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Physical Review C

Hot drops: The mean field approximation for statistical and collective phenomena in nuclei

George Mantzouranis, Hans–Christian Pauli

Published Oct 1, 1980• 9 citations
Publisher

Abstract

A summation of selected graphs in the perturbation expansion for the statistical operator and the operator of state leads to a simplified equation of motion for the nuclear many-body problem. It contains as a special limit the time-dependent Hartree-Fock and the quantum-statistical master equation. In a pictorial way one could label the approach as "Hartree-Fock plus collision terms." The formalism allows one to understand the progression of the nucleus from the boiling of the particle-hole excitations to the collective state of motion. The work is restricted presenting the formalism. Various applications are possible and indicated, but apart from an illustrative example no concrete comparison with the experiment has been performed.NUCLEAR REACTIONS Formalism developed for unifying collective and single-particle aspects; mean field theory, Hartree-Fock plus collision terms.

DOI
10.1103/physrevc.22.1550

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A summation of selected graphs in the perturbation expansion for the statistical operator and the operator of state leads to a simplified equation of motion for the nuclear many-body problem. It contains as a special limit the time-dependent Hartree-Fock and the quantum-statistical master equation. In a pictorial way one could label the approach as "Hartree-Fock plus collision terms." The formalism allows one to understand the progression of the nucleus from the boiling of the particle-hole excitations to the collective state of motion. The work is restricted presenting the formalism. Various applications are possible and indicated, but apart from an illustrative example no concrete comparison with the experiment has been performed.NUCLEAR REACTIONS Formalism developed for unifying collective and single-particle aspects; mean field theory, Hartree-Fock plus collision terms.
In the time-dependent Hartree-Fock approximation, the fermions are assumed to interact only through the mean field and the collisions between particles are neglected. We formulate an extended time-dependent Hartree-Fock approximation which incorporates particle collisions due to the residual interaction, with the usual time-dependent Hartree-Fock approximation as the collisionless limit. It is obtained by a truncation of the Martin-Schwinger hierarchy at the second level and by using a simple representation of the one-body Green's function in terms of time-dependent occupation numbers. The final set of coupled equations consists of a modified time-dependent Hartree-Fock equation and a master equation for the occupation numbers. These results are physically transparent and describe properly the physics of the collision process. They may also be simple enough to be of practical use to study heavy-ion collisions or the dynamics of other fermion systems. Furthermore, as the configuration-space analog of the quantum Boltzmann equation, many important results concerning statistical dynamics are obtained. Concepts such as entropy, temperature, and local and thermal equilibrium can be quantitatively introduced. The well-known HHH theorem that entropy never decreases can be readily recovered. With the collision term explicitly exhibited, the macroscopic equations (equations of continuity, momentum flux, and energy) and their associated conservation theorems can also be derived. Analytic solutions for the master equation for simple cases lead to new "level crossing" formulas having characteristics distinctly different from the Landau-Zener level-crossing formula and illuminate the salient features as to how a nonequilibrium fermion system approaches thermal equilibrium.NUCLEAR REACTIONS Extension of time-dependent Hartree-Fock approximation. Collisions between particles. Master equation for occupation probabilities. Entropy, temperature, and thermal equilibrium. HHH theorem. Analytic solution of the master equation. New types of level crossing formula.
The excitation of a many-body system by a one-body perturbation is considered. The stationary phase approximation to a functional-integral representation of the many-body evolution operator shows that the optimal mean field for describing the final expectation values of one- and two-body operators is that given by the time-dependent Hartree-Fock method. The forced Lipkin model is considered as a test of the mean-field approximation.NUCLEAR REACTIONS Mean-field approximation, inclusive observables. Time-dependent Hartree-Fock.
A general method for a microscopic description of nuclear dynamical problems is discussed. The method is based on a functional integral representation for the many-body time evolution operator U(tf,ti)U({t}_{f},{t}_{i})U(tf​,ti​). In the stationary phase limit, a time-dependent mean-field approximation for the matrix elements of UUU is obtained. Using standard procedures this allows for extraction of quantum mechanical information about bound states, tunneling, or scattering phenomena in a many-nucleon system. The approximation represents a natural generalization of the time-dependent Hartree-Fock method.NUCLEAR STRUCTURE, NUCLEAR REACTIONS derived mean-field approximation for matrix elements of $\mathrm{exp}(\ensuremath{-}iHt)$ from exact functional integral; discussed relation to static HF, TDHF, and scattering calculations.
A nonperturbative method is developed for calculating the excitation of a many-body system by a time-dependent Hamiltonian. The stationary-phase approximation to a functional-integral representation of the interaction-picture many-body evolution operator results in a mean-field approximation to the SSS matrix which is asymptotically time independent. A one-body temporally nonlocal evolution equation defines the stationary mean-field configurations. The general method and character of the stationary solutions are illustrated by application to the forced harmonic oscillator and forced Lipkin model. Potential applications to realistic nuclear and atomic scattering situations are discussed.NUCLEAR REACTIONS Mean-field approximation, many-body SSS matrix, time-dependent Hartree Fock.
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
A classical theory of large amplitude, adiabatic collective motion, developed and applied previously to test examples, is transcribed into the language of time-dependent Hartree-Fock theory in order to initiate the application of the new method to problems in nuclear physics. The formulation which emerges can be characterized as a generalized cranking theory, in the sense that the cranking operator cannot be chosen arbitrarily, as in the conventional formulation of this method, but is instead fully constrained by the formalism. A procedure for obtaining approximate solutions to the new equations is described and illustrated with several simplified many-body models. It is inferred that traditional cranking calculations can serve as starting points for more realistic models. This paper also includes additional discussion concerning the calculation of collective masses and the problem of local stability of collective motion, not covered in our previous work.