Highly versatile laboratory X-ray scattering instrument enabling (nano-)material structure analysis on multiple length scales by covering a scattering vector range of almost five decades Open Access

X-ray scattering and diffraction techniques^1–6 are widely used for the nondestructive analysis of micro- and nanostructures in matter, and particularly for investigations on the arrangement of atoms in crystal structures. These techniques are applicable to virtually all types of hard and soft matter and give statistically significant information. The photons of the incident X-ray beam interact with the electrons in the sample, and the intensity of the scattered X-rays is measured as a function of the scattering angle 2θ. The angular dependence of the coherent, elastic scattering intensity is related to the spatial distribution of the scattering centers. From well-ordered, crystalline materials, Bragg scattering is observed at discrete 2θ values,⁴ whereas in the case of poorly ordered materials, diffuse scattering prevails.^1,7 The length scale d that is probed in an X-ray scattering experiment depends on the covered 2θ range as well as on the used wavelength λ of radiation. It is given by d = 2π/q, where q denotes the scattering vector [q = 4π λ⁻¹ sin(θ)]. In classical powder X-ray diffraction (XRD), d is known as the Bragg spacing. It indicates the distance between the diffracting lattice planes, which corresponds to a diffraction peak that is observed at a given q-value.

X-ray diffraction (XRD), also referred to as wide-angle X-ray scattering (WAXS), is most commonly measured using a 2-circle goniometer with a Cu X-ray source (λ ≈ 1.54 Å) and in a 2θ range of typically 10–100°. This technique covers d-spacings approximately between 1 and 10 Å, which is comparable with the interatomic distances. It is mainly used to identify and quantify crystalline phases, to estimate the degree of crystallinity, to determine crystal lattice parameters, to estimate the size of nanocrystallites, and for crystal structure refinement and determination.⁴ From the reciprocal relationship between q and d, it is obvious that with decreasing q-values, increasingly larger Bragg spacings are probed. Crystals formed by organic molecules (and proteins in particular) have unit cell dimensions typically ranging between tens and hundreds of Angstroms. Therefore, diffraction peaks are observed at smaller q-values, and measurements must be extended to smaller 2θ angles (well below 10° with Cu radiation). Small-angle X-ray scattering (SAXS) is a well-established technique for the analysis of structures, shapes, and dimensions on a length scale of typically 1–100 nm.^5,8–14 Applications include the characterization of nanoparticle systems, colloids, surfactants, protein solutions, polymers, liquid crystals, nanocomposites, and porous materials. SAXS measurements are performed in the very close vicinity to the direct beam, with a smallest accessible 2θ value typically around 0.1° or slightly below. The ultra-small-angle X-ray scattering (USAXS) technique¹⁵ enables measurements even closer to the direct beam, with a smallest scattering angle below 0.01° 2θ (q_min < 0.007 nm⁻¹). It thus enables access to Bragg spacings of up to several micrometers and can be used to investigate particle systems with particle dimensions in the range of hundreds of nanometers and even micrometers. The q-range that is covered by the USAXS technique also overlaps at the low-q side with the range that is accessible with static (visible) light scattering (SLS) measurements¹⁶ and thus bridges the gap between SAXS and SLS.^17,18 The combination of data obtained from SLS, USAXS, SAXS, and WAXS measurements, thus covering a q-range extending over several decades, has proven very powerful for the analysis of multilevel (hierarchical) structures in complex materials, such as polymers, polymer nanocomposites, fibers, aggregated nanoparticle systems, colloids, colloidal crystals, and porous materials.^19–27 Other closely related, complementary scattering techniques are small-angle neutron scattering (SANS),^5,10–12 ultra-small-angle neutron scattering (USANS),^28,29 and dynamic light scattering (DLS).³⁰ 

The total scattering (atomic pair distribution function; PDF) technique^3,7,31 enables the extension of XRD (WAXS) measurements further toward the higher end of the q-scale, with a q_max of at least 150–200 nm⁻¹. It is used to study the local atomic structure in (partially) disordered materials, such as nanocrystalline, semicrystalline, and amorphous samples. The technique takes both Bragg and diffuse scattering into account and provides information not only about the long-range (>1 nm) atomic ordering but also about the short-range atomic ordering in materials. The method is performed in several steps.^3,7 First, the diffraction pattern of the sample is corrected for background signal (using a separate diffraction measurement of an empty container), Compton scattering, detector dead-time, absorption, diffraction geometry, polarization, and instrumental aberrations; then the corrected X-ray diffraction data are scaled to electron units and the structure function is calculated. Finally, the structure function is Fourier transformed to obtain the atomic pair distribution function: G(r) = 4πr (ρ(r) − ρ₀). To ensure a good quality of the derived atomic PDF and to reduce artifacts (e.g., termination ripples), it is essential that high-quality experimental data with enough counting statistics are available up to high q-values. Only quite recently it was shown by Farrow et al.³² that the modeling and interpretation of total scattering data from nanoparticles can be supported by the additional information about the size, shape, and structure that is obtainable from SAXS measurements. The combination of total scattering and SAXS measurements, and the corefinement of the obtained data (concept of complex modeling),³² therefore seems very promising.

