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

International Tables for Crystallography (2022). Vol. I. Early view chapter
https://doi.org/10.1107/S1574870721006844

X-ray focusing methods for X-ray absorption spectroscopy

Alexandre S. Simionovici,a* Laurence Lemelleb and Christian G. Schroerc,d

aInstitut des Sciences de la Terre, UGA/CNRS, Observatoire des Sciences de l'Univers de Grenoble, CS 40700, Grenoble, France,bLGL–TPE, ENS de Lyon, Université Claude Bernard/CNRS, 46 Allée d'Italie, 69364 Lyon, France,cDESY Photon Science, Notkestrasse 85, 22607 Hamburg, Germany, and dFachbereit Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
Correspondence e-mail:  alexandre.simionovici@univ-grenoble-alpes.fr

X-ray focusing methods for X-ray absorption spectroscopy (XAS) are an essential but controversial endeavour, as XAS was devised as a way to average out chemical environment effects over large numbers of atoms. However, focusing becomes necessary when one attempts to study speciation over essential but minute spatial phases, extrapolating the method towards low numbers of atoms. Following progress with X-ray sources and detectors, focusing tools completed the picture, allowing the extraction of seminal speciation results from phases of a few micrometres down to a few tens of nanometres. To do so, one must use high-intensity focused X-ray beams such as those produced by third- and fourth-generation synchrotron sources and track their spatial stability in time to fractions of their footprint, but also delve into sample-damage constraints, as required by the high flux densities employed. The section addresses the various sources of focused beam instabilities and the most frequently used focusing devices for XAS.

Keywords: focusing; Kirkpatrick–Baez mirrors; zone plates; capillaries.

1. Introduction

Ever since the `founding fathers' of the X-ray absorption spectroscopy (XAS) field, E. A. Stern, D. E. Sayers and F. W. Lytle, laid out the main lines of this revolutionary characterization method (Sayers et al., 1971[link]; Stern, 1974[link]; Lytle et al., 1975[link]; Stern et al., 1975[link]), it has been one of the preferred analytical methods for solving complex structures, with an ever-increasing precision, complexity and span of applications. Hereafter, we will use the term X-ray absorption fine structure (XAFS) to address the way that an atom absorbs the incident X-rays at or near the binding energy of a core level. XAFS covers X-ray absorption near-edge structure (XANES), the 50–100 eV region above the core-level ionization energy, and extended X-ray absorption fine structure (EXAFS), the continuing region up to 500–1000 eV. This specific behaviour is a function of its physical/chemical state, such as for example in an environment with various atomic neighbours, in a magnetic field or at a certain temperature. XAS requires the probing and averaging out of the energy-dependent signal over a large ensemble of atoms.

XAFS was initially performed using X-ray sources in the laboratory, using a double-crystal spectrometer mounted inside a diffractometer, as shown by F. Lytle in 1963 (Lytle, 1963[link]). In the years following the inception of the modern era of XAFS, several synchrotron-radiation sources (SRS) fielded dedicated endstations and even beamlines, and the field quickly advanced, as witnessed by the impressive number of studies performed. The first XAFS synchrotron beamline was installed by Stern, Sayers, Lytle, Eisenberger, Kincaid and Hunter on the SPEAR ring at Stanford University (Eisenberger et al., 1974[link]). Today, with at least 40 operational synchrotron-radiation sources, XAFS has been a success story in the history of modern science, as testified by the development of IXAS, the International X-ray Absorption Society (http://www.ixasportal.net/ixas/ ). The synchrotron-radiation sources operating today are second-, third- or fourth-generation machines optimized for the generation of synchrotron light, mainly in the IR, UV, VUV and X-ray regimes of the electromagnetic spectrum. For a general overview of the field, check the http://www.lightsources.org/ website and download the synchrotron user's guide at http://xdb.lbl.gov/ . Suffice to say that XAFS is nowadays routinely performed on many beamlines at the more than 40 operational SRS worldwide.

We are continuously reducing the probed sample volumes to account for ever smaller structural patterns in materials, Earth and planetary, environmental and life sciences. Extracting information from minute sample regions implies decreasing the number of atoms in the beam, and this can be performed by focusing the beam. Using focusing devices, one reduces the sample volume impinged on by the beam by orders of magnitude compared with the traditional unfocused mode and must ensure that the stability of the beam position on the sample is compatible with its resolution; this means that during XAFS energy scans, the beam position should not vary by more than a fraction of the beamsize. This is a key requirement which affects the whole measurement and we will thoroughly address it here. Currently, at several third-generation SRS the beamspots in the 1–20 keV energy range are in the submicrometre range, down to several tens of nanometres. In order to extract meaningful results from such small spots, the thickness of these samples cannot be much larger than the spatial resolution of the probing beam, otherwise indefinite results, obtained from scanning several different features superimposed in the depth of the sample, will always not be resolved. As the sample thickness is confined to micrometres or tens of nanometres, the conditions for using the transmission mode of detection of XAFS are quite far from the optimal conditions,[\mu (E) \cdot t = - \ln \left({{I \over {{I_0}}}} \right) \simeq 1, \eqno(1)]where μ is the absorption coefficient, t is the sample thickness and I0 and I are the beam intensities before and after the sample, respectively. Therefore, we will mainly address the fluorescence collection mode, which allows the collection of XAFS from both very dilute and highly concentrated samples. The statistics associated with these measurements are simply a function of the number of atoms probed by the focused beam and the fluence of the beam (the product of the beam intensity and the acquisition time). Beam intensities can sometimes be increased in specific focusing setups at modern SRS, but this does not necessarily imply better statistics, as it is controlled by the acquisition time inasmuch as no sample damage is produced.

