ASAXS - In Depth -Part 1 - the basics                               

A more in depth look at the mathematics and procedures involved with Anomalous Small-Angle X-ray Scattering.

Small angle x-ray scattering (SAXS) was first observed during the 1930’s and has since been widely used for the structural characterisation of solid, and fluid, materials in the nanometer range. During a SAXS experiment, the sample is irradiated by a well defined, monochromatic x-ray beam and the scattered photons observed. When a non-homogeneous medium is irradiated, structural information of the scattered particles can be derived from the intensity distribution of the scattered beam at very low angles. Using SAXS it is possible to study both mono-disperse and poly-disperse systems. In the case of mono-disperse systems one can determine the size and shape of the scatterers, whereas for poly-disperse systems a size distribution can be calculated under the assumption that all the scatterers have the same shape. Since the work of William and Lawrence Bragg in 1913, the laws concerning the diffraction of x-rays by crystals have been well known. This work led to the relationship known as Bragg’s law.

PIC RIGHT: The 7T-MPW-SAXS beamline at BESSY. The picture shows the extendible x-ray light tube with the 2D gas detector at its left hand end.

For small-angle scattering, it is usual to work in reciprocal space which is a set of imaginary points constructed in such a way that the direction of a vector, q, from one point to another coincides with the direction of a normal to the real space planes ad the separation of those points (absolute value of the vector) is equal to the reciprocal of the real interplanar distance, which in our case is, d.

The relationship between momentum space, q, and real space, d, is:

Now by expansion using Euler’s identity one arrives at the formalism:

In the small angle approximation cos(q.d)=1 and sin(q.d)=q.d, giving for positive small angles:

where n=1,2,3.... I we substitute d from the above equation into the Bragg equation we arrive at:

In a x-ray scattering experiment, the scattered intensity is experimentally determined as a function of the scattering vector (whose magnitude is given by the momentum transfer equation above. The scattering intensity obtained from a small-angle scattering experiment can be expressed as,

where I0 is the primary beam intensity produced by a beam of cross-sectional area A, delta omega is the solid angle subtended at the sample by the detector, delta rho squared in the scattering contrast between phases within the sample, and I `(q) is the scattering function characterising the sample. The scattering function, I `(q) is a product of two further functions, the form factor, F(q) and the structure factor, S(q). The form factor is dependent on the shape of the phases, whilst the structure factor depends on the way the phases interact. Let us now consider the case of particles embedded in a matrix. The contrast is between the particles and the matrix, so the form factor will be that of the particles. If these particles a homogeneous spheres of radius, r, the form factor has been calculated by Lord Rayleigh in 1911 to be

The structure factor would describe the inter-particle interactions between scatterers, and is a function of local order and inter-particle potentials. If the system is dilute, i.e. the particles are far away from each other and without spatial correlation, then S(q)=1 over the whole q range. For such a dilute system it is assumed that the intensities are simply added to give the total scattering pattern, but with increasing concentration there is an increasing effect from inter-particle interferences. This interference comes from two sources: a pure geometric influence (impenetrability of the particles) and Coulombic interactions. The inter-particle interactions cause a change in the shape of the scattering curve, particularly at low-q, with the appearance of a low-q maximum. For an extended list and discussion of structure factors refer to the paper by Pedersen `Analysis of Small-angle Scattering Data from Polymeric and Colloidal Systems: Modelling and Least-squares Fitting.' Advances in Colloid and Interface Science 70, 171-201 (1997).

As I mentioned earlier, it is possible to obtain information regarding the size and shape of the scatterers within the sample. This information can be obtained by looking at the low-q and high-q limits of the scattering patterns. In the limit of low-q, qr << 1, it is possible to obtain information regarding the particle (scatterer) size. This is known as Guinier’s approximation and can be obtained by expanding the trigonometric terms in the Form factor as a series.

Substituting back into the equation for the scattered intensity, yeilds

However since the q4r4 term is very small, we can neglect it, writing instead

which is more commonly written as

since                                               The form of this equation can be adapted for any arbitrary shaped particle using the radius of gyration in place of r. The radius of gyration, Rg for a spherical object is

So, the particle size can be obtained by fitting either a straight line to the low-q limit of the scattering pattern represented on a log(I(q)) vs q2 graph, or by fitting an exponential to the Guinier part (qr << 1) on a log-log plot.

The particle shape can be obtained by looking at the high-q tail of the scattering pattern. In the limit of high-q, ( qr >> 1) the scattering pattern is representative of the surface properties of the particles. This is known as the Porod approximation, and can be applied when qr >>1. In this limit

and

This gives the scattered intensity to be

Porod’s approximation states that the scattered intensity, in the limit of high-q, should fall off as a function of q^(-4). This, of course, assumes that the particles are smooth spheres (since we started with the form factor for smooth spheres). Surface dimensions other than smooth will give exponents less than 4.