International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 7.5, p. 666

## Section 7.5.2. Counting modes

A. J. C. Wilsona

aSt John's College, Cambridge CB2 1TP, England

### 7.5.2. Counting modes

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Whatever the radiation, in both single-crystal and powder diffractometry, the integrated intensity of a reflection is obtained as a difference between a counting rate averaged over a volume of reciprocal space intended to include the reflected intensity and a counting rate averaged over a neighbouring volume of reciprocal space intended to include only background. If these intentions are not effectively realized, there will be a systematic error in the measured intensity, but in any case there will be statistical fluctuations in the counting rates. The two basic modes (Parrish, 1956) are fixed-time counting and fixed-count timing. In the first, counts are accumulated for a pre-determined time interval, and the variance of the observed counting rate is proportional to the true (mean) counting rate. In the second, on the other hand, the counting is continued until a pre-determined number of counts is reached, and the variance of the observed counting rate is proportional to the square of the true counting rate. Put otherwise, the relative error in the intensities goes down inversely as the square root of the intensity for fixed-time counting, whereas it is independent of the intensity for fixed-count timing. Each mode has advantages, depending on the purpose of the measurements, and numerous modifications and compromises have been proposed in order to increase the efficiency of the use of the available time. References to some of the many papers are given in Section 7.5.7.

In principle, probability distributions can be determined for any postulated counting mode. In practice, they become complicated for all but the simplest modes; this is true even for the single measurement of the total counting rate or the background counting rate, but is even more pronounced for the distribution of their difference (the reflection-only rate). For most crystallographic purposes, however, it is only necessary to know the mean (to correct for bias, if present) and the variance (for the estimation of weights in refinement processes, see Part 8 ) of the distribution function.

### References

Parrish, W. (1956). X-ray intensity measurements with counter tubes. Philips Tech. Rev. 17, 206–221.