A Variational Principle for Heterogeneous Resonance Capture as a Basis for Approximation Methods

A variational principle for resonance capture in heterogeneous reactors has been developed. The functional becomes the exact resonance integral when the flux is exact, and in general the functional also has the convenient form of an explicit resonance integral multiplied by a correction factor. A reasonable trial function for the adjoint is selected, which allows explicit, interpretable expressions to be derived for the correction factor when trial functions corresponding to the various currently used approximations are inserted. When solutions of Chernick-Rothenstein type equations are used for trial functions, the correction factor is unity. The inexactness in these equations is detectable only with higher-order approximations to the adjoint function. The correction factor for other approximations then furnishes a measure of the error as compared to exact solutions of C-R equations as a standard. Several applications are discussed.