This section presents the fact that focused XAFS is at variance with the requirement for averaging out over a large ensemble of atoms for XAS, but that we can however extract pertinent results with poorer statistics from micro/nano-samples. It explains why focusing the beam makes the beam-plus-sample system dependent on the spatial stability during the measurement and any variations of either the beam parameters or the sample position translate into measuring outside the zone aimed for and producing `noise' or, even worse, ambiguous signal. This is the crux of the problem: once our `rifle' (the synchrotron, monochromator and focusing device) is primed, the `bullet' (a photon of optimal energy/intensity/polarization) is selected and our `hunting grounds' (sample environment: pressure, temperature, magnetic field etc. and optimized detector system) are prepared, one must aim for the right `game' (sample), which must be hunted (analysed) in an efficient, nondestructive and reproducible way. We will succinctly address the sources of noise, then move on to specific focusing devices in order to produce an overview of the field, allowing new or infrequent users of XAFS to better select their rifle, bullet and hunting grounds in order to have a fruitful hunt (not being actual hunters, we will not delve further into this comparison).

2. Sources of XAFS instability

The main parameters affecting the performance of an XAFS focused-beam experiment are the storage-ring lattice with its electron-beam parameters and respective X-ray sources (bending magnets or insertion devices), an eventual mirror (for higher energy photon rejection and to lower the thermal downstream load), the monochromator energy-selecting device, the focusing device itself and the sample environment comprising the sample-holder drives (for scanning a sample in front of the beam), the monitors (for measuring the intensity of the beam before and after the sample) and the specific detectors for the three XAFS regimes specified above.

To summarize, they can be classified as

  • (i) electron-beam parameters and stability due to storage-ring fluctuations and energy scans,

  • (ii) specific X-ray sources (hereafter only high-intensity insertion devices will be surveyed) and stability of the associated beamline optics (slits, harmonics rejection mirrors etc.) due to thermal and mechanical fluctuations when performing energy scans,

  • (iii) X-ray beam parameters and stability due to monochromator energy scans,

  • (iv) X-ray beam parameters and stability due to the focusing device,

  • (v) sample heterogeneity and stability.

2.1. Storage-ring stability

The storage rings of operational synchrotron-radiation sources span four decades of technological advances and thus are very different, but one main parameter describes their performance: the charged particle beam emittance. Modern storage rings operate with either electrons (the majority) or positrons, so hereafter the term `electron' will exclusively be used when addressing the charged particle beams. The emittance is an intrinsic property of a charged particle beam and is a measure of the spread of the beam in position and momentum space. In the storage rings of SRS, emittance is conserved, following Liouville's theorem in Hamiltonian mechanics dealing with trajectories in multi-parameter phase space. Thus, at any point in the storage ring the electron-beam transverse size times its angular divergence will be invariant as a function of the energy. This means that one can only reduce the diameter of the beam by increasing its divergence, and conversely. The emittance is measured in metres × radians and its unit is metres. At the European Synchrotron Radiation Facility (ESRF) in Grenoble, France the emittance figures were 4 nm horizontal and 4 pm vertical, while after the improvement arising from the Extra Brilliant Source (EBS) ring upgrade in 2018–2020 these values were greatly reduced to 110 pm horizontal by 5 pm vertical, making it in effect the first fourth-generation light source.

The standard re-injection mode at some SRS is sometimes accompanied by a total X-ray beam stop, as the re-injection takes a few minutes and is performed with the front ends closed. For all downstream beamline components (slits, mirrors and monochromators) which require cooling when taking the beam, a great deal of instability occurs when the X-ray beams are switched back on and the optical components experience temperature gradients. This is a major cause of problems in long scans such as X-ray fluorescence (XRF) or EXAFS scans when attempting to go through the refill and, without exception, the position and angular divergence of the beam after re-injection, whether through a focusing system or direct, is changed, thus forfeiting the continuation of a long XAFS scan. In Fig. 1[link] three popular re-injection modes are presented, with their inherent instabilities. The standard 12 h period refill with the front end open (continuous refill) ensures that the critical components are kept under the X-ray beam at all times, but the period immediately after re-injection is fairly unstable in intensity and in electron-beam position and divergence. Downstream devices which critically depend on the beam properties are obviously affected by these instabilities and pass them on to the measuring system. The continuous re-injection mode keeps the beam components under X-rays during re-injection, which removes a first-order instability effect. However, during the 12 h period the beam intensity decreases, depending on the storage-ring packet mode (single bunch, few packets or multibunch), to 15–30% in some cases, which necessarily induces thermal gradient walks on sensitive downstream devices. In order to keep the injected current at a maximum, decrease the vertical beam emittance and reduce beam instabilities, the `top-up' re-injection mode was adopted at the ESRF in April 2016, with a period of approximately 20 min, which changed to one hour after the EBS upgrade.

[Figure 1]

Figure 1

Synchrotron re-injection modes at the ESRF.

Instabilities still occur immediately after re-injection, but these may gradually be removed by an optimized lattice tuning mode.

2.2. Insertion-device energy scans

Insertion devices, unlike bending magnets, deliver a non­continuous spectrum of photon lines as a function of the energy. Undulators emit a spectrum of harmonics, which are integer multiples of the first harmonic (fundamental) of lowest energy. One chooses a specific harmonic of energy E with a bandwidth of 1–2% ΔE/E, which is then passed downstream to the optical elements and to the monochromator. In order to perform an XAFS energy scan, several strategies of scanning the undulator harmonic energy are used, as presented in Fig. 2[link].

[Figure 2]

Figure 2

Undulator energy scan by gap opening.

In order to match the energy of the monochromator to the maximum of the nth undulator harmonic, the gap of the undulator must be opened up, lowering the on-axis magnetic field and shifting the position of the harmonic towards higher energies. Thus, the monochromator bandwidth (in green in Fig. 2[link]) always samples the maximum of the harmonic, which slightly decreases over a XANES scan of, for example, 100 eV. The undulator gap can either be opened at every energy step, which is time-consuming, or every few eV, which is sometimes preferred. Alternatively, the undulator can be scanned by setting the gap motors in a continuous mode matched to the monochromator motors, and the XAFS signal is recorded `on the fly'. The simplest operation can be performed at a fixed undulator gap, thus having the monochromator scan only the width of the harmonic, which is frequently sufficient for a XANES scan (100 eV).

Some undulator magnetic jaws can be moved differentially from one end of the device to the other, producing a variable gap opening throughout the undulator length. This mode is called `taper' and allows broadening of the harmonics, albeit with a moderate loss in intensity. In this way, the width of the harmonic is increased by a factor of 50–100%, allowing a 300–400 eV XAFS scan by simply scanning the monochromator, as shown in Fig. 3[link].

