Volterra integral equation of the first kind

In animal studies of the effect of the deposition of radioactive debris in the lung, use is made of very small spheres (about 10 μm diameter) of ZrO2 impregnated with plutonium, an α-particle emitter. A quality control problem that arises in the manufacture of these spheres is that of determining the radial density distribution of plutonium throughout the sphere. It turns out that the sphere diameters are shorter than the α-particle range in ZrO2 (about 20 μm), so that, with a knowledge of the energy--path length relation in ZrO2, one may obtain information on the plutonium distribution by measuring the energy spectrum of emitted α particles. An idealized description of an experimental setup is given. Indeed, a similar instrument has already been built, and a numerical method of obtaining the Pu distribution has been developed. However, the method exhibited numerical instabilities. The purpose of this paper is to examine a stable calculation of the distribution. 3 figures, 2 tables