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

Ultradilute systems

Steve M. Healda*

aAdvanced Photon Source, Building 435E, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA
Correspondence e-mail:

A fundamental strength of XAFS is the ability to measure ultradilute components of a sample. With the advent of high-flux beamlines, the ultradilute limit is currently determined more by the detectors that are used than by the available flux. After a description of the fundamental limits and detector types, the discussion concentrates on the most effective use of current detectors and the prospects for improvements.

Keywords: ultradilute systems.

1. Introduction

As described in Chantler (2022[link]), with the use of fluorescence detection X-ray absorption fine-structure (XAFS) measurements can be made on very dilute components of a sample. This is an advantage of the technique and allows measurements of the structure and chemistry of doped materials, heavy-metal pollutants in the environment and metallo­proteins at biologically relevant concentrations, to give just a few examples. With increasingly powerful X-ray beamlines the concentration limits can be expected to continue to improve. However, as will be discussed in this chapter, the limits are determined more by the detectors available rather than by the incident flux. After a consideration of the factors limiting the signal to noise, the discussion will concentrate on the most effective use of current detectors and the prospects for improvements in the future.

2. The effective count rate

The fluorescence signal is always accompanied by some background that will degrade the signal to noise. A strength of fluorescence detection is that the detector can isolate the fluorescence line from the element of interest, rejecting the background from scattering or fluorescence from other components. No detector is ideal, however, and there can be sources of background at the same energy as the fluorescence. In the ultradilute regime these backgrounds can still overwhelm the signal even for a high-quality detector.

The main goals of any detector system are to maximize the amount of fluorescence signal collected from the sample while minimizing the background that contaminates the signal. These two goals are often competing, and there is a tradeoff between them. A useful concept in evaluating and optimizing detector schemes is the effective count rate (Heald, 2015[link]; Stern & Heald, 1979[link]). If the fluorescence counting rate is Nf and the background counting rate is Nb, then for an integrating detector the signal to noise will be the same as if there is no background, and there is an effective fluorescence counting rate of[N_{\rm e} = {{N_{\rm f}}\over {1 + (N_{\rm b}/N_{\rm f}).}} \eqno(1)]In the ultradilute range the background can be many times the signal, and this equation emphasizes the importance of separating the signal from the background. An energy-resolving detector can improve the effective counting rate by reducing the background that is counted. However, this can come at the expense of reduced signal counts, either through inefficiencies in the detector or through restrictions on the total counting rate.

3. Sources of background

From the effective count-rate expression, it is obvious that it is important to reduce the background that is detected along with the fluorescence signal. There are two important sources of background: scattering, both elastic and inelastic, and fluorescence from other components of the sample. Typically, the scattering cross sections are a few percent of the absorption. Therefore, for dilute components with concentrations of parts per million or lower, the scattering signal will be much stronger than the fluorescence. Just detecting the total emission signal with, for example, an ion chamber is not an effective method. There needs to be some type of energy-resolving detector. These will be discussed in the next section. Here, we consider the characteristics of the background in more detail.

Fig. 1[link](a) shows a high-resolution (∼1 eV) measurement of the scattering from SiO2 as an example (Heald, 2015[link]). There is a sharp elastic peak and lower energy inelastic scattering that is dominated by Compton scattering, but it also has other contributions as seen by the jump at the O K-edge energy loss. This is an X-ray Raman signal. Fig. 1[link](b) shows a different measurement of background as measured by a solid-state detector (SSD) with typical energy resolution (∼200 eV). The elastic and Compton contributions now appear as a broad asymmetric peak with a tail to lower energy. This tail will extend into the fluorescence line, giving rise to a background signal. For K edges the Kα lines are about 10% lower in energy than the K edge. The vertical lines in Fig. 1[link](b) show a typical energy window or region of interest (ROI) that would be used to record the fluorescence of copper when the beam energy (elastic line) is just above the edge. In this particular case the background signal for the SSD would be about 1.5% of the elastic scattering peak. Part of this background is true inelastic scattering and part of it comes from the tail of the detector resolution function. For a detector with improved energy resolution the data in Fig. 1[link](a) can be used to estimate the background. To collect the entire Kα1 and Kα2 signals, the energy window needed is about 20 eV. The inelastic background would then only be 0.02% of the elastic scattering.

