On the absolute detection efficiency of a cylindrical scintillation phosphor in the case of an extended plane γ-ray source

Abstract

It is the aim of the present article to establish some practical formulae permitting an accurate evaluation of the absolute detection efficiency T( E) of a cylindrical scintillation phosphor for extended plane γ-ray sources. After an introductory discussion in which the special case of a circular disk source of radius R appears to be of particular interest, the efficiency for such a source, represented by a complicated triple integral, is developed as a power series of R by means of an elementary transformation method. This method is simple but it entails such extensive amounts of calculations that it is only used to obtain a preliminary formula exact up to the R 2-term. However, a more advanced study of certain peculiar characteristics of the integral representation of the disk source efficiency leads to another method offering the possibility of an extension of the mentioned formula to all powers of R at the cost of a minimum of intermediate steps. As it appears to be sufficient for practical purposes, this method is applied to establish a fourth order formula for the circular disk source. In the next section, the entire formalism is generalized to the case of an arbitrarily shaped plane source and the final result is a formula expressing T( E) as a simple polynomial in terms of a point source efficiency and its first four partial derivatives with respect to the crystal radius ν regarded as a variable. After a few other considerations, the paper ends with a discussion of the numerical aspects of the problem under study. A set of new formulae permitting an easy evaluation of point source efficiencies to within a high degree of precision is presented. Among them, there is a new numerical integration procedure applicable to cases where the variation of the integrand is approximately exponential. Finally, it is shown how the previously mentioned partial derivatives can be simply obtained by means of a finite difference method. In this way, it appears that the computation of five properly chosen point source efficiencies suffices to evaluate the detection efficiencies for an infinite variety of extended sources. This conclusion is illustrated by means of a typical numerical example.