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. 2.1, pp. 28-30

Section 2.1.4.1. Basic design principles and instrument geometry considerations

A. Kerna*

aBruker AXS, Östliche Rheinbrückenstrasse 49, Karlsruhe 76187, Germany
Correspondence e-mail: arnt.kern@bruker-axs.de

2.1.4.1. Basic design principles and instrument geometry considerations

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X-ray scattering data are generally recorded in what is virtually the simplest possible manner, where the scattered intensity is measured by a detector mounted at some distance from the specimen. This is illustrated in Fig. 2.1.1[link], where a narrow, essentially monochromatic beam illuminates a small spherical specimen. For a rotating single crystal, the diffracted beams point in discrete directions in space as given by Bragg's law for each lattice vector dhkl (Fig. 2.1.1[link]a). For an ideal powder consisting of a virtually unlimited number of randomly oriented crystallites, the diffracted beams will form concentric cones (`Debye cones') with a semi-apex angle of 2θ, representing all randomly oriented identical lattice vectors dhkl (Fig. 2.1.1[link]b). Note that in contrast to a single crystal, an ideal powder does not need to be rotated to obtain a complete powder diffraction pattern.

[Figure 2.1.1]

Figure 2.1.1 | top | pdf |

Diffraction of X-rays by (a) a rotating single crystal and (b) an ideal powder. The scattered intensity may be measured by a detector placed on the detector circle.

Most instruments are built around a central specimen and consist of the following beam-path components, the numbering of which is consistent with the mounting positions shown in Fig. 2.1.2[link]:

  • (1) X-ray source;

    [Figure 2.1.2]

    Figure 2.1.2 | top | pdf |

    The basic design principle of modern diffractometers. Currently available instruments are built around a centrally mounted specimen and represent an X-ray optical bench with mounting positions for any (1) X-ray sources, (2) incident-beam optics, (3) specimen stages, (4) diffracted-beam optics and (5) detectors. The 2θ position of the scattered-X-ray optical bench refers to the 2θ angle of the Debye cone shown in bold in Fig. 2.1.1[link](b).

  • (2) incident-beam optics;

  • (3) goniometer base or specimen stage;

  • (4) diffracted-beam optics;

  • (5) detector.

The directions of the incident and diffracted beams (also called `primary' and `secondary' beams) form the diffraction plane (also called the `equatorial plane' or `scattering plane'). The goniometer base can be mounted horizontally (horizontal diffraction plane) or vertically (vertical diffraction plane). The direction perpendicular to the equatorial plane is known as the axial direction. The detector circle (also called the `goniometer circle' or `diffractometer circle') is defined either by the centre of the active window of a stationary detector, or, in most cases, by a detector moving around the specimen, and is coplanar to the diffraction plane. The 2θ angle of both the diffracted beam in Fig. 2.1.1[link](a) and the Debye cone in Fig. 2.1.1[link](b) (shown in bold) refers to the 2θ position of the diffracted-beam X-ray optical bench in Fig. 2.1.2[link]. It is obvious from Figs. 2.1.1[link] and 2.1.2[link] that, in principle, diffraction from single crystals and (ideal) powders can be measured using the same instrument.

An instrument design with a centrally mounted specimen has the important advantage that it implicitly allows the operation of one and the same instrument in both Bragg–Brentano and Debye–Scherrer geometry, depending on the beam divergence chosen. The actual instrument geometry is thus a function of the actual beam propagation angle (divergent, parallel or convergent), making the X-ray optics the most important part of any instrument-geometry conversion. The relationship between the two geometries and their implementation in a single instrument using an incident-beam X-ray optical bench is illustrated in Fig. 2.1.3[link].

[Figure 2.1.3]

Figure 2.1.3 | top | pdf |

Transformation between the Bragg–Brentano and Debye–Scherrer geometries using a incident-beam X-ray optical bench. SR: flat specimen, reflection mode; SC: capillary specimen, transmission mode; ST: flat specimen, transmission mode. The actual instrument geometry is a function of the actual beam-propagation angle, making the X-ray optics the most important part of any instrument-geometry conversion. (a) Divergent beam: Bragg–Brentano geometry, (b) divergent beam: Bragg–Brentano geometry extended by an incident-beam monochromator. (c) Convergent beam: focusing Debye–Scherrer geometry, (d) parallel beam: Debye–Scherrer geometry. Transformation is achieved by mounting the X-ray tube and pre-aligned optical components at pre-defined positions of the optical bench. None of the figures are to scale.

