Perturbation theory bilinear functionals

The development and application of generalized perturbation theory (GPT) has made considerable progress since its introduction by Usachev (i(S). Usachev developed GPT for a ratio of linear flux functionals in critical systems. Gandini 39) extended GPT to the ratio of linear adjoint functionals and of bilinear functionals in critical systems. Recently, Stacey (40) further extended GPT to ratios of linear flux functionals, linear adjoint functionals, and bilinear functional in source-driven systems. A comprehensive review of GPT for the three types of ratios in systems described by the homogeneous and the inhomogeneous Boltzmann equations is given in the book by Stacey (41). In the present review we formulate GPT for composite functionals. These functionals include the three types of ratios mentioned above as special cases. The result is a unified GPT formulation for each type of system. 

Generalized-function formulations of GPT for homogeneous systems are the source of sensitivity functions for different integral parameters Equation (189) for reactivity worths, and Eq. (162) for ratios of linear and bilinear functionals. The first-order perturbation theory expression for reactivity [Eq. (132)] can also be used for sensitivity studies. 

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