Tables for
Volume I
X-ray absorption spectroscopy and related techniques
Edited by C. T. Chantler, F. Boscherini and B. Bunker

International Tables for Crystallography (2023). Vol. I. Early view chapter


Grant Bunkera*

aDepartment of Physics, Illinois Institute of Technology, Chicago, IL 60616, USA
Correspondence e-mail:

A brief, high-level overview of methods for the study of solutions by XAFS is presented, with description of potential pitfalls and how to avoid them. Fluorescence self-absorption effects are explained in terms of a `sampling depth' and the case of nonuniform density versus depth is treated.

Keywords: solutions.

1. Introduction

Much of the utility of X-ray absorption fine structure (XAFS) lies in its applicability to a wide variety of samples in different states, for example solid, liquid and gas, including many of relevance to biology, chemistry, medicine, catalysis, materials science and metallurgy. The essential characteristic of solutions is that they are defined as homogeneous mixtures. This is an idealization, and therefore an approximation to real substances. As described below, a sample may appear to be homogeneous for some measurements (for example XAFS at one energy) and inhomogeneous for others (for example XAFS at a different energy).

Solutions may be measured in transmission mode, fluorescence mode, electron yield or other experimental detection modes. To carry out measurements, specially constructed sample chambers are often required because of the wide range of possible samples and measurement energies. Designing such chambers may require some design ingenuity by the experimenter. Overviews of XAFS experimental methods and some approaches to cell designs are given, for example, in Stern & Heald (1983[link]), Lee et al. (1981[link]), Koningsberger & Prins (1988[link]), Bunker (2010[link]), Van Bokhoven & Lamberti (2016[link]) and Iwasawa et al. (2017[link]). This article is intended to complement discussion in Di Cicco (2022[link]) and Heald (2023[link]).

2. Sample inhomogeneity

By definition a solution is a homogeneous mixture of two or more substances, which can be any combination of (gas, liquid, solid) dissolved in (gas, liquid, solid). In practice, samples are often assumed to be homogeneous, which may or may not be the case, and it can be important to validate this assumption by other means, for example microscopy.

A sample must be treated as inhomogeneous if variations in its composition (or orientational order) occur on a length scale that is comparable to the spatial variations of the beam intensity and polarization (the most important of which is beam size). If the beam is much larger than the spatial variations, it averages over the inhomogeneities; if the inhomogeneities (inverse gradients) are much larger than the size of the beam, the absorption characteristics do not appreciably vary over the size of the beam. If this homogeneity condition is not met, the measured spectra will be sensitive to beam motion, as well as to time-dependent changes in the sample, so the acquired spectra may be noisy, have artifacts or simply may not represent the form of the sample that is desired. Typical problems include radiation damage, precipitation, aggregation or changes in the oxidation state of proteins, or X-ray-induced photolysis of water in liquid aqueous samples, which creates bubbles. These problems can often be reduced by flowing a liquid sample, by rastering the beam over an extended sample to continuously expose undamaged material to the beam, by the use of cryoprotection and radioprotectants, or by other means.

The penetration depth of the X-rays through the sample is also a key parameter: it defines the relevant length scale along the direction of beam transport. As shown below, spatial variations in density versus depth are not necessarily a problem if the sample is compositionally homogeneous.

3. Calculations

3.1. Uniform composition and density

To orient ourselves, let us first consider a typical experiment: a planar slab fluorescence sample of thickness τ, with the incident beam impinging on the sample at an angle θ with respect to the surface, which then penetrates to depth x, is absorbed by a specific element s in layer thickness dx, and produces fluorescence at energy Ef which exits at angle φ. For a complete calculation the detected counts should average this expression over the solid angle of the fluorescence detector; for simplicity, we also suppress other factors such as the fluorescence yield which are not of interest for our purposes here.