[Figure 1]

Figure 1

Two views of the scattering from an SiO2 substrate near a 90° scattering angle: (a) with a crystal analyzer with ∼1 eV resolution and (b) with a silicon drift detector with ∼200 eV resolution and an incident beam energy of 9000 eV.

For a synchrotron source, the polarization dependence of the signals can also be used to suppress the background. The elastic and Compton scattering both have a cos2θ dependence, where θ is the emission angle relative to the polarization direction. The scattering is suppressed for a detector oriented along the beam polarization direction. The fluorescence emission is isotropic with an angular dependence determined only by the escape probability from the sample. Of course, to obtain a reasonable signal the detector must detect a finite solid angle letting in a scattering contribution. Fig. 2[link] shows a calculation of the angular dependence of the scattering and fluorescence signals for the typical flat-plate sample geometry.

[Figure 2]

Figure 2

Calculation of the angular dependence of the scattering (green circles) and fluorescence (red squares) signals for the detector geometry illustrated in the inset. The calculation assumes that the X-ray polarization is along the z axis and includes the absorption of the signals as they escape the samples. The signals are normalized to the total scattering and fluorescence, respectively.

So far, the background discussion applies to the case of a dilute component in a matrix without significant fluorescence. In complex samples there can be other fluorescence lines that could overlap with the lines of interest. Their background contribution will critically depend on the detector resolution. As seen in Fig. 1[link](b), even lines that can nominally be resolved by the detector can have a tail that extends into the signal ROI if they are very strong. The background fluorescence can have another important impact. Most detectors will have a maximum count rate above which their performance degrades. Background fluorescence components can use up this count rate, limiting the solid angle and the usable signal that can be detected.

4. Detection methods and their characteristics

There are three general types of detectors: filter–slit systems, electronic energy-resolving detectors and diffraction-based detectors. These can often be combined for better performance. The simplest detector is the filter–slit system (Stern & Heald, 1979[link]), as illustrated in Fig. 3[link]. The filter material is selected such that it has an absorption edge above the fluorescence line but below the scattering signal. For a K-edge-based filter the change in the absorption coefficient at the edge can be as great as sevenfold. Since the attenuation depends exponentially on the absorption coefficient, the scattering can be strongly suppressed with a small loss in fluorescence. The slits are required since when the filter absorbs a scattered photon it can be re-emitted as filter fluorescence. This degrades the background reduction. As shown in Fig. 3[link], appropriate slits can block much of the filter fluorescence solid angle, while passing most of the fluorescence. Filter-based detectors have the advantage that they do not need an energy-resolving detector and can be combined with ion-chamber or diode-based detectors with essentially unlimited count rates. They are also easy to scale up to large collection solid angles. Their primary disadvantage is a background-rejection factor limited by the properties of the filter material. As will be seen, this limits their effectiveness when samples become very dilute. However, it is easy to combine them with the other detector types for improved performance.

[Figure 3]

Figure 3

The top image illustrates the principle of the filter–slit detector. Fluorescence from the sample (blue arrows) is relatively unimpeded, while the scattering is highly absorbed by the filter. Filter fluorescence (red dashed arrows) is restricted in solid angle by the slits. The bottom image shows an example of a two-dimensional slit for use with a focused beam.

Electronic energy-resolving detectors include scintillation detectors, proportional counters, solid-state detectors based on semiconductors such as silicon and germanium, and superconductor-based systems. The most widely used are silicon and germanium solid-state detectors (Cramer et al., 1988[link]; Jaklevic et al., 1977[link]; Warburton, 1986[link]; Strüder et al., 1998[link]). As shown in Fig. 1[link], they can provide sufficient energy resolution to separate the fluorescence and scattering, and using silicon drift technology with modern signal processing a single detector can handle up to several million photons per second (Barkan et al., 2015[link]). Detectors with up to 400 detector elements have been developed (Ryan et al., 2010[link]). However, to date the detectors with many elements have been based on less sophisticated detectors and electronics, and the individual elements have lower count-rate capabilities than state-of-the-art silicon drift detectors.

As discussed in Section 3[link], for the typical solid-state detector there is still significant leakage of the background into the fluorescence channel. The energy resolution of these detectors is limited by the properties of the materials and is unlikely to be improved. To obtain better energy resolution, detectors based on superconductors have been developed (Day et al., 2003[link]; Eckart et al., 2012[link]). These can achieve resolutions approaching 1 eV, but to date have very limited count rates (∼100 Hz). Their widespread use will depend on developing methods for multiplexing large numbers of detector elements while maintaining the extremely low temperatures needed for their operation.