[Figure 3]

Figure 3

Undulator tapering effect.

2.3. Monochromator stability

A monochromator is a critical device for XAFS measurements. It selects (monochromatizes) the beam energy by reflection from one or more crystals, to a value E ± ΔE (keV). The well known Bragg law governs the reflectivity of the crystals as a function of the crystal materials and the angle of incidence of the primary X-ray beam on the crystals,[2d\,({\rm \AA}) \cdot \sin (\theta) = n \cdot \lambda\, ({\rm \AA}) = n \cdot {{12.398} \over {E\,{\rm (keV)}}} \eqno(2)]where d is the crystal lattice spacing, θ is the angle of incidence of X-rays on the lattice planes, λ is the beam wavelength, n is the diffraction order and E is the beam energy. Most frequently, a system of two parallel crystals is used in the `fixed-exit' mode in order to ensure the exit of the monochromatic beam at the same height regardless of the energy.

For an XAFS scan one must span the range of angles between θ1 and θ2, a total angular range of Δθ. At the same time, in order to follow the `fixed-exit' requirement, the position of the second crystal must be gradually translated by Δx to intercept the beam as a function of the Bragg angle. These two movements are sometimes accompanied by a few other movements in the crystal tilt, fine piezodriver rotations etc. for optimal tuning. The result of these movements is visualized in Fig. 4[link] as an overall deviation from the desired position by the quantities ɛθ and ɛx in angular divergence and position, respectively. These deviations are generally monotonic, accompanying the rotation of the Bragg axis, which explains why the monochromator XAFS energy (angular) scans are preferably performed in the same direction to avoid backlash. Suffice to say that these deviations in both angle and vertical position will be passed onto the focusing device downstream. In practice, the utmost care must be exercised to quantify these displacements and a compensation curve correcting for them should be implemented for all energy scans, which are usually reproducible.

[Figure 4]

Figure 4

The origin of fixed-exit monochromator instabilities.

Many synchrotron beamlines nowadays use a diaphragm-like device such as a pinhole, a double-slit system or even a single refractive lens placed downstream from the monochromator to define a smaller aperture source. This then becomes a secondary source, imaged by the subsequent focusing device into a smaller focused beam. This is obviously performed at the expense of beam intensity, but it has the beneficial effect of reducing the beam-position deviation ɛx after passing through this diaphragm.

2.4. Focusing-device stability

All of the focusing devices treated hereafter have a vertical × horizontal acceptance which is smaller than the size of the pre-focused beams (≥0.5 mm) produced by any insertion device. Thus, they are less sensitive to beam-position movement as long as it is within their reduced incident acceptance (0.1–0.3 mm). The angular divergence of the beam, however, seriously affects the position of the focused beam on the sample. Focusing devices are either `inline', where the focused beam is produced in the same direction as the incident beam, or `off-axis', where the beam is deviated by reflection or refraction. The angular deviations of the beam will translate after off-axis focusing into variations of the position of the beam, and this is the major source of instability when performing an XAFS scan. The respective sensitivities to the inline and off-axis types of beam deviations introduced above will be treated separately for each focusing geometry.

2.5. Sample-environment stability

When performing XAFS scans the samples are kept at fixed positions with respect to the beam, but to perform X-ray imaging vertical/horizontal `step-by-step' or `continuous' movements of the sample are executed. The `static' stability of a setup depends on outside parameters such as the ground vibrations at the beamline location, the experimental hutch temperature and humidity, with their specific frequencies, which must be assessed before operation by means of accelerometer or interferometer devices. The `dynamic' stability of moving devices and its repeatability must then be calibrated. The precision and stability of the sample stages are the key parameters here, and currently setups using traditional stepper motors with micrometre resolutions are being replaced by piezoelectric actuators capable of nanometre resolutions. For `on-the-fly' continuous movement scanning, the specific range of speeds achievable and the response times of each drive must be compatible with the required acquisition mode, and the choice of equipment is quite ample here, allowing high-performance setups.

At dedicated state-of-the-art XAFS beamlines, voltage-to-frequency fast electrometers monitor the effective beam instabilities prior to/during each scan, and the acquisition times are tailored in order to avoid local noise frequencies introduced by the synchrotron radiofrequency injection (7, 13 Hz) or by concrete slab vibrations (1–10 Hz). In this way, dwell times on the sample are optimized and XAFS signals, which are necessarily frequency-driven, are greatly improved.

3. Focusing devices

After a long period at the beginning of the 20th century when X-rays were considered to be nearly impossible to focus, even by scientists such as Röntgen, a new era started in the 1990s with the first application of diffractive optics focusing at both the LURE and ESRF SRS (Chevallier et al., 1995[link], 1996[link]). Successively, a series of optics developments by the group led by Aristov at IMT-RAS, Chernogolovka, Russia yielded Bragg–Fresnel lenses and Fresnel zone plates, which achieved micrometre focusing at several SRS beamlines (Erko et al., 1996[link]).

SRS deliver high-flux, tunable energy X-ray beams, which can be used to probe the K, L or M atomic edges of most elements in one of the standard collection modes: transmission, fluorescence and electron yield. The high flux available for experiments in a naturally pre-collimated beam, such as those of insertion devices on synchrotron beamlines, allowed new types of experiments, namely those using highly focused beams. All focusing devices operate under similar conditions, imaging the incident X-ray source onto the sample in a strongly demagnifying geometry. The smaller the source size, the smaller the focused beam spot should be. All X-ray sources in operation, bending magnets and insertion devices (wigglers and undulators), can be demagnified by an optical focusing device, but the most efficient sources, benefitting from both small source sizes and natural collimation that does not require beam losses using slits, are the undulators, which provide the highest fluxes in the smallest beam spots. During the past 30 years, the synchrotron community has moved from beams of a few millimeters to a few micrometres and then to a few nanometres. The point-focusing devices were successively simple crossed slits, Bragg–Fresnel lenses, Fresnel–Laue lenses, capillaries, Fresnel zone plates (FZPs), waveguides, compound refractive lenses (CRLs) and Kirkpatrick–Baez (K-B) double-mirror devices. Other devices have recently appeared producing line foci, such as multilayer Laue lenses (MLLs), but in principle these are the main types of focusing devices that are still used today. Hereafter, we will focus on the following devices: FZPs, MLLs, CRLs, capillaries and K-B mirrors.

