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, pp. 122-123

## Section 2.5.2.3. Sample space and goniometer geometry

B. B. Hea*

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

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#### 2.5.2.3.1. Sample rotations and translations in Eulerian geometry

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In a 2D-XRD system, three rotation angles are necessary to define the orientation of a sample in the diffractometer. These three rotation angles can be achieved either by a Eulerian geometry, a kappa (κ) geometry or another kind of geometry. The three angles in Eulerian geometry are ω (omega), ψ (psi) and ϕ (phi). Fig. 2.5.7(a) shows the relationship between rotation axes (ω, ψ, ϕ) in the laboratory system XL, YL, ZL. The ω angle is defined as a right-handed rotation about the ZL axis. The ω axis is fixed in the laboratory coordinates. The ψ angle is a right-handed rotation about a horizontal axis. The angle between the ψ axis and the XL axis is given by ω. The ψ axis lies on XL when ω is set at zero. The ϕ angle defines a left-handed rotation about an axis on the sample, typically the normal of a flat sample. The ϕ axis lies on the YL axis when ω = ψ = 0. In an aligned diffraction system, all three rotation axes and the primary X-ray beam cross at the origin of the XL, YL, ZL coordinates. This cross point is also known as the goniometer centre or instrument centre.

 Figure 2.5.7 | top | pdf |Sample rotation and translation. (a) Three rotation axes in laboratory coordinates; (b) rotation axes (ω, ψ, ϕ) and sample coordinates.

Fig. 2.5.7(b) shows the relationship and stacking sequence among all rotation axes (ω, ψ, ϕ) and the sample coordinates S1, S2, S3. ω is the base rotation; all other rotations and translations are on top of this rotation. The next rotation above ω is the ψ rotation. The next rotation above ω and ψ is the ϕ rotation. The sample coordinates S1, S2, S3 are fixed to the sample regardless of the particular sample orientation given by the rotation angles (ω, ψ, ϕ). The ϕ rotation in the goniometer is intentionally chosen as a left-handed rotation so that the diffraction vectors will make a right-hand rotation observed in the sample coordinates S1, S2, S3.