Section 11.2.6.1. Forming the standard profiles

In order to apply profile-fitting methods, the first requirement is to derive a `standard' profile that accurately represents the true reflection profile. Although analytical functions can be used, it is difficult to define a simple function that will cope adequately with the wide variation in spot shapes that can arise in practice. Most programs therefore rely on an empirical profile derived by summing many different spots. The optimum profile is that which provides the best fit to all the contributing reflections, i.e. that which minimizes where is the profile value for the pixel, is the observed background-corrected count at that pixel for reflection h, is a scale factor and is a weight for the pixel of reflection h. The summation extends over all reflections contributing to the profile. The weight is given by and from Poisson statistics is the expectation value of the counts at pixel j, and is given by After Rossmann (1979), the summation integration intensity can be used to derive a value for : In equations (11.2.6.3) and (11.2.6.4), as the profile values are not yet determined, a preliminary profile derived, for example, from simple summation of strong reflections used in the detector-parameter refinement can be used, which will give acceptable weights for use in equation (11.2.6.1).

This method of deriving the standard profile is only appropriate for fully recorded reflections. However, in many cases there will be very few or no fully recorded reflections on each image. In such cases the profile is determined by simply adding together the background-corrected pixel counts from all contributing reflections. In the program MOSFLM (Leslie, 1992), the profiles are determined using reflections on, typically, ten or more successive images, so that partials will be summed to give the correct fully recorded profile for the majority of the contributing reflections. Tests carried out using standard profiles derived using only fully recorded reflections and equation (11.2.6.1), or using both fully recorded and partially recorded reflections and simple summation, give data of the same quality as judged by the merging statistics.

The reflection profile changes across the face of the detector, due to obliquity of incidence, changes in the projected diffracting volume and geometric factors. In the MOSFLM program, this variation is accommodated by determining several standard profiles (typically nine or 25) for different regions of the detector. When evaluating the profile-fitted intensity for a given reflection, a weighted sum of the nearest standard profiles is calculated to provide the best estimate of the true profile at that position on the detector. For the central regions of the detector there will be four contributing profiles, while at the edges there will be between one and three. The weights assigned to each profile vary linearly with the distance from the reflection to the centres of the regions used in determining the standard profiles. An alternative procedure used in DENZO (Otwinowski & Minor, 1997) is to evaluate a new profile for each reflection based on spots lying within a pre-specified radius.