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Physical Review C

Quantum corrections to the potential energy for large amplitude collective motion

Niels R. Walet, Abraham Klein, G. Do Dang

Published Jan 1, 1992• 12 citations
Publisher

Abstract

We discuss the problem of inclusion of quantum corrections in the semiclassical theory of adiabatic large amplitude collective motion for many-fermion systems. We concentrate on deriving a formula for the leading quantum correction to the classical collective potential energy function in this theory. This is an extension of the usual calculation of the quantum corrections to the static Hartree-Fock energy using the random phase approximation. The answer can be expressed in terms of those solutions of a local random phase approximation that describe oscillations orthogonal to the collective surface. Because of the strict enforcement of the Pauli principle, however, the answer differs from the usual quasiboson approximation, yielding the correct ground-state corrrelation energy for a static solution to the Hartree-Fock equations. The result is applied, approximately, to help improve a previous treatment of the low energy spectrum of $^{28}\mathrm{Si}$.

DOI
10.1103/physrevc.45.249

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arXiv:nucl-th/0001056•Progress of Theoretical Physics•May 1, 2000•63 citations
We discuss the problem of inclusion of quantum corrections in the semiclassical theory of adiabatic large amplitude collective motion for many-fermion systems. We concentrate on deriving a formula for the leading quantum correction to the classical collective potential energy function in this theory. This is an extension of the usual calculation of the quantum corrections to the static Hartree-Fock energy using the random phase approximation. The answer can be expressed in terms of those solutions of a local random phase approximation that describe oscillations orthogonal to the collective surface. Because of the strict enforcement of the Pauli principle, however, the answer differs from the usual quasiboson approximation, yielding the correct ground-state corrrelation energy for a static solution to the Hartree-Fock equations. The result is applied, approximately, to help improve a previous treatment of the low energy spectrum of 28Si^{28}\mathrm{Si}28Si.
A finite many-particle system can have collective states for which the off-diagonal matrix elements of certain one-particle operators are of the same order of magnitude as the diagonal elements. In such cases it is suggested that the random-phase approximation is in need of generalization. Examples are the uniform translational motion of any system and the rotational motion of deformed nuclei. The generalization is suggested after a review and critical analysis of the Hartree-Fock approximation. The model single-particle wave functions of the latter are replaced by wave functions in a space labeled both by the particle variables and by the quantum numbers of the collective motion. These generalized amplitudes are defined field-theoretically, and a self-consistent scheme for their calculation is obtained from the equations of motion. In addition to the self-consistent potential defined in the enlarged space, the energies of the excited states also turn out to be given by a natural self-consistency requirement. The new calculational scheme is first applied to a systematic restudy of the random-phase approximation where the self-consistency requirement on the energies has previously been overlooked. As a first characteristic application we obtain without "pushing" the total mass of a system in uniform translation, and a reinterpretation of the Hartree-Fock average field.
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.
A theory of large amplitude collective motion of a many-particle system is presented, which is relevant, for example, to nuclear fission. The theory is a combination of techniques used in many areas of physics and mathematics. The starting point is the application of the time-dependent Schrödinger equation to generate invariant subspaces of the Hamiltonian in the Hartree–Fock approximation. This is a generalization of the group-theoretical device of generating orbits of a group in the construction of reduced representations. It is shown how solutions of the time-dependent Schrödinger equation can be expressed as instantaneous stationary states of a constrained static Hamiltonian. Thus contact is made with the traditional cranking models and constrained Hartree–Fock theories of large amplitude collective motion. The collective motion is quantized using the Hill–Wheeler–Griffin method of generator coordinates in a basis of generalized coherent states. One is thereby able to exploit much of the theory of harmonic oscillator coherent states, which have been so successfully used in the quantum theory of the laser. The resulting Schrödinger equation for the collective dynamics is expressed both in the Bargmann representation and in the more familiar Schrödinger representation. It is shown that solution of the Schrödinger equation in the small amplitude harmonic approximation reproduces the well-known RPA result. A pilot calculation for 28Si shows that application in large amplitude is also feasible.
Collective excitations are one of the most common and interesting features of many-body systems. Of particular interest are those collective modes which can be interpreted in terms of vibrations. Nuclei show a large variety of such modes, both low-lying and high-lying. In particular the giant dipole resonance is due to the coherent motion of protons against neutrons. Its analogue in metal clusters is the dipole plasmon excitation, due to the oscillation of electrons against the positive ions. The Random Phase Approximation (RPA) is extensively used as a microscopic theory to study the basic properties of these collective excitations. In the derivation of RPA use is made of the Quasi Boson Approximation (QBA). It consists in replacing the expectation values in the correlated ground state with those in the uncorrelated (Hartree-Fock) one. Strictly related to QBA is the feature of RPA of predicting a harmonic spectrum. On the other hand, the existence of anharmonicities in the multiphonon spectra of nuclei and their relevance in various physical processes are well established. Overcoming QBA has represented the starting point of many attempts aiming at improving RPA. A recently developed approach in this line will be presented and discussed. A very nice property of RPA is to satisfy Energy Weighted Sum Rules (EWSR). However, some violations are present in several extensions of RPA. We will show how to cure this shortcoming.
An adiabatic approximation to the selfconsistent collective coordinate method is formulated in order to describe large amplitude collective motions in nuclei with pairing correlations on the basis of the time-dependent Hartree-Fock-Bogoliubov equations of motion. The basic equations are presented in a local harmonic form which can be solved in a manner similar to that of the quasiparticle RPA equations. The formalism guarantees the conservation of nucleon number expectation values. An extension to the multi-dimensional case is also discussed.