Corrected optimum design of experiments on exponential absorption using more than one absorber thickness

Abstract

A corrected general expression is obtained for the statistical error in the absorption coefficient measured in an experiment on exponential absorption with n observations, making an allowance for a background measurement for each observation. For n = 2 the design is optimised without any constraint on the weighting of the points. The sensitivity to different backgrounds for the two points is computed. With equally spaced values of absorber thickness, and with equal weighting on each point, the values of thickness interval and all the counting times are optimised for n = 2−7 with backgrounds of zero and 0.001–100 times the count rate for no absorber. For zero and very large backgrounds simplified formulae are obtained and optimised for any value of n, and approximate optimum values are found for any background at any value of n. The sensitivities of the errors in the absorption coefficient are investigated for variations from optimum values of absorber thickness and of background counting times, with compensating changes in counting times with absorbers. The consequential departures from equal weighting for these variations of thickness and counting times are also computed. In consideration of a possible shortage of rare absorber materials, the minimum errors for optimum counting times are computed for fixed absorber thicknesses down to one quarter of the optimum thicknesses, for n = 3 and 7 and relative backgrounds of zero and 0.001–100.