Monday, June 10, 2013

Understanding the static structure factor

The structure factor \(S(q)\) is an important quantity for characterizing disordered systems of particles, like colloids. Its significance comes from the fact that it can be directly measured in a light scattering experiment and is related to other quantities that characterize a system's microscopic arrangement and inter-particle interactions. However, it's difficult to learn about it the context of disorder because it's primarily used in crystallography. Crystals are far from being disordered.

In this post, I'll explore the nature of the static structure factor, which is something of an average structure factor over many microscopic configurations of a material. A dynamic structure factor describes the statistics of a material in time as well as space.

The structure factor of a disordered material can be measured by illuminating the material with a beam of some type of radiation (usually X-rays, neutrons, or light). The choice of radiation depends on the material. It is also important that the material should not scatter the incident beam too strongly because the structure function is typically found in singly-scattered radiation (see the first Born approximation for a discussion about a related concept). If the material is multiply scattering, the information about the material's structure, which is largely carried by the singly scattered light, is washed out.

In a measurement, the sample is usually placed at the center of rotation of a long rotating arm. A detector for the radiation is placed at the opposite end of the arm. The arm is rotated about this axis and the intensity of the scattered radiation as determined by the detector is recorded as a function of the angle. This data set essentially contains the structure factor, but must be transformed and corrected for, accordingly.

First, the structure factor is usefully represented as a function of the scattering wave number, \(q\), and not as a function of the scattering angle. In optics, \(q\) is usually given by
\[ q = \frac{4 \pi n}{\lambda} \sin( \theta / 2) \]

where \(n\) is the refractive index of the background material (usually a solvent like water) and \(\lambda\) is the wavelength of the light. \(\theta\) is the scattering angle.

Additionally, the structure factor must be corrected for a large number of confounding factors, such as scattering from the sample cell and radiation frequency-dependent detectors. A classic paper that details all these corrections to find \(S(q)\) in a neutron scattering experiment is given here.

Once the structure factor is found in an experiment, it may be Fourier transformed numerically to give the radial distribution function, \(g(r)\) (see Ziman for the proper conditions for which this applies) of particles. This function gives the probability of finding a particle at a radial distance from another particle in the system. Many important thermodynamic properties are related to \(g(r)\). Importantly, the pair-wise interaction potential between any two particles is related to \(g(r)\), and the pair-wise interaction determines many macroscopic system properties.

The structure factor as \(q\) (or equivalently the scattering angle) goes to zero is also an important quantity in itself. \(S(0)\) is equal to the macroscopic density fluctuations of particles in the medium (see Ziman, Section 4.4, p. 130). But density fluctuations can be calculated from thermodynamics and leads to the isothermal compressibility of a material.

In the language of optics, which I'll stick to for the rest of this post, the density fluctuations would correspond to large regions of refractive index variations across the sample.

This leads to an interesting problem, the resolution of which reminds me of the fallibility in taking some models too literally: for a homogeneous and non-scattering optical material, like a very nice piece of glass, the value of the density fluctuations in the refractive index are essentially zero (this is true because the disorder in a glass is at a length scale that is much smaller than the wavelength of light). This means that \(S(0) = 0\). At the same time, the scattered intensity in the type of experiment measured above is directly proportional to the structure factor:
\[I(q) \sim S(q).\]
So, if I illuminate a nice piece of glass with a laser beam, and I know that \(S(0)\) is equal to zero, the above expression means that there should be no scattered intensity in the forward direction. But this is a silly conclusion, because when I do this experiment in the lab I see the laser beam shining straight through the glass! In other words, \(I(0)\) is not zero.

The problem is that this expression is for the scattered intensity. In random media, we often talk about the scattered light and the ballistic light. The latter of these two is not scattered but directly transmitted through the material as if the material were not there. So, even though no light is scattered into the forward direction, there is still the ballistic, unscattered beam, that is passing straight through the sample.

Most small angle light scattering experiments measure as close as they can to \(q=0\) and extrapolate to the structure function's limiting value. \(S(0)\) can't actually be measured. But, it's determination is important for materials with significant long-range order, such as those near a phase transition, because the small angles correspond to large distances, due to their inverse Fourier relationship.

One can also engineer a material to not transmit any light into the forward direction. To do this, \(S(q)\) must be zero AND there must be no ballistic light passing through the material. This can be achieved with a crystal that diffracts all the light into directions other than the forward direction, such as a blazed grating.

On a final note, the structure factor can sometimes be related to important material properties beyond the radial distribution function. Ziman says in section 4.1, pg. 126 that the direct correlation function (which measures interactions between pairs of particles) can be derived directly from the structure factor. This correlation function is related to the Percus-Yevick model for liquids.