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. 41-43

Section 2.1.6.3.2. Diffractive X-ray optics

A. Kerna*

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

2.1.6.3.2. Diffractive X-ray optics

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Single crystals or highly textured polycrystals (mosaic crystals) represent effective beam conditioners by allowing the spectral bandwidth as well as the X-ray beam divergence to be modified. When they are placed at a specific angle with respect to the incident and diffracted beams, according to Bragg's law, only a small spectral bandwidth will be transmitted depending on the divergence of the incident beam and the rocking angle (mosaic spread) of the crystal. Higher harmonics (λ/2, λ/3,…) are diffracted as well, but can be successfully suppressed by using materials with small higher-order structure factors and via energy discrimination by the detector. Depending on the application, a crystal monochromator can be either used as a spectral filter (`monochromator'), typically used in the incident beam, or as an angular filter (`analyser'), typically used in the diffracted beam to restrict the angular acceptance of the detector.

It is likely that all monochromators currently employed in laboratory X-ray diffractometers are of the reflective type (`Bragg geometry'). Transmission-type monochromators (`Laue geometry') play no role in laboratory powder diffraction. Two designs are in common use and are described below: (a) single-reflection monochromators and (b) multiple-reflection monochromators.

2.1.6.3.2.1. Single-reflection monochromators

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The most common types of single-reflection monochromators are illustrated in Figs. 2.1.16[link] and 2.1.17[link]. Flat crystals (Fig. 2.1.16[link]) are used in parallel-beam geometry and curved crystals in focusing geometries (Fig. 2.1.17[link]). A beam reflected from a flat crystal with the reflecting lattice planes parallel to its surface (symmetric cut) is nearly parallel (Fig. 2.1.16[link]a). If the crystal is cut at an angle to the reflecting lattice planes (asymmetric cut), then the beam will be expanded (Fig. 2.1.16[link]b), or compressed if reversed (Fankuchen, 1937[link]). Monochromators can be curved (Johann, 1931[link]) or curved and ground (Johannson, 1933[link]), and may be cut symmetrically (Fig. 2.1.17[link]a) or asymmetrically (Fig. 2.1.17[link]b). The latter has the particular advantage of providing different focal lengths for the incident and diffracted beam. A shortened incident beam allows the monochromator to be mounted closer to the X-ray source to capture a larger solid angle of the emitted beam. If the diffracted-beam focusing length is sufficiently large, then the instrument geometry can be converted between the Bragg–Brentano and the focusing Debye–Scherrer geometries by shifting the monochromator crystal and the X-ray source along the incident-beam X-ray optical bench (see Section 2.1.4.1[link] and Fig. 2.1.3[link]).

[Figure 2.1.16]

Figure 2.1.16 | top | pdf |

Illustration of flat single-reflection monochromators. (a) Symmetrically cut crystal, (b) asymmetrically cut crystal with an angle γ between the reflecting lattice planes and the crystal surface.

[Figure 2.1.17]

Figure 2.1.17 | top | pdf |

Illustration of curved and ground single-reflection monochromators. Only the central beam is shown for clarity. (a) Symmetrically cut crystal, (b) asymmetrically cut crystal with two different focal lengths a and b.

The most commonly used monochromator crystal materials are germanium and quartz, which have very small mosaic spreads and are able to separate the Kα1/Kα2 doublet. In contrast to germanium and quartz crystals, graphite and lithium fluoride have large mosaic spreads and thus high reflectivity, but cannot suppress Kα2. In principle, any of these monochromators can be mounted in the incident as well as the diffracted beam; the choice mostly depends on the purpose of the monochromator. Germanium and quartz monochromators are typically used as incident-beam monochromators to produce pure Kα1 radiation. Graphite (focusing geometries) and lithium fluoride (parallel-beam geometry) are often used as diffracted-beam monochromators to suppress fluorescence radiation. Germanium and quartz can also be used as diffracted-beam monochromators, but are usually not because of their lower reflectivity. Where mounting of diffracted-beam monochromators is difficult or impossible, which is specifically true for one- and two-dimensional detector applications, curved graphite monochromators are frequently used as incident-beam monochromators.

The use of diffracted-beam monochromators – at least in powder X-ray diffraction – is declining steeply because of the geometric incompatibility issues with one- and two-dimensional detector systems (which, since 2010, have been sold with more than 90% of all diffractometers; see Section 2.1.3.2[link]). With the recent improvements of energy-discrimination capabilities for silicon micro-strip detectors, the need for diffracted-beam monochromators will further diminish (see Section 2.1.7.2.3[link]).

