Statistical fluctuations in the ionization yield and their relation to the electron degradation spectrum

Abstract

Various indices that characterize the statistical fluctuations in the ionization yield for electron incidence are discussed. The indices include the variance, the moments, the central moments, and the cumulants, all defined in terms of the probability $P(T_0,j)$ that an electron of kinetic energy $T_0$ causes, upon complete slowing down, exactly $j$ ion pairs. Like the mean ionization yield, each of the indices is expressible as an integral over the current energy $T$ of electrons. The integrand is a product of the Spencer-Fano degradation spectrum $y(T_0,T)$ for source energy $T_0$ with another factor, and displays contributions from different values of $T$ to the quantity of interest. It has been shown for the first time that the cumulant of any order (including the variance as a special case) receives contributions from different $T$ nearly the same way as the mean ionization yield. Therefore, $T_0$ greatly exceeding the first ionization threshold, every cumulant is roughly proportional to $T_0$, just as the mean yield is. This leads to a generalization of the Fano factor. Thus the present findings reinforce the importance of the cumulant notion in statistical physics in general. Numerical work on molecular hydrogen as an example substantiates the general conclusion and predicts the value 0.30 for the Fano factor for an electron of initial energy 10 keV.