For a homogeneous sample the differential probability of this process is [\exp[-\mu(E) x \csc(\theta)] \mu_{\rm s}(E) \csc(\theta) \exp[-\mu(E_{\rm f})x \csc(\varphi)]\, {\rm d}x, \eqno(1)]where μ(E) and μ(Ef) are the linear attenuation coefficient (absorption plus scattering) at energies E and Ef, respectively, and μs(E) is the absorption coefficient of the element of interest. This can be more usefully written as μs(E)csc(θ)exp[−x/Δ(E)] dx in terms of an energy-dependent function which we will call the sampling depth,[\Delta(E) = [\mu(E)\csc(\theta)+\mu(E_{\rm f})\csc(\varphi)]^{-1}. \eqno(2)]Integrating over the sample thickness τ gives[\csc(\theta) \mu_{\rm s}(E) \Delta(E)\{1-\exp[-\tau/\Delta(E)]\}. \eqno(3)]

For τΔ(E) (the thick limit), the exponential term drops to zero and we simply obtain fluorescence signal proportional to csc(θ)μs(E)Δ(E). For a thin sample τΔ(E), and a first-order Taylor expansion of the exponential tells us that the fluorescence signal is proportional to μs(E)τcsc(θ), as expected. Self-absorption (or `overabsorption') effects occur when μs is not small compared with the total absorption coefficient μ(E), so that the energy dependence of the sampling depth Δ(E) is strongly affected by the absorption from the element of interest. In such a case, if μs(E) increases (say, over an absorption edge) then Δ(E) decreases, which compensates, and, among other things, compresses strong absorption features. When designing fluorescence experiments the sampling depth Δ(E) above the absorption edge should be calculated to find the relevant length scale over which the sample must be homogeneous if it is to be considered a solution.

This simple model is appropriate for a slab geometry; more elaborate geometries and effects of packing fraction and particle shape and size distribution have been worked out in Tannazi (2004[link]). A common error to avoid in practice is to assume that a material is homogeneous, when in fact the element of interest may be concentrated in particles or volumes that are not small compared with Δ(E) [for purposes of estimation, the angle factors in Δ can be taken as csc(θ) ≃ csc(φ) ≃ 1/21/2].

3.2. Uniform composition with non-uniform density

This analysis can be extended to account for variations in density ρ(x) as a function of depth x, with very simple results. In this case, for a compositionally homogeneous system, we write μ(E) = ρ(x)σT(E), where σT(E) is the total of the absorption and scattering cross sections of the sample matrix and σs(E) is the absorption cross section. Here, we will use mass units so that (for example) the cross section has units of cm2 g−1, in which case the density ρ would be in g cm−3.

Repeating the analysis above, we find that the attenuation factor exp[−μ(E)x] is replaced by exp[−σT(E)R(x)], where the mass thickness [R(x) = \textstyle\int_0^x\rho(x')\,{\rm d}x'], which, in our units, has dimensions of g cm−2. Following the previous analysis, since by hypothesis we have assumed that the composition of the sample is uniform (whereas the density varies in an arbitrary manner with depth x), we see that the differential probability can be written [\sigma_{\rm s}(E) \rho(x) \csc(\theta) \exp[-R(x)/D(E)]\,{\rm d}x, \eqno (4)]where D(E) = [(σT(E)csc(θ) + σT(Ef)csc(φ)]−1. D(E) bears the same relation to the sampling depth Δ(E) as the mass thickness R(x) does to the depth x, and can be called the `mass-sampling depth'. Integrating over the sample thickness gives the simple result[\csc(\theta) \sigma_{\rm s}(E) D(E) \{1-\exp[-R(\tau)/D(E)]\}. \eqno(5)]The reader can show that these expressions agree with those in the previous discussion for uniform density.

