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. 56-60

Section 2.2.4. Diffractometers

A. Fitcha*

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

2.2.4. Diffractometers

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Most powder-diffraction beamlines are angle dispersive, operating with monochromatic radiation. When scanning a detector arm or employing a curved position-sensitive detector (PSD), detection is normally in the vertical plane because the polarization of the radiation in the plane of the synchrotron orbit means there is very little effect on the intensities due to polarization. By contrast, if diffracting in the horizontal plane, the projection of the electric vector onto the direction of the diffracted beam means that the intensity is reduced by a factor of cos2 2θ, going to zero at 2θ = 90°, and so horizontal detection is less useful unless working at hard energies when 2θ angles are correspondingly small. In addition, for the highest angular resolution, the natural beam divergence in the vertical plane is usually lower than in the horizontal plane, particularly if the instrument has a bending magnet or wiggler as its source.

In general, diffractometers are heavy-duty pieces of equipment and are designed to have excellent angular accuracy while working with substantial loads. A high degree of mechanical accuracy is required to match the high optical accuracy inherent in the techniques employed. The calibration of the incident wavelength and any 2θ zero-point error is best done by measuring the diffraction pattern from a sample such as NIST standard Si (640 series), each of which has a certified lattice parameter (see Chapter 3.1[link] ). It is also good practice to measure the diffraction pattern of a standard sample regularly and whenever the instrument is realigned or the wavelength changed, to be sure that everything is working as expected.

Monochromatic instruments can have an analyser crystal or long parallel-foil collimators in the diffracted beam (a so-called parallel-beam arrangement), or can scan a receiving slit, or possess a one- or two-dimensional PSD, similar to Debye–Scherrer or Laue front-reflection geometry. Instruments equipped with a PSD can collect data much faster than those with a scanning diffractometer, so are exploited especially for time-resolved measurements. They may also have advantages for rapid data collection if the sample is sensitive to radiation, or be helpful if the sample is prone to granularity or texture to assess the extent of the problem.

Instruments can also be equipped with a sample changer, allowing measurements on a series of specimens, perhaps prepared by systematically changing the conditions of synthesis or the composition in a combinatorial approach. The use of beam time can be optimized with minimal downtime due to interventions around the instrument, and with the possibility to control the data acquisition remotely if desired.

2.2.4.1. Parallel-beam instruments

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Cox et al. (1983[link], 1986[link]), Hastings et al. (1984[link]) and Thompson et al. (1987[link]) described the basic ideas behind these instruments via their pioneering work at CHESS (Cornell, USA) and NSLS (Brookhaven, USA). The highly collimated monochromatic incident beam is diffracted by the sample and passes via a perfect analyser crystal [such as Si or Ge(111)] to the detector. The analyser crystal defines a very narrow angular acceptance for the diffracted radiation, determined by its Darwin width. The combination of the collimation of the incident radiation, its highly monochromatic nature and the stringent angular acceptance defines the instrument's excellent angular resolution. The detector arm supporting the analyser is scanned through the desired range of 2θ angles either in a step-scan mode or continuously, reading out at very short intervals the electronic modules that accumulate the detector counts.

To be transmitted by the analyser crystal, a photon must be incident on the crystal at the correct angle θa that satisfies the Bragg condition. The analyser crystal defines therefore a true direction (2θ angle) for the diffracted beam irrespective of where in the sample it originates from. This removes a number of aberrations that affect diffractometers with a scanning slit or PSD where the 2θ angle is inferred from the position of the slit or detecting pixel. Thus, with a capillary specimen, peak widths are independent of the capillary diameter, so a fat capillary of non-absorbing sample can be used to optimize diffracted intensity, and any modest misalignment of the sample from the diffractometer axis, or specimen transparency or surface roughness for flat-plate samples, does not lead to shifts in the peak positions. Modest movement of the sample with temperature changes in a furnace etc. does not cause shifts in peak positions. These instruments are therefore highly accurate, and are ideal for obtaining peak positions for indexing a diffraction pattern of a material of unknown unit cell (the first step in the solution of a structure from powder data), or following the evolution of lattice parameter with temperature etc. For flat samples, the θ/2θ parafocusing condition does not need to be satisfied to have high resolution. The peak width does not therefore depend on sample orientation, which is useful for measurements of residual strain by the sin2 ψ technique or for studying surfaces and surface layers by grazing-incidence diffraction. Interchange between capillary and flat-plate samples can easily be done as required without major realignment of the instrument. The stringent acceptance conditions also help to suppress parasitic scattering originating from sample-environment windows etc. and inelastic scattering such as fluorescence and Compton scattering.

On the other hand, at any 2θ angle only a tiny fraction of the diffracted photons can be transmitted by an analyser crystal, so this is a technique that consumes a lot of photons, and the high incident flux is essential to keep scan times to reasonable values. To overcome this, at least to some extent, Hodeau et al. (1998[link]) devised a system of multiple analyser crystals, with nine channels mounted in parallel, each separated from the next by 2° (Fig. 2.2.10[link]). In effect, as the detector arm is scanned, nine high-resolution powder-diffraction patterns are measured in parallel, each offset from the next by 2°. If the data from the channels are to be combined, which is the usual procedure, the detectors must be calibrated with respect to each other, in terms of counting efficiency and exact angular offset, by comparing regions of the diffraction pattern scanned by several detectors (Wright et al., 2003[link]). A multianalyser system speeds up data collection significantly and can be found in various modified forms at a number of powder-diffraction beamlines (e.g. Lee, Shu et al., 2008[link]).

[Figure 2.2.10]

Figure 2.2.10 | top | pdf |

Multianalyser stage, nine channels separated by 2°, devised by Hodeau et al. (1998[link]), originally installed on the BM16 bending-magnet beamline at the ESRF with Ge(111) analyser crystals. With an undulator source, the greatly increased flux allows use of Si(111), which has a narrower Darwin width (by a factor of ∼2.4) and thus improved 2θ resolution, but with a lower fraction of the diffracted radiation accepted.

The multianalyser approach is best suited to capillary samples because of the axial symmetry of the arrangement. With flat plates in reflection, only one detector can be in the θ/2θ condition where the effect of specimen absorption (for a sufficiently thick sample) is isotropic. Corrections must therefore be made to the intensities from the other channels (Lipson, 1967[link]; Koopmans & Rieck, 1968[link]). For a capillary, choosing the wavelength and the diameter allows absorption to be kept to an acceptable value. Maximum diffracted intensity is expected at μr = 1 (where μ is the linear absorption coefficient and r the radius of the capillary), and below this value simple absorption corrections can be applied (Hewat, 1979[link]; Sabine et al., 1998[link]). A value of μr greater than 1.5 begins to degrade the quality of the pattern significantly. If a sample with high absorption is unavoidable, such as when working close to an absorption edge of an element, e.g. the K edge of Mn at 6.539 keV (1.896 Å), then it can be preferable to stick a thin layer of sample on the outside of a 1-mm-diameter capillary. The shell-like nature of the sample has no effect on the peak shape or resolution because of the use of analyser crystals.

Capillaries also have the advantage that preferred orientation can be significantly less as compared to a flat sample, where there is a tendency for crystallites to align in the surface layers, especially if compressed to hold the powder in place. Spinning or otherwise moving the sample is necessary, whether capillary or flat plate, to increase the number of crystallites appropriately oriented to fulfil the Bragg condition and avoid a spotty diffraction pattern, the likelihood of which is exacerbated by the highly collimated nature of the incident radiation.

2.2.4.1.1. Angular resolution

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Various authors (e.g. Sabine, 1987a[link],b[link]; Wroblewski, 1991[link]; Masson et al., 2003[link]; Gozzo et al., 2006[link]) have discussed the resolution of a synchrotron-based diffractometer equipped with a double-crystal monochromator and an analyser crystal. The most usual setting of the diffracting crystals, ignoring any mirrors or other optical devices, is non-dispersive, alternatively described as parallel or (1, −1, 1, −1).

The approach developed by Sabine (1987a[link],b[link]) involves modelling the vertical divergence of the source and the angular acceptance of the monochromator and analyser crystals as Gaussian distributions with the same full width at half-maximum (FWHM) as the real distributions, and considering a powder as a crystal with an infinite mosaic spread. The rocking curve of the analyser crystal (equivalent to rocking 2θ) is given by[&I(\beta) = \int\int {\rm d}\alpha\,{\rm d}\delta \exp\left\{-\left[\left({\alpha \over\alpha_m^\prime}\right)^2 +2\left({\delta - \alpha\over \Delta_m^\prime}\right)^2 + \left({b\delta+\alpha-\beta\over \Delta_a^\prime}\right)^2\right]\right\},]where[b=\tan\theta_a/\tan\theta_m -2\tan\theta/\tan\theta_m.]Here α represents the vertical divergence from the source, δ is the difference between the Bragg angles of a central ray reflected from the monochromator at the angle θm and of another ray at angle [\theta_m^\prime] such that [\delta = \theta_m^\prime -\theta_m], and θa is the Bragg angle of the analyser crystal. The terms [\alpha_m^\prime], [\Delta_m^\prime] and [\Delta_a^\prime] are related to the FWHM of the Gaussians representing the vertical divergence distribution or the Darwin widths of the monochromator and analyser crystals, αm, Δm and Δa, respectively, with[\alpha_m^\prime=\alpha_m/2(\ln 2)^{1/2},\quad \Delta_m^\prime=\Delta_m/2(\ln 2)^{1/2},\quad \Delta_a^\prime= \Delta_a/2(\ln 2)^{1/2}.]

From the above equation, the intrinsic FWHM of the Gaussian-approximated peaks of the powder-diffraction pattern can be obtained as[\eqalignno{\Delta^2(2\theta)&=\alpha_m^2\left({\tan\theta_a\over\tan\theta_m}-2{\tan\theta\over\tan\theta_m}+1\right)^2 +{\textstyle{1\over 2}}\Delta_m^2\left({\tan\theta_a\over\tan\theta_m}-2{\tan\theta\over\tan\theta_m}\right)&\cr&\quad + \Delta_a^2.&(2.2.2)}]

Note that the true peak shape is not Gaussian, and a pseudo-Voigt (e.g. as described by Thompson et al., 1987[link]), Voigt (e.g. Langford, 1978[link]; David & Matthewman, 1985[link]; Balzar & Ledbetter, 1993[link]) or other function modelled from first principles (e.g. Cheary & Coelho, 1992[link]; Ida et al., 2001[link], 2003[link]) is usually better. Examples of FWHM curves calculated from equation (2.2.2)[link] are plotted in Fig. 2.2.11[link] at three wavelengths. Differentiating the Bragg equation gives Δd/d = −cot θ Δ(θ), where θ is in radians.

[Figure 2.2.11]

Figure 2.2.11 | top | pdf |

Δ(2θ) calculated from equation (2.2.2)[link] for a beamline with a double-crystal Si(111) monochromator, an Si(111) analyser (Δm = Δa and θm = θa) and an FWHM vertical divergence of 25 µrad at λ = 0.4 Å (solid line: Δm ≃ 8.3 µrad, θm = 3.6571°), λ = 0.8 Å (dashed line: Δm ≃ 16.6 µrad, θm = 7.3292°) and λ = 1.2 Å (dotted line: Δm ≃ 25.2 µrad, θm = 11.0319°).

Gozzo et al. (2006[link]) have extended the formulation of Sabine to include the effects of collimating and focusing mirrors in the overall scheme. Axial (horizontal) divergence of the beam between the sample and the detector causes shifts and broadening of the peaks, as well as the well known low-angle peak asymmetry due to the curvature of the Debye–Scherrer cones. Sabine (1987[link]b), based on the work of Hewat (1975[link]) and Hastings et al. (1984[link]), suggests the magnitude of the broadening, B(2θ), due to horizontal divergence Φ can be estimated via[B(2\theta) = (\textstyle{1\over 4}\Phi)^2 (\cot 2\theta + \tan \theta_a),]where B and Φ are in radians. This value is added to Δ(2θ).

2.2.4.1.2. Hart–Parrish design

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A variant of the parallel-beam scheme replaces the analyser crystal with a set of long, fine Soller collimators (Parrish et al., 1986[link]; Parrish & Hart, 1987[link]; Parrish, 1988[link]; Cernik et al., 1990[link]; Collins et al., 1992[link]) (Fig. 2.2.12[link]). The collimators define a true angle of diffraction, but with lower 2θ resolution than an analyser crystal because their acceptance angle is necessarily much larger and so the transmitted intensity is greater. They are not particularly suitable for fine capillary specimens, as the separation between foils may be similar to the capillary diameter, resulting in problems of shadowing of the diffracted beam. However, they are achromatic, and so do not need to be reoriented at each change of wavelength, which may have advantages when performing anomalous-scattering studies around an element's absorption edge. Unlike an analyser crystal, however, they do not suppress fluorescence. Peak shapes and resolution can be influenced by reflection of X-rays from the surface of the foils, or any imperfections in their manufacture, e.g. if the blades are not straight and flat. The theoretical resolution curve of such an instrument can be obtained from equation (2.2.2)[link] by setting tan θa to zero and replacing the angular acceptance of the analyser crystal Δa with the angular acceptance of the collimator Δc.

[Figure 2.2.12]

Figure 2.2.12 | top | pdf |

Schematic representation of a parallel-beam diffractometer of the Hart–Parrish design. The collimators installed on Stations 8.3 and 2.3 at the SRS Daresbury (Cernik et al., 1990[link]; Collins et al., 1992[link]) had steel blades 50 µm thick, 355 mm long, separated by 0.2 mm spacers, defining a theoretical opening angle (FWHM Δc) of 0.032° and a transmission of 80%.

2.2.4.2. Debye–Scherrer instruments

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The simplest diffractometer has a receiving slit at a convenient distance from the sample in front of a point detector such as a scintillation counter. The height of the slit should match the capillary diameter, or incident beam height for flat plates. A slightly larger antiscatter slit near the sample should also be employed to reduce background. The detector arm is scanned and a powder pattern recorded. This arrangement can be used for narrow capillary samples on lower-flux sources, avoiding the loss of intensity that use of an analyser crystal entails. The resolution is largely determined by the opening angle defined by the capillary and the receiving slit. Despite the simplicity of such an instrument, high-quality high-resolution data can be obtained.

For much faster data acquisition, a one-dimensional (1D) PSD or an area detector can be employed. Any sort of 1D detector with an appropriate number of channels, channel separation, efficiency, count rate (in an individual channel and overall) and speed of read out can be employed. Technology evolves and detectors make continual progress in performance. At the time of writing the most advanced 1D detector is the Mythen module developed by the Swiss Light Source (SLS). Mythen modules are based on semiconducting silicon technology and have 1280 8-mm-wide strips with a 50 µm pitch (64 × 8 mm2). They can be combined to form very large curved detectors such as that on the powder diffractometer of the materials science beamline at the SLS (Fig. 2.2.13[link]). This detector consists of 24 modules, 30 720 channels, set on a radius of 760 mm, covering 120° 2θ. Detector elements are therefore separated by ∼0.004°. The whole detector can be read out in 250 µs. Being Si based, its efficiency falls off above 20–25 keV, where the absorbing power of Si falls to very small values. Nevertheless, at intermediate and low energies a full powder-diffraction pattern for structural analysis can be measured in just seconds, or even faster if the intention is to follow a dynamic process.

[Figure 2.2.13]

Figure 2.2.13 | top | pdf |

(a) 120° Mythen detector box, containing helium, mounted on the powder diffractometer of the materials science beamline at the Swiss Light Source. (b) Multianalyser detector stage. (c) Capillary spinner. (Bergamaschi et al., 2009[link], 2010[link].)

Two-dimensional (2D) detectors are generally flat, so cannot extend to the same 2θ values as a curved multistrip detector unless scanned on a detector arm. This is possible, but usually a short wavelength is used with a fixed detector. This allows an adequate data range to be recorded, particularly if the detector is positioned with the direct beam (2θ = 0) near an edge. A 2D detector records complete or partial Debye–Scherrer rings, which increases the counting efficiency with respect to scanning an analyser crystal by several orders of magnitude. In addition, if the rings do not appear smooth and homogeneous, this indicates problems with the sample, such as preferred orientation or granularity, both of which can seriously affect diffraction intensities when measuring just a thin vertical strip. Detectors that have been used are diverse and include image plates, though these have slow read out, charge-coupled devices (CCDs) or Si-based photon-counting pixel detectors used for single-crystal diffraction or protein crystallography (e.g. Broennimann et al., 2006[link]), and medical-imaging detectors, which are designed for hard-energy operation. Examples include the CCD-based Frelon camera, developed at the ESRF (Labiche et al., 2007[link]), and commercially available large flat-panel medical-imaging detectors up to 41 × 41 cm2, based on scintillator-coated amorphous silicon, which have been exploited at speeds of up to 60 Hz for selected read-out areas (Chupas, Chapman & Lee, 2007[link]; Lee, Aydiner et al., 2008[link]; Daniels & Drakopoulos, 2009[link]).

Note that a 2D detector can be used as a 1D detector by applying a mask and reading out only a narrow strip, which can enhance the rate of data acquisition. For CCD chips, the electronic image can be rapidly transferred to pixels behind the masked part of the detector from where it can be read out while the active area is re-exposed. Translating an image plate behind a mask is a simple way of acquiring a series of diffraction patterns for following a process with modest time resolution.

These instruments are vulnerable to aberrations that cause systematic shifts in peak positions, such as misalignment of the capillary or surface of the sample from the diffractometer axis, and specimen transparency, which also affects the peak width and shape. The peak width also depends on whether a flat sample is in the θ/2θ condition, or on the diameter of a capillary sample, etc. Focusing the incident beam onto the detector decreases the peak width, as fewer pixels are illuminated compared to using a highly collimated incident beam. PSDs are much more open detectors than those behind an analyser crystal or set of slits, so are more susceptible to background and parasitic scatter from sample environments etc. However, the speed and efficiency of data acquisition usually outweigh such concerns.

2.2.4.3. Energy-dispersive instruments

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The broad, continuous spectrum from a wiggler or bending magnet is suitable for energy-dispersive diffraction (EDD). Here, the detector is fixed at an angle 2θ and the detector determines the energy, [epsilon], of each arriving photon scattered by the sample (Fig. 2.2.14[link]). The energy [keV] can be converted to d-spacing [Å] via[d \simeq 12.3984 / 2\varepsilon \sin\theta. ]

[Figure 2.2.14]

Figure 2.2.14 | top | pdf |

Schematic representation of an energy-dispersive diffraction arrangement.

The detector usually consists of a cryogenically cooled semiconducting Ge diode. An absorbed X-ray photon promotes electrons to the conduction band in proportion to its energy. By analysing the size of the charge pulse produced, the energy of the photon is determined. The powder-diffraction pattern is recorded as a function of energy (typically somewhere within the range 10–150 keV, depending on the source) via a multichannel analyser (MCA). Instruments may have multiple detectors, at different 2θ angles covering different ranges in d-spacing (Barnes et al., 1998[link]), or arranged around a Debye–Scherrer ring, as in the 23-element semi-annular detector at beamline I12 at Diamond Light Source (Korsunsky et al., 2010[link]; Rowles et al., 2012[link]).

Prior to performing the EDD experiment, the detector and MCA system must be calibrated, e.g. by measuring signals from sources of known energy, such as 241Am (59.5412 keV) or 57Co (122.06014 and 136.4743 keV) at hard energies, and/or from the fluorescence lines of elements such as Mo, Ag, Ba etc. The 2θ angle also needs to be calibrated if accurate d-spacings are desired. This should be done by measuring the diffraction pattern of a standard sample with known d values.

The detector angle is typically chosen in the range 2–6° 2θ and influences the range of d-spacings accessible via the term 1/sin θ, i.e. the lower the angle, the higher the energy needed to access any particular d. Normally, the range of most interest should be matched to the incident spectrum, taking account also of sample absorption and fluorescence, to produce peaks with high intensity. More than one detector at different angles can also be employed. Energy-sensitive Ge detectors do not count particularly fast, up to 50 kHz being a typical value compared to possibly 1–2 MHz with a scintillation detector. Hence they are relatively sensitive to pulse pile-up and other effects of high count rates (Cousins, 1994[link]; Laundy & Collins, 2003[link]; Honkimäki & Suortti, 2007[link]), particularly if the synchrotron is operating in a mode with a few large electron bunches giving very intense pulses of X-rays on the sample.

The energy resolution of the detector is of the order of 2%, which dominates the overall resolution of the technique. Its main uses are where a fixed geometry with penetrating X-rays is required, e.g. in high-pressure cells, for in situ studies (Häusermann & Barnes, 1992[link]), e.g. of chemical reactions under hydrothermal conditions (Walton & O'Hare, 2000[link]; Evans et al., 1995[link]), electrochemistry (e.g. Scarlett et al., 2009[link]; Rijssenbeek et al., 2011[link]; Rowles et al., 2012[link]), or measurements of residual strain (Korsunsky et al., 2010[link]). Owing to the use of polychromatic radiation, the technique has very high flux on the sample and can be used for high-speed data collection, following rapid processes in situ. However, accurate modelling of the intensities of the powder-diffraction pattern for structural or phase analysis is difficult because of the need to take several energy-dependent effects into account, e.g. absorption and scattering factors, the incident X-ray spectrum, and the detector response. Nevertheless, examples where this has been successfully carried out have been published (e.g. Yamanaka & Ogata, 1991[link]; Scarlett et al., 2009[link]).

A higher-resolution variant of the energy-dispersive technique can be performed by using a standard detector behind a collimator at fixed 2θ scanning the incident energy via the monochromator. The Hart–Parrish design with long parallel foils is suitable. Such an approach has been demonstrated in principle (Parrish, 1988[link]), but is rarely used in practice. The advantage is to be able to measure data of improved d-spacing resolution, as compared to using an energy-dispersive detector, from sample environments with highly restricted access. In principle, as a further variant, white incident radiation could be used with scanning of θa, the angle of the analyser crystal, and associated detector at 2θa, all at fixed 2θ.

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