ASAXS - In Depth -Part 4 - deconvolution of the
ASAXS patterns.
As previously explained, in order to model an anomalous, or regular, small angle scattering pattern one must have some information regarding the type of sample under consideration. In small angle scattering the observed scattering pattern can be described as being proportional to a product of the scattering contrast, the form factor and the structure factor. There are some other variables which are important to obtain an absolute intensity scaling, e.g. beam current, transmission, particle volume etc, however these factors do not change the overall scattered lineshape, and so will not be discussed here.
The particle form factor , in essence describes the shape of the particles. in the following formalisms of the form factor it is assumed that the particles are randomly orientated within the sample such that the theoretical form factor has to be averaged over all orientations. The simplest form factor is that of homogeneous sphere of radius, r.
where F(q,r) is the particle form factor and q is the scattering vector.
PIC LEFT: A graphical representation of the scattering pattern produced by a single isolated homogeneous spherical particle. the green line represents a power law envelope of q^(-4).
PIC RIGHT: A graphical representation of the scattering produced by a set of homogeneous spheres with a size distribution (ignoring inter-particle interactions). Note the smearing of the minima as the size distribution widens. (The steps seen at high-q are due to computation errors and are not representative at a physical scattering event.
One can see in the figure to the right that the introduction of a size distribution causes a smearing of the minima as the distribution widens. However, this smearing does not cause a change in the envelope function.
The structure factor describes the inter-particle interactions and there are only a few cases for which it can be calculated analytically. Some of the analytically derived structure factors can be found in 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).
Before we can model the scattering patterns we must deal with the problem of ASAXS contribution deconvolution. This can be done in a variety of ways, but in essence it is a set of simultaneous equations with three unknowns (the desired ASAXS contributions).
We have three energies, with f0, f ` and f `` different for each energy, so we have the matrix equation
[A](x)=(b), where [A] is a square matrix, and (x) and (b) are column vectors, which can be solved in a variety of ways.
