Small-angle scattering

Small-angle scattering (SAS) is a scattering technique based on deflection of collimated radiation away from the straight trajectory after it interacts with structures that are much larger than the wavelength of the radiation. The deflection is small (0.1-10°) hence the name small-angle. SAS techniques can give information about the size, shape and orientation of structures in a sample.

Small-angle scattering (SAS) is a powerful technique for investigating large-scale structures from 10 Å up to thousands and even several tens of thousands of angstroms. The most important feature of the SAS method is its potential for analyzing the inner structure of disordered systems, and frequently the application of this method is a unique way to obtain direct structural information on systems with random arrangement of density inhomogeneities in such large-scales.

Currently, the SAS technique, with its well-developed experimental and theoretical procedures and wide range of studied objects, is a self-contained branch of the structural analysis of matter. Reflecting these situations, the international meeting on the small-angle scattering studies have been held in every three years. SAS can refer to:

• Small angle neutron scattering (SANS)
• Small angle X-ray scattering (SAXS)

Applications

Small-angle scattering is particularly useful because of the dramatic increase in forward scattering that occurs at phase transitions, known as critical opalescence, and because many materials, substances and biological systems possess interesting and complex features in their structure, which match the useful length scale ranges that these techniques probe. The technique provides valuable information over a wide variety of scientific and technological applications including chemical aggregation, defects in materials, surfactants, colloids, ferromagnetic correlations in magnetism, alloy segregation, polymers, proteins, biological membranes, viruses, ribosome and macromolecules. While analysis of the data can give information on size, shape, etc., without making any model assumptions a preliminary analysis of the data can only give information on the radius of gyration for a particle using Guinier's equation.

Theory

Continuum description

SAXS patterns are typically represented as scattered intensity as a function of the magnitude of the scattering vector $q=4\pi \sin (\theta ) / \lambda$. Here $2\theta$ is the angle between the incident X-ray beam and the detector measuring the scattered intensity, and $\lambda$ is the wavelength of the X-rays. One interpretation of the scattering vector is that it is the resolution or yardstick with which the sample is observed. In the case of a two-phase sample, e.g. small particles in liquid suspension, the only contrast leading to scattering in the typical range of resolution of the SAXS is simply Δρ, the difference in average electron density between the particle and the surrounding liquid, because variations in ρ due to the atomic structure only become visible at higher angles in the WAXS regime. This means that the total integrated intensity of the SAXS pattern (in 3D) is an invariant quantity proportional to the square Δρ². In 1-dimensional projection, as usually recorded for an isotropic pattern this invariant quantity becomes $\int I(q)q^2\,dx$, where the integral runs from q=0 to wherever the SAXS pattern is assumed to end and the WAXS pattern starts. It is also assumed that the density does not vary in the liquid or inside the particles, i.e. there is binary contrast.

SANS is described in terms of a neutron scattering length density.

Porod's law

Main article: Porod's law

At wave numbers that are relatively large on the scale of SAS, but still small when compared to wide-angle Bragg diffraction, local interface intercorrelations are probed, whereas correlations between opposite interface segments are averaged out. For smooth interfaces, one obtains Porod's law:

$I(q) \sim Sq^{-4}$

This allows the surface area S of the particles to be determined with SAS. This needs to be modified if the interface is rough on the scale q⁻¹. If the roughness can be described by a fractal dimension d between 2-3 then Porod's law becomes:

$I(q) \sim S' q^{-(6-d)}$

Scattering from particles

Small-angle scattering from particles can be used to determine the particle shape or their size distribution. A small-angle scattering pattern can be fitted with intensities calculated from different model shapes when the size distribution is known. If the shape is known, a size distribution may be fitted to the intensity. Typically one assumes the particles to be spherical in the latter case.

If the particles are dispersed in a solution and they are known to be monodisperse, all of the same size, then a typical strategy is to measure different concentrations of particles in the solution. From the obtained SAXS patterns one can extrapolate to the intensity pattern one would get for a single particle. This is a necessary procedure that eliminates the concentration effect, which is a small shoulder that appears in the intensity patterns due to the proximity of neighbouring particles. The average distance between particles is then roughly the distance 2π/q*, where q* is the position of the shoulder on the scattering vector range q. The shoulder thus comes from the structure of the solution and this contribution is called the structure factor. One can write for the small-angle X-ray scattering intensity:

$I(q) = P(q)S(q) ,$

where

• $I(q)$ is the intensity as a function of the magnitude $q$ of the scattering vector
• $P(q)$ is the form factor
• and $S(q)$ is the structure factor.

When the intensities from low concentrations of particles are extrapolated to infinite dilution, the structure factor is equal to 1 and no longer disturbs the determination of the particle shape from the form factor $P(q)$. One can then easily apply the Guinier approximation (also called Guinier law, after André Guinier), which applies only at the very beginning of the scattering curve, at small q-values. According to the Guinier approximation the intensity at small q depends on the radius of gyration of the particle.

An important part of the particle shape determination is usually the distance distribution function $p(r)$, which may be calculated from the intensity using a Fourier transform

$p(r) = \frac{r^2}{2\pi^2}\int_0^\infty I(q)\frac{\sin qr}{qr}q^2dq.$

The distance distribution function $p(r)$ is related to the frequency of certain distances $r$ within the particle. Therefore it goes to zero at the largest diameter of the particle. It starts from zero at $r = 0$ due to the multiplication by $r^2$. The shape of the $p(r)$-function already tells something about the shape of the particle. If the function is very symmetric, the particle is also highly symmetric, like a sphere. The distance distribution function should not be confused with the size distribution.

The particle shape analysis is especially popular in biological small-angle X-ray scattering, where one determines the shapes of proteins and other natural colloidal polymers.

History

Small-angle scattering studies were initiated by André Guinier (1937). Subsequently, Peter Debye, Otto Kratky, Günther Porod, R. Hosemann and others developed the theoretical and experimental fundamentals of the method and they were established until around 1960. Later on, new progress in refining the method began in the 1970s and is continuing today.

Organisations

As a 'low resolution' diffraction technique, the worldwide interests of the small-angle scattering community are promoted and coordinated by the Commission on Small-Angle Scattering of the International Union of Crystallography. There are also a number of community-led networks and projects. One such network, CanSAS - the acronym stands for Collective Action for Nomadic Small-Angle Scatterers, emphasising the global nature of the technique, champions the development of instrumental calibration standards and data file formats. It has also created a Web Portal to a variety of SAS-related resources.

Textbooks

• André Guinier, Gerard Fournet: Small-angle scattering of x-rays. New York: John Wiley & Sons (1955)
• O. Glatter, Otto Kratky (eds.): Small Angle X-ray Scattering. London: Academic Press (1982). Out of print, available online.