Hartmut M. Hofmann[1]

Institute for Theoretical Physics, University of Erlangen-Nurnberg,

Erlangen, Germany

Textures given at the
Workshop on Nuclear Reaction Data and
Nuclexir Reactors: Physics, Design and Safety
Trieste, 25 February — 28 March 2002

LNS0520001

We describe in some detail the refined resonating group model and its application to light nuclei. Microscopic calculations employing real­istic nuclear forces are given for the reaction 3He(n, p). The extension to heavier nuclei is briefly discussed.

1 Introduction

The neutron standard cross sections cover a wide range of target masses from hydrogen to uranium. The high mass range is characterized by many over­lapping resonances, which cannot be understood individually. In contrast, the few-nucleon regime is dominated by well-developed, in general, broad resonances. The interpolation and to less extend extrapolation of data re­lies heavily on R-matrix analysis. This analysis has to fit a large number of parameters related to position and decay properties of resonances. Due to the limited number of data and their experimental errors, any additional input is highly welcome. Except for neutron scattering on the proton any of the standard cross sections involve few to many nucleon bound states. These many body systems can no more be treated exactly. The best model to treat scattering reactions of such systems proved the resonating group model (RGM) in its various modifications. Therefore we begin with a dis­cussion of the RGM.

The solution of the many-body problem is a long standing problem. The few-body community developed methods, which allow an exact solution of few-body problems, via sets of integral equations. In this way the 3-body problem is well under control, whereas the 4-body problem is still in its infancy. Hence, for systems containing four or more particles one has to rely on approximations or purely numerical methods. One of the most successful methods is the resonating group model (RGM), invented by Wheeler [1] more than 50 years ago in molecular physics. The basic idea was a resonant jump of a group of electrons from one (group of) atom(s) to another one.

This seminal idea sets already the framework for present day calculations: Starting from the known wave function of the fragments, the relative wave function between the fragments has to be determined e. g. via a variational principle. The basic idea, however, also sets the minimal scale for the cal­culation: a jump of a group of electrons needs at least two different states per fragment leading to coupled channels. Hence, an RGM calculation is basically a multi-channel calculation, which renders immediately the tech­nicality problem. This essential point of any RGM calculation is the key to an understanding of the various realisations of the basic idea. Besides the most simple cases, for which even exact solutions are possible, the RGM is always plagued with necessary, huge numerical efforts. Therefore, a dis­cussion about the various approaches has to be given. In most applications of the RGM till now, the evaluation of the many-body r-space integrals is the largest obstacle. It can only be overcome by using special functions, essentially Gaussians, for the internal wave functions of the fragments. Two basically different methods are well developed: One uses shell model tech­niques to perform the integration over the coordinates of the known internal wave functions leading to systems of integro-differential equations, whose kernels have to be calculated analytically. The other expands essentially all wave functions in terms of Gaussian functions and integrates over all Ja­cobian coordinates leading to systems of linear equations, whose matrices can be calculated via Fortran-programs. Since the latter is more suited for few-body systems and I’m more familiar with it, I will concentrate on this so-called refined resonanting group model (RRGM) introduced by Hacken — broich [2]. As detailed descriptions of the first method exist [3], I will not discuss it. I will, however, compare the advantages and disadvantages of both methods at various stages.

In order to allow the reader to find further applications of the RRGM, I will try to generalize the formal part from the nuclear physics examples I will give later on. Therefore I will first discuss the variational principle for the determination of the relative motion wave function. I will then demonstrate how the r-space integrals are calculated in the RRGM. The next two chapters deal with the treatment of the antisymmetriser and the evaluation of spin — isospin matrix elements. The last chapter, dealing with formal developments, demonstrates how the wave function itself is used by the evaluation of matrix elements of electric transition operators.

A chapter on various results from nuclear physics illustrates various points of the formal part and helps to understand the final part on possible exten­sions and also on the limitations of the model. Part of the work is already described previously [4]. Some repetition cannot be avoided in order to keep this article self-contained, so I will refer sometimes to ref. [4] for details.