Variational treatment of electron degradation and yields of initial molecular species

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Abstract
We present a general theory of electron degradation and of the yields of initial species such as ions and excited states that appear in matter as a result of irradiation with electrons. We give variational expressions for observable quantities such as the degradation spectrum, the mean yields, and the yield fluctuations. A systematic analysis using variational principles reveals, in greater detail than ever before, relations among major analytical methods including the Fowler method, the Spencer-Fano method, and the method of Knipp et al. Each of these methods represents a different angle of approach to the same problem and leads to the same prediction for the mean yield of any initial species, for a fixed set of electron cross-section data. Among our findings it is noteworthy that the knowledge of the Spencer-Fano degradation spectra for various source-electron energies enables one to calculate by quadrature not only the mean but also the statistical fluctuations of the yield of any initial species. Furthermore, when one introduces small changes in the cross-section data (because of new information or upon consideration of chemical impurities in the medium), one may express the ensuing modifications of any observable quantity in the form of a perturbation expansion. Then, every term in the perturbation series is calculable again from the Spencer-Fano degradation spectra for various source energies for the unperturbed problem. In this respect and many others, the Spencer-Fano degradation spectrum is the most basic element in the solution of the degradation problem. Finally, our mathematical analysis of various difference-integral equations (including nonlinear equations) may be of interest in areas other than electron degradation. Therefore, the present paper includes remarks on the physical meaning of adjoint operators and on other points belonging to general mathematical physics.