Another approach to improved energy resolution is to use crystal spectrometers to analyze the signal. There have been a variety of approaches to collect large solid angles while maintaining the strict angle alignment dictated by the Bragg equation (Bergmann & Cramer, 1998[link]; Hastings et al., 1979[link]; Heald et al., 2012[link]; Kirkland et al., 1995[link]; Kropf et al., 2003[link]; Marcus et al., 1980[link]; Mattern et al., 2012[link]; Pease et al., 2000[link]; Welter et al., 2005[link]). Crystal detectors can be classified into two general types: focusing and nonfocusing. Focusing detectors such as Rowland circle-based analyzers collect and focus the fluorescence into a small spot. By placing an aperture at the focus they can achieve near-perfect rejection of the background. The primary background is then the inelastic scattering at the same energy as the fluorescence. Nonfocusing detectors such as those based on log spiral shaped crystals require a large area detector that is more difficult to shield from the background. Nevertheless, they can achieve sufficient background rejection to be a significant improvement over solid-state detectors.

The downside of crystal-based systems is the difficulty in collecting a large solid angle. For example, a common analyzer crystal comprises a 100 mm diameter silicon or germanium wafer spherically bent to operate at about 1 m from the sample close to backscattering. Such a crystal only collects about 0.06% of the emission solid angle. Also, it can be difficult to match the crystal energy resolution to that needed. Common silicon or germanium crystal reflections have resolutions that only collect a fraction of the fluorescence line. One approach is to strongly bend the crystals to expand the energy range, but this reduces the reflection efficiency. Mosaic crystals such as lithium fluoride or graphite can have a wider energy range, but at the expense of peak reflectivity.

The largest solid-angle crystal detectors are based on the log spiral nonfocusing geometry. A commonly used variant based on using Laue diffraction collects about 1–2% of the solid angle (Kropf et al., 2003[link]). However, the diffraction efficiency is only 10–30%. A graphite-based analyzer using Bragg geometry has been made that collects about 17% of 4π with an estimated efficiency of 20% (Pease et al., 2000[link]). However, the energy resolution was only about 4%. A disadvantage of these detectors is their limited energy range. A single bent crystal can handle only a few nearby fluorescence lines.

5. Optimizing fluorescence detection

The optimum choice of detector depends primarily on the concentration of the element of interest. The detectable concentration also depends on the matrix material. A low concentration in a highly absorbing matrix will be more difficult to detect since the X-ray penetration is small and fewer atoms will be excited to fluoresce. A good way to compare different types of samples is by their background-to-signal ratio as measured with a non-energy-resolving detector such as an ion chamber. When the ratio is small (∼10 or less), the filter–slit detector is a good choice. It provides adequate background rejection, collects a good solid angle and has no count-rate limitations. As the background increases, solid-state detectors (SSDs) become a better choice with their superior rejection of the background. However, on high-flux beamlines they can reach their total count-rate limits, resulting in the need to reduce the collection solid angle. Note that when the background is dominated by scattering, reducing the solid angle to limit the count rate is preferred over attenuating the incoming beam, since then the polarization can be used to reduce the background ratio. When the sample becomes very dilute and the solid angle is reduced there may not be enough remaining fluorescence photons for a good signal when the total count-rate limit of the detector is reached. It then becomes desirable to reduce the background-to-signal ratio reaching the detector. Either filter slits or crystal analyzers can be used to reduce the background reaching the solid-state detector. A detailed comparison of detectors is given in Heald (2015[link]) for the case in which scattering dominates the background. There, it is shown that adding a filter–slit system to an SSD can extend its effectiveness to background ratios exceeding 1000. Just as importantly, the filter reduces the maximum count rate that the SSD needs to handle, allowing the collection of a larger solid angle. Without the filter, the SSD system might need to handle total count rates exceeding 109. With filters, current multi-element SSDs that can handle count rates of a few times 107 can approach the performance of higher count-rate detectors at high dilutions. See the solid curves in Fig. 4[link].