In the following, we will treat focusing devices based on a crucial criterion for XAFS, which is their chromatic properties, i.e. their operation as a function of the incident beam energy. They can be grouped into chromatic devices, for which the position of the focal spot changes with the energy of the incoming photon, and achromatic devices, which are independent of the energy.

3.1. Chromatic devices

These are the most frequently used devices nowadays, as they are monolithic, inline focusing devices that are easy to align and to operate. Their chromaticity, however, is a strong drawback, as the position of their focal spot changes over the range of a few millimetres to a few centimetres for an XAFS energy scan. The lens must therefore be moved longitudinally to compensate for the chromatic shift (Cotte et al., 2007[link]; focus tracking), and the sample (spot tracking) is laterally moved in the focal plane to compensate for side deviations. These movements are a source of errors and instabilities. Mechanically, it is next to impossible to move objects by several millimeters while conserving the position of the focused beam within a beamspot of a few micrometres or even tens of nanometres.

3.2. Fresnel zone plates

Fresnel zone plates (FZPs) are diffractive devices produced by electron beam lithography composed of a large number of alternating X-ray-opaque l-phase-shifting and transparent radial zones. FZPs are circular diffraction gratings of increasing zone densities as a function of the radius, working in transmission. For a rigorous description of FZP optics and their properties, see Attwood (1999[link]) and Michette (1986[link]). Using the notation of Attwood (1999[link]), one can model an FZP using only a few parameters.

In imaging mode (Fig. 5[link]), an FZP lens demagnifies the image of a monochromatic source of X-ray wavelength λ located at a distance p into a point P at a distance q using the well known thin-lens formula[{1 \over f} = {1 \over p} + {1 \over q}, \eqno(3)]where f the focal length of the lens in the first order of diffraction, given by[f \simeq {{4N(\Delta r)^{2}}\over {\lambda}}. \eqno(4)]where N is the number of zones of increasing radii but equal surfaces and Δr is the width of the last zone. Alternatively, an online FZP calculation site is available, thanks to J.  Vila-Comamala (https://sites.google.com/site/vilacomamala/fresnel-zone-plate-calculator ), allowing the calculation of focal distances, depths of focus and numbers of zones. Using these simple formulae, one can calculate focal distances for higher (weaker) diffraction orders as[f_m = {{f}\over{m}}. \eqno (5)]

[Figure 5]

Figure 5

Fresnel zone plate in imaging mode.

These diffraction orders produce weak but large-diameter spots at the desired first-order focal spot where the sample is located, as in Fig. 6[link]. A small-diameter order-sorting aperture (OSA) is routinely used in conjunction with a central beamstop to partially block the higher orders.

[Figure 6]

Figure 6

(a) Fresnel zone plate focusing orders; (b) focused X-rays in the sample plane.

Without the central beamstop, when scanning heterogenuous samples featuring hotspots of high concentrations of elements, the third and fifth orders of diffraction produce weak, large surface backgrounds far from the first-order focused spot size of a few tens or hundreds of nanometres, which can be seen as wings of several micrometres in width. The knife-edge method is used to estimate the beam sizes, which consists of scanning a highly absorbing or fluorescing polished straight edge of a sample across the beam and plotting the transmitted or fluorescent intensity. As shown in Fig. 7[link], using the first derivative of this intensity as a function of the vertical/horizontal position in the focal spot, the full-width half-maximum (FWHM) of the beam and the horizontal/vertical lateral position of the beamspot in the focal plane are obtained. As a faster method and for beamspots of a few micrometres a CCD camera is used, which expands the micrometric beamspot to a larger and easier to measure spot and then uses center of mass (COM) image processing to directly extract the beamspot position. If no particular corrections are performed, and the undulator harmonic is centred on the K edge energy of, for example, iron, the beam-position displacement stays within a 1–2 µm range for a 100 eV energy scan (Bonnin-Mosbah et al., 2001[link]) while the beamspot FWHM increases, consequently decreasing the flux on the sample by up to 20%. These variations are directly dependent on the FZP parameters.

[Figure 7]

Figure 7

Knife-edge method of estimating focus lateral position and FWHM.

As an example, using a tungsten FZP from ZonePlates Ltd, UK with 600 zones, D = 240 µm and Δr = 100 nm, scanning the energy at the Fe K edge between 7.05 and 7.35 keV in 100 steps of 0.3 eV produces a shift in the focal position of 5.8 mm for a depth of focus of about 113 µm. For the Si(111) monochromator, this scan corresponds to an angular range of 0.69° (15.6–16.29°). Consequently, one must compensate for this shift by simultaneously moving the FZP and the θ angle of the monochromator, so-called active `focus tracking'. As previously stated, moving the FZP by 5800 µm while conserving a lateral focal position to within 0.05 µm is impossible, as it requires an FZP translation stage of unusual rigidity and precision over a large span of movement.

In Fig. 8[link], we show (i) the focused beam position in the focal plane during an energy scan between 7.05 and 7.35 keV with active `focus tracking' of the FZP, using a camera in the focal plane in beam-position monitor mode. The beam moves within an envelope of 1 µm in the horizontal direction and 2 µm in the vertical direction, which is larger than the spot size of 0.3 × 0.7 µm (vertical × horizontal). This displacement results from the combined contribution of the monochromator angular scan (vertical axis mainly) and the translation of the FZP along the beam axis (horizontal axis). To decouple these effects, we repeated the scan (ii) with the FZP fixed along a beam axis optimized for 7.2 keV and (iii) at a 7.2 keV fixed energy, with the FZP moving along the beam axis as in `focus tracking' mode. These scans confirm that the main source of focused beam displacements is the translation of the FZP along the beam axis. Consequently, the parallelism of the translation with respect to the beam direction and its angular errors (roll, pitch and yaw) are critical parameters which limit the XAFS scan precision.