For a given thickness, the quantity R(τ) is just a number; all of the details of the density variation are integrated over. For thick samples, if R(τ) ≫ D(E), the exponential drops out and we have csc(θ)σs(E)D(E). For small mass-thickness samples R(τ) ≪ D(E), and performing a first-order Taylor expansion tells us that the fluorescence signal is proportional to csc(θ)σs(E)R(τ). This shows us that for a compositionally homogeneous sample, density variations do not necessarily cause more difficulties than are present in the uniform density case (lateral density gradients, however, would cause problems from coupling to beam lateral intensity gradients, as mentioned above). If the sample is compositionally inhomogeneous this will not in general be true, however.

3.3. Transmission

Inhomogeneities in transmission measurements induce the well known `thickness effects' which are covered elsewhere in this volume. Here, we simply add the observation (Bunker, 1984[link], 2010[link]) that variations in mass thickness in a sample (even with a uniform X-ray beam profile) cause systematic non­linearities in the dependence of the measured absorption on the true absorption μ. It is simple to show that (μx)meas = [-\ln\textstyle\int_0^\infty P(x) \exp{(-\mu x)}\,{\rm d}x] = [\mu \bar x - \mu^2\sigma^2/2 + \ldots], where [\bar x] and σ2 are the mean and the variance of the distribution in mass thickness P(x) (normalized), respectively; the coefficients of powers of μ are the cumulants of P(x).

4. Experimental equipment

4.1. Cells and windows

The rest of this article is oriented towards implementation. The basic challenge posed by fluids is that they flow, so they must be contained in a cell that lets X-rays pass through the sample (in transmission mode) or lets incident X-rays in and fluorescence X-rays out of the sample (in fluorescence mode). In transmission mode, it is also important that the path length of the beam through the sample is constant over the beam profile, otherwise the spectra will be distorted: the measured spectra are a nonlinear function of the true spectrum. For this reason, the X-ray-transparent windows should be approximately flat and parallel. The windows can be made of many materials, but the X-ray absorption from the windows should be minimized (absorption lengths can easily be calculated or looked up), the windows should not contain significant amounts of the element being measured, they should be fairly rigid (to prevent excessive bowing or deformation) and they should be chemically unreactive to the liquid (and also when irradiated by the beam).

Cell designs can be very simple, but conditions often require more complex designs. Some specific examples of liquid sample-cell designs are given in Marcos et al. (1994[link]) and those for high temperature and pressure use by Testemale et al. (2005[link]); other cell designs are reviewed in Iwasawa et al. (2017[link]). A very simple and inexpensive solution cell (Fig. 1[link]) consists of a sheet of plastic material (for example PTFE, polycarbonate or acrylic) or machinable ceramic of the desired sample thickness, with a slot milled through it, onto which the polyimide (or BoPET, polypropylene or PTFE) films are glued (or clamped by aluminium plates with screws onto compliant gaskets to improve reusability) onto the surface on both sides of the plastic. Filling holes, which are large enough to insert a syringe needle into, are then drilled perpendicularly through the material at each end of the slot so liquid can be introduced at one end and air can leave at the other. These can be sealed by various means. The materials used should not contain significant amounts of the element being measured, especially when measuring dilute fluorescence samples, because these will contribute to stray background; even if the beam does not directly hit the cell, X-rays scattered from air and sample will. The materials should be experimentally checked for such contamination at each absorption edge of interest by measuring blanks, i.e. solutions with negligible amounts of the element of interest. Such sample cells can also be produced by 3D printing, but the abovementioned concerns about trace contamination will remain.

[Figure 1]

Figure 1

Simple and inexpensive solution cell with plastic film X-ray windows. Holes allow the use of a syringe for filling and emptying.