[Figure 4]

Figure 4

Calculation of the effective count rate for dilute arsenic in a sample matrix containing 10% iron in SiO2 (dashed curves) and for arsenic in a pure SiO2 matrix (solid curves). Two different count-rate limits are considered for the solid-state detector. The background refers to the scattering background and is a proxy for the arsenic concentration. At each background (concentration) the filter thickness and detector solid angle is optimized with the constraint of the total count rate possible for the detector.

Also shown in Fig. 4[link] is an example of a situation that is often encountered in environmental samples, a dilute pollutant (arsenic in this example) accompanied by an element such as iron that gives a large fluorescence background signal. In this case the iron signal is below the filter edge, but because it is well below the arsenic energy the filter can still provide a preferential attenuation. However, in this situation the count-rate capability of the detector becomes more important. As can be seen in the figure, there is a large improvement as the maximum count rate of the detector is increased to 108 from 107.

The background-to-signal level depends on several factors. Table 1[link] shows a few measured values to provide a basis for estimation. Obviously, as the concentration is reduced the ratio increases. It will also increase as the detection solid angle increases (see Fig. 2[link]) and as the scattering power of the matrix increases. Currently practical limits are in the range of a few hundred, depending on the amount of radiation that the sample can withstand and the amount of time that one is willing to devote to a sample.

Table 1
Some measured background-to-signal ratios

The background in this case refers only to the scattering contribution and does not include lower energy fluorescence signals such as, for example, calcium fluorescence in the glass sample.

ElementConcentrationMatrixSolid angle (% of 4π)Background-to-signal ratio
Fe 1 mmol Water 10 20
Cu 55 µmol Water 1 200
As 34 mg kg−1 Glass 2 25
72% SiO2, 14% Na2O, 12% CaO and 2% Al2O3.

If crystal detectors can be developed that collect large solid angles they could offer improved performance over SSDs and filters, especially at very high background ratios of >1000. The inelastic scattering contribution at the fluorescence energy then becomes significant and improved energy resolution is helpful. They may also be the best choice when the background is not solely due to scattering but comes from fluorescence of the matrix elements, as for the example in Fig. 4[link]. For complex samples with many elements there may be an actual overlap of fluorescence lines in an SSD. For example, the Kβ line of an element is at nearly the same energy as the Kα line of the next heaviest element. Similarly, the L lines of heavy elements can overlap with the K lines of transition metals. In these cases, it may become essential to use a high-resolution crystal detector to separate lines that may only be a few tens of electronvolts apart.

6. Prospects for improvement

As the discussion above illustrates, there are two routes to improved detectors: (i) improve the count-rate limits of solid-state detectors or (ii) improve the collection efficiency of high-resolution detectors such as crystal analyzers. In the near-term, it is quite reasonable to expect high count-rate silicon drift detector arrays with larger numbers of detector elements. The main impediment is the cost of the detectors and the sophisticated signal-processing electronics along with data-analysis procedures capable of handling large numbers of detectors whose characteristics may not be identical. Since each detector can operate at >106 Hz, it is not unreasonable to expect total count rates approaching 108 in the future. The second approach to higher count rates is taken by the Maia detector. It uses less sophisticated detector technology and less expensive integrated electronics to allow large numbers of detectors at a more reasonable price per detector (Ryan et al., 2010[link]). Current designs are optimized for use on X-ray microprobes and are aimed more at collecting a large solid angle rather than working on ultradilute samples, but similar technology could be used to make detectors that can also approach total count rates of 108.

Even with higher count rates there will still be applications where better energy resolution is needed, either to separate closely spaced fluorescence lines or to reach the ultimate diluteness. There is no fundamental limit to enhancing the collection angle of crystal-based detectors. For inelastic scattering applications, detectors with many tens of backscattering crystals have been developed. However, to date practical engineering issues have precluded the production of crystal detectors that combine high efficiency, large solid angles and the flexibility to cover a wide energy range.

As mentioned, superconducting detectors are another promising avenue for development. They already reach the desired energy resolution. The issue is improving the count rate. To date, the emphasis in signal processing has been the ultimate energy resolution. Potentially, the count rate can be improved by optimizing the signal processing for somewhat poorer energy resolution. Multiplexing technology is also improving, potentially allowing of the order of 1000 detector elements. However, combining both methods is unlikely to give total count rates of much greater than 106. They will almost certainly need to be combined with filters or crystal analyzer front ends to make the total count rate tractable.


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