Counting statistics

Parrish, W.; Langford, J. I.

doi:10.1107/97809553602060000578

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

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International Tables for Crystallography (2006). Vol. C. ch. 2.3, pp. 64-65

Section 2.3.3.6. Counting statistics

W. Parrish^a ^‡ and J. I. Langford^b

^a IBM Almaden Research Center, San Jose, CA, USA, and ^bSchool of Physics & Astronomy, University of Birmingham, Birmingham B15 2TT, England

2.3.3.6. Counting statistics

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X-ray quanta arrive at the detector at random and varying rates and hence the rules of statistics govern the accuracy of the intensity measurements. The general problems in achieving maximum accuracy in minimum time and in assessing the accuracy are described in books on mathematical statistics. Chapter 7.5 reviews the pertinent theory; see also Wilson (1980). In this section, only the fixed-time method is described because the fixed-count method takes too long for most practical applications.

Let $[\bar N]$ be the average of N, the number of counts in a given time t, over a very large number of determinations. The spread is given by a Poisson probability distribution (if $[\bar N]$ is large) with standard deviation $[\sigma = \bar N ^{1/2}. \eqno (2.3.3.7)]$ Any individual determination of N or the corresponding counting rate n (= N/t) will be subject to a proportionate error $[\varepsilon]$ which is also a function of the confidence level, i.e. the probability that the result deviates less than a certain percentage from the true value. If Q is the constant determined by the confidence level, then $[\varepsilon = Q/N ^{1/2}, \eqno (2.3.3.8)]$ where Q = 0.67 for the probable relative error $[\varepsilon_{50}]$ (50% confidence level) and Q = 1.64 and 2.58 for the 90 and 99% confidence levels $[(\varepsilon_{90},\ \varepsilon _{99}),]$ respectively. For a 1% error, N = 4500, 27 000, 67 000 for $[ \varepsilon _{50}]$ , $[\varepsilon _{90}]$ , $[\varepsilon _{99}]$ , respectively. Fig. 2.3.3.6 shows various percentage errors as a function of N for several confidence levels.

Figure 2.3.3.6| top | pdf |

Percentage error as a function of the total number of counts N for several confidence levels.

In practice, there is usually a background count [N_B] . The net peak count $[N_{P+B}- N_B = N_{P-B}]$ is dependent on the P/B ratio as well as on $[N_{P+B}]$ and [N_B] separately. The relative error $[\varepsilon _D]$ of the net peak count is $[\varepsilon _D = {[(N_{P+B} \varepsilon _{P+B})^2 + (N _N \varepsilon _B) ^2] ^{1/2} \over N_{P-B}}, \eqno (2.3.3.9)]$ which shows that $[\varepsilon _D]$ is similarly influenced by both absolute errors $[N_{P+B} \varepsilon _{P+B}]$ and $[N_B \varepsilon _B]$ . The absolute standard deviation of the net peak height is $[ \sigma _{P-B} = (\sigma ^2_{P+B} + \sigma ^2 _B) ^{1/2} \eqno (2.3.3.10)]$ and expressed as the per cent standard deviation is $[\sigma _{P-B} = {(N_{P+B} + N _B) ^{1/2} \over N_{P-B}} \times 100. \eqno (2.3.3.11)]$ The accuracy of the net peak measurement decreases rapidly as the peak-to-background ratio falls below 1. For example, with N_B = 50, the dependence of $[\sigma_{P-B}]$ on P/B is $[\halign{# \quad&\quad#\cr \hbox{$P/B$} &\hbox{${\sigma_{P-B}}$ (\%)}\cr \kern3pt0.1 &\kern1pt205 \cr \kern2.2pt1 &\kern6pt24.5 \cr \kern-3pt10 &\kern10.6pt4.9 \cr \kern-8pt100 &\kern11pt1.43 \cr}]$ It is obviously desirable to minimize the background using the best possible experimental methods.

References

Wilson, A. J. C. (1980). Relationship between `observed' and `true' intensity: effect of various counting modes. Acta Cryst. A36, 929–936.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 2.3, pp. 64-65