International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 2.5, p. 123

Section 2.5.2.4. Diffraction-vector transformation

B. B. Hea*

aBruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA
Correspondence e-mail: bob.he@bruker.com

2.5.2.4. Diffraction-vector transformation

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2.5.2.4.1. Diffraction unit vector in diffraction space and sample space

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In 2D-XRD data analysis, it is crucial to know the diffraction-vector distribution in terms of the sample coordinates S1, S2, S3. However, the diffraction-vector distribution corresponding to the measured 2D data is always given in terms of the laboratory coordinates XL, YL, ZL because the diffraction space is fixed to the laboratory coordinates. Fig. 2.5.8[link] shows the unit vector of a diffraction vector in both (a) the laboratory coordinates XL, YL, ZL and (b) the sample coordinates S1, S2, S3. In Fig. 2.5.8[link](a) the unit vector hL is projected to the XL, YL and ZL axes as hx, hy and hz, respectively. The three components are given by equation (2.5.5)[link]. In order to analyse the diffraction results relative to the sample orientation, it is necessary to transform the unit vector to the sample coordinates S1, S2, S3. Fig. 2.5.8[link](b) shows the same unit vector, denoted by hs projected to S1, S2 and S3 as h1, h2 and h3, respectively.

[Figure 2.5.8]

Figure 2.5.8 | top | pdf |

Unit diffraction vector in (a) the laboratory coordinates and (b) the sample coordinates.

2.5.2.4.2. Transformation from diffraction space to sample space

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The transformation of the unit diffraction vector from the laboratory coordinates XL, YL, ZL to the sample coordinates S1, S2, S3 is given by[{{\bf{h}}_{{s}}} = {\bi{A}}{{\bf{h}}_{{L}}}, \eqno(2.5.10)]where A is the transformation matrix. For Eulerian geometry in matrix form, we have[\eqalignno{&\left [{\matrix{ {{h_1}} \cr {{h_2}} \cr {{h_3}} \cr } } \right] = \left [{\matrix{ {{a_{11}}} & {{a_{12}}} & {{a_{13}}} \cr {{a_{21}}} & {{a_{22}}} & {{a_{23}}} \cr {{a_{31}}} & {{a_{32}}} & {{a_{33}}} \cr } } \right]\left [{\matrix{ {{h_x}} \cr {{h_y}} \cr {{h_z}} \cr } } \right] &\cr &= \left [\matrix{- \sin \omega \sin \psi \sin \varphi \hfill & \cos \omega \sin \psi \sin \varphi \hfill & -\cos \psi \sin \varphi \cr \quad - \cos \omega \cos \varphi \hfill &\quad - \sin \omega \cos \varphi \hfill\cr\cr \sin \omega \sin \psi \cos \varphi \hfill & - \cos \omega \sin \psi \cos \varphi \hfill & \cos \psi \cos \varphi \hfill\cr \quad - \cos \omega \sin \varphi\hfill & \quad - \sin \omega \sin \varphi \hfill\cr\cr{ - \sin \omega \cos \psi } \hfill & {\cos \omega \cos \psi } \hfill &{\sin \psi } \hfill }\right]&\cr&\times\left [\let\normalbaselines\relax\openup3pt\matrix{ - \sin \theta \cr\cr - \cos \theta \sin \gamma \cr\cr - \cos \theta \cos \gamma } \right]. &\cr &&(2.5.11)}]In expanded form:[\eqalignno{{h_1} &= \sin \theta (\sin \varphi \sin \psi \sin \omega + \cos \varphi \cos \omega) + \cos \theta \cos \gamma \sin \varphi \cos \psi\cr &\quad- \cos \theta \sin \gamma (\sin \varphi \sin \psi \cos \omega - \cos \varphi \sin \omega)\cr {h_2} &= - \sin \theta (\cos \varphi \sin \psi \sin \omega - \sin \varphi \cos \omega) &\cr&\quad- \cos \theta \cos \gamma \cos \varphi \cos \psi &\cr&\quad+ \cos \theta \sin \gamma (\cos \varphi \sin \psi \cos \omega + \sin \varphi \sin \omega)\cr {h_3} &= \sin \theta \cos \psi \sin \omega - \cos \theta \sin \gamma \cos \psi \cos \omega - \cos \theta \cos \gamma \sin \psi \cr &&(2.5.12)}]In addition to the diffraction intensity and Bragg angle corresponding to each data point on the diffraction ring, the unit vector hs{h1, h2, h3} provides orientation information in the sample space. The transformation matrix of any other goniometer geometry, such as kappa geometry (Paciorek et al., 1999[link]), can be introduced into equation (2.5.10)[link] so that the unit vector hs{h1, h2, h3} can be expressed in terms of the specified geometry. All equations using the unit vector hs{h1, h2, h3} 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.

2.5.2.4.3. Transformation from detector space to reciprocal space

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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[link]; Mudie et al., 2004[link]; Smilgies & Blasini, 2007[link]; Schmidbauer et al., 2008[link]). 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 hs{h1, h2, h3} and the magnitude of the scattering vector is given by [2\sin \theta /\lambda ], so that the scattering vector corresponding to a pixel is given by[{\bf{H}} = {{2\sin \theta } \over \lambda }{{\bf{h}}_{{s}}}.\eqno(2.5.13)]The 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 Scholar
Mudie, 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








































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