Continuum description

SAS patterns are typically represented as scattered intensity as a function of the magnitude of the scattering vector ${\displaystyle q=4\pi \sin(\theta )/\lambda }$. Here ${\displaystyle 2\theta }$ is the angle between the incident beam and the detector measuring the scattered intensity, and ${\displaystyle \lambda }$ 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 ${\textstyle \int I(q)q^{2}\,dx}$, 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.

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:

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

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:

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

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.

Form factor of a spherical nanoparticle analyzed through a Huygens-Fresnel-Feynman arrow-adding construction ^[3].

If the particles are in solution and known to have uniform size dispersity, then a typical strategy is to measure different concentrations of particles in the solution. From the SAXS patterns obtained 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:

$${\displaystyle I(q)=P(q)S(q),}$$

where

• ${\displaystyle I(q)}$ is the intensity as a function of the magnitude ${\displaystyle q}$ of the scattering vector
• ${\displaystyle P(q)}$ is the form factor
• and ${\displaystyle S(q)}$ 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 ${\displaystyle P(q)}$. 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.^[4]

An important part of the particle shape determination is usually the distance distribution function ${\displaystyle p(r)}$, which may be calculated from the intensity using a Fourier transform^[5]

$${\displaystyle p(r)={\frac {r^{2}}{2\pi ^{2}}}\int _{0}^{\infty }I(q){\frac {\sin qr}{qr}}q^{2}dq.}$$

The distance distribution function ${\displaystyle p(r)}$ is related to the frequency of certain distances ${\displaystyle r}$ within the particle. Therefore, it goes to zero at the largest diameter of the particle. It starts from zero at ${\displaystyle r=0}$ due to the multiplication by ${\displaystyle r^{2}}$. The shape of the ${\displaystyle p(r)}$-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.^[4] 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.

Conference history

• 2024, XIX, Taipei, ROC Taiwan
• 2022, XVIII, Campinas, Brazil
• 2018, XVII, Traverse City, Michigan, US
• 2015, XVI, Berlin, Germany
• 2012, XV, Sydney, Australia
• 2009, XIV, Oxford, UK
• 2006, XIII, Kyoto, Japan
• 2002, XII, Venice, Italy
• 1999, XI, Upton, New York, US
• 1996, X, Campinas, Brazil
• 1993, IX, Saclay, France
• 1990, VIII, Leuven, Belgium
• 1987, VII, Prague, Czechoslovakia
• 1983, VI, Hamburg, Germany
• 1980, V, Berlin, Germany
• 1977, IV, Gatlinburg, Tennessee, US
• 1973, III, Grenoble, France
• 1970, II, Graz, Austria
• 1965, I, Syracuse, New York, US

Awards

Several awards are presented at the international conference.