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 2.5.2.4.2. Transformation from diffraction space to sample space

B. B. Hea*

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

2.5.2.4.2. 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.

References

Paciorek, W. A., Meyer, M. & Chapuis, G. (1999). On the geometry of a modern imaging diffractometer. Acta Cryst. A55, 543–557.Google Scholar








































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