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. 667

Section 7.5.4. Fixed-count timing

A. J. C. Wilsona

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

7.5.4. Fixed-count timing

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The probability of a time t being required to accumulate N counts when the true counting rate is ν is given by a Γ distribution (Abramowitz & Stegun, 1964[link], p. 255): [ p(t)\,{\rm d}t = [(N-1)!]^{-1}(\nu t)^{N-1}\exp (-\nu t)\,{\rm d}(\nu t). \eqno (7.5.4.1)]The ratio N/t is a slightly biased estimate of the counting rate ν; the unbiased estimate is (N − 1)/t. The variance of this estimate is [\nu ^{2}/(N-2)], or, nearly enough for most purposes, [(N-1)^{2}/(N-2)t^{2}]. The differences introduced by the corrections −1 and −2 are generally negligible, but would not be so for counts as low as those proposed by Killean (1967[link]). If such corrections are important, it should be noticed that there is an ambiguity concerning N, depending on how the timing is triggered. It may be triggered by a count that is counted, or by a count that is not counted, or may simply be begun, independently of the incidence of a count. Equation (7.5.4.1)[link] assumes the first of these.

Equation (7.5.4.1)[link] may be inverted to give the probability distribution of the observed counting rate νo instead of the probability distribution of the time t: [\eqalignno{ p(\nu _{o})\,{\rm d}\nu _{o} &= [(N-1)!]^{-1}[\nu (N-1)/\nu _{o}]^{N-1} \cr &\quad \times \exp \{-(N-1)\nu /\nu _{o}\}\,{\rm d}[\nu _{o}/(N-1)\nu] . & (7.5.4.2)}]There does not seem to be any special name for the distribution (7.5.4.2)[link]. Only its first (N − 1) moments exist, and the integral expressing the probability distribution of the difference of the reflection and the background rates is intractable (Wilson, 1980[link]).

References

Abramowitz, M. & Stegun, I. A. (1964). Handbook of mathematical functions. National Bureau of Standards Publication AMS 55.
Killean, R. C. G. (1967). A note on the a priori estimation of R factors for constant-count-per-reflection diffractometer experiments. Acta Cryst. 23, 1109–1110.
Wilson, A. J. C. (1980). Relationship between `observed' and `true' intensity: effect of various counting modes. Acta Cryst. A36, 929–936.








































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