
In the context of the boson expansion theory, it is usually the case that the bosonization of single-boson (fermion) states is ignored. Although this is tolerable to some extent in cold systems, it causes serious difficulties in the treatment of thermal ensembles where the single-boson (fermion) density of states plays an important role. In the framework of the thermo-field dynamics it is shown how extended forms of the Holstein-Primakoff mapping for both bosons and fermions can lead to consistent thermal-boson expansions. Applications to the O(N)O(N)O(N) anharmonic oscillator and the Lipkin model are presented.
We illustrate some of the concepts introduced in a recent paper devoted to the formulation of a self-consistent theory of large amplitude collective motion at finite excitation energy. We apply these concepts to a Hamiltonian with the symmetry of $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ (a doubled Lipkin model), in which the latter, exemplifying the slow, collective degrees of freedom, interacts with a set of harmonic oscillators representing the fast, noncollective variables. We pass to a mean-field approximation, which implies a collisionless regime, but show as well how the formalism may be extended to include the limit of collisions that establish instantaneous local equilibrium. In the representation of natural orbitals, the variables split into classical canonical sets and occupation numbers of the orbitals. Careful attention is paid to the problem of eliminating the intrinsic canonical coordinates from the equations of motion in such a way (white noise approximation) as to yield dissipative equations of motion for the collective variables. For the case of local equilibrium, the relaxation problem requires in addition an equation relating the slow change in temperature to the slow change of the collective variables. The two limiting relaxation scenarios are studied numerically.
Based on recent progress in the application of thermo field dynamics (TFD) [1] to thermal many-body systems, several authors [2, 3, 4] have considered a consistent thermal boson expansion (TBE). In this regard two points of view have been taken [2]. The first (path I) consists of a bosonization of the original degrees of freedom of the system, substituting for these ideal boson images. The thermalization is then achieved by doubling those newly introduced bosons as according to TDF prescription. The second possibility (path II) proceeds on the other hand via a thermalization of the system by doubling the original degrees of freedom and a subsequent bosonization of the entire new system. The two paths do not lead to the same results. Moreover, when applied to the Lipkin model, a closer look shows that the choice of path I implies that