# Small-angle X-ray scattering

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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°). This angular range contains information about the shape and size of macromolecules, characteristic distances of partially ordered materials, pore sizes, and other data. 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.Glatter & Kratky] USAXS (ultra-small angle X-ray scattering) can resolve even larger dimensions.

SAXS and USAXS belong to a family of X-ray scattering techniques that are used in the characterization of materials. In the case of biological macromolecules such as proteins, the advantage of SAXS over crystallography is that a crystalline sample is not needed. NMR methods encounter problems with macromolecules of higher molecular mass (> 30000-40000). However, owing to the random orientation of dissolved or partially ordered molecules, the spatial averaging leads to a loss of information in SAXS compared to crystallography.

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. Not only particles, but also the structure of ordered systems like lamellae, and fractal-like materials can be studied. 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.

AXS instruments

In a SAXS instrument a monochromatic beam of X-rays is brought to a sample from which some of the X-rays scatter, while most simply go through the sample without interacting with it. The scattered X-rays form a scattering pattern which is then detected at a detector which is typically a 2-dimensional flat X-ray detector situated behind the sample perpendicular to the direction of the primary beam that initially hit the sample. The scattering pattern contains the information on the structure of the sample.

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 non-scattered beam that merely travels through the sample must be blocked, "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 "focusing" the beam, but this is not easy when dealing with X-rays and was previously not done except on synchrotrons where large bent mirrors can be used. This is why most laboratory small angle devices rely on collimation instead.

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

# 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 &mdash; only those photons are allowed to pass that happen to fly in the right direction &mdash; the scattered intensity is small and therefore the measurement time is in the order of hours or days in case of very weak scatterers. If focusing optics like bent mirrors or bent monochromator crystals or collimating and monochromating optics like multilayers are used, measurement time can be greatly reduced. Point-collimation allows to determine the orientation of non-isotropic systems (fibres, sheared liquids).
# 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. Line collimation is becoming less and less used in small-angle X-ray scattering due to the increasing number of synchrotron sources which are all point sources, and due to the availability of more powerful X-ray laboratory sources in combination with new multilayer optics.

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 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 &Delta;&rho;, 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 &Delta;&rho;2. In 1-dimensional projection, as usually recorded for an isotropic pattern this invariant quantity becomes $int I\left(q\right)q^2,dx$, 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.

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:

:: $lim_\left\{q ightarrow infty\right\} I\left(q\right) propto Sq^\left\{-4\right\}$

This allows the surface area S of the particles to be determined with 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:

:: $lim_\left\{q ightarrow infty\right\} I\left(q\right) propto S\text{'} q^\left\{-\left(6-d\right)\right\}$

Thus if plotted logarithmically the slope of ln(I) versus ln(q) would vary between -4 and -3 for such a surface fractal. Slopes less negative than -3 are also possible in fractal theory and they are described using a volume fractal model in which the whole system can be described to be self similar mathematically although not usually in reality in the nature.

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 2&pi;/"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:

::$I\left(q\right) = P\left(q\right)S\left(q\right) ,$where
*$I\left(q\right)$ is the intensity as a function of the magnitude $q$ of the scattering vector
*$P\left(q\right)$ is the form factor
*and $S\left(q\right)$ 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 from factor $P\left(q\right)$. One can then easily apply the Guinier approximation (also called Guinier law), 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 $p\left(r\right)$, which may be calculated from the intensity using a Fourier transform [Feigin & Svergun, s. 40.]

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

The distance distribution function $p\left(r\right)$ 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 $p\left(r\right)$-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.D.I. Svergun & M.H.J. Koch. Small-angle scattering studies of biological macromolecules in solution. "Rep. Progr. Phys." 2003 66, 1735-1782.] 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.

* Biological Small angle X-ray scattering (SAXS)
* X-ray scattering techniques (SAXS instrumentation manufacturers)
* GISAXS

References

*Glatter, O. & Kratky, O., eds. Small Angle X-ray Scattering. Academic Press, 1982. ISBN 0-12-286280-5 [http://physchem.kfunigraz.ac.at/sm/Software.htm Available online]
* L.A. Feigin & D.I. Svergun: Structure Analysis by Small-Angle X-Ray and Neutron Scattering. New York: Plenum Press, 1987. ISBN 0-306-42629-3 [http://www.embl-hamburg.de/ExternalInfo/Research/Sax/reprints/feigin_svergun_1987.pdf Online version]

Notes

* [http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=pubmed&dopt=Abstract&list_uids=12464319 Advances in structure analysis using small-angle scattering in solution]
* [http://www.ncbi.nlm.nih.gov/pubmed/17284163 Recent review of the SAXS technique, with an emphasis on structure reconstruction]
* [http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=pubmed&dopt=Abstract&list_uids=10354416 Restoring low resolution structure of biological macromolecules from solution scattering using simulated annealing]
* [http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=pubmed&dopt=Abstract&list_uids=12496082 Addition of missing loops and domains to protein models by x-ray solution scattering]
* [http://www-ssrl.slac.stanford.edu/conferences/workshops/scatter2006/talks/pople_saxs_workshop_060522.pdf#search=%22%22introduction%20to%20SAXS%22%22 SAXS at a Synchrotron]

Examples of SAXS beamlines

* [http://hasylab.desy.de/facilities/doris_iii/beamlines/b1/index_eng.html Anomalous small-angle x-ray scattering (ASAXS), Hamburg]
* [http://www.elettra.trieste.it/experiments/beamlines/saxs/ SAXS Beamline at Elettra, Trieste, Italy]
* [http://hasylab.desy.de/facilities/doris_iii/beamlines/bw4/index_eng.html BW4 Beamline at Desy, Hamburg, Germany]
* [http://www.srs.ac.uk/srs/ Synchrotron Radiation Source, Daresbury, England ]
* [http://www.esrf.eu/ European Synchrotron Radiation Facility, Grenoble, France]
* [http://www.lnls.br/principal.asp?idioma=2&conteudo=118&opcaoesq=/ D11A beamline at Brazilian Synchrotron Light Laboratory]
* [http://www.delta.uni-dortmund.de/index.php?id=2/ BL9 beamline of DELTA, Technical University of Dortmund, Germany]

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