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. 2.3, pp. 64-65
|
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 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
is large) with standard deviation
Any individual determination of N or the corresponding counting rate n (= N/t) will be subject to a proportionate error
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
where Q = 0.67 for the probable relative error
(50% confidence level) and Q = 1.64 and 2.58 for the 90 and 99% confidence levels
respectively. For a 1% error, N = 4500, 27 000, 67 000 for
,
,
, respectively. Fig. 2.3.3.6
shows various percentage errors as a function of N for several confidence levels.
In practice, there is usually a background count . The net peak count
is dependent on the P/B ratio as well as on
and
separately. The relative error
of the net peak count is
which shows that
is similarly influenced by both absolute errors
and
. The absolute standard deviation of the net peak height is
and expressed as the per cent standard deviation is
The accuracy of the net peak measurement decreases rapidly as the peak-to-background ratio falls below 1. For example, with NB = 50, the dependence of
on P/B is
It is obviously desirable to minimize the background using the best possible experimental methods.
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