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. 119-121

Section 2.5.2.1. Diffraction space and laboratory coordinates

B. B. Hea*

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

2.5.2.1. Diffraction space and laboratory coordinates

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2.5.2.1.1. 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.

2.5.2.1.2. Diffraction-vector cones in laboratory coordinates

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Fig. 2.5.3[link](b) shows the diffraction-vector cone corresponding to the diffraction cone in the laboratory coordinate system. C is the centre of the Ewald sphere. The diffraction condition can be given by the Laue equation as[{{{\bf{s}} - {{\bf{s}}_0}} \over \lambda } = {{\bf{H}}_{hkl}},\eqno(2.5.2)]where s0 is the unit vector representing the incident beam, s is the unit vector representing the diffracted beam and Hhkl is the reciprocal-lattice vector. Its magnitude is given as[\left| {{{{\bf{s}} - {{\bf{s}}_0}} \over \lambda }} \right| = {{2\sin \theta } \over \lambda } = \left| {{{\bf{H}}_{hkl}}} \right| = {1 \over {{d_{hkl}}}}, \eqno(2.5.3)]in which dhkl is the d-spacing of the crystal planes (hkl). It can be easily seen that it is the Bragg law in a different form. Therefore, equation (2.5.2)[link] is the Bragg law in vector form. In the Bragg condition, the vectors s0/λ and s/λ make angles θ with the diffracting planes (hkl) and Hhkl is normal to the (hkl) crystal plane. In order to analyse all the X-rays measured by a 2D detector, we extend the concept to all scattered X-rays from a sample regardless of the Bragg condition. Therefore, the index (hkl) can be removed from the above expression. H is then a vector which takes the direction bisecting the incident beam and the scattered beam, and has dimensions of inverse length given by [2\sin \theta /\lambda ]. Here 2θ is the scattering angle from the incident beam. The vector H is referred to as the scattering vector or, alternatively, the diffraction vector. When the Bragg condition is satisfied, the diffraction vector is normal to the diffracting lattice planes and its magnitude is reciprocal to the d-spacing of the lattice planes. In this case, the diffraction vector is equivalent to the reciprocal-lattice vector. Each pixel in a 2D detector measures scattered X-rays in a given direction with respect to the incident beam. We can calculate a diffraction vector for any pixel, even if the pixel is not measuring Bragg scattering. Use of the term `diffracted beam' hereafter in this chapter does not necessarily imply that it arises from Bragg scattering.

For two-dimensional diffraction, the incident beam can be expressed by the vector s0/λ, but the diffracted beam is no longer in a single direction, but follows the diffraction cone. Since the direction of a diffraction vector is a bisector of the angle between the incident and diffracted beams corresponding to each diffraction cone, the trace of the diffraction vectors forms a cone. This cone is referred to as the diffraction-vector cone. The angle between the diffraction vector and the incident X-ray beam is 90° + θ and the apex angle of a vector cone is 90° − θ. It is apparent that diffraction-vector cones can only exists on the −XL side of the diffraction space.

For two-dimensional diffraction, the diffraction vector is a function of both the γ and 2θ angles, and is given in laboratory coordinates as[{\bf{H}} = {{{\bf{s}} - {{\bf{s}}_0}} \over \lambda } = {1 \over \lambda }\left [{\matrix{ {\cos 2\theta - 1} \cr { - \sin 2\theta \sin \gamma } \cr { - \sin 2\theta \cos \gamma } \cr } } \right].\eqno(2.5.4)]The direction of the diffraction vector can be represented by its unit vector, given by[{{\bf{h}}_{{L}}} = {{\bf{H}} \over {\left| {\bf{H}} \right|}} = \left [{\matrix{ {{h_x}} \cr {{h_y}} \cr {{h_z}} \cr } } \right] = \left [{\matrix{ { - \sin \theta } \cr { - \cos \theta \sin \gamma } \cr { - \cos \theta \cos \gamma } \cr } } \right],\eqno(2.5.5)]where hL is a unit vector expressed in laboratory coordinates and the three components in the square brackets are the projections of the unit vector on the three axes of the laboratory coordinates, respectively. If γ takes all values from 0 to 360° at a given Bragg angle 2θ, the trace of the diffraction vector forms a diffraction-vector cone. Since the possible values of θ lie within the range 0 to 90°, hx takes only negative values.








































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