Rotation of the crystal around the Z-axis has the effect of simply rotating the diffraction pattern about Z. The crystal's Z orientation therefore only affects the positions of the diffracted spots on a diffraction image. Rotations about X and Y however also affect the intensities of so called partially recorded reflections. For accurate refinement of the X and Y rotations information must be derived from the intensities of partial reflections which enter the diffraction condition at the beginning and end of the oscillation range and therefore lie on the Ewald sphere at these orientations. If the exact fraction of these partially recorded intensities which have been observed can be ascertained then this provides a powerful means of refining the orientational parameters. Least-squares refinement based on partiality is made by minimising the function ,
where the weight is given by
Here, and are the intensity and error of a partially recorded reflection, is its fractional partiality, defined as the fraction of the total integrated intensity which has been observed, is the average intensity of all equivalent reflections and is a parameter estimating the minimum expected positional error of the measured intensities and is typically set at a fraction of the detectors pixel size .
The quality of the refinement is assessed by a Goodness of Fit defined as
where is the weighted deviance between the observed and predicted values of the positions and fractional partialities of intensities and is an estimate of the total expected error in the observed data, where
Individual contributions to the overall error model arise from , which is defined above, and is the expected error in the determination of the fractional partiality (typical values are of ). The value accounts for the increase in the positional uncertainty of reflections which are just on the edge of the blind region and can therefore appear as elongated diffraction maxima. If is the angle between the crystal oscillation axis and the straight line connecting the image centre and the measured spot, then
A value of has been found to provide a suitable estimate of the error introduced as a result of the blind region.