### Abstract:

A method of treating resonance captures for few energy group
calculations is developed using the Narrow Resonance-Infinite
Absorber approximation to the resonance integral. The application
of this method is not restricted to this approximation, however,
as other approximations to the resonance integral could also be
used.
The approach presented defines average parameters based on
assuming that both the total dilute resonance integral and the
total effective resonance integral are the sums over the individual
resonances in the energy range under consideration. It is shown
then that if the effective resonance integral equation for a single
average resonance has the same functional form as the single resonances
using the above definition for the total dilute resonance
integral, then all parameters have been determined. The factor
remaining to be calculated from these definitions is the averaged
resonance peak height. This parameter, which is also in the form
of a sum over the individual resonances, is then studied in some
detail to see if any simpler form can be found.
The first approach is to examine several simple cases representing
basic types of resonance size combinations using only two
resonances. These studies lead to considering several simplifications
to the equation for the average resonance peak height,
including a series form, a harmonic mean form, and several series
corrections to harmonic mean. The range of applicability is examined
and it is shown that none of these forms has universal
applicability.
Upon generalizing these approximations to include more resonances,
several more forms based on correcting the mean value are
postulated. Also based on this work, a functional form requiring
curve fitting techniques is suggested. This curve fitting approach
is not pursued here, however.
These generalized equations for N resonances are applied to
the resonances of several real isotopes. It is shown that the
assumptions made in developing the approximations based on two
resonances and then expanded to N resonances can be correlated
with the results of applying these equations to the more complicated
case of many resonances. This then forms a basis for predicting
the behavior of group-averaged resonance parameters.