As laboratory X-ray sources invariably produce divergent beams, the `natural' instrument geometry is self-focusing, `automatically' leading to the Bragg–Brentano geometry as shown in Fig. 2.1.3[link](a). In this geometry the angle of both the incident and the diffracted beam is θ with respect to the specimen surface. The X-ray-source-to-specimen and the specimen-to-detector distances are equal. The diffraction pattern is collected by varying the incidence angle of the incident beam by θ and the diffracted-beam angle by 2θ. The focusing circle is defined as positioned tangentially to the specimen surface. The focusing condition is fulfilled at the points where the goniometer circle intersects the focusing circle, and thus requires measurements in reflection mode.

The Bragg–Brentano geometry may be extended by an incident- or a diffracted-beam monochromator. In the case of an incident-beam monochromator as shown in Fig. 2.1.3[link](b), the focus of the X-ray source is replaced by the focus of the monochromator crystal. This involves mounting the monochromator crystal (and the X-ray source) a certain distance away along the incident-beam X-ray optical bench, as given by the focusing length of the monochromator crystal (the dotted line in Fig. 2.1.3[link]b). For a diffracted-beam monochromator or mirror, the geometry shown in Fig. 2.1.3[link](b) can be thought of as reversed (simply consider the X-ray source and detector switching their positions).

The conversion from Bragg–Brentano to Debye–Scherrer geometry involves the mounting of some kind of optics designed to convert the divergent beam coming from the X-ray source into a focusing or parallel beam; this is shown in Figs. 2.1.3[link](c) and (d), respectively.

In a focusing Debye–Scherrer geometry setup, as shown in Fig. 2.1.3[link](c), the divergent beam coming from the X-ray source is normally focused on the detector circle (for highest resolution) by means of an incident-beam monochromator or a focusing mirror. The focusing circle is identical to the detector circle and the focusing condition requires measurements in transmission mode. When employing an incident-beam monochromator with sufficient focusing length, then conversion between the Bragg–Brentano geometry and the focusing Debye–Scherrer geometry involves a shift of the monochromator crystal (and the X-ray source) along the incident-beam X-ray optical bench (note the identical focusing length of the monochromator shown in Figs. 2.1.3[link]b and c).

For a parallel-beam setup, as shown in Fig. 2.1.3[link](d), parallelization of the divergent beam coming from the X-ray source may be achieved by different means, such as collimators (classic Debye–Scherrer geometry) or reflective optics such as mirrors or capillaries. In principle, the X-ray source and the detector may be placed at any distance from the specimen, as there are no focusing requirements. As a consequence, measurements can be performed in both reflection and transmission mode.

In a simplified scheme, conversion between the geometries discussed above involves repositioning of the X-ray source, together with mounting of X-ray optics with suitable beam divergence. To make this possible, the incident-beam optical X-ray bench offers the necessary predefined mounting positions including relevant translationary and rotationary degrees of freedom.

An important aspect directly related to the choice of the instrument geometry is the geometric compatibility with position-sensitive detectors. In contrast to Debye–Scherrer geometry, large line and area detectors may not be used in Bragg–Brentano geometry. This is an important limitation of the latter, as the focusing circle does not coincide with the detector circle and has a 2θ-dependent diameter, as illustrated in Fig. 2.1.4[link]. As a consequence, the diffracted beam is only focused on a single point of the goniometer circle, as shown in Fig. 2.1.5[link]. However, small position-sensitive detectors with an angular coverage of not more than about 10° 2θ are used with great success, as defocusing can be ignored at diffraction angles larger than about 20° 2θ if high angular accuracy and resolution are not required. For measurements at smaller 2θ angles, or for highest angular accuracy and resolution, the active window size of a position-sensitive detector may be reduced by means of slits and/or electronically down to a point, allowing the use of this detector as a point detector.

[Figure 2.1.4]

Figure 2.1.4 | top | pdf |

Bragg–Brentano geometry. The focusing circle, given for two different angles 2θ, is tangential to the specimen surface. Its diameter is given by the intersections between the detector and the focusing circles and is thus 2θ dependent.

[Figure 2.1.5]

Figure 2.1.5 | top | pdf |

Bragg–Brentano geometry. While all diffracted beams focus on the (variable-diameter) focusing circle (here shown for two beams), focusing on the detector circle is only achieved at the X-ray source and detector positions (located at the intersections between the detector and the focusing circles). This prevents the use of larger position-sensitive detectors because of defocusing, as indicated by the hypothetical large position-sensitive detector represented by the bold grey line.








































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