[Figure 8]

Figure 8

FZP focus position (µm) in the sample plane for a scan between 7.05 and 7.35 keV.

Fortunately, these relatively small mechanical contributions (1–2 µm) are reproducible and can be calibrated in a lookup table as a function of the energy. The strategy consists of compensating for these effects by shifting the sample in the plane perpendicular to the beam axis to `follow' the focused beam movement, the so-called `spot-tracking' mode, which provides an improved stability for XAFS measurements on small features in the beam resolution range. Obviously, all of these beam displacements are dependent on the beam energy and the energy scan width, so lower edge energies and reduced scan widths as for XANES (ΔE ≤ 100 eV) produce the best results.

As an example of XAFS scans performed on submicrometre diluted samples, in Fig. 9[link] we present S K-edge XANES scans performed on the NASA Stardust mission samples from the Wild 2 comet (Flynn et al., 2006[link]; Zolensky et al., 2006[link]) to extract the valence states of sulfur, an important element in the mineralogy of extraterrestrial samples. Here, high-velocity grains travelling at about 6 km s−1 were collected in the vicinity of Mars in a very low density (0.05 g cm−3) aerogel (SiO2) collector. The ID21 beamline of the ESRF delivered a 300 nm focused beam for XAFS at the S K edge in spot-tracking mode, which allowed precise scans of the solid terminal grain (diameter of ≤2 µm) within the larger cone of the cometary grain dispersed in the aerogel capture medium.

[Figure 9]

Figure 9

(a) XRF map of the sulfur distribution in a stardust cometary grain; (b) sulfur oxidation states of the track entry (sulfate) and terminal particle (sulfide).

The first-order focusing efficiency of FZPs reaches 40% and decreases at higher energy, but a new way of increasing the efficiency has been demonstrated by Di Fabrizio et al. (1999[link]). Using multilevel blazed zones, which are much more difficult to manufacture, they reached 56% efficiency with a four-level nickel FZP at 7 keV and opened the possibility of reaching 90% efficiencies at higher energies for a 40 nm focal spot quaternary FZP (Di Fabrizio et al., 2001[link]).

The diffraction-limited focus size of FZPs is roughly given by the width of the outermost zone (Attwood, 1999[link]). Reducing the outermost zone width to reduce the X-ray spot size and thus increasing the spatial resolution is increasingly challenging, as zone structures with smaller and smaller zone widths but constant height have to be fabricated.

3.3. Multilayer Laue lenses

One way to overcome the limitations of current lithographic fabrication techniques is to generate the Fresnel zones in a multilayer stack and then cut the lens structure out of the multilayer by focused ion-beam milling (Kang et al., 2006[link]; Yan et al., 2010[link], 2014[link]). The diffractive optics obtained in this way focus the beam in one dimension, and two such optics need to be aligned in crossed geometry to generate a point focus. These so-called multilayer Laue lenses (MLLs) have the same working principle as FZPs; however, much finer zone structures and extreme aspect ratios can be obtained in this way (Kang et al., 2006[link], 2008[link]; Yan et al., 2010[link], 2014[link]). As the zone structures reach the single-digit nanometre range they become optically thick, requiring tilted zones that locally fulfil the Bragg condition for the given zone structures (Kang et al., 2006[link]; Yan et al., 2010[link], 2014[link]). Ideal MLLs can in principle focus X-rays to subnanometre dimensions (Schroer, 2006[link]; Yan et al., 2007[link]). However, at present their performance is still limited by the fabrication technology. The smallest beam size reached so far was 8 nm (Morgan et al., 2015[link]). As these optics are chromatic (in the same way as FZPs), scanning the energy for local XAS experiments is very challenging, as both crossed optics have to be aligned individually with respect to the sample and the depth of focus is very small for these small beams.

3.4. Compound refractive lenses

Refractive lenses for hard X-rays were first introduced in the mid-1990s (Tomie, 2010[link]; Snigirev et al., 1996[link]) and are now widespread as inline optics for nanofocusing at SRS (Schroer et al., 2005[link]) and X-ray free-electron lasers (Schropp et al., 2013[link]). They are suitable to focus X-rays with energies in the range from 5 keV to well above 100 keV. In refractive lenses, the weak refraction of hard X-rays in matter is compensated by a large number of strongly curved individual lenses stacked behind each other (Snigirev et al., 1996[link]; Lengeler et al., 1999[link]). As the refractive index is slightly smaller than 1, the individual lenses need to be concave to achieve focusing. In order to minimize the attenuation inside the lens, the lens material should be as X-ray transparent as possible, i.e. made of elements with low atomic number (Lengeler et al., 1999[link]), such as beryllium (Schroer et al., 2002[link]), polymers (Nazmov et al., 2004[link]) or silicon (Schroer, Kuhlmann, Hunger et al., 2003[link]). The strong curvature of the individual lenses implies an aspherical lens shape in order to avoid spherical aberrations. For most lens designs, the individual lenses are optically thin, requiring a parabolic shape (Lengeler et al., 1999[link]). Only for single lenses with a thickness comparable to their focal length (Schroer et al., 2002[link]) is the deviation of the asphere from the parabola significant. Parameters for and properties of refractive X-ray lenses and the microbeams and nanobeams generated by them can be calculated using several online tools (http://crl.desy.de/crlcalc/ , https://www.rxoptics.de/design-parameters/ ).

The focal length of refractive lenses f is given in the thin lens approximation by (Lengeler et al., 1999[link])[f = {R \over {2N\delta}}, \eqno(6)]where R is the radius of curvature of the lens surfaces, N is the number of individual lenses in the compound lens and δ is the decrement of the index of refraction for the hard X-rays in the lens material1. The refractive-index decrement[\delta = {{{N_{\rm A}}} \over {2\pi}}{r_0}\rho {\lambda^2}{{Z + f'(E)} \over A} \eqno(7)]depends quadratically on the wavelength λ of the X-rays, introducing a significant chromaticity. Here, NA is Avogadro's number, r0 is the classical electron radius, ρ is the mass density, Z + f′(E) is the real part of the atomic form factor of the lens material in the forward direction and A is the atomic mass of the lens material. To date, the smallest hard X-ray beams generated with refractive lenses lie in the region of 50 nm (Schroer et al., 2005[link], 2011[link], 2013[link]).

