Refinements in the computation of collision probabilities

Generally collision probabilities, P/sub II/, are computed by the Carlvik method using equally spaced azimuthal angles, and the P/sub II/'s are adjusted so that the probabilities sum to unity. It will be shown that for many XY cells efficiency can be substantially improved through use of double-Gauss azimuthal angles, and through an alternative adjustments procedure. Collision probabilities (which we'll refer to when convenient, as ''CP's'') have been very useful in a wide range of 1D and 2D transport calculations. Unfortunately, however, as K. Smith has pointed out, the computation of multidimensional collision probabilities is, in many cases, quite expensive. Consider, for example, the 1D one-group cell configuration whose physical parameters are exhibited. In such a 1D cell, collision probabilities can be computed very cheaply and accurately via E functions. For future reference essentially exact collision probabilities have been determined for this cell, and all the P/sub II/'s are shown. But suppose these same collision probabilities had to be computed by typical 2D methods incorporated, for example, in Argonne's CPXY code. With 40 angles, and a line spacing of 0.1cm, such a computation consumes 112 seconds of CPU time on the IBM 33A. Yet the P/sub II/'s in the clad regions are still in error by more than 10%. It is difficult to say just how accurate these P/sub II/'s must be in practice, but one would need a good deal of justification before one could accept collision probabilities much coarser than these. It seems clear, then, that the computation of multigroup collision probabilities, especially for fine-group fast-reactor calculations, is difficult enough to merit some consideration