Abstract
A theoretical treatment of the fluctuations in the ionization yield in finite regions of a medium under electron irradiation is presented, and its salient features are discussed vis-\`a-vis the existing theory for the completely space-independent case. It is shown that unlike in the latter situation, the integral equations for the moments of the probability distribution are not adjoint to the integral equation for the flux, though their corresponding integro-differential forms are. For a uniformly distributed source, the Spencer-Fano equation again turns out to be adjoint to the integral equations for the moments; however, in contrast to the space-independent case, a knowledge of the degradation spectra is not sufficient to determine the moments in this case. Finally, to characterize the fluctuations in any volume $v$, a volume-dependent Fano factor is introduced. The theory indicates that this factor should initially increase with $v$ from a value of 1 for small $v$, pass through a maximum, and then approach the conventional Fano factor, which, as is well known, is generally much less than unity.
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