Tables for
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. 119-120

Section Diffraction cones in laboratory coordinates

B. B. Hea*

aBruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA
Correspondence e-mail: Diffraction cones in laboratory coordinates

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Fig. 2.5.3[link](a) describes the geometric definition of diffraction cones in the laboratory coordinate system XL, YL, ZL. The laboratory coordinate system is a Cartesian coordinate system. The plane given by XL and YL is the diffractometer plane. The axis ZL is perpendicular to the diffractometer plane. The axes XL, YL and ZL form a right-handed rectangular coordinate system with the origin at the instrument centre. The incident X-ray beam propagates along the XL axis, which is also the rotation axis of all diffraction cones. The apex angles of the cones are determined by the 2θ values given by the Bragg equation. The apex angles are twice the 2θ values for forward reflection [(2\theta\leq90^\circ)] and twice the value of 180° − 2θ for backward reflection [(2\theta\,\gt\,90^\circ)]. For clarity, only one diffraction cone of forward reflection is displayed. The γ angle is the azimuthal angle from the origin at the six o'clock direction with a right-handed rotation axis along the opposite direction of incident beam (−XL direction). A given γ value defines a half plane with the XL axis as the edge; this will be referred to as the γ plane hereafter. The diffractometer plane consists of two γ planes at γ = 90° and γ = 270°. Therefore many equations developed for 2D-XRD should also apply to conventional XRD if the γ angle is given as a constant of 90° or 270°. A pair of γ and 2θ values represents the direction of a diffracted beam. The γ angle takes a value of 0 to 360° for a complete diffraction ring with a constant 2θ value. The γ and 2θ angles form a spherical coordinate system which covers all the directions from the origin of sample (instrument centre). The γ–2θ system is fixed in the laboratory system XL, YL, ZL, which is independent of the sample orientation and detector position in the goniometer. 2θ and γ are referred to as the diffraction-space parameters. In the laboratory coordinate system XL, YL, ZL, the surface of a diffraction cone can be mathematically expressed as[y_L^2 + z_L^2 = x_L^2\tan ^22\theta, \eqno(2.5.1)]with [{x_L} \ge 0] or [2\theta \le 90^\circ ] for forward-diffraction cones and [{x_L} \,\lt\, 0] or [2\theta\, \gt \,90^\circ ] for backward-diffraction cones.

[Figure 2.5.3]

Figure 2.5.3 | top | pdf |

The diffraction cone and the corresponding diffraction-vector cone.

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