International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C. ch. 7.4, pp. 661-663

Section 7.4.4.2. Incident beam and sample

P. Suorttic

7.4.4.2. Incident beam and sample

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An ideal diffraction experiment should be viewed as an X-ray optical system where all the parts are properly matched for the desired resolution and efficiency. The impurities of the incident beam are the wavelengths and divergent rays that do not contribute to the signal but scatter from the sample through the various processes mentioned above. The propagation of the X-ray beam through the instrument is perhaps best illustrated by the so-called phase-space analysis. The three-dimensional version, which will be used in the following, was introduced by Matsushita & Kaminaga (1980a[link],b[link]) and was elaborated further by Matsushita & Hashizume (1983[link]). The width, divergence and wavelength range of the beam are given as a contour diagram, which originates in the X-ray source, and is modified by slits, monochromator, sample, and the detection system. The actual five-dimensional diagram is usually given as three-dimensional projections on the plane of diffraction and on the plane perpendicular to it and the beam axis, and in most cases the first projection is sufficient for an adequate description of the geometry of the experiment.

The limitations of the actual experiments are best studied through a comparison with the ideal situation. A close approximation to the ideal experimental arrangement is shown in Fig. 7.4.4.1[link] as a series of phase-space diagrams. The characteristic radiation from a conventional X-ray tube is almost uniformly distributed over the solid angle of 2π, and the relative width of the Kα1 or Kα2 emission line is typically Δλ/λ = 5 × 10−4. The acceptance and emittance windows of a flat perfect crystal are given in Fig. 7.4.4.1(b)[link]. The angular acceptance of the crystal (Darwin width) is typically less than 10−4 rad, and, if the width of the slit s or that of the crystal is small enough, none of the Kα2 distribution falls within the window. Therefore, it is sufficient to study the size and divergence distributions of the beam in the λ(Kα1) plane only, as shown in Fig. 7.4.4.1(c)[link]. The beam transmitted by the flat monochromator and a slit is shown as the hatched area, and the part reflected by a small crystal by the cross-hatched area. The reflectivity curve of the crystal is probed when the crystal is rotated. In this schematic case, almost 100% of the beam contributes to the signal. The typical reflection profile shown in Fig. 7.4.4.2[link] reveals the details of the crystallite distribution of the sample (Suortti, 1985[link]). The broken curve shows the calculated profile of the same reflection if the incident beam from a mosaic crystal monochromator had been used (see below).

[Figure 7.4.4.1]

Figure 7.4.4.1| top | pdf |

Equatorial phase-space diagrams for a conventional X-ray source and parallel-beam geometry; x is the size and x′ = dx/dz the divergence of the X-rays. (a) Radiation distributions for two wavelengths, λ1 and λ2, at the source of width Δx, and downstream at a slit of width ±s1. (b) Acceptance and emittance windows of a flat perfect crystal, where the phase-space volume remains constant, AwaΔλ = EweΔλ, and the (x′, λ) section shows the reflection of a polychromatic beam (Laue diffraction). (c) Distributions for one wavelength at the source, flat perfect-crystal monochromator, sample (marked with the broken line), and the receiving slit (RS); z is the distance from the source.

[Figure 7.4.4.2]

Figure 7.4.4.2| top | pdf |

Reflection 400 of LiH measured with a parallel beam of Mo Kα1 radiation (solid curve). The broken curve shows the reflection as convoluted by a Gaussian instrumentation function of 2σ = 0.1° and θ(α2) − θ(α1) = 0.13°, which values are comparable with those in Fig. 7.4.4.4[link].

The window of acceptance of a flat mosaic crystal is determined by the width of the mosaic distribution, which may be 100 times larger than the Darwin width of the reflection in question. This means that a convergent beam is reflected in the same way as from a bent perfect crystal in Johann or Johansson geometry. Usually, the window is wide enough to transmit an energy band that includes both [K\alpha_1] and [K\alpha_2] components of the incident beam. The distributions of these components are projected on the (x, x′, λ1) plane in Fig. 7.4.4.3[link]. The sample is placed in the (para)focus of the beam, and often the divergence of the beam is much larger than the width of the rocking curve of the sample crystal. This means that at any given time the signal comes from a small part of the beam, but the whole beam contributes to the background. The profile of the reflection is a convolution of the actual rocking curve with the divergence and wavelength distributions of the beam. The calculated profile in Fig. 7.4.4.2[link] demonstrates that in a typical case the profile is determined by the instrument, and the peak-to-background ratio is much worse than with a perfect-crystal monochromator.

[Figure 7.4.4.3]

Figure 7.4.4.3| top | pdf |

Equatorial phase-space diagrams for two wavelengths, λ1 (solid lines) and λ2 (broken lines), projected on the plane λ = λ1. The monochromator at z = 200 mm is a flat mosaic crystal, and a small sample is located at z = 400 mm, as shown by the shaded area. The reflected beams at the receiving slit are shown for the (+, +) and (+, −) configurations of the monochromator and the sample.

An alternative arrangement, which has become quite popular in recent years, is one where the plane of diffraction at the monochromator is perpendicular to that at the sample. The beam is limited by slits only in the latter plane, and the wavelength varies in the perpendicular plane. An example of rocking curves measured by this kind of diffractometer is given in Fig. 7.4.4.4[link]. The [K\alpha_1] and [K\alpha_2] components are seen separately plus a long tail due to continuum radiation, and the profile is that of the divergence of the beam.

[Figure 7.4.4.4]

Figure 7.4.4.4| top | pdf |

Two reflections of beryllium acetate measured with Mo Kα. The graphite (002) monochromator reflects in the vertical plane, while the crystal reflects in the horizontal plane. The equatorial divergence of the beam is 0.8°, FWHM.

In the Laue method, a well collimated beam of white radiation is reflected by a stationary crystal. The wavelength band reflected by a perfect crystal is indicated in Fig. 7.4.4.1(b)[link]. The mosaic blocks select a band of wavelengths from the incident beam and the wavelength deviation is related to the angular deviation by [\Delta\lambda/\lambda=\cot\theta\Delta\theta]. The angular resolution is determined by the divergences of the incident beam and the spatial resolution of the detector. The detector is not energy dispersive, so that the background arises from all scattering that reaches the detector. An estimate of the background level involves integrations over the incident spectrum at a fixed scattering angle, weighted by the cross sections of inelastic scattering and the attenuation factors. This calculation is very complicated, but at any rate the background level is far higher than that in a diffraction measurement with a monochromatic incident beam.

References

First citation Matsushita, T. & Hashizume, H. (1983). Handbook of synchrotron radiation, Vol. I, edited by E. E. Koch, pp. 261–314. Amsterdam: North-Holland.Google Scholar
First citation Matsushita, T. & Kaminaga, U. (1980a). A systematic method of estimating the performance of X-ray optical systems for synchrotron radiation. I. Description of various optical elements in position-angle space for ideally monochromatic X-rays. J. Appl. Cryst. 13, 465–471.Google Scholar
First citation Matsushita, T. & Kaminaga, U. (1980b). A systematic method of estimating the performance of X-ray optical systems for synchrotron radiation. II. Treatment in position-angle-wavelength space. J. Appl. Cryst. 13, 472–478.Google Scholar
First citation Suortti, P. (1985). Parallel beam geometry for single-crystal diffraction. J. Appl. Cryst. 18, 272–274.Google Scholar








































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