A change in X-ray energy ΔE results in a change in focal length Δf of about[{{\Delta f}\over f} = 2{{\Delta E}\over {E}}. \eqno (8)]

A detailed analysis of the chromatic effects of thick refractive lenses is given in Seiboth et al. (2014[link]). Compared with an FZP, the focal length of which scales only with 1/λ, compound refractive X-ray lenses are twice as chromatic, with the focal length scaling with 1/λ2 (see equation 6[link] and the λ dependence of δ in equation 7[link]). This explains the factor of two in equation (8[link]). Kinoform (Aristov, Grigoriev, Kuznetsov, Shabelnikov, Yunkin, Weitkamp et al., 2000[link]; Evans-Lutterodt et al., 2003[link]) and adiabatically focusing refractive X-ray lenses (Schroer & Lengeler, 2005[link]) are not well suited for micro-XAS experiments, as they are designed for a specific energy and show strong aberrations away from the design energy. In X-ray scanning microscopy, the X-ray source is imaged onto the sample position in a strongly demagnifying geometry. This implies that the focal length f is much smaller than the source-to-lens distance L1 (fL1) and that the image (lens-to-sample) distance L2 is slightly larger than f. In this case,[{{\Delta {L_2}} \over {{L_2}}} \simeq {{\Delta f} \over f}. \eqno(9)]

When ΔL2 exceeds the depth of focus of the nanofocused beam, the resolution of a scanning XAS experiment is affected. The longitudinal depth of focus dl for a diffraction-limited X-ray nanobeam created by refractive lenses is given by [d_{\rm l} = \alpha {\lambda \over {2{{({\rm NA})}^2}}} = {{{d_{\rm t}}} \over {\rm NA}}, \eqno(10)]where α ≃ 0.752, NA is the numerical aperture of the optics in the given geometry, λ is the wavelength of the X-rays and dt is the transverse size of the Airy disc (Nazmov et al., 2004[link]; Schroer et al., 2013[link]; Aristov, Grigoriev, Kuznetsov, Shabelnikov, Yunkin, Hoffmann et al., 2000[link]). Due to the parabolic shape of refractive X-ray lenses, the aperture function of this optics is a truncated Gaussian, resulting in an Airy disc shape that is also nearly Gaussian (Schroer et al., 2013[link]).

For scanning XAS experiments, the X-ray energy is varied in an interval ΔE around the absorption edge of the element of interest. The scanning microscope based on refractive X-ray lenses should be aligned such that the specimen is in focus for the energy at the centre of the scanned energy interval. If dlΔL2, the focal spot size is nearly unaffected by the energy scan, despite the chromaticity. Solving this inequality for dl, using equations (8)[link], (9)[link] and (10)[link], yields a lower limit of[d_{\rm l} \ge \alpha \cdot \lambda \cdot {L_2} \cdot {{\Delta E} \over E}. \eqno(11)]

For a given energy scan and focusing geometry, this equation can be used to estimate the best resolution achievable in scanning microscopy with refractive lenses.

As an example, for a XANES scan ΔE = 80 eV around the Cu K edge (E = 8995 eV) with an image distance of L2 = 300 mm, the spatial resolution cannot be better than about half a micrometre. For nanofocusing lenses (Schroer et al., 2005[link]; Nazmov et al., 2004[link]), which typically have much shorter focal lengths, for example L2 = 30 mm, the spatial resolution is limited to 170 nm for the above XANES scan. Chromaticity of the lenses becomes more pronounced for EXAFS scans. For an EXAFS scan with a range of ΔE = 1000 eV at the Cu K edge at an image distance of L2 = 300 mm the spatial resolution is limited to about 2 µm, and to about half a micrometre for L2 = 30 mm. If a higher spatial resolution is to be achieved in a scanning XAS microscopy experiment with refractive lenses, the optics-to-sample distance needs to be adapted as a function of energy.

Refractive X-ray optics have been used in scanning microscopy with XANES contrast (Schroer, Kuhlmann, Günzler et al., 2003[link]). In this experiment, a catalytic microreactor (a reactor capillary containing a CuO/ZnO catalyst) was scanned in tomographic mode (Fig. 10[link]a), recording a full XANES spectrum at the Cu K edge at each position of the scan using a fast scanning monochromator. For each X-ray energy in the XANES spectrum, a slice through the microreactor was reconstructed (Fig. 10[link]b), yielding a full XANES spectrum at each position in the reconstruction (Fig. 10[link]c). A set of reference spectra were fitted to the local spectra, revealing the concentrations of copper in different oxidation states (Fig. 10[link]d). For the microbeam in this experiment, with a lateral size of about 10 µm, the depth of focus was larger than the shifts ΔL2 in the focal position due to the chromaticity of the optics.

[Figure 10]

Figure 10

(a) Scheme of the capillary microreactor imaged by XANES tomography. (b) Tomographic reconstruction of a slice through the capillary at a given X-ray energy. Similar reconstructions are available for each energy in the XANES spectrum. Therefore, a full XANES spectrum is reconstructed at each position in the reconstruction. (d) Concentrations of copper in different oxidation states as obtained from fitting reference spectra to the XANES tomographic reconstruction.

Scanning XAS micro-imaging based on refractive X-ray lenses has not been widely used so far. A significant limitation is the chromaticity of the optics, which either limits the spatial resolution or will require tracking of the focal plane with changing energy. With increasing spatial resolution, smaller features will be able to be resolved, such as catalyst nanoparticles (Grunwaldt & Schroer, 2010[link]). However, these smaller features exhibit a small absorption contrast, making it difficult to record the XAS signal in transmission mode. In this case, fluorescence detection will become more important (Schroer et al., 2010[link]). Alternatively, modern scanning coherent X-ray diffraction imaging techniques can exploit the resonant scattering signal (Beckers et al., 2011[link]; Hoppe et al., 2013[link]) to obtain chemical information about the sample by measuring the real part f′(E) of the anomalous atomic form factor. A major advantage of this imaging technique is that chromatic aberrations of the optics are automatically corrected for by ptychographic reconstruction (Hoppe et al., 2013[link]).