Powder XRD is normally done with a goniometer system in the classical Bragg-Brentano reflection geometry.⁴ Alternatively, for samples which are sufficiently transparent for X-rays, transmission geometry can be used. The latter is particularly useful when measurements must be extended toward lower angles, which often is the case for organic materials. Modern XRD systems support measurements both in reflection and transmission geometries. For SAXS measurements in the very close vicinity to the intense direct beam, a monochromatic, very narrow, and highly collimated X-ray beam is needed. Also, parasitic scattering must be minimized to achieve a low background scattering signal, which is essential for the measurement of weakly scattering samples. Therefore, an evacuated beam path is used to eliminate air scattering and a low-noise X-ray detector is essential for the measurement of weakly scattering samples. Also, large distances (several meters) between the X-ray source and the sample, as well as between the sample and detector, are often necessary to achieve a good small-angle and point-to-point resolution. Due to these demanding requirements for the experimental setup, SAXS measurements are traditionally performed on single-purpose lab instruments or on dedicated beamlines at synchrotron radiation facilities.⁶ Modern SAXS instruments also enable extension of the measurement range toward higher scattering angles, for the acquisition of complementary WAXS data. We have recently presented a new concept of instrumentation for high-performance SAXS/WAXS measurements,³³ which makes use of a goniometer-based XRD system, as widely used for powder X-ray diffraction and related techniques. The instrument is based on a modular concept and can be easily configured with various X-ray optics and sample stages, thus enabling not only for SAXS and WAXS measurements, but also for a variety of X-ray diffraction techniques.

The ultimate small-angle resolution that is required for USAXS measurements can be achieved by using multiple reflections from single crystals, both for beam collimation and in analyzer optics. Such a setup, which was first proposed by Bonse and Hart,³⁴ has the advantage that it can be very compact. On the other hand, the very high resolution in 2θ comes with a trade-off in intensity. Also, this setup uses a tight beam collimation only in the equatorial direction, whereas in the axial direction, the beam is strongly elongated and divergent. Whereas at some synchrotron beamlines, there is a possibility to acquire both SAXS and USAXS data,^35–39 laboratory instruments for USAXS have traditionally been dedicated to this single technique.^40–45 Only very recently a concept has been developed that enables a large laboratory SAXS instrument also for USAXS measurement by making use of motorized USAXS optical modules that can be moved in and out of the beam path.⁴⁵ 

Although total scattering measurements are traditionally done at dedicated beamlines at synchrotron radiation facilities,⁴⁶ it has been demonstrated that laboratory X-ray diffraction instruments can be used for performing in-house PDF measurements, by using high-energy X-ray sources and suitable X-ray optics.^47–54 The possibility and benefits of using a synchrotron beamline for SAXS/WAXS as well as for total scattering experiments were described by Daniels et al.⁵⁵ On the other hand, laboratory SAXS/WAXS instruments that are also capable of performing total scattering measurements have not been reported to date.

As outlined above, the USAXS and total scattering techniques are closely related and complementary to SAXS and WAXS. When combined, these techniques enable the analysis of complex structures in hard and soft matter, covering multiple length scales. However, to the best of our knowledge, none of the available SAXS/WAXS instruments can also be used both for total scattering and USAXS measurements, thus covering this very wide q-range. In this paper, we describe how the compact goniometer-based laboratory instrument, that we introduced earlier as a platform for high-performance SAXS/WAXS measurements,³³ can also be configured for total scattering and USAXS measurements. The accessible q-range can thus be expanded to a range spanning over almost five decades. The high data quality that can be achieved in these applications will be demonstrated on some well-defined test samples.

The general concept of the floor-standing X-ray scattering platform (Empyrean Nano Edition, Malvern Panalytical, The Netherlands) is already explained elsewhere.³³ In summary, the instrument (see the supplementary material, Fig. S1) is based on a high-resolution goniometer with interfaces for the attachment of various X-ray optics, sample stages, and detectors. These modules can be quickly and easily exchanged for many different applications. Prealigned interfaces on the goniometer platform and on the modules enable highly reproducible mounting, simply by fixing a single screw (see the supplementary material, Fig. S2). Therefore, realignment is not required when switching between different instrument configurations. The optical modules are tagged and automatically recognized by the system when mounted on the instrument. The vertical goniometer has two concentric circles, omega, and 2theta, with a radius of 240 mm. An enclosure with wide-opening doors and see-thru windows ensures radiation safety as well as easy access to the experimental setup. The raw measurement data together with all information about the instrument configuration and the scan parameters are stored in an open, XML-based file format.³³ 

The three main applications on the instrument that is described here are small- and wide-angle X-ray scattering (SAXS/WAXS), ultra-small angle X-ray scattering (USAXS), and total scattering for the determination of the atomic pair distribution function (PDF method). Details of the instrument configurations for these applications are given in Subsections II B 1–II B 3. Although being outside the scope of this paper, it is noteworthy that the same instrument platform can additionally be configured for a whole range of other applications, including powder X-ray diffraction, microdiffraction, thin film techniques [e.g., grazing incidence diffraction, X-ray reflectivity, small-angle X-ray scattering under grazing incidence (GISAXS), and high-resolution diffraction], residual stress and texture analysis, and computed tomography. Changing between the setups for these different applications is guided by a software wizard and can typically be done within 5–10 min.

The details of the experimental setup for SAXS and WAXS are already reported elsewhere.³³ In short, a sealed X-ray tube with a copper anode (λ ≈ 1.54 Å and E ≈ 8.04 keV for CuKα) is used, in combination with a focusing X-ray mirror, an evacuated beam path (ScatterX⁷⁸; Malvern Panalytical), and a hybrid pixel area detector (GaliPIX^3D or PIXcel^3D, Malvern Panalytical) (Fig. 1). The vacuum path contains antiscatter devices, a sample capsule, and a semitransparent beam stop. Sample holders for powders, liquids, and solids, that can be inserted in the vacuum path, have been developed (see the supplementary material, Fig. S3). For liquid samples, a temperature-controlled capillary holder (T = 5–70 °C) is available. The sample is located exactly in the center of the goniometer, and its distance to the detector is only 240 mm. The detectors has a very small pixel size (60 × 52 µm² for GaliPIX^3D and 55 × 55 µm² for PIXcel^3D) and a very high spatial resolution (equal to the size of one pixel), coupled with a high count rate capability and a very low noise level. With this configuration, measurements are normally done by using a line-shaped X-ray beam. However, a modification of this setup also enables 2D SAXS and 2D WAXS measurements with a pencil beam.^33,56 SAXS/WAXS measurements can be performed in a continuous range of scattering angles from approximately 0.08° up to 78° 2θ [q = 0.06–51.4 nm⁻¹, where q denotes the scattering vector q = 4π λ⁻¹ sin(θ), and λ denotes the wavelength of radiation]. The FWHM of the direct beam at the detector position is about 0.035° 2θ (0.025 nm⁻¹ in terms of q). SAXS measurements can be performed with a static detector, whereas a 2θ scan of the detector around the sample enables measurements at the higher angles in the WAXS range. Typical SAXS measurement times are in the range of 1–60 min, depending on the sample. SAXS data reduction and analysis are done using the EasySAXS software (Malvern Panalytical).³³ 

The angular resolution of a conventional SAXS setup that makes use of collimating X-ray optics and a set of slits or pinholes, can be improved by increasing the sample-to-detector distance, thus enabling for USAXS measurements. Whereas at synchrotron radiation facilities, some (very) long beamlines with large detectors and two-dimensional collimation have been built for this purpose,^38,57 for a lab instrument, this is not feasible due to limited lab space and low intensity. A compact, dedicated USAXS lab instrument has been developed by Bonse and Hart,³⁴ which uses multibounce, channel-cut perfect crystals for both beam collimation and high-resolution analyzer optics. On the multifunctional instrument platform presented here, a similar approach is used, as described in more detail below and shown schematically in Fig. 2.

A sealed Cu X-ray source is used in line focus orientation. For the shaping and monochromatization of the incident beam, a 4-bounce Ge(220) symmetrical Bartels monochromator⁵⁸ is mounted on the omega arm of the goniometer, behind the X-ray tube. It consists of two channel-cut Ge crystals in an antiparallel, dispersive (+, −) (−, +) arrangement. The width of each crystal is 12 mm. The multiple reflections from these high-quality crystals result in a highly monochromatic beam that is strongly collimated (angular divergence of 12 arcsec) and very narrow in the equatorial direction, but divergent and elongated in the axial direction. The beam that exits the monochromator is parallel to the incident beam, and there is no translational offset between them. At the sample position, the X-ray beam has a size of approximately 60 µm (vertically) × 14 mm (horizontally), and the photon flux is about 2.0 × 10⁶ ph/s. The benefit of using a Bartels monochromator for the horizontal beam collimation at a synchrotron beamline for USAXS experiments was recently also reported by Sztucki et al.⁵⁹ 

A 3-bounce Ge(220) channel-cut analyzer crystal in a nondispersive (−, +, −) arrangement relative to the incident beam monochromator is mounted on the 2theta arm of the goniometer and functions as high-resolution receiving optics. It has a very narrow angular acceptance of 12 arcsec and therefore enables the measurement of weak scattering in very close vicinity to the direct beam. A point detector is mounted behind the analyzer. A 0D proportional counter, filled with a xenon-methane mixture, or alternatively, a 1D strip detector or a 2D pixel detector (PIXcel^1D or PIXcel^3D, Malvern Panalytical) based on hybrid pixel photon counting technology, was used. In a USAXS experiment, these 1D or 2D detectors are operated in the 0D detection mode, in which all photons that reach the active detector area are summed up. A slit with an opening of 0.2 mm in the equatorial direction is inserted between the analyzer crystal and the detector. A compact stage (Fig. 3, top) was developed onto which different sample holders, designed for powders, liquids, and solids (see the supplementary material, Fig. S3), can be reproducibly mounted. A capillary holder enables the measurement of samples under controlled temperature, within a range of 5 and 70 °C. All these holders also fit in the vacuum path (ScatterX⁷⁸, Fig. 1) that is used for SAXS and WAXS. The sample stage is attached to the central interface of the goniometer and ensures that a mounted sample is positioned exactly in the center of the goniometer. The distances between the X-ray source and the sample and from the sample to the detector are both 240 mm. The beam path for USAXS may either be horizontal (Fig. 3, top) or vertical (Fig. 3, bottom). In the latter case a transmission spinner is used as a sample stage, which can optionally be combined with an automatic sample changer for up to 45 samples.

The same optical components that are used for USAXS in a transmission geometry are routinely also used for high-resolution, thin film characterization techniques on the same instrument, but then in a reflection geometry. Examples are measurements of reciprocal space maps and rocking curves from epitaxial layers and of high-resolution X-ray reflectivity from thin films.^58,60–62

USAXS measurements of the scattered intensity as a function of 2θ can be done in two ways:⁴³ (i) by rotating the analyzer crystal around its own axis, together with the detector (possibly coupled with a translational movement), or (ii) by rotating the analyzer crystal with the detector around the 2θ sample axis. Here, we use the second approach. The high-precision and high-resolution goniometer allows for a smallest step size of 0.0001° 2θ, which is well-suited to achieve an appropriate point-to-point resolution for very fine details that may be present in a USAXS curve. Intensity data are measured sequentially as a function of 2θ. Each data point can be measured either with the same acquisition time, or until a certain number of counts is reached. It is also possible to perform a scan in which the 2theta arm is moved at constant speed and measured intensity data are continuously assigned to the corresponding scattering angles. This type of “continuous scan” (as opposed to “step scan”) program is already being used for powder X-ray diffraction and related techniques with our X-ray diffraction instruments (PW1820, Philips, the Netherlands; and all successor instruments) for more than 25 years. Compared to a step scan, the continuous scan mode enables shorter measurement times without compromising the data quality. The same approach was recently also implemented under the name “fly-scan” method at the USAXS beamline of the Argonne Photon Source.³⁹ As with SAXS experiments, the direct beam is included in each USAXS measurement. It is an accurate reference for q = 0 and enables the determination of the absorption by a sample, which must be considered when doing background-correction.

The direct beam profile that is obtained with this USAXS setup is shown in Fig. 4. The FWHM Δq is approximately 0.0025 nm⁻¹ (0.0035° 2θ), which is in good agreement with the theoretical Darwin width for the (220) reflection from a flat germanium crystal [0.0036° 2θ for CuKα radiation (https://www.chess.cornell.edu/users/calculators/x-ray-calculations-darwin-width)]. The FWHM with the USAXS setup is approximately ten times smaller as compared to the FWHM with the SAXS setup. Due to the multiple reflections on the channel-cut germanium crystals, the tails of the direct beam are very effectively reduced and air scattering is largely rejected, resulting in a steep intensity decrease: relative to its peak intensity at 0° 2θ; at q = 0.0039 nm⁻¹ (0.0055° 2θ) the intensity of the direct beam is already reduced by a factor of 10³, at q = 0.0093 nm⁻¹ (0.013° 2θ) by 10⁵, and at q = 0.018 nm⁻¹ (0.025° 2θ) by a factor of 10⁶. The overall signal-to-background ratio is about 3 × 10⁶. The smallest accessible scattering vector q_min is about 0.0036 nm⁻¹ (0.0050° 2θ). An even narrower direct beam profile and thus a better resolution could be achieved by making use of, e.g., the (220) reflections from a Si crystal; however, due to the concomitant loss of intensity, such setup is not preferable. For comparison, Fig. 4 also shows the direct beam profile that was obtained upon replacing the analyzer crystal by a receiving slit with an opening of 0.1 mm. With this setup, the profile is significantly broader (FWHM 0.022 nm⁻¹; 0.031° 2θ) and the intensity dynamic range (4 × 10²) is much reduced, because significant air scattering can then enter the detector.

In practice, the USAXS setups shown in Fig. 3 are most suited for measurements up to a q_max of 0.5–1.0 nm⁻¹, depending on the sample. Measurements at higher q are better done with the SAXS configuration. As the USAXS data are acquired with a line collimation setup, only isotropic samples can be studied, and the concomitant slit smearing effects must be considered in data analysis. USAXS data reduction and analysis is done using the EasySAXS software (Malvern Panalytical).³³ 

Typical USAXS measurement times are in the range of 30 min to several hours, depending on the sample under investigation. Rather than a single long measurement, multiple short measurements are normally performed and subsequently summed up. By adding an evacuated beam path to our experimental setup, we expect that the intensity could be increased by about 40%, whereas the background signal would probably not be significantly lowered further. An easier way to increase the intensity is by using a more compact setup: with a reduced distance between the sample and the detector (185 mm instead of 240 mm), we obtained a 25% intensity increase without sacrificing resolution. A further intensity increase by about a factor of 8 could be achievable by using a precollimating multilayer mirror in front of the Bartels monochromator.⁶¹ We also found that a hybrid incident beam monochromator, an optical module that houses a parabolic mirror and a 2-bounce channel-cut Ge(220) crystal, enables faster USAXS measurements, however with trade-offs in small-angle resolution and in background suppression.

The total scattering technique^3,7 is often applied to samples in which the atomic structure is (partially) disordered, as, e.g., in nanocrystalline, semicrystalline, and amorphous samples. From the measured Bragg scattering and the underlying diffuse scattering data, the atomic pair distribution function (PDF) is calculated by a Fourier transformation. Scattering data must be available up to highest possible q-values in order to avoid artefacts and loss of resolution in the deduced PDF.⁴⁹ Furthermore, the inherently weak scattering intensities, especially at the higher q-values, must be measured with good counting statistics. The underlying instrument background signal must be measured separately and carefully subtracted. Just like with the other scattering techniques, a low and featureless background signal will therefore greatly improve the data quality. Until recently such demanding measurements could only be done at synchrotron radiation facilities.

As we have shown previously, good quality total scattering data can also be measured on an optimized lab instrument,^48,63 and the deduced PDFs compare well with those obtained from synchrotron radiation data.⁵⁴ The transmission setup (Fig. 5) uses a sealed X-ray tube with a silver (λ = 0.560 Å and E = 22.1 keV for AgKα) or molybdenum (λ = 0.709 Å and E = 17.4 keV for MoKα) target for producing X-ray radiation in a line-shaped primary beam. An elliptically shaped, multilayer X-ray mirror focuses the X-ray beam in the equatorial direction on the detector plane. Alternatively, a set of slits may be used to define a larger beam, which may be beneficial for less-absorbing samples that can be measured at higher thicknesses. Samples are usually prepared in capillaries or confined in transmission holders between thin polymer foils and mounted on a spinner stage. Samples with high coefficient of absorption can be also measured in reflection geometry using the Bragg-Brentano parafocusing arrangement. A hybrid pixel detector (GaliPIX^3D, Malvern Panalytical)³³ with a CdTe sensor is attached to the 2θ arm of the goniometer platform and used in 1D detection mode, like a strip detector. It measures the angle-dependent scattering signal in a 2θ scan around the sample. Due to its 100% quantum efficiency for hard X-rays and relatively large size, the required measurement times could be reduced up to 10–15 times for the same quality of the data as compared to using other detector technologies found in laboratory XRD instruments (e.g., scintillation detectors or detectors based on thin Si sensors). Effective data collection is achieved by a variable counting time strategy⁵⁰ that allocates more time for the measurements of the weaker signals, particularly at the highest angles. The compact form factor of the detector enables measurements up to a maximum 2θ angle of 145°, which corresponds to a maximum scattering vector q_max of 21.5 Å⁻¹ with Ag radiation and 16.9 Å⁻¹ with Mo radiation. Several antiscatter devices are inserted in the beam path to block parasitic scattering and thus to ensure a low and featureless background signal. Typical measurement times are in the range of 1–24 h, depending on the type and amount of sample. Different modifications of the setup are possible which allow for measurements at nonambient temperatures ranging from 80 K with an Oxford Cryostream to 1200 K with the Anton-Paar HTK 1200N chamber. Analysis of the data is facilitated by the software HighScore Plus⁶⁴ which provides functionality for data merging, background subtraction, corrections for Compton, and multiple scattering among others. This software also calculates the experimental structure function and pair distribution function using an algorithm based on the established GudrunX program⁶⁵ and is optimized for analysis of data measured on laboratory X-ray diffractometers.

In the following, we will demonstrate the instrument performance in different scattering techniques on a set of well-defined test samples.

The SAXS and WAXS capabilities of the instrument, using both line- and point-collimation setups, have already been demonstrated elsewhere in detail^33,56 on a variety of samples. Particularly, it has been shown that good resolution and sensitivity can be achieved also for very weakly scattering samples (e.g., dilute protein solutions, liposomes) and that deduced results are in good agreement with those from dedicated SAXS instruments. Therefore, only one additional application example is briefly presented here.

Figure 6 shows SAXS data from an aqueous dispersion of spherical silica particles (2.2 vol. %; nominal diameter 136 nm; microParticles GmbH, Germany) that were measured with the line-collimation setup. The data were corrected for the background that was determined by a measurement of the identical capillary filled with water. Subsequently, the data were also corrected (“desmeared”) for effects related to the finite dimensions of the X-ray beam.³³ In the thus obtained scattering curve, more than ten distinct, narrowly spaced oscillations could be well-resolved, indicating that the particles have a very well-defined size and shape. This also demonstrates the good resolution that can be achieved, even for such large particles having a size well above 100 nm, which is quite remarkable for such a compact SAXS setup with a sample-to-detector distance of 240 mm. From a model fit (not shown here) to the SAXS data, the average particle diameter could be determined to 131 nm and the size polydispersity to 3.5%. The determined particle diameter correlates well with the hydrodynamic diameter of 136 nm as obtained by dynamic light scattering (DLS; Zetasizer Ultra, Malvern Panalytical). The polydispersity value of 12.7% obtained by DLS is clearly an overestimation, which is inherent to this technique.¹⁴ Figure 7 shows 2D SAXS data measured from the same sample, after drying at elevated temperatures, this time using the point-collimation setup. In order to be able to observe the fine features of the pattern at these very small scattering angles, this measurement was done without insertion of a beam stop. Several concentric rings could be resolved, as well as an azimuthal variation of the intensity along these rings, having a 6-fold symmetry (Fig. 8). Also, distinct Bragg spots with a 6-fold symmetry are observed on the innermost rings, which indicates that the particles form a colloidal crystal. WAXS data that were measured from the dried sample are given in the supplementary material (Fig. S4). Only broad humps are observed in the WAXS data, indicating that the particles are amorphous.

The aqueous silica dispersion that was measured with the SAXS configuration (see Sec. III A) was subsequently also measured with the USAXS setup. Upon background-correction, desmearing, and appropriate scaling of the intensities, a good agreement of the SAXS and USAXS data is observed within the q-range of mutual overlap (Fig. 6). It is thus not surprising that the particle diameter (132 nm) and the size polydispersity (3.5%) deduced from the USAXS data is in very good agreement with the SAXS results (131 nm and 3.5%, respectively). In Fig. 9, it is demonstrated that good USAXS data can also be obtained from colloidal silica dispersions (2.2 vol. %; microParticles GmbH, Germany) of significantly larger particles, with nominal diameters ranging between 235 nm and 1550 nm. Here, the measured raw data are displayed, before background-subtraction and without desmearing. Symmetrical scans around and including the direct beam were performed. The data can subsequently be mirrored and merged for further analysis. Of course, it would also be enough to measure just one side of the scattering patterns. For all samples, including the 1550 nm particles, the fine structures due to the narrow particle size distributions could be well-resolved by choosing scans with appropriate ∆q sampling. As expected, with increasing particle size, an increase in the oscillation frequency is observed.⁸ 

In the following, exemplary USAXS data analysis will be demonstrated on one of these samples. Figure 10 (top) shows the data measured from the aqueous dispersion of spherical silica particles, with a nominal particle diameter of 350 nm and a nominal size polydispersity of 5.7%. A characteristic, bell-shaped scattering curve is observed at the very smallest scattering angles, as expected for an ensemble of isolated, nonaggregated particles.⁸ Several distinct oscillations are well resolved at the higher angles, which points to a narrow particle size distribution. The background signal measured from the capillary filled with the dispersion medium (solid line in Fig. 10, top) is only significant in the immediate vicinity of the direct beam and at the highest q-values. As the direct beam profile is included in each USAXS measurement, the attenuation of the X-ray beam by the sample can be determined, and this effect is considered in the background correction procedure. The background-corrected intensities (Fig. 10, bottom) decay approximately with q⁻³, as expected according to Porod’s law for slit-smeared data.⁸ Note that the intensity could be observed within a dynamic range of at least four decades (at least five decades in the case of the desmeared data). Figure 10 (bottom) also demonstrates that meaningful USAXS data could be obtained down to a q_min of 0.0035 nm⁻¹ (0.0050° 2θ).

The particle size distribution D_v(R) (Fig. 11), as determined by the indirect Fourier transformation procedure,^10,33 points to a volume-average particle diameter of 350 nm and to a size polydispersity of 5.1%, in very good agreement with the nominal values. The experimental data are well-fitted (solid line in Fig. 10, bottom) by the back-transform of this distribution curve that also includes data smearing effects due to the line collimation setup. The radius of gyration R_g, estimated from a Guinier plot (ln I vs q², inset of Fig. 10, bottom)⁸ of the data at the smallest q-values, is 134 nm. For spherical particles, R_g relates to the particle radius R by R_g² = 3/5 R². The thus determined particle diameter 2R is 346 nm, which is consistent with the result obtained by the indirect FT procedure (350 nm; Fig. 11), for which the full scattering curve was used. Furthermore, the pair distance distribution function p(r), which is the real space counterpart to the intensity distribution I(q), was calculated using an indirect FT procedure. The p(r) function (see the supplementary material, Fig. S5) is overall symmetrical, as expected for spherical particles. It becomes zero at an intraparticle distance of r = 345 nm, which can be interpreted as the maximum particle dimension D_max, in this case the particle diameter. This value is consistent with the result from particle size distribution analysis (350 nm, Fig. 11). R_g, derived from the first moment of the p(r) function,⁹ is 133 nm. This is in good agreement with the value obtained from the Guinier plot (134 nm; inset of Fig. 10, bottom). Also, based on the USAXS data, the specific surface area (SSA) of the particles could be determined. The calculation was done in two different ways: (i) based on the particle size distribution and (ii) from the ratio of the Porod constant and the scattering invariant. Both approaches gave consistent results, namely, 7.47 m²/g and 7.96 m²/g, respectively. Further details of the SSA analysis are given in the supplementary material (Fig. S6).

All USAXS data displayed in Fig. 9 were analyzed in the same way as demonstrated above for the 350 nm particles. The derived particle sizes are in good agreement with the nominal values that were provided by the supplier of these samples (see Table I). In the case of the two samples with the largest particle size, partial formation of a loose sediment was visually observed. For the 832 nm and 1550 nm particles, a first correlation peak (indicated by arrows in Fig. 9) appears at q = 0.0071 nm⁻¹ and at 0.0039 nm⁻¹, respectively. The corresponding Bragg spacing d is 884 nm (1610 nm), which is just slightly larger than the particle diameter that was estimated to 865 nm (1570 nm) based on the frequency of the higher oscillations in the scattering curve. Note that particularly for the 832 nm particles, the distinct correlation peak at a scattering angle as low as 0.010° 2θ is very well resolved and clearly separated from the direct beam. Higher-order structure factor peaks are also present on top of the oscillations from the single particle form factors. This is typically observed for hard-sphere fluids at higher concentrations,⁸ due to particle-particle interactions. It indicates that, in the sediment, the particles have a local, short-range order.

For comparison, we also determined the particle size in the same (but diluted) colloidal silica samples by using static light scattering (SLS; Mastersizer 3000, Malvern Panalytical) and dynamic light scattering (DLS; Zetasizer Ultra, Malvern Panalytical). The obtained volume-average mean particle diameters are given in Table I. Overall, these values derived from light scattering are in good agreement with the USAXS result. Note that from DLS measurements, a hydrodynamic particle size is derived, which also includes a thin hydration shell around the particle. Therefore, with DLS, often a slightly larger particle size is obtained as compared to SAXS or USAXS, particularly when the particles are charged.^66,67 An example of measured DLS data and of the derived particle size distribution curve is given in the supplementary material (Fig. S7).

In Fig. 12, it is demonstrated that the q-dependent scattering curves from the USAXS and SLS measurements can readily be appended and that there is a significant mutual overlap. USAXS can thus bridge the gap between SAXS/WAXS and SLS.¹⁸ 

Figure 13 shows USAXS data from powders that were obtained upon drying selected silica dispersions (with nominal particle diameters of 136 nm, 235 nm, and 560 nm) at an elevated temperature. For all samples, several distinct Bragg peaks are observed on top of the oscillations due to the particle form factor. The relative positions of these peaks indicate that the particles are arranged in ordered, face-centered-cubic (fcc) lattice structures with derived lattice constants a of 192, 356, and 792 nm, respectively. In an fcc lattice, the nearest-neighbor distance d_fcc is given by (⁠$a/2$⁠). As shown in Table II, the thus calculated d_fcc values are consistent with the particle diameters that were determined from the USAXS data of the corresponding liquid dispersions (Figs. 6 and 9). Previously, we have already shown that the same instrument configured for SAXS can be used to study the formation of colloidal crystals from smaller nanoparticles.^33,56 Note that this type of analysis on powdered or optically opaque samples is not possible with light scattering techniques, whereas USAXS and SAXS are well-suited due to the much higher penetration depth of X-rays in matter.

The silica sample with the nominal particle diameter of 136 nm (see Sec. III A) was dried, and the resulting powder was prepared in a glass capillary with 1 mm diameter and length around 25 mm. A total scattering experiment was performed using a sealed tube with an Ag anode, focusing multilayer mirror, and the GaliPIX^3D detector. Variable counting time strategy was used for data collection in which the 2theta range 3°–40° was measured for 1 h, 40°–80° for 3 h, 80°–120° for 7 h, and finally 120°–145° for 11 h resulting in a total measurement time of 22 h. In addition to the measurement of the sample, an empty capillary was measured with similar conditions for background subtraction. The results of this experiment are shown in Fig. 14 and demonstrate that the investigated silica particles are amorphous as they show only broad scattering features and no sharp diffraction peaks. The reduced structure function F(q) and the atomic pair distribution function G(r) shown in Fig. 15 were calculated from the background-corrected data.

As can be observed from Fig. 15(a), the F(q) structure function shows broad features over the whole q-range, which again demonstrates the presence of only short-range atomic ordering typical for amorphous materials. The atomic pair distribution function G(r) in Fig. 15(b) shows a well-defined peak around 1.6 Å, which can be associated with the first Si–O distance. The oscillations in the PDF disappear around at 12 Å indicating the extent of structural coherence in this material. Both F(q) and G(r) are in good qualitative agreement with previous reports^68,69 based on synchrotron experiments.

This example can be used to demonstrate also the effect of the available q-range in the total scattering experiment. Figure 16 shows the calculated pair distribution functions obtained from the reduced structure function in Fig. 15(a) by successively reducing the value of q_max (upper limit of the q-range) used in the calculation. As can be observed, reduction of the q-range results in broader PDF features, which in turn leads to less accurate determination of interatomic distances and potentially to overlap of close distances. By using q_max = 170 nm, the main features of the PDF can be still distinguished, although they are somewhat broader compared to the case of q_max = 215 nm⁻¹. For q_max = 80 nm⁻¹, the broadening of the PDF features is more severe and close distances cannot be distinguished from each other. This makes the structural analysis with such data unreliable and may lead to wrong interpretation. The values of 170 nm⁻¹ and 80 nm⁻¹ were selected as they correspond to the maximum achievable q_max when using Mo or Cu radiation, respectively. The real space resolution of the PDF can be estimated as Δr = π/q_max,⁷⁰ which gives indication of the smallest details that can be theoretically resolved for a given q_max value. In addition to the q-range, the PDF is also affected by the statistical noise in the data. Good counting statistics must be achieved over the whole range of measurement. Note that, in the experiment shown in Fig. 14, half of the total measurement time is used for the range 120°–145° 2theta, which corresponds to the q-range between 195 and 215 nm⁻¹.

Using the SAXS/WAXS setup on the instrument platform reported here, we previously presented a detailed characterization of nearly monodisperse NaYF₄ nanocrystals in colloidal dispersions, precipitated powders, and colloidal crystals.⁵⁶ When doped with luminescent lanthanide ions, NaYF₄ shows strong upconversion emission.⁷¹ Here we show the results from a total scattering experiment that was performed on the precipitated powder of spherical NaYF₄ nanoparticles having an average particle diameter of 9.0 nm and a polydispersity of 5.8%, as previously determined by SAXS⁵⁶ and an average crystallite size of 8.2 × 8.8 nm, as derived from the WAXS data.⁵⁶ Experiments were performed using the same conditions as described in Sec. III C 1 with the exception that the sample was prepared in a 0.3 mm capillary and the total measurement time of the sample was 14 h. The collected scattering data of the NaYF₄ and the empty capillary are shown in Fig. 17.

Unlike the silica particles reported in Sec. III C 1, the NaYF₄ nanoparticles show well defined Bragg peaks indicating a crystalline arrangement with a hexagonal atomic structure.⁷² Due to the nanometric dimensions of the particles, the diffraction peaks are broader than those exhibited by microcrystalline powders. The presence of Y atoms in this material creates relatively high absorption for Ag radiation, and it also creates fluorescence radiation with energy corresponding to the K emission lines of the Y atoms. To reduce the amount of fluorescence radiation collected by the detector, the energy levels of the detector were adjusted accordingly. The calculated reduced structure function F(q) and pair distribution function G(r) of NaYF₄ are shown in Fig. 18.

In this case, the G(r) shows well defined peaks at short distances; however, the amplitude of the peaks gradually decreases toward extended interatomic distances. This behavior is typical for nanocrystalline materials and shows the reduced number of atomic neighbors located at distances close to the size of the crystalline grains. Figure 19 shows a fitting of the experimental PDF with the structure of NaYF₄ and using the program PDFgui.⁷³ The structural model used for the fitting⁷² has a space group P-6. In order to emphasize the behavior at large distances, the PDF in the range 50–100 Å was scaled with a factor 10. As can be observed in the figure, the last PDF peak that is consistent with the structural model appears at a distance of around 88 Å. This is the longest interatomic distance that can be reliably observed in this material and is in good agreement with the average particle diameter of 9.0 nm as determined previously by SAXS.⁵⁶ Also, in this case, the determined F(q) and G(r) of NaYF₄ are in good qualitative agreement with the published data of similar materials based on synchrotron measurements.⁷⁴ 

The modular concept of the instrument offers a lot of flexibility for incremental improvements of individual modules, for instrument upgrades, and for the development of new modules for entirely new applications. Whereas we already demonstrated a good performance of the instrument with the setups for the different scattering techniques, there is certainly room for further improvement. For example, by replacing the sealed X-ray tube by a microfocus X-ray source and suitable 2D optics, a significantly higher photon flux could be achieved in 2D SAXS and 2D WAXS measurements. The USAXS performance could be further improved by combining a 4-crystal monochromator with a precollimating mirror, by a further reduction of the sample-to-detector distance, and by using an evacuated beam path.

A highly versatile, compact laboratory instrument that enables a variety of X-ray scattering techniques for the structural and dimensional characterization of matter on multiple length scales has been presented. A gapless range of scattering angles 2θ, spanning from 0.0050 to 145°, and of scattering vectors q from 0.0036 to 215 nm⁻¹ (with corresponding Bragg spacings d from 1700 nm to 0.03 nm), can be accessed (Fig. 20). To the best of our knowledge, no other single instrument enables X-ray scattering experiments in such a wide q-range, covering almost five orders of magnitude.

On a length scale of approximately 1–100 nm, SAXS allows us to study the size distribution, internal structure, and shape of nanoparticles and macromolecules and to determine characteristic repeat distances in nanostructured materials. The technique also enables the investigation of interparticle and intermolecular interactions and spatial correlations at higher sample concentrations. Larger length scales in the range of hundreds of nanometers, and even up to several micrometers, become accessible with the USAXS technique. It can be used to characterize larger particles as well as to investigate higher-order structures of molecules and in particle assemblies. To the other end, WAXS measurements at higher scattering angles are essential for the identification and quantification of crystalline phases that may be present and to estimate the size and strain in nanocrystallites. With the total scattering (atomic PDF) technique, the local atomic structures in (partially) disordered materials can be studied. The combination of all scattering techniques thus enables the investigation of complex structures on the atomic and nano- and meso- (length) scales, which is essential for a better understanding of and control over the macroscopic properties of a given material. For example, the spontaneous formation of mesoscopic structures from smaller building blocks (self-assembly) is frequently observed in soft matter.¹³ The complex structures and interactions across multiple length scales that are observed in such materials can only be revealed by combining the results that are obtained from different experimental techniques.

The flexibility and high performance of this novel and compact X-ray scattering platform presented here is mainly due to these key features: (i) a high-resolution goniometer with kinematic mounts, (ii) a modular concept of prealigned and quickly exchangeable X-ray optics and sample stages, and (iii) a low-noise detector with a very good spatial resolution and 100% quantum efficiency not only for Cu radiation but also for hard radiation. In SAXS experiments, the X-ray source and the detector are usually kept stationary. With the high-resolution goniometer, the accessible q-range can be significantly expanded, both toward very small and very high scattering vectors. A setup based on high-resolution crystal optics allows for the sequential acquisition of USAXS data, by performing a 2θ scan with a 0D detector. 2θ scans toward the higher angles enable WAXS and total scattering experiments. The use of this instrument is not restricted to transmission experiments, but it can also be configured in the classical Bragg-Brentano reflection geometry for powder diffraction, in a grazing incidence setup for thin film analysis, for high-resolution X-ray diffraction on epitaxial layers, or for microdiffraction, to name a few. The modular design concept of the instrument allows for incremental improvements as well as to add more applications in the future.

See the supplementary material for images about the instrument platform, the kinematic mounting of sample stages and optical modules, photos of the sample holders that are used for USAXS, SAXS, and WAXS measurements, WAXS data from the silica particles having a nominal diameter of 136 nm, and the p(r) function and the specific surface area (SSA) derived from the USAXS data of the silica particles having a nominal diameter of 350 nm, together with dynamic light scattering (DLS) results that were obtained from the same sample.

We thank the Science and Technology Facilities Council (STFC, Swindon, UK), the Rutherford Appleton Laboratory, and Dr. A. K. Soper for the collaboration and the adaption of GudrunX and enabling its integration into the Malvern Panalytical HighScore Plus software. We also gratefully acknowledge Professor M. Haase and C. Homann (Univ. Osnabrück, Germany) for having provided the NaYF₄ upconversion nanocrystals as a test sample for total scattering measurements.