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 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. 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. SAS can refer to small angle neutron scattering (SANS) or small angle X-ray scattering (SAXS). 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. SAS patterns are typically represented as scattered intensity as a function of the magnitude of the scattering vector q = 4 π sin ⁡ ( θ ) / λ {displaystyle q=4pi sin( heta )/lambda } . Here 2 θ {displaystyle 2 heta } is the angle between the incident beam and the detector measuring the scattered intensity, and λ {displaystyle lambda } is the wavelength of the radiation. 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 SAS is simply Δρ, the difference in average scattering length density between the particle and the surrounding liquid, because variations in ρ due to the atomic structure only become visible at higher angles. This means that the total integrated intensity of the SAS 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 ∫ I ( q ) q 2 d x {displaystyle int I(q)q^{2},dx} , where the integral runs from q=0 to wherever the SAS pattern is assumed to end and the diffraction 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. SAXS is described in terms of the electronic density where SANS is described in terms of a neutron scattering length density. 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: 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: 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.