Statistical Error in Absorption Experiments

In certain exponential absorption experiments, notably measurements of cross sections by transmission, it is important to achieve minimum statistical error in a limited time or to minimize the counting time required to measure the absorption coefficient with a preassigned accuracy. The conditions required to attain these ends, i.e., the geometry for optimum transmission, and the best apportionment of counting times among the incident and transmitted beams and background, have been investigated for a wide range of relative backgrounds (${10}^{-3\mathrm{}}$ to ${10}^2$), and for two geometries: I. Beam area fixed, absorber thickness alone is varied. II. Beam area and absorber thickness are both disposable parameters, while the total amount of absorber intercepting the beam remains fixed. In both cases the incident flux density and the background rate are assumed constant. The optimum transmissions are shown to be, in general, considerably smaller than those commonly used in absorption experiments. Thus, in Case I, a useful rule is to employ a transmission of about 0.1 for low backgrounds, 0.2 for moderate backgrounds, and 0.3 for high backgrounds. The following have also been determined: (a) minimum statistical error for a given total counting time, (b) statistical error and the best distribution of counting times for non-optimum geometry, and (c) sensitivity of the accuracy or total counting time to deviations from optimum transmission.