Standard uncertainty of angular positions and statistical quality of step-scan intensity data

In step-scan diffraction measurements, the diffraction angle 2 theta is an observation with standard uncertainty u(2 theta). By the law of uncertainty propagation, u(2 theta), typically 0.001 < u(2 theta) < 0.004 degrees, affects the standard uncertainty u(total)(y) of the intensity y at each step 2 theta(i), depending on the local slope y'(i) = dy/d(2 theta)\(2 theta i) by u(total)(2)(y(i)) = u(Poisson)(2) + [y'(i)u(2 theta)](2), where u(Poisson) = (y(i))(1/2) is the conventional Poisson statistics. For the intensity y at 2 theta of steepest slope, u(total)(y) is given by u(total)(2)(y) = u(Poisson)(2)(1 + v(2)), where v = 2u(2 theta)y(0)(1/2)/h is the ratio of y'(i)u(2 theta) and u(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(total)(2)(Int) = u(Poisson)(2) (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'(i)u(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.