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. 2.2, pp. 54-55

Section 2.2.3.1. Monochromator

A. Fitcha*

aESRF, 71 Avenue des Martyrs, CS40220, 38043 Grenoble Cedex 9, France
Correspondence e-mail: fitch@esrf.fr

2.2.3.1. Monochromator

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The monochromator is a crucial optical component in any angle-dispersive powder-diffraction beamline, and consists of one or a pair of perfect crystals (e.g. Beaumont & Hart, 1974[link]), Fig. 2.2.7[link], set to a particular angle to the incident beam, θm, that transmits by diffraction wavelengths that satisfy the Bragg equation, nλ = 2d(hkl) sin θm, where d(hkl) is the lattice spacing of the chosen reflection. Note that photons from higher-order reflections can also be transmitted, corresponding to wavelengths λ/n, depending on the structure factor of the nth-order reflection and its Darwin width, but these can be eliminated by use of a mirror (see Section 2.2.3.2[link]), or by adjusting the electronic acceptance windows of the detector system, if possible. They can also be suppressed to some extent by slightly detuning the second crystal from the first, because the Darwin width of a higher-order reflection is narrower than that of a lower-order reflection, and is thus more seriously affected by the mismatch between Bragg angles.

[Figure 2.2.7]

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Double-crystal monochromator arrangement.

For a given reflection, a crystal does not transmit a unique single wavelength but a narrow distribution. The width of the distribution, δλ, is determined by the effective divergence of the incident beam Ψ (which corresponds to a range of values for θm) and the Darwin width of the reflection, ω, at the chosen wavelength. The energy resolution of a monochromator crystal can be estimated via[\delta\varepsilon/\varepsilon = \delta\lambda/\lambda = \cot\theta_m(\Psi^2+\omega^2)^{1/2}.]

With a highly collimated beam incident on a crystal and with a narrow Darwin width, high energy resolution is achieved. The Darwin width of a reflection can be calculated from dynamical theory [Zachariasen (1945[link]); Chapter 5.1[link] of International Tables for Crystallography, Volume B (Authier, 2006[link])] via[\omega = {2r_e\lambda^2 \over \pi V}|F({\bf h})|{ K\over \sin 2\theta_m},]where re is the classical electron radius (∼2.818 fm), V is the volume of the unit cell, F(h) is the structure factor and K the polarization factor (1 for reflection in the vertical plane, cos 2θm for the horizontal plane). Thus for Si(111), with d(111) = 3.1356 Å and F(h) ≃ 59, a Darwin width of about 8.3 µrad is obtained at 31 keV (λ = 0.4 Å). With an effective beam divergence of say 25 µrad (delivering a beam 1.1 mm high at 44 m from the source), an energy resolution of 4.8 × 10−4 is obtained. Even better energy resolution can be obtained by increasing the collimation of the beam before the monochromator, e.g. with a curved mirror.

Energy resolution is an important quantity to control. Its value needs to be known when modelling powder-diffraction peak shapes via a fundamental-parameters approach, and it affects the angular resolution of the powder-diffraction pattern, broadening the peaks as 2θ increases, as can be seen by differentiating the Bragg equation to yield[{\delta\lambda\over\lambda}=\cot\theta\, \delta\theta\quad {\rm or}\quad \delta\theta = {\delta\lambda\over\lambda}\tan\theta.\eqno(2.2.1)]Thus powder-diffraction peaks broaden towards higher 2θ angles because of this effect.

Silicon is a common choice for a monochromator; it forms large, perfect single crystals, with dimensions of cm if required, has appropriate mechanical, diffraction and thermal properties, and can resist prolonged exposure to an intense radiation source. A monochromator crystal absorbs a large fraction of the energy incident upon it, and hence must be cooled. Even when cooled, the high power density (tens or even more than a hundred W mm−2 at normal incidence) can cause local heating of the surface, which leads to distortion of the lattice planes via thermal expansion. This degrades the performance, as a heat bump increases the range of θm values, broadening the energy band transmitted by the crystal. With a double-crystal arrangement, this bump cannot be matched by the second crystal, which has a much lower heat load so is flat, with the result that photons from the first crystal are not transmitted by the second, thus losing intensity from the monochromatic beam. By cooling to cryogenic temperatures, the thermal expansion of Si can be reduced to a very small value, going through zero at around 120 K (Bilderback, 1986[link]; Glazov & Pashinkin, 2001[link]) and thereby alleviating the heat-bump problem. Thus cryogenically cooled monochromators can be found at high-performance synchrotron beamlines. Other crystals employed as monochromators are germanium and diamond, the latter in transmission because of the small size of available diamond crystals.

Although a monochromator assembly can employ only one crystal, for example deflecting the beam horizontally into a side branch of a beamline, a double-crystal arrangement (Fig. 2.2.7[link]) is more usually used to conserve the direction of the beam from the storage ring. This can exploit either a channel-cut crystal or two crystals, with a number of adjustments in the position and orientation of the second crystal to allow it to be aligned optimally to transmit the wavelength envelope defined by the first crystal. In some cases, the second crystal can be bent sagittally to focus X-rays horizontally onto the sample. Although this increases the divergence of the beam arriving at the sample and so affects the 2θ resolution of the powder pattern, it can lead to a significant increase in intensity, and is useful to capture more radiation from a horizontally divergent source such as a bending magnet or wiggler.

References

Authier, A. (2006). Dynamical theory of X-ray diffraction. International Tables for Crystallography, Vol. B, Reciprocal Space, 1st online ed., ch. 5.1. Chester: International Union of Crystallography.Google Scholar
Beaumont, J. H. & Hart, M. (1974). Multiple Bragg reflection monochromators for synchrotron X radiation. J. Phys. E Sci. Instrum. 7, 823–829.Google Scholar
Bilderback, D. H. (1986). The potential of cryogenic silicon and germanium X-ray monochromators for use with large synchrotron heat loads. Nucl. Instrum. Methods Phys. Res. A, 246, 434–436.Google Scholar
Glazov, V. M. & Pashinkin, A. S. (2001). The thermophysical properties (heat capacity and thermal expansion) of single-crystal silicon. High Temp. 39, 413–419.Google Scholar
Zachariasen, W. H. (1945). Theory of X-ray Diffraction in Crystals. Dover Publications Inc.Google Scholar








































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