Relationship between `observed' and `true' intensity; effect of various counting modes

Abstract

be neglected and the familiar formula for shortrange-order diffuse scattering (see e.g. Guinier, 1963, p. 269) is obtained. For values of a much greater than this many such diffuse curves corresponding to higher values of P must be included in the summation. Each of these will represent successively broader more diffuse peaks as r 2p approaches zero. The factor (k 2 t72)P/P! eventually goes to zero as P increases for any tr but for values of a ~_ 1 many terms must be included.References BARTLETT, M. S. (1967). J. R. Stat. Soc. A, 130, 467-477. BESAG, J. (1974). J. R. Stat. Soc. B, 36, 192-225. BR~i, MER, R. (1975 Expressions for the probability P(Ro) that a reflexion of 'true' intensity R will have an observed value R o (possibly negative) are obtained for four counting modes: fixed-time counting, equal and unequal times for total and background; fixed-count timing, equal and unequal counts for total and background. The distributions have a positive excess and are in general skew, though the skewness may be zero for particular choices of unequal times (counts). Deviations from the normal distribution with the same mean and variance may be considerable for IRol ~_ 0 and for IRol large, and may possibly be significant in some applications even for R o ~_ R. This apparent conflict with the central limit theorem is reconciled. 0567-7394/80/060929-08501.00In both single-crystal and powder diffractometry the integrated intensity of a reflexion is obtained as the difference between a counting rate averaged over a region 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 the intentions are not effectively realized there will be a systematic error in the measured intensity, but the present concern is not with such systematic errors but with statistical fluctuations in the intensity as observed. Although an intensity can never be really negative, it is not uncommon for the measured background counting rate to be higher than the measured reflexion-plus-background rate, giving an

Parametric Modelling of the Crystalline Microstructure of the MCM41-Type Mesoporous Silica Modified with Derivatives of Alkyls

“…First, for each pattern, the background was determined as a monotonically decreasing smoothed natural cubic spline function to approximate together the manually selected fragments of the pattern, and then the background was subtracted (as illustrated in Figure 5). After subtracting the background, each pattern was represented with the small step of diffraction angle as a natural cubic spline by applying optimal smoothing [65] with an error estimate derived from quantum count statistics of X-ray intensity measurements [66][67][68]. The optimal smoothing was performed so that the root-mean-square difference between the vector of measured background-corrected X-ray intensities and the vector of corresponding smoothing spline was equal to the root-mean-square estimate of the error in the experimental pattern.…”