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, pp. 122123
Section 2.5.2.3.1. Sample rotations and translations in Eulerian geometry^{a}Bruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA 
In a 2DXRD 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 X_{L}, Y_{L}, Z_{L}. The ω angle is defined as a righthanded rotation about the Z_{L} axis. The ω axis is fixed in the laboratory coordinates. The ψ angle is a righthanded rotation about a horizontal axis. The angle between the ψ axis and the X_{L} axis is given by ω. The ψ axis lies on X_{L} when ω is set at zero. The ϕ angle defines a lefthanded rotation about an axis on the sample, typically the normal of a flat sample. The ϕ axis lies on the Y_{L} axis when ω = ψ = 0. In an aligned diffraction system, all three rotation axes and the primary Xray beam cross at the origin of the X_{L}, Y_{L}, Z_{L} coordinates. This cross point is also known as the goniometer centre or instrument centre.

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 S_{1}, S_{2}, S_{3}. 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 S_{1}, S_{2}, S_{3} 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 lefthanded rotation so that the diffraction vectors will make a righthand rotation observed in the sample coordinates S_{1}, S_{2}, S_{3}.