3.5. Achromatic devices

These devices are those allowing the easiest and most stable XAFS scans with focused beams. The position of their focal spots does not depend on the energy so they are fixed devices, minimizing beamspot displacements. The monochromators located upstream of the achromatic devices will be the major source of instabilities.

Capillaries are inline devices that will produce larger beamspots than K-B mirrors and their focal distances will be quite small to achieve high resolutions. This is a limitation for setups requiring a specific sample environment (cryostats, ovens, high-pressure cells, magnetic fields etc.). As they have a relatively small acceptance, their output intensity will also be smaller than that of K-B mirrors. The efficiencies of both devices are high, as they rely on glancing incidence reflection, at up to 90%.

4. Capillaries

Pioneered in the late 1980s by the group of M. A. Kumakhov at the Institute of Röntgenoptics in Moscow, Russia (Kumakhov & Komarov, 1990[link]), capillary concentrators and lenses are achromatic devices based on one or more total external reflections of X-rays under grazing-incidence conditions, as is customary for X-rays, inside hollow glass tubes.

Polycapillaries are made of numerous internal tubes, guiding the X-rays by multiple reflections off their walls. They are recommended for moderate focusing setups (Kumakhov, 2000[link]) and they exhibit the largest focused intensities, albeit for larger focal spots and larger focal distances. If a single tube is used, one can achieve high-resolution beamspots by single reflections inside the tube, at the detriment of the focused intensity. Indeed, monocapillaries have low acceptance; their aperture is a few tens of micrometres, thus requiring pre-collimated beams such as those delivered by synchrotron insertion devices to achieve usable intensities. They focus the X-rays either by a single-bounce reflection or by multiple-bounce reflections. Single-bounce monocapillaries can be ellipsoidally shaped for high focusing but with a focal distance that is moderate, while multiple-bounce capillaries are tapered for the highest achievable resolution and smallest focusing distance (Bilderback et al., 1994[link]). Ellipsoidal-shaped capillaries are recommended for X-ray imaging, with the X-ray source placed at one of their foci and the sample at the other (Bilderback, 2003[link]).

All are highly efficient optics (20–90%; Huang et al., 2014[link]; MacDonald, 2010[link]) based on X-ray reflection, which can be modelled using the formula of total reflection (Wang et al., 1996[link]) at critical grazing incidence θc:[\theta_{\rm c}\,({\rm rad}) \cong {{30\,({\rm keV})} \over {{E_{\rm c}}\,({\rm keV})}}. \eqno(12)]

For simple calculations of physical parameters, one is directed to the Cornell group website (http://glasscalc.chess.cornell.edu/imageprof.shtml ) or to the papers by Vincze et al. (1995[link]) and Chen et al. (1994[link]). Several companies or groups manufacture either monocapillaries or polycapillaries for use on synchrotron beamlines or with X-ray sources in the laboratory: X-Ray Optical Systems, Horiba, Australian X-ray Capillary Optics Pty Ltd and the Cornell CHESS group mentioned previously. Table 1[link] presents the main parameters of the three capillary types (Fig. 11[link]) to guide the users in their feasibility choices.

Table 1
Guide for capillary choices

OpticsBeamspot (µm)Focal distance (mm)Divergence (mrad)GainComments
Single-bounce monocapillary 0.2–100 0.2–150 1–10 1–103 Small aperture/flux
Multi-bounce monocapillary 0.1–50 0.001–0.01 1–10 1–103 Focus too close
Polycapillary 10–100 1–100 ≤20° 102–104 Moderate resolution
[Figure 11]

Figure 11

Capillary types.

Polycapillaries are very tolerant optics for moderate focusing and fairly high fluxes (Yan & Gibson, 2002[link]). If one uses a so-called pink beam, that is a full harmonic of an undulator spectrum, of energy width ΔE ≃ 1.5%·E, high fluxes of up to 1 × 1012 photons s−1 can generally be achieved at third-generation SRS for XRF and XAFS measurements. Silversmit et al. (2009[link]) successfully used a polycapillary with an acceptance of 5 mm, a length of 50 mm, a focal distance of 3.6 mm and a focal spot in the range 12–16 µm placed after a non-fixed-exit double-crystal sagittal focusing monochromator. Given the large incident beam size (0.5 × 2 mm) and the small vertical displacement as a function of the energy (200 µm), the capillary lens was kept in the beam regardless of the energy (Fig. 12[link]).

[Figure 12]

Figure 12

(a) Polycapillary focal spot displacement and (b) efficiency over the Zn K-edge XAS region.

Over the large energy range 7–12 keV, the vertical displacement was only 2.7 µm versus a beamsize of approximately 15 µm. EXAFS scans are routinely performed in a 1 keV range, and at 7 keV, the energy range of the highest beam shifts, the total vertical shift was 1.3 µm, roughly 10% of the focused beam size, despite the non-fixed-exit monochromator full beam vertical displacement of 190 µm. The authors measured the polycapillary (PC) transmission as a function of the energy using standards (nickel, copper and zinc foils and ZnO powder) and only applied this correction to their spectra (PC, corrected). This effective simple procedure successfully accounted for combined monochromator and polycapillary position and efficiency variations as a function of the energy. An EXAFS spectrum taken on a pure metallic zinc foil (the worst case for their standards), and thus less sensitive to focal spot displacements but sensitive to efficiency variations, is shown in Fig. 13[link], validating this simple approach.

[Figure 13]

Figure 13

Effect of polycapillary (PC) focusing variations simply corrected for transmission on a zinc foil EXAFS spectrum.

Thanks to their larger acceptance, polycapillaries show a great suitability for XAFS in the few tens of micrometres to few micrometres beamspot range, with moderate fluxes in the range 1 × 108 to 1 × 1010 photons s−1. Their efficiencies are reasonable (20–45%) and they are easy to align using a double-angle gimbal device coupled to a double-translation drive. They are relatively impervious to radiation damage, at least at these moderate fluxes. The overall effect of their presence is to render stability to the setup, compensating for non-fixed-exit monochromator beamspot displacements as well as lateral displacements.

