Small-angle X-ray scattering (SAXS) is an X-ray-based characterization method that provides detailed structural and physical information at orders of c. 1-100 nm.^[1]It can be viewed as an extension to the operating scale of X-ray diffraction (XRD), as well as a complement to modern microscopy technique. Like XRD, SAXS is based on the scattering of X-rays by heterogeneities in electron density. Fields of application range from characterizing glass and liquid crystals, to proteins and mesoporous materials. This is an introduction to the technique, with a focus on the background of the technique and its uses, for the sake of giving a general understanding of this tool to the reader. Multiple sources are included for deeper studying, of which especially the books of Guinier et Fournet, Glatter et Kratky, and Schnablegger et Singh go in-depth in describing the technique and its fundamentals.

Background and Theory

There are two main ways in which X-rays may interact with matter: absorption and scattering.^[2]The latter may be imagined as each scattering atom acting as the emitter of spherical waves when illuminated by an X-ray beam. The waves originating from each atom then interfere to produce diffraction patterns that depend on the orientation and distance of atoms relative to each other. In effect, the so-called "primary beam" illuminating the sample is scattered at different intensities into different directions.^[3]This is the basis of many techniques, such as XRD, a fundamental technique of crystallography. XRD is generally used to observe well-ordered crystalline matter, in which lattice spacings are often at a scale similar to the X-ray wavelengths (for example, the common CuKα X-ray line has a wavelength of 1.54 Å). This is advantageous in that the produced diffraction angles are rather large and thus easier to resolve. In a way, SAXS takes the same methodology further by focusing on the scattering at small angles (made possible by sufficient advances in resolution of the scattering pattern).^[4]

The fundamental relation describing X-ray diffraction in crystalline matter, Bragg's law, shows that the diffraction angle varies inversely with the separation of diffracting lattice plane (with d equal to interplanar distance, θ to diffraction angle, n to an integer, and λ to wavelength): \[ \text{sin(}\theta\text{)}=\frac{n\,\lambda}{2d\,} \] This suggests a connection between very small diffraction angles and larger-scale structures. For example, with a Cu Kα radiation source spacings of 100 and 1000 Å (10 and 100 nm) lead to diffraction angles of 0.45 and 0.045 degrees, respectively. Another approach considers the description of diffraction patterns in terms of reciprocal space. At small angles, the amplitude of diffracted radiation is primarily controlled by terms in electronic density with a periodicity that is large compared to the X-ray wavelength. Going further, by defining a "form-factor" that describes where a particle is and is not, it can be proven that small-angle scattering depends primarily on the exterior form and dimensions of a particle. Based on these calculations, it is clear that scattering of X-rays at small angles conveys information on larger spacings than ordinary XRD, taking the observer from e.g. typical inorganic crystal lattices to such compounds as synthetic polymers and proteins. ^[4]

Experimentally, Guinier (1955) refers to Krishnamurti^[5]and Warren^[6]as among the first to observe intense continuous scattering at small angles from materials with heterogeneity at the scale of 10's or 100's of X-ray wavelengths. Examples of materials studied by them include fine-grain carbon and carbon blacks. Warren also later reported on the structure of glass, deducing via an early form of SAXS that glass is a continuous amorphous mass absent of 10-100 Å structural variation; the glass sample produced no small-angle scattering under X-ray illumination, but fine-grained silica gel produced multiple sharp peaks.^[7]. Guinier's thesis from 1939 is often considered seminal to the development of SAXS. In it he reported the presence of nanometer-scale copper platelets that formed in a copper-aluminium alloy^[8]; so-called Guinier-Preston zones that would be observed via electron microscopy some 60 years later.^[9]

Figure 1. SAXS patterns of graphene oxide sheet dispersions (average thickness 0.8 nm, average lateral width 810 nm) with successive volume fractions of 0.38, 0.91 and 1.5%. Adapted from Xu et Gao (License: CC BY-NC-SA 3.0).^[10]

Examples of Applications

Owing to its wide range of observable scales and properties, SAXS is applied for a wide variety of purposes. A few short examples of applications are collected here:

• Proteins: SAXS can be used as a sensitive detector to follow protein aggregation and crystallization.^[2] The simple size and shape of biomacromolecules have also been investigated.^[3]
• Cancer cells, SAXS can be used to follow the degradation of supramolecular collagen fibrils to detect cancer invasion.^[2]

• Self-assembled nanostructures: The evaporation-driven orientation and self-assembly of cellulose nanocrystals into larger-scale lamellae have been followed using SAXS, among others the orientation of the rod-like crystals was investigated.^[13]

• Nanocomposites
• Liquid crystals
• Mesoporous materials
• Membranes
• Minerals

Further reading on Guinier

Guinier's work can be followed throughout his doctorate and book, with Ravy providing a complete historic of the formers' career.^[9] For those interested in historical stories in science, Guinier has also contributed a short reminiscing in 1962.^[14]

References