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]

Practice and Interpretation

In practice, SAXS measures electron density fluctuations rather than electron density itself. This is because "background" scattering from any matter other than the structures of interest need to be subtracted from the signal. By tweaking solvents, swelling and other parameters, different structures can be contrasted for selective analysis.[2][3]

Accurate interpretation of SAXS patterns is quite complex, with the nonintuitive relations between scattering patterns and different properties of the sample mixed into a single curve. According to Glatter and Kratky (1982), the main issue in interpretation is often finding the correct class of possible models to compare experimental results with, which allows working within a certain framework.[3] A full review of interpretation would warrant a textbook's worth of text, but some general aspects will be covered here.

SAXS experiments generally measure intensity as a function of q, "length of the scattering vector" or "momentum transfer", which is essentially a measure of scattering angle normalized with regards to the used wavelength. The dimension of q is one over length.[2]

\[ q=\frac{4\pi}{\lambda}\text{sin(}\theta\text{)} \]

As stated before, small diffraction angles correspond with larger-scale structures. By choosing to analyze a specific range of q, different aspects of the system may be studied. This is sometimes referred to as "q-window", a "window" of a certain diameter into realspace. In the high q domain, this window is very small, and the curve contains information on contrasted interfaces.  At intermediate q values, the intensity plot contains information on single particles, and at low values information is obtained on structures of multiple particles. As such, the choice of q range is crucial to focus on certain aspects of a system. Ultimately, this range is limited by geometry and technical limitations. [2][11]

A single particle with a given shape and of suitable size for SAXS produces a "form factor" P(q), an interference pattern characteristic to that shape. It may be considered as the sum of waves produces by electrons/atoms in a particle under illumination. In practice, the observed scattering pattern corresponds to the form factor of a given one particle only if (1) the sample is monodisperse and (2) the particles are far away from each other. When particles are closely packed, the interparticle atomic separations may also reach a scale that produces signals in SAXS. The interference pattern arising from neighboring particles is called the "structure factor" S(q). It contains information about the relative positions of particles, and it may overlap with the form factor if interparticle and intraparticle separations are similar in scale. The form factor and structure factors can be thought of as signals obtained with smaller or larger q-windows, respectively.[2] Sometimes, the domain of the SAXS curve corresponding to particle size is referred to as the "Guinier domain" or "Guinier knee", the region corresponding to particle shape as the "Fourier domain", and the region related to specific surface as the "Porod domain" (Figure 2). For example, based on the asymptote of the Porod region the specific surface can be estimated (within certain limits of the asymptote).[2][12]

Parameters that may be obtained from SAXS include[2]:

  • Radius of gyration (which may be used to calculate dimensions if the structure can be assumed or is known)
  • Surface per volume
  • Molecular weight
  • Particle structure: globular, cylindrical, lamellar, core-shell, aggregate, etc.
  • Particle interactions
  • Degree of orientation
  • Degree of crystallinity

SAXS, XRD, and Microscopy

SAXS may be viewed as an extension of XRD; the applied principles are very similar, but the scale of observed structures shifts from around nanometers or less to tens or hundreds of nanometers. When a very large variety of observation scales is needed, the two may be complementary.

Compared to microscopy, SAXS does not show specific details present in the sample. The results are instead representative of the sample as a whole, with the downside of ambiguity. This can be especially challenging with polydisperse or polymorphous samples. Preparation artifacts are generally scarcer when using SAXS, compared to microscopy. From the comparison of the two technologies, it is quite apparent that they can complement each other well.[2]

Figure 2. Illustration of different domains in SAXS intensity curves. The domain in the lower frame corresponds with the so-called "Fourier domain"; the shape of the scattering pattern in this region contains information on particle morphology. Adapted from Zhang et al.[12](License: CC BY-NC-ND 4.0).

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]


1. 1
Boldon et al., Review of the fundamental theories behind small angle X-ray scattering, molecular dynamics simulations, and relevant integrated application, Nano Reviews, 2015, 6, 25661 (10.3402/nano.v6.25661)
2. 1 2 3 4 5 6 7 8 9 10

Schnablegger, Singh, The SAXS Guide - Getting acquainted with the principles, Anton Paar GmbH, Austria, 2013

3. 1 2 3 4

Glatter, Kratky, Small angle x-ray scattering, Academic Press, London, 1982

4. 1 2

Guinier, Fournet, Small-Angle Scattering of X-rays, Wiley, New York, 1955

5. 1

Krishnamurti, Studies in X-Ray Diffraction. Part I: The Structure of Amorphous Scattering. Part II: Colioidal Solutions and Liquid Mixtures, Indian J. Phys., 1930, 5, 473-500

6. 1

Warren, X-Ray Diffraction Study of Carbon Black, J. Chem. Phys., 1934, 2, 551-555

7. 1

Warren, X-ray Determination of the Structure of Liquids and Glass, Journal of Applied Physics, 1937, 8, 645-654

8. 1

Guinier, La diffraction des rayons X aux très petits angles : application à l'étude de phénomènes ultramicroscopiques, Annales de Physique, 1939, 11, 161-237

9. 1 2

Ravy, André Guinier (1911–2000): a physicist among crystallographers, Physica Scripta, 2015, 90, 038001

10. 1

Xu et Gao, Graphene chiral liquid crystals and macroscopic assembled fibres, Nature Communications, 2011, 2, 571

11. 1

NIMBE - CEA: Nanosciences et Innovation pour les Matériaux la Biomédecine et l'Energie - Commisariat à l'énergie atomique et aux énergies alternatives, What is measured in a Small Angle X-ray Scattering (SAXS) ?, 2013,

12. 1 2
Zhang et al., Novel X-Ray and Optical Diagnostics for Studying Energetic Materials: A Review, Engineering, 2020, 6, 992-1005
13. 1
Tardy et al., Exploiting Supramolecular Interactions from Polymeric Colloids for Strong Anisotropic Adhesion between Solid Surfaces, Advanced Materials, 2020, 32, 1906886
14. 1

Guinier (Contrib.), Ewald (Edit.), Personal Reminiscences, Fifty Years of X-ray Diffraction, N.V.A Oosthoeks's Uitgeversmaatschappij, Utrecht, Netherlands, 1962

  • No labels