International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 2.5, p. 123
Section 2.5.2.4. Diffraction-vector transformation^{a}Bruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA |
In 2D-XRD data analysis, it is crucial to know the diffraction-vector distribution in terms of the sample coordinates S_{1}, S_{2}, S_{3}. However, the diffraction-vector distribution corresponding to the measured 2D data is always given in terms of the laboratory coordinates X_{L}, Y_{L}, Z_{L} because the diffraction space is fixed to the laboratory coordinates. Fig. 2.5.8 shows the unit vector of a diffraction vector in both (a) the laboratory coordinates X_{L}, Y_{L}, Z_{L} and (b) the sample coordinates S_{1}, S_{2}, S_{3}. In Fig. 2.5.8(a) the unit vector h_{L} is projected to the X_{L}, Y_{L} and Z_{L} axes as h_{x}, h_{y} and h_{z}, respectively. The three components are given by equation (2.5.5). In order to analyse the diffraction results relative to the sample orientation, it is necessary to transform the unit vector to the sample coordinates S_{1}, S_{2}, S_{3}. Fig. 2.5.8(b) shows the same unit vector, denoted by h_{s} projected to S_{1}, S_{2} and S_{3} as h_{1}, h_{2} and h_{3}, respectively.
The transformation of the unit diffraction vector from the laboratory coordinates X_{L}, Y_{L}, Z_{L} to the sample coordinates S_{1}, S_{2}, S_{3} is given bywhere A is the transformation matrix. For Eulerian geometry in matrix form, we haveIn expanded form:In addition to the diffraction intensity and Bragg angle corresponding to each data point on the diffraction ring, the unit vector h_{s}{h_{1}, h_{2}, h_{3}} provides orientation information in the sample space. The transformation matrix of any other goniometer geometry, such as kappa geometry (Paciorek et al., 1999), can be introduced into equation (2.5.10) so that the unit vector h_{s}{h_{1}, h_{2}, h_{3}} can be expressed in terms of the specified geometry. All equations using the unit vector h_{s}{h_{1}, h_{2}, h_{3}} in this chapter, such as in data treatment, texture analysis and stress measurement, are applicable to all goniometer geometries provided that the unit-vector components are generated from the corresponding transformation matrix from diffraction space to the sample space.
Reciprocal-space mapping is commonly used to analyse the diffraction patterns from highly oriented structures, diffuse scattering from crystal defects, and thin films (Hanna & Windle, 1995; Mudie et al., 2004; Smilgies & Blasini, 2007; Schmidbauer et al., 2008). The equations of the unit-vector calculation given above can also be used to transform the diffraction intensity from the diffraction space to the reciprocal space with respect to the sample coordinates. The direction of the scattering vector is given by the unit vector h_{s}{h_{1}, h_{2}, h_{3}} and the magnitude of the scattering vector is given by , so that the scattering vector corresponding to a pixel is given byThe three-dimensional reciprocal-space mapping can be obtained by applying the normalized pixel intensities to the corresponding reciprocal points. With various sample orientations, all pixels on the detector can be mapped into a 3D reciprocal space.
References
Hanna, S. & Windle, A. H. (1995). A novel polymer fibre diffractometer, based on a scanning X-ray-sensitive charge-coupled device. J. Appl. Cryst. 28, 673–689.Google ScholarMudie, S. T., Pavlov, K. M., Morgan, M. J., Hester, J. R., Tabuchi, M. & Takeda, Y. (2004). Collection of reciprocal space maps using imaging plates at the Australian National Beamline Facility at the Photon Factory. J. Synchrotron Rad. 11, 406–413.Google Scholar
Paciorek, W. A., Meyer, M. & Chapuis, G. (1999). On the geometry of a modern imaging diffractometer. Acta Cryst. A55, 543–557.Google Scholar
Schmidbauer, M., Schäfer, P., Besedin, S., Grigoriev, D., Köhler, R. & Hanke, M. (2008). A novel multi-detection technique for three-dimensional reciprocal-space mapping in grazing-incidence X-ray diffraction. J. Synchrotron Rad. 15, 549–557.Google Scholar
Smilgies, D.-M. & Blasini, D. R. (2007). Indexation scheme for oriented molecular thin films studied with grazing-incidence reciprocal-space mapping. J. Appl. Cryst. 40, 716–718.Google Scholar