2.1.6.3.2.2. Multiple-reflection monochromators

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Multiple-reflection monochromators can reduce the wavelength dispersion Δλ /λ significantly more than single-reflection monochromators. Multiple-reflection monochromators are often made of monolithically grooved single crystals and are also known as channel-cut monochromators (Bonse & Hart, 1965[link]). In Fig. 2.1.18[link] an overview is given of the most common channel-cut monochromator types; for a detailed discussion see e.g. Hart (1971[link]) and Bowen & Tanner (1998[link]). Successive reflection of the X-ray beam at the channel walls by the same lattice planes causes a strong reduction of the X-ray intensity contained in the tails of the beam. Depending on the number of reflections, multiple-reflection monochromators are denoted as two-bounce, three-bounce etc. channel-cut monochromators. The Bartels monochromator (Bartels, 1983[link]) comprises two two-bounce channel-cut crystals. For Cu radiation, such a monochromator results in a wavelength spread which is less than the natural line width of the Cu Kα1 line. The most commonly used crystal material is germanium, which delivers higher intensity than silicon, using the 400, 220, or 440 reflections. Crystals may be cut symmetrically or asymmetrically. In Table 2.1.5[link] several types of germanium channel-cut monochromators are compared in terms of divergence and intensity.

Table 2.1.5| top | pdf |
Comparison of divergence and intensity for several types of germanium channel-cut monochromators

In each case, the monochromator is coupled with a graded multilayer providing 3 × 109 counts per second at <0.028° beam divergence. The values in parentheses denote the percentage of intensity diffracted by the respective monochromator crystals.

Type(hkl)Divergence (°)Intensity
Two-bounce 220, symmetric <0.0052 5.0 × 107 (∼1.5%)
Two-bounce 220, asymmetric <0.0085 3.3 × 108 (∼10%)
Two-bounce 400, asymmetric <0.0045 4.8 × 107 (∼1.5%)
Four-bounce 220, symmetric <0.0035 6.5 × 106 (∼0.2%)
Four-bounce 220, asymmetric <0.0080 2.7 × 107 (∼1%)
Four-bounce 440, symmetric <0.0015 2.2 × 105 (∼0.075%)
[Figure 2.1.18]

Figure 2.1.18 | top | pdf |

Illustration of multiple-reflection monochromators. (a) Symmetrically cut two-bounce channel-cut monochromator, (b) asymmetrically cut two-bounce channel-cut monochromator for beam compression, or, if reversed, for beam expansion, (c) symmetrically or asymmetrically cut four-bounce channel-cut monochromator, (d) symmetrically cut three-bounce channel-cut monochromator.

Switching between different channel-cut monochromators is extremely easy these days and can be accomplished without the need for any tools and without realignment. This is also true for cases where a beam offset is introduced, e.g. by switching between two- and four-bounce channel-cut monochromators. In sophisticated instruments such an offset can be compensated fully automatically by a software-controlled motor.

The combination of different types of channel-cut monochromators in both the incident and diffracted beam allows the construction of advanced diffractometer configurations with extremely high resolution capabilities. It should be emphasized that laboratory X-ray diffractometers can have identical optical configurations to diffractometers operated at synchrotron beamlines. The important and obvious difference, however, is the extremely low flux coming from laboratory X-ray sources, which is further diminished by each reflection in a channel-cut monochromator (Table 2.1.5[link]). While such configurations work perfectly for strongly scattering single-crystal layers in thin films, for example, analysis of ideal powders is normally not possible.

References

Bartels, W. J. (1983). Characterization of thin layers on perfect crystals with a multipurpose high resolution X-ray diffractometer. J. Vac. Sci. Technol. B, 1, 338–345.Google Scholar
Bonse, U. & Hart, M. (1965). Tailless X-ray single crystal reflection curves obtained by multiple reflection. Appl. Phys. Lett. 7, 238–240.Google Scholar
Bowen, D. K. & Tanner, B. K. (1998). High Resolution X-ray Diffractometry and Topography. London: Taylor & Francis.Google Scholar
Fankuchen, I. (1937). A condensing monochromator for X-rays. Nature (London), 139, 193–194.Google Scholar
Hart, M. (1971). Bragg-reflection X-ray optics. Rep. Prog. Phys. 34, 435–490.Google Scholar
Johann, H. H. (1931). Die Erzeugung lichtstarker Röntgenspektren mit Hilfe von Konkavkristallen. Z. Phys. 69, 185–206.Google Scholar
Johannson, T. (1933). Über ein neuartiges, genau fokussierendes Röntgenspektrometer. Z. Phys. 82, 507–528.Google Scholar








































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