The optimal cell design is part of the larger system design, which requires consideration of the edge energy, sample concentration, beam dimensions (focused or not?) and available detector technology (is a large exit window needed to collect fluorescence or can the detector be used in vacuum?). The best choice of window materials will depend on the nature of the sample, such as the edge energy, thickness and reactivity/sensitivity of the sample. For first-row transition-metal elements, common window material choices include polyimide, PTFE and polypropylene films, low-Z ceramics such as boron nitride, diamond films or graphite. For higher energy edges many other choices are available. The reactivity of window materials with the liquids (for example buffer or organic solvent) should be tested in the laboratory before carrying out synchrotron experiments. Low-absorption X-ray windows are commercially available for laboratory-based X-ray fluorescence analysis, and they can be put to use, but the vastly higher X-ray intensities from synchrotron-radiation sources, especially from wigglers and undulators, tend to damage the windows much more rapidly than expected in a laboratory. Composite layered materials can be useful, for example nonreactive coatings on structurally reinforced materials for large windows.

It should be mentioned that it is possible to measure a free horizontal liquid surface without using any entrance window by deflecting the incident beam downward onto the liquid using an X-ray mirror or a diffracting optic.

4.2. Cryostats, furnaces and anaerobic environments

A common approach for measuring biological samples, such as enzyme solutions, is to rapidly freeze-quench them and then keep them frozen in a cryostat, in which the sample is thermally isolated from the outside (for example by a vacuum, closed-cell foam or aerogel insulation) but kept in good thermal contact with a thermally regulated cryogenic cold element (a cold finger or exchange gas). This has several benefits: the frozen sample is solid and so may not require any windows on the sample cell at all, radiation damage may be suppressed by limiting the mobility of photoproducts such as radicals, and the spectra are also improved by reduction of the thermal Debye–Waller factors. A complication of this approach, depending on the available instrumentation, is that operation in a vacuum is usually required for adequate thermal isolation and to prevent ice buildup on the sample when samples are changed. Another is the need to reduce ice-crystal formation to prevent diffractive artifacts in the measured spectra. Suitable instrumentation and procedures are normally available at beamlines that often perform this kind of measurement. It is possible to build in optical monitoring systems to provide in situ measurement of sample integrity. It is also possible to place certain types of detectors within the vacuum environment itself so the need for large-area exit windows in fluorescence mode can be avoided.

Instead of cooling aqueous samples for measurement, it may be instructive to heat samples, for example solutions of metals, alloys and eutectics, to melt them and measure these substances in liquid form. In analogy to the cryogenic apparatus, the sample can be thermally isolated from the outside (for example by a vacuum, closed-cell ceramic foam or aerogel insulation) but heated by conduction or radiation from a hot thermally regulated element. The choice of sample-cell windows is governed by the considerations above: mechanical stability against pressure gradients at the operating temperature and nonreactivity with the sample.

A vacuum environment also can be adapted for some types of anaerobic liquid samples; alternatively, the cell can be purged with gases from which oxygen has been removed. In this case a very slightly elevated pressure relative to atmospheric may be helpful to prevent contamination.

4.3. In situ measurements

In situ (in place) or operando (in operation) experiments are experiments in which a chemical reaction or other process is carried out as the XAFS is measured in real time. These often involve substances in liquid form and may be considered as solutions. In addition to the considerations and equipment described above, some ancillary instrumentation, such as cyclic voltammetric apparatus, requires electrodes for monitoring and driving the reaction that also are an integral part of the measurement. In some cases the window material and the electrode material may serve dual functions.

Measurements of enzyme or chemical solution kinetics can be studied in flow systems (stopped-flow or continuous) to explore the structural changes as a function of time during reactions following excitation by rapid mixing or flash photolysis. These measurements may involve capillary flow cells (Wallen et al., 1996[link]), but the design considerations described above still apply.

5. Summary

Solutions are defined as homogeneous mixtures, solid, liquid or gas, which is only approximately true. For XAFS experiments one must define the appropriate length-scale metrics that are relevant for quantifying homogeneity. Here, we define a `sampling depth' that is useful for uniform density and composition and a `mass-sampling depth' for uniform composition but varying density. We recommend that these metrics be used for experimental design. Other considerations regarding beam intensity or polarization inhomogeneities are also discussed. The article concludes with practical considerations for the design of experimental apparatus.


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