Statistical Theory of Nuclear Collision Cross Sections. II. Distributions of the Poles and Residues of the Collision Matrix

The relationship between the statistical properties of the parameters defining the $R$ matrix and the distributions and correlations of the poles and residues of the statistical collision matrix are explored by means of some limited numerical computations involving models for reactions in the presence of large numbers of competing strongly absorbed channels. The results shed light on the distributions of resonance energies and widths, and on the relationship between the partial width to spacing ratios and the channel transmission coefficients. The calculations also yield substantial channel-channel and resonance-resonance correlations in the complex amplitudes which define the collision-matrix pole residues. These are important for their effects on average cross section and fluctuation calculations. It is found that the investigated statistical relationships depend on the choice of $R$-matrix boundary conditions, and the implications of this for the choice of boundary conditions are discussed.