Perturbation theory deep-penetration problems

A primary objective of this work is to provide the general theoretical foundation for different perturbation theory applications in all types of nuclear systems. Consequently, general notations have been used without reference to any specific mathematical description of the transport equation used for numerical calculations. The formulation has been restricted to time-independent and linear problems. Throughout the work we describe the scope of past, and discuss the possibility for future applications of perturbation theory techniques for the analysis, design and optimization of fission reactors, fusion reactors, radiation shields, and other deep-penetration problems. This review concentrates on developments subsequent to Lewins review (7) published in 1968. The literature search covers the period ending Fall 1974. 

Generalized perturbation theory for two special cases of composite functionals are presented and discussed in some detail GPT for reactivity (Section V,B), and GPT for a detector response in inhomogeneous systems (Section V,E). The GPT formulation for reactivity is equivalent to a high-order perturbation theory, in the sense that it allows for the flux perturbation, GPT for a detector response in inhomogeneous systems 42, 43) is, in fact, the second-order perturbation theory known from other derivations I, 44, 45). These perturbation theory formulations provide the basis for new methods for solution of deep-penetration problems. These methods are reviewed in Section V,E,2. 

Perturbation theory for detector response in inhomogeneous systems provides the foundation for efficient and interesting methods for the solution of deep-penetration problems characterized by a large source-detector distance and a localized geometrical irregularity. 

The advent of perturbation theory for altered systems (see Section V,F) opens a new field for the application of perturbation theory—the field of perturbation sensitivity studies. This is the study of changes in effects of perturbations, or system alterations, caused by uncertainties or variations in input parameters. Examples are (1) the uncertainty of the change in an integral parameter (such as the breeding ratio) resulting from design variations due to uncertainties in cross sections, (2) nonlinear effects of cross-section uncertainties, and (3) the effects of data uncertainties, approximations in computational models, or design variations on the detector response in a deep-penetration problem that is solved with a flux-difference or an adjoint-difference method (see Section V,E). 

Perturbation theory and techniques are coming of age. They provide increasing support for the design and analysis of nuclear systems, and for the evaluation of nuclear data. This is evidenced by the large number of perturbation theory based computer codes developed within the last few years. These trends characterize the new codes (1) the extension of conventional perturbation techniques to multidimensional systems and to high-order approximations of the Boltzmann equation (2) the development of methods for implementing new perturbation theory formulations, such as the generalized perturbation theory formulations and (3) the application of perturbation theory formulations to new fields, such as sensitivity studies and the solution of deep-penetration problems. 

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