[MUSIC] An extremely useful way to characterize a large system of particles is to look at something known as the pair distribution function. The pair distribution function is simply the probability of finding a particle with index 1 at r1 and particle 2 at r2. Now this can be valuated by integrating over all the other particles 3 through n. Now, the actual index itself, 1 and 2, are dummy and so we don't actually care about the indices or the labels of these particles. Hence evaluating this as a joint probability density, which weights the probability by the total number of such combinations, which is simply given by N times N minus 1. Now, the pair distribution density is simply given by the equation. Now more generally we can write the Nth particle joint probability distribution as. Now for an isotropic liquid, the first joint probability distribution is simply given by the density of that medium. Now in the case of a non-interacting system, such as an ideal cast for instance, The second joint probability density is simply given by the square of the density. This simply says that you can invoke a mean field approximation with the average particle density simply given by the density of the median. Now an easy way to quantify interactions present in a liquid for instance is to measure correlations through the pair distribution function. Now it's frequently useful to normalize this with aid of a non-interacting system that is an ideal cast. Now for an isotropic fluid the pair distribution function only depends on the distance between the two particles and not their actual locations themselves. So hence, the actual functional dependence only depends on r and not on r1 and r2 respectively. Now the pair distribution function gives the average density of particles at a given distance r, given a tagged particle is at the origin. Now we've learned that liquids are interesting in that neither the potential energy nor the kinetic energy dominates. Now a really crucial way to distinguish different liquids is from a measurement of their pair distribution function. Now immediately we might ask the question, well how do we calculate it? The most efficient way to do this is actually to carry an experimental measurement known as x-ray diffraction. Now let's consider once again, the same situation, a collection of n particles in a liquid. Now to probe distances of the order of one angstrom, we irradiate the sample with x-rays or neutrons. Now our discussion here we will assume x-ray scattering. Now a beam of wavelength lambda illuminates the sample. The incident beam is coherent, that is all the incoming photons are in phase. Now due to a difference between the refractive index of the particles and their surrounding, each particle acts as a scattering center. Now let's compare the path lengths for scattering between two different particles. Now the position of the ith and jth particle, which are going to act as scattering centers are located at ri and rj respectively. Now the difference in the path length of the two photons