X-ray intensity fluctuation spectroscopy

Abstract

At the outset let me stress that X-ray intensity fluctuation spectroscopy (XIFS) is a diffraction technique. As such, the intuition and expertise that you have developed for diffraction carries over to this new technique. The tendency in this chapter is to emphasize the aspects that are different from conventional diffraction but results learned from conventional diffraction will be called upon as needed. Wave phenomena are very prevalent in nature. Essential features of wave behaviour are the effects of interference and diffraction. For example, most textbooks on X-rays derive the X-ray diffraction by considering X-rays as planes waves and calculating the constructive and destructive interferences of these waves diffracting from atoms. In spite of the predominance of interference phenomena fundamental to wave phenomena, we are accustomed to the lack of such phenomena when dealing with conventional light. In order to describe whether or not one needs to consider these effects one should discuss the coherence of wave sources. Just the mention of coherence almost universally brings to mind lasers. Although lasers are intrinsically coherent sources of light, coherence effects can be seen from any source of waves. As an introduction to intensity fluctuation spectroscopy (IFS), let me first describe "speckle". Speckle is an effect, often seen in laser light, that results when coherent light reflects or scatters diffusely off disordered material. The intensity at each spot in the "image" is the result of light from many different points in the disordered materials. The essentially random path lengths of the light from these points to the given spot in the image leads to the light being the sum of rays with a random set of phases. However, since the light is coherent, the phase of the resulting light, even though it is randomly distributed from point to point, has a definite value at each point. Where the phases add destructively, it results in a dark spot and where they add constructively a bright spot. This is the origin of the speckled appearance of the image. It is a good thing for us that conventional light sources are incoherent, since speckle could be quite annoying in everyday life. As an introduction to the mathematics needed to deal with speckle and coherence, I would like to give an alternative derivation 1. Assume we have a set of N particles, randomly distributed throughout some volume. The scattering I(q) at wavevector q from this system is (ignoring constants) (equation presented) where fj is the scattering factor for each particle. The last equation has been specialized to identical particles, f0 . From this equation it is easy to see that the average intensity (averaged over particle positions) is N times the scattering of each particle (equation presented) To see why there is "speckle", one looks at the standard deviation of the intensity. First, calculate the variance. For identically distributed values, this will be the number of objects times the variance of each object. Since the phase factor above varies between plus and minus one we can estimate the variance of each object as (equation presented) We can now conclude that the scattering from N randomly placed identical particles is I(q) I(q) and this varies from zero to about twice the average. A detailed calculation of the distribution of intensities shows, for perfectly coherent light, that the most probable intensity is actually zero! It is easy to numerically simulate the above ideas. Figure 1a shows a randomly generated set of diamond shaped particles 2 distributed in a plane. Figure 1b shows the 2D fast Fourier transform (FFT) of these particles. One of the striking features in the image is that the speckles have a characteristic size. Since speckle results from the sum of cosines in equation {1}, the sharpest features are due to particles that are the furthest apart. Hence the speckle size is given by one over the size of the scattering region. For comparison, the FFT of a single particle is shown in figure 1c and the FFT of points replacing the particles in figure 1d. (figure presented) As helped by figure 2, close inspection shows that to a high degree of approximation the speckle pattern is the product of the single pinhole and point position scattering. (figure presented) Reviewing the above arguments it is difficult to imagine why we don't see speckle more often in everyday life. We will leave this as a puzzle until the next section. With this relatively detailed introduction to speckle it is now easy to see the origin of intensity correlation spectroscopy. If the particles shown in figure 1a where to fluctuate in position, the interference pattern in figure 1b would also fluctuate. Intensity fluctuation spectroscopy is the technique of measuring these intensity fluctuations and relating them to the kinetics of materials undergoing diffraction. One of the powerful features of IFS is that it gives direct access to measuring thermodynamic fluctuations even in equilibrium systems. This is a technique that was first developed for light scattering in the late sixties and is often called dynamic light scattering. Recently, the technique has been extended to X-rays (which is abbreviated as XIFS). X-rays have the advantage over light that most things are transparent to X-rays, the problems of multiple scattering are not as pronounced or non-existent, and the shorter wavelengths probe smaller distances. Their prime disadvantage is that X-ray sources are not as intense as laser light sources. This chapter will first describe how to characterize the coherence of a beam and then how coherence effects diffraction. This is followed by a general discussion of kinetics in materials and some examples of the technique. © 2006 Springer.