International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 7.5, pp. 667668

There have been many papers on optimizing counting times for achieving different purposes, and all optimization procedures require some knowledge of the distributions of counts or counting rates; often only the mean and variance of the distribution are required. It is also necessary to know the functional relationship between the quantity of interest and the counts (counting rates, intensities) entering into its measurement. Typically, the object is to minimize the variance of some function of the measured intensities, say . General statistical theory gives the usual approximation where cov(I_{i}, I_{j}) is the covariance of I_{i} and I_{j} if , and is the variance of , , if i = j. There is very little correlation^{2} between successive intensity measurements in diffractometry, so that cov(I_{i}, I_{j}) is negligible for . Equation (7.5.7.1) becomes These equations are strictly accurate only if F is a linear function of the I's, a condition satisfied for the integrated intensity, but for few other quantities of interest. In most applications in diffractometry, however, the contribution of each I_{j} is sufficiently small in comparison with the total to make the application of equations (7.5.7.1) and (7.5.7.2) plausible. Any proportionality factors (Section 7.5.1) can be absorbed into the functional relationship between F and the I_{j}'s.
The object is to minimize σ^{2}(F) by varying the time spent on each observation, subject to a fixed total time It is simplest to regard the total intensity and the background intensity as separate observations, so that in (7.5.7.2) the sum is over n `background' and n `total' observations. With expressed as a counting rate, its variance is [equation (7.5.3.8)], so that (7.5.7.2) becomes where for brevity G has been written for . The variance of F will be a minimum if, for any small variations of the counting times , subject to the constancy of the total time T. There is thus the constraint These equations are consistent if for all j where k is a constant determined by the total time T: The minimum variance is thus achieved if each observation is given a time proportional to the square root of its intensity. A little manipulation now gives for the desired minimum variance The minimum variance is found to be a perfect square, and the standard uncertainty takes a simple form.
Here, the optimization has been treated as a modification of fixedtime counting. However, the same final expression is obtained if the optimization is treated as a modification of fixedcount timing (Wilson, Thomsen & Yap, 1965).
Space does not permit detailed discussion of the numerous papers on various aspects of optimization. If the time required to move the diffractometer from one observation position to another is appreciable, the optimization problem is affected (Shoemaker & Hamilton, 1972, and references cited therein). There is some dependence on the radiation (Xray versus neutron) (Shoemaker, 1968; Werner, 1972a,b). A few other papers of historical or other interest are included in the list of references, without detailed mention in the text: Grant (1973); Killean (1972, 1973); Mack & Spielberg (1958); Mackenzie & Williams (1973); Szabó (1978); Thomsen & Yap (1968); Zevin, Umanskii, Kheiker & Panchenko (1961).
References
Grant, D. F. (1973). Singlecrystal diffractometer data: the online control of the precision of intensity measurement. Acta Cryst. A29, 217.Killean, R. C. G. (1972). The a priori optimization of diffractometer data to achieve the minimum average variance in the electron density. Acta Cryst. A28, 657–658.
Killean, R. C. G. (1973). Optimization of scan procedure for singlecrystal Xray diffraction intensities. Acta Cryst. A29, 216–217.
Mack, M. & Spielberg, N. (1958). Statistical factors in Xray intensity measurements. Spectrochim. Acta, 12, 169–178.
Mackenzie, J. K. & Williams, E. J. (1973). The optimum distribution of counting times for determining integrated intensities with a diffractometer. Acta Cryst. A29, 201–204.
Shoemaker, D. P. (1968). Optimization of counting times in computercontrolled Xray and neutron singlecrystal diffractometry. Acta Cryst. A24, 136–142.
Shoemaker, D. P. & Hamilton, W. C. (1972). Further remarks concerning optimization of counting times in singlecrystal diffractometry: rebuttal to Killean; consideration of background counting and slewing times. Acta Cryst. A28, 408–411.
Szabó, P. (1978). Optimization of the measuring time in diffraction intensity measurements. Acta Cryst. A34, 551–553.
Thomsen, J. S. & Yap, F. Y. (1968). Simplified method of computing centroids of Xray profiles. Acta Cryst. A24, 702–703.
Werner, S. A. (1972a). Choice of scans in Xray diffraction. Acta Cryst. A28, 143–151.
Werner, S. A. (1972b). Choice of scans in neutron diffraction. Acta Cryst. A28, 665–669.
Wilson, A. J. C., Thomsen, J. S. & Yap, F. Y. (1965). Minimization of the variance of parameters derived from Xray powder diffractometer line profiles. Appl. Phys. Lett. 7, 163–165.
Zevin, L. S., Umanskii, M. M., Kheiker, D. M. & Panchenko, Yu. M. (1961). K voprosu o difraktometricheskikh priemah pretsizionnyh izmerenii elementarnyh yacheek. Kristallografiya, 6, 348–356.