Eigenvalue Sensitivity Theory for Resonance-Shielded Cross Sections

A method is presented to compute sensitivity coefficients for the eigenvalue of a critical assembly, including implicit effects associated with changes in resonance-shielded multigroup cross sections. Two alternative approaches, based on a forward and an adjoint solution, respectively, are developed to determine the effect of perturbations on the weight function used in group averaging of resonance cross sections. The forward method uses an automated methodology to compute the flux derivative with respect to various cross-section processing parameters. The adjoint method introduces adjoint equations for a multigroup cross-section functional and presents adjoint slowing-down equations for two common methods of resonance self-shielding. Expressions are presented for sensitivity coefficients of self-shielded group cross sections. These sensitivity coefficients are combined with conventional eigenvalue sensitivity coefficients to obtain a general perturbation expression for the multiplication factor. An example application determines the sensitivity of the critical eigenvalue to hydrogen density changes in a homogeneous sphere containing low-enriched uranium. It is shown that changes in ²³⁸U-shielded cross sections caused by perturbations in hydrogen concentrations are important components in the overall eigenvalue sensitivity coefficient, which is predicted well by the developed method.