5. Kirkpatrick–Baez double mirrors

Currently, K-B mirrors (Fig. 14[link]), which were invented in 1948 by P. Kirkpatrick and A. V. Baez (Kirkpatrick & Baez, 1948[link]), are the most straightforward systems for XAS and yield the highest intensity focused beams. They are based on total reflection at or below the energy-dependent critical grazing-incident angle described before, which varies over the optics. The mirrors are elliptically shaped and are sometimes coated with multilayers such as molybdenum or tungsten with boron carbide (B4C) to increase their reflectivities. However, as multilayer mirrors have a non-negligible ΔE/E bandpass (a few percent), they are not fully EXAFS compatible.

[Figure 14]

Figure 14

Kirkpatrick–Baez double-mirror focusing setup.

The mirrors are either fixed by construction or bent by flexure mechanisms at the appropriate curvatures so that the two respective focal distances coincide. Up to eight precision stepper motors are used to ensure bending, rotation and translation in order to optimize the parameters over an energy range. The lengths of the two mirrors are variable from 70 to 200 mm and their surface quality is very high. For ultimate precision in the elliptical cylinder shape, the mirrors are `figured' by optical polishing or ion milling. The final shape routinely has figure errors of a few nanometres, and even these can be decreased by novel ion-figuring techniques to sub­nanometre errors.

For XRF applications, multilayer substrates are used with bandwidths of a few percent, thus precluding the use of monochromators, in pink beams. These multilayers are `graded'; that is, the widths of the layers are gradually changed across the mirror faces to compensate for the change in incidence angle. At the nano-imaging station of the ESRF (Morawe et al., 2015[link]; Fig. 15[link]), the mirrors are figured down to subnanometre shape errors by ion-milling processing and the beamspots at 17 keV reach the 20 nm range with pink beam fluxes of up to 2 × 1012 photons s−1, while at 33.6 keV an unprecedented 13 nm spot with fluxes of about 6 × 109 photons s−1 was obtained (Da Silva et al., 2017[link]). Recently, Mimura and coworkers (Mimura et al., 2009[link]; Motoyama & Mimura, 2015[link]), using a 20 keV double-mirror adaptive optics setup, showed focusing down to 7 nm at the SPring-8 synchrotron, but no analytical applications were envisioned in the short term. For XRF imaging in the 17–30 keV energy range, after the ESRF upgrade it is expected that the beam emittance, which has decreased by a factor of more than 30, will allow a diffraction-limited spot of about 7 nm to be reached. However, for XAFS, which is sensitive to all of the sources of instabilities mentioned earlier, our current high-resolution measurements on the nano-analysis beamline of the ESRF at the Ni K edge are in the 50 nm range, but are confined to XANES and with a compensation for displacement as shown in Fig. 16[link]. The compensation is performed by using a `lookup table' of vertical and lateral displacements of the beamspot as a function of the energy, using the knife-edge method in both dimensions.

[Figure 15]

Figure 15

Kirkpatrick–Baez submicrometre setup at the ESRF.

[Figure 16]

Figure 16

High-energy effect of K-B compensation.

The corrections are applied here in `sample-tracking' mode by using the lookup table and they are highly effective, as can be seen in Fig. 16[link]. They decrease the total vertical displacement (the largest displacement due to the vertical monochromator design) from 113 to 13 nm, which is adequate for a focal spot of approximately 50 nm. Using a K-B device on a synchrotron low-beta section (small source size, moderate divergence), when scanning the energy in a 300 eV range at the Fe K edge, one obtains displacements in the focal lateral position of about 250–400 nm, as shown in Fig. 17[link].

[Figure 17]

Figure 17

K-B focus position (µm) in the sample plane for a scan between 7.05 and 7.35 keV.

These displacements are mainly due to monochromator divergence shifts, as mirrors are highly sensitive in angular variations. A change in the incident angle of Δθ is doubled in size at the same distance and as the monochromator is a fixed-exit vertical diffracting monochromator, the largest walk in angular range will be that produced by horizontal translation of the second crystal.

6. Conclusions

Inline, chromatic focusing optics can readily be used in a simple setup to obtain high-resolution XAFS from moderately diluted samples. Even easier to field are achromatic capillaries, which feature high resolution at the detriment of sample environment space, but show a gain in stability. The best working devices so far have been K-B mirrors, which achieve high resolutions and high fluxes. Being off-axis focusing devices, their stability is sensitive to beam divergence from the storage ring and monochromators. The focal spot lateral displacements are more challenging and have to be finely monitored and compensated.

Depending on the XAFS study specifics, one should be able to match the focusing system to the synchrotron-radiation parameters, and then to the sample environment. Solutions exist, but they are sometimes difficult to install and operate, and both their cost and time of deployment are different. By reading this chapter, users will have a first walkthrough of the field and should be able to make experimental choices for the required measurement protocols.

It is increasingly the case that XAFS is directed towards the submicrometre range, and this is a region where sample radiation damage is of concern. Future studies of nano-samples will have to deal with strategies to diminish radiation damage, particularly on the new beamlines at modern sources such as the ESRF after the low-emittance/high-flux upgrade. XAFS samples are sometimes prone to irradiation effects which alter their local chemistry, such as photo-oxidation or photo-reduction. Ways to reduce these effects are under study by using cryostats and by performing time-controlled scans where fractional irradiation is implemented, allowing samples to `recover'. Sample-charge buildup must be avoided by implementing non-insulating sample holders or by flooding the sample surface with either electron or ion unfocused fluxes to compensate for charge changes. It is expected that XAS will continue to advance, benefitting from new focusing optics, high-intensity SRS or free-electron laser X-ray beams and faster and more efficient detectors, towards a region limited by physical laws and counting statistics to only a few nanometres, spearheading a novel view of the chemical environment of an atom.

Acknowledgements

The authors gratefully acknowledge Dr M. Salomé, ESRF for sharing ID21 data and for critical reading.

References

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