Instrument-model refinement in normalized reciprocal-vector space for X-ray Laue diffraction

Abstract

A simple yet efficient instrument-model refinement method for X-ray diffraction data is presented and discussed. The method is based on least-squares minimization of differences between respective normalized (i.e. unit length) reciprocal vectors computed for adjacent frames. The approach was primarily designed to work with synchrotron X-ray Laue diffraction data collected for small-molecule single-crystal samples. The method has been shown to work well on both simulated and experimental data. Tests performed on simulated data sets for small-molecule and protein crystals confirmed the validity of the proposed instrument-model refinement approach. Finally, examination of data sets collected at both BioCARS 14-ID-B (Advanced Photon Source) and ID09 (European Synchrotron Radiation Facility) beamlines indicated that the approach is capable of retrieving goniometer parameters (e.g. detector distance or primary X-ray beam centre) reliably, even when their initial estimates are rather inaccurate.

Keywords: Laue diffraction; X-ray diffraction; data processing; instrument models; refinement.

Figure 1

(a) Schematic representation of a typical Laue data set consisting of diffraction frames measured at various goniometer setting angles (i.e. each frame is collected with the sample rotated by a certain increment, Δφ). A single φ-angle spindle axis is assumed for simplicity. This example data set consists of N frames; the cross indicates the frame’s centre. (b) Overlay of two adjacent frames showing changes in the positions of selected spots due to the horizontal sample rotation (Δφ = 1°): green solid spots – frame No. 1, red empty spots – frame No. 2 (for an overlay of two frames see the supporting information; displacements, spot sizes and shapes are exaggerated).

Figure 2

Schematic representation of two selected adjacent-frame pairs of normalized reciprocal vectors with the same hkl indices [see e.g. Fig. 1 ▸(b)] reconstructed from spot positions, goniometer setting angles and other instrument model parameters (e.g. detector distance, primary beam position etc.). Left panel: imperfect IM parameters (the respective vectors do not overlap); right panel: ideal IM parameters (the reconstructed vectors overlap perfectly after the least-squares minimization of vector differences with respect to IM parameters).