Is there a fundamental upper limit for the significance I/σ(I) of observations from X-ray and neutron diffraction experiments?
The answer is yes. A fundamental limit exists, which is not strongly applicable to individual reflections but to a sufficiently large set of reflections such as any set for structure determination. The limit originates from Poisson statistics which gives a minimum (average) error. The proposed limit of significance and a proposed decrease in significance due to data processing are also tested by monitoring W = <I(hkl)(1/2)>/<sigma(I(hkl))> for raw and for Bragg data. Since Poisson statistics are the lower limit for the experimental standard uncertainties, it is expected that W < 1 for raw and Bragg data, and that W decreases upon data processing. W also gives a measure of systematic errors in the experimental data as W ≈ 1 characterizes pure Poisson data, W >> 1 is physically impossible for sufficiently large data sets of unmerged reflections and W < 1 describes the contamination with systematic errors. Systematic differences depending on the software used to process the data were found. Also, the frequency distributions in particular of σ(I) values change considerably depending on the data-processing software used. We have no explanation for these differences in the distributions of σ(I), which lead to distinct changes in the frequency distribution of the significances I/σ(I) compared with the raw data. Another consequence of Poisson statistics is that lower limits also exist for the agreement factors, the internal agreement factor and the goodness of fit. These limits depend on the moments <1/I(o)>, <I(o)>, <I(o)(2)> and <I(o)(1/2)> of the observed set of intensities I(o) about the origin. These agreement factors are theoretically attainable when no systematic sources of error apply. They may be used in future to construct further measures of systematic error in experimental data.
Laboratory of Crystallography, University of Bayreuth, Universitätsstrasse 30, 95447 Bayreuth, Germany. julian.henn@uni-bayreuth.de
Julian
J
Henn
Kathrin
K
Meindl