Abstract
In step-scan diffraction measurements, the diffraction angle 2θ is an observation with standard uncertainty u(2θ). By the law of uncertainty propagation, u(2θ), typically 0.001 < u(2θ) < 0.004°, affects the standard uncertainty u_{\rm total}(y) of the intensity y at each step 2\theta_i, depending on the local slope y'_i = \left. {{\rm d}y}/{{\rm d}(2\theta)}\right|_{2\theta_{i}}, by u_{\rm total}^2(y_i) = u^2_{\rm Poisson}+[y'_iu(2\theta)]^2, where u_{\rm Poisson} = ({y_i})^{1/2} is the conventional Poisson statistics. For the intensity y at 2θ of steepest slope, u_{\rm total}(y) is given by u_{\rm total}^2(y) = u^2_{\rm Poisson}(1+v^2), where v = 2u(2\theta){y_0}^{1/2}/h is the ratio of y'_iu(2\theta) and u_{\rm Poisson}, y_0 is the peak intensity and h the full width at half-maximum of the profile. The error of the intensities at individual steps modifies also the standard uncertainty of the integrated intensity: u_{\rm total}^2(\rm Int) = u^2_{\rm Poisson}({\rm Int}) (1+ {v^2}/{2}). As v depends on y_0^{1/2}/h, it is evident that the importance of the correction increases with increasing count rates and decreasing line width. In most practical cases, y'_iu(2\theta) contributes a multiple of Poisson statistics to the standard uncertainty of intensity. It will be shown that with a realistic weighting scheme the \chi^2 as well as the Durbin–Watson test become more meaningful.
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