Chapter 3.45. Bandwidth and divergence

The X-ray beams that are produced by double-crystal monochromators are not perfectly monochromatic, even if the harmonics are eliminated. The beams still have a small spread in energy, the bandwidth, which should normally be of the order of an electronvolt or less. The quality of X-ray absorption fine-structure (XAFS) spectra is significantly affected by the energy bandwidth of the X-ray beam. Using a beam with an excessively large bandwidth will broaden out near-edge and pre-edge features, distort the shape of X-ray absorption near-edge structure (XANES) features (and derivative spectra), shift apparent edge energies and reduce extended X-ray absorption fine-structure (EXAFS) amplitudes at low k. When comparing spectra, it is important that the energy resolution of their measurements should be approximately the same or that suitable corrections are made to account for the different resolutions. A simple way to verify the adequacy of energy resolution and other instrumental characteristics is to measure the same reference sample at the start of each experimental run, especially a sample with known prominent and sharp absorption features. Additional background material can be found in Stern & Heald (1983), Koningsberger & Prins (1988) and Bunker (2010).

XAFS spectral features, even bound states nominally of a single energy, have an intrinsic energy broadening owing to the finite core-hole lifetime; this is distinct from any instrumental effects. It is necessary to keep this in mind when using experimental spectral peaks or other features to assess instrumental broadening. It also puts limits on the energy resolution needed for an experiment: there is little utility in striving for a monochromator resolution that is much less than the lifetime broadening.

The lifetime broadening is described by Heisenberg's time–energy uncertainty relation ΔEΔT ≥ ℏ/2, which relates the energy width ΔE of a quantum state and its lifetime ΔT. In XAFS measurements the relevant lifetime is that of the core hole (i.e. the vacancy in the initial state, for example 1s for K edges), which is filled by an electron from a higher energy level, resulting in the emission of a fluorescence photon or the ejection of Auger electrons. Core-hole lifetime values have been tabulated by Krause & Oliver (1979) and are plotted (for K edges) versus atomic number in Fig. 1. It can be seen that the core-hole width increases rapidly with the atomic number Z of the central atom, and especially so for Z = 30 (zinc) and above, for which it increases approximately as Z⁴. The K-shell width in electronvolts can usefully be approximated (Bunker, 2010) by the polynomial 1.0(Z/42) + 2.1(Z/42)² − 4.6(Z/42)³ + 6.0(Z/42)⁴ (here 42 refers to the atomic number of molybdenum, not the meaning of life etc.).

Figure 1 Core-hole level widths from Krause & Oliver (1979).

For L edges the core-hole level width increases less rapidly with Z, with an exponent of approximately 1.75, and in a more complicated manner. For this and other reasons, the L edges of elements of high atomic number are more often measured in XAFS than are K edges. Using deconvolution procedures, it is possible to remove the core-hole broadening from the spectra of high-Z elements at high energies (for example the Au K edge), but doing so requires very low-noise data and it is not common practice.

The monochromators used for XAFS are based on Bragg diffraction, usually employing silicon, germanium or other crystals which, for our purposes, can be treated as perfect. The diffraction properties of such perfect crystals are best described by dynamical diffraction theory (Warren, 1990; Zachariasen, 1967; James, 1962; Authier, 2001; Batterman & Cole, 1964), in which all orders of multiple scattering of the X-ray beam within the crystal are accounted for.

The wavelengths λ_n (where n is a positive integer) that are diffracted from a crystal are related to the d-spacing of the relevant crystal planes (indexed by the integers h, k, l), and the incidence angle θ_B onto those planes, by Bragg's law: nλ = 2d_hklsinθ_B. Using E = hc/λ for photons (where E is the X-ray photon energy, h is Planck's constant and c is the speed of light), we have E = sinθ_B = nhc/2d_hkl. This equation (which is an approximation) shows that a spread in the angle of the beam hitting the crystal results in a spread in the energy of the diffracted beam. Taking the logarithm and differentiating gives |ΔE|/E = |cotθ_BΔθ|, where Δθ is the angular spread. The cotangent factor is minimum at 90°, so choosing a reflection that diffracts at an angle nearly perpendicular to the Bragg planes gives optimal energy resolution.

There are several contributions to Δθ that result from convoluting angular distributions which originate from several sources; some of these are described below. Mathematically, the cumulants (specific combinations of the power moments of various orders) of the several distributions being convoluted simply add together. One corollary is that the total squared angular width (second cumulant) is the sum of the squared widths from the various sources, i.e. the widths add in quadrature. This is common knowledge for Gaussian distributions, but it is also true for any distributions that have convergent cumulant expansions. Here, we consider only low-order moments for which integrals are convergent.

Here, we confine our attention to the most common type of double-crystal monochromator used for XAFS, with two crystals in a parallel (nondispersive) configuration and with the crystal surface cut parallel to the relevant Bragg planes. In this case the first crystal diffracts a narrow range of energies and angles, and the second crystal essentially acts as a mirror by diffracting the resulting beam in a direction nearly parallel to the incoming beam onto the first crystal. The energy resolution of the beam produced by such a monochromator depends on the angular width of the incoming beam, the intrinsic angular reflection width of the crystals and the extent of monochromator detuning (i.e. a small misalignment between the two crystals).

The intrinsic width depends on the composition and structure of the crystals, the crystal reflection (hkl) that is used and the Bragg angle (or energy). The angular spread of the X-ray beam hitting the first crystal depends on geometric parameters such as the projected source size, the collimation of the incoming beam, the distance from the source to the first crystal and the angular width of the slits that define the beam. If a bent (for example collimating) mirror is installed upstream of the monochromator this will strongly affect the angular spread of the beam incident on the first crystal. An incorrectly bent or thermally distorted upstream mirror can seriously degrade the energy resolution. Conversely, a collimating mirror with the correct shape can correct for the angular divergence of the source and thereby improve the resolution. Finally, thermally distorted or incorrectly bent monochromator crystals may also degrade the resolution.

Bragg's law gives a very good approximation to the angle at which photons of a given wavelength will diffract, but dynamical diffraction theory gives a more accurate and complete description. In particular, dynamical diffraction theory shows that there are slight deviations from the Bragg's law equation, and it shows that X-rays at a given wavelength will diffract from a crystal over a small, but nonzero, range of angles near the result of Bragg's law, called the Darwin width. The finite angular acceptance of such diffraction peaks is one source of the angular spread that results in a finite energy spread in the beam. This can be studied by measuring the intensity of the transmitted beam when rotating the second crystal of a two-crystal monochromator (parallel configuration) while holding the first one fixed. The resulting `rocking curve' is a convolution of the reflectivity curves of the two crystals. Because of the finite angular acceptance, the reflectivity of a crystal can usefully be represented as a function of both the angle and the wavelength (or alternatively the energy). In such a representation, DuMond diagrams (DuMond, 1937) are useful for understanding the properties of various configurations in which there are multiple crystals and slits.

The energy bandwidth of X-ray beams can be directly measured with the use of supplementary instrumentation, or alternatively through indirect methods based on computations derived from sample measurements. The detectors that are used for XAFS typically do not have sufficient energy resolution for this purpose.

De Jonge et al. (2004) have a described a method of quantifying the bandwidth in an X-ray beam by comparing the apparent mass-attenuation coefficients of foils of different thicknesses. This is an extension and refinement of earlier observations by Parratt et al. (1957) regarding the effect of the tails of the monochromator resolution function on measured spectra, which is one cause of `thickness effects'. At the absorption edge, there is a sharp rise in absorption as a function of energy; the smearing effect of the finite bandwidth has a greater effect on the spectra of thick foils than those of thin foils. Quantifying these observations provided de Jonge and coworkers with an estimate of the bandwidth, under the assumption of a Gaussian spectral distribution.

With suitable rotation stages, pinhole slits and detector, diffraction from a crystal using a high-order reflection can be used to directly measure the energy bandwidth of the X-ray beam. This is optimally performed in a nearly backscattering geometry with θ_B ≃ 90°. Fig. 2 shows reflectivity curves at 8 keV for silicon reflections (111), (311) and (333) calculated using the XCRYSTAL module of XOP 2.4 (Sánchez Del Río & Dejus, 2011). The higher order reflections provide better angular resolution and energy resolution than the lower order reflections.

Figure 2 Reflectivity curve at 8 keV for Si(111) (solid line), Si(311) (dashed line) and Si(333) (dotted line) reflections. The higher order reflections have better angular resolution and energy resolution.

An alternate approach is to use reference samples with sharp absorption features, such as KMnO₄ or KCrO₄, which have intense pre-edge transitions in their K-edge spectra that are associated with bound states, and which are broadened slightly by the core-hole lifetime. If high-resolution reference spectra can be obtained (for example using the methods below) then estimates of the additional instrumental broadening found under routine conditions can be deduced using a numerical convolution procedure by determining how much numerical broadening is needed to make the spectra best coincide. In principle, the monochromator resolution function could be extracted by solving an integral equation, but this is not common practice. A similar numerical convolution/matching process can be performed on derivatives of the spectra in the absence of any sharp absorption peaks, but the data must have low noise.

For this approach to work one must have access to high-resolution spectra of reference samples. These could be measured with a monochromator with high-index reflection, or a four-crystal monochromator in a dispersive configuration, but such instrumentation is not commonly available. One approach to generating a high-resolution spectrum to use for calibration is to make measurements using an allowed high-order reflection of the monochromator and to use an absorber to eliminate the low-energy component of the beam. For example, a Si(111) monochromator set to 4 keV will transmit 12 keV photons from the 333 reflection if the crystals are fully aligned. Because of the roughly 1/E³ behavior of absorption coefficients between edges, an absorber (for example aluminium) that has half of the absorption length at 12 keV will have approximately 13.5 absorption lengths at 4 keV, which would attenuate the fundamental by a factor of roughly about 10⁶ while only reducing the 12 keV component by 40%. This provides ample suppression of the undesired low-energy component.

This approach does require the monochromator to go to energies that are only a fraction (for example 1/2 or 1/3) of the energy of interest, however. If one needs to work closer to the low-energy limit of the beamline then there are other approaches. The double-crystal monochromator of the beamline can be adjusted to improve the resolution in order to generate a higher resolution reference measurement. For a double reflection, as shown in Fig. 3, the two single-crystal reflectivity curves are multiplied together, which causes the tails of the resulting reflectivity curve to decrease more rapidly than a single reflection. As pointed out by DuMond (1937), if the crystals are then `detuned', i.e. intentionally slightly misaligned, as shown in Fig. 4, the central part of the resulting transmission curve will have a narrower angular width (full width at half maximum; FWHM) and one can obtain a correspondingly reduced energy bandwidth, in this case a fivefold reduction of the central peak. The resulting curve retains a broad baseline absorption, however, which may pose a computational nuisance.

Figure 3 Transmission through double reflection with crystals in alignment. The full width at half maximum is 7.15 arcsec (4.85 µrad).

Figure 4 Double reflection with crystals misaligned by 6.6 arcsec (32 µrad). The full width at half maximum is 1.35 arcsec.

The dependence of energy resolution on various beamline settings can be deduced by measuring spectra as a function of those parameters, such as the vertical slit apertures, that define the beam hitting the sample. The estimated resolution can be tabulated and parameterized to aid in beamline setup.

The divergence of X-ray beams, unlike the energy resolution, is essentially geometrical in nature, although it can be affected by X-ray optics such as mirrors. Bending magnets and wigglers produce a fan of radiation in the horizontal plane over an angle that is typically at least a few milliradians, while collimation in the vertical direction is typically of the order γ = E_beam/mc², which is of the order of 100 µrad; E_beam is the electron storage-ring energy and mc² is the rest energy of the electron. Undulator beams are typically collimated to angles of the order of 100 µrad in both the horizontal and vertical directions.

If the flux is sufficiently low (or the beam is attenuated), direct optical imaging using phosphor screens, 1D or 2D diode arrays or X-ray area detectors can be used to profile the beams. By themselves, such methods do not define the direction of a beam at a given upstream location, however. To do so, one can place a narrow slit at this location, which can then be followed by some type of imaging device downstream. Alternatively, two sets of motorized XY slits at upstream and downstream locations in a beamline can be useful for profiling the beam, with detection by ionization chambers. XY maps of the transmitted beam by scanning slits at both upstream and downstream locations define a collection of ray directions and starting positions, which is essentially what one wishes to measure: the XY location of each small portion of the beam at the first slit and two angles describing its direction. Measuring such a map with two slits requires a four-dimensional scan; doubling the resolution multiplies the acquisition time by 16, so continuous data acquisition during slit scans can be helpful. Fortunately, the counting time can be minimal because the flux is substantial. Scanning a single slit with continuous-scan data acquisition and then differentiating the result may improve scan speeds, instead of coordinating pairs of slits.

For very precise angle measurements, of the order of 100 µrad (20 arcsec) or less, crystal analyzers can be used, as long as one knows the energy composition of the beam. This may require a suitable goniometer or rotation stage and detector; diffraction using thin crystals in Laue geometry can also be useful. If little is known about the energy content of the beam, the fact that the crystal reflectivity depends on both angle and energy can make analysis using diffraction methods nontrivial. Polarization effects may further complicate the analysis. A less-sensitive angle analyzer can be fabricated using a variety of materials (for example float glass or a metal surface) with a sufficiently flat surface by using the fact that, for a given energy, total external reflection has a well defined cutoff angle.