Static path approximation in deformed nuclei.

Abstract

The static path approximation to the partition function of a nucleus is studied for Hamiltonians with quadrupole-quadrupole interactions. The static path approximation yields a higher level density than the finite-temperature Hartree theory due to a better treatment of the deformation degrees of freedom. The level density is further increased when the nucleus has a static deformation. Comparing with the exact partition function of the Elliott SU(3) model, the static path approximation is found to be remarkably accurate and far superior to finite-temperature Hartree theory. We have recently suggested that the statistical mechanics of nuclei might be better described by a certain approximation in a functional integral formulation than by the conventional finite-temperature Hartree-Pock theory see also Ref. 3. The functional integral we employ is the Hubbard-Stratonovich representation of the partition function, which is a path integral in the space of single-particle potential fields. The approximation, introduced to condensed matter physics in Ref. 4, is to integrate only over static potential fields; we call this the static path approximation (SPA). It makes a tractable theory for dealing with the hundred or so particles that constitute a nucleus, provided the residual interaction is represented by a few separable terms. The computational diKculty would only be somewhat larger than required for systematic finite-temperature Hartree calculations, but since this is already large it is important to test the static path approximation on simplified models before proceeding with the numerical program. In our previous work we investigated the theory using a pairing Hamiltonian, and found very encouraging results. Deformations and rotational degrees of freedom are also an important aspect of nuclear dynamics, and in this work we want to see how they afFect the statistical properties of nuclei. Empirically, deformed nuclei exhibit higher level densities due to the rotational degrees of freedom: ' We will show in the next section qualitatively how this enhancement emerges from the path-integral formulation. It will also be shown that the SPA level density is higher than that of Hartree theory even when the nucleus is spherical. In order to test the approximation, it is necessary to compare with a model that can be exactly solved. The partition function for Elliot's SU(3) model can be evaluated for various shells of interest, and we use this as our testing ground. The details are presented in the third section. We find that the static path approximation describes the level density very well, unlike the Hartree approximation, which we also consider for comparison purposes. However, the static path approximation does not produce the lowest part of the energy spectrum. This is the major efFect of the truncation of the paths, that the highly correlated low states are efFectively absent from the partition function. In Sec. IV we apply the SPA to a more realistic Hamiltonian, with single-particle wave functions and energies appropriate for the 3 -60 mass region. This demonstrates the feasibility of the method for a more realistic situation than we could achieve with the SU(3) model, where the single-particle levels are all degenerate. We also find here that the level density in SPA is much higher than in Hartree theory.