Small-angle X-ray scattering

Small-angle X-ray scattering (SAXS) is a small-angle scattering (SAS) technique where the elastic scattering of X-rays (wavelength 0.1 ... 0.2 nm) by a sample which has inhomogeneities in the nm-range, is recorded at very low angles (typically 0.1 - 10°). In this angular range, information about the shape and size of macromolecules, characteristic distances of partially ordered materials, pore sizes and the like is contained. SAXS is capable of delivering structural information of macromolecules between 5 and 25 nm, of repeat distances in partially ordered systems of up to 150 nm. USAXS (ultra-small angle X-ray scattering) can resolve even larger dimensions.

The advantage over crystallography is that the samples need not be crystalline, the measurement is non-destructive and NMR methods encounter problems with macromolecules of higher molecular mass (> 30000-40000). However, owing to the random orientation of dissolved or partially ordered molecules there occurs spatial averaging which leads to a loss of information.

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Applications

SAXS is used for the determination of the microscale or nanoscale structure of particle systems in terms of such parameters as averaged particle sizes, shapes, distribution, and surface-to-volume ratio. The materials can be solid or liquid and they can contain solid, liquid or gaseous domains (so-called particles) of the same or another material in any combination. The method is accurate, non-destructive and usually requires only a minimum of sample preparation. Applications are very broad and include colloids of all types, metals, cement, oil, polymers, plastics, proteins, foods and pharmaceuticals and can be found in research as well as in quality control. The X-ray source can be a laboratory source or Synchrotron light which provides a higher X-ray flux.

SAXS instruments

The major problem that must be overcome in SAXS instrumentation is the separation of the weak scattered intensity from the strong main beam. The smaller the desired angle, the more difficult this becomes. The problem is comparable to one encountered when trying to observe a weakly radiant object close to the sun, like the sun's corona. Only if the moon blocks out the main light source does the corona become visible. Likewise, in SAXS the main beam must be blocked after its interaction with the sample, without blocking the closely adjacent scattered radiation. Most available X-ray sources produce divergent beams and this compounds the problem. In principle the problem could be overcome by focussing the main beam, but this is not easy when dealing with X-rays, because lenses are virtually non-existent. This is why most practical small angle devices must rely on collimation instead.

SAXS instruments can be divided into two main groups: point-collimation and line-collimation instruments:

1) point-collimation instruments have pinholes that shape the X-ray beam to a small circular or elliptical spot that illuminates the sample. Thus the scattering is centro-symmetrically distributed around the primary X-ray beam and the scattering pattern in the detection plane consists of circles around the primary beam. Owing to the small illuminated sample volume and the wastefulness of the collimation process -only those photons are allowed to pass that happen to fly in the right direction- the scattered intensity is small and therefore the measurement time is in the order of hours or days. Point-collimation allows to determine the orientation of non-isotropic systems (fibres, sheared liquids).

2) line-collimation instruments confine the beam only in one dimension so that the beam profile is a long but narrow line. The illuminated sample volume is much larger compared to point-collimation and the scattered intensity at the same flux density is proportionally larger. Thus measuring times with line-collimation SAXS instruments are much shorter compared to point-collimation and are in the range of minutes to hours. This disadvantage is that the recorded pattern is essentially an integrated superposition (a self-convolution) of many pinhole adjacent pinhole patterns. The resulting smearing can be removed using deconvolution methods based on Fourier transformation, but only if the system is isotropic.

Porod's law

SAXS patterns are typically represented as scattered intensity as a function of the scattering vector q=4π.sin(θ)/λ. One interpretation of this vector is that of a resolution or yardstick with which the sample is observed. In the case 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 1D projection, as usually recorded for a isotropic pattern this invariant = ʃI(q).q².dq, 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.

In the transitional range at the high resolution end of the SAXS pattern the only contribution to the scattering come from the interface between the two phases and the intensity should drop with the fourth power of q if this interface is smooth. This a consequence of the fact that in this regime any other structural features, e.g. interference between one surface of a particle and the one on the opposite side, are so random that they do not contribute. This is known as Porod's law:

lim_q->high I = S.q^-4

This allows the surface area S of the particles to be determined for SAXS. However, since the advent of fractal mathematics it has become clear that this law requires adaptation because the value of the surface S may itself be a function of the yardstick by which it is measured. In the case of a fractally rough surface area with a dimensionality d between 2-3 Porod's law becomes:

lim_q->high I = S'.q^-(6-d)

Thus if plotted logarithmically the slope of ln(I) versus ln(q) would vary between -4 and -3. Slopes less negative than -3 are also possible in fractal theory but require a volume fractal rather than a surface fractal and in a sense that would represent a single phase system (e.g. a solution with polymer molecules in it) rahter than a two phase system.

See also

• Biological Small angle X-ray scattering (SAXS)

External links and references

• Advances in structure analysis using small-angle scattering in solution
• Restoring low resolution structure of biological macromolecules from solution scattering using simulated annealing
• Addition of missing loops and domains to protein models by x-ray solution scattering
• SAXS instrumentation
• SAXS applications & laboratory equipment
• SAXS / WAXS applications & instrumentation
• SAXS at a Synchrotron
• Dedicated SAXS & multi-purpose XRD/SAXS Instrumentation
• SAXS/WAXS/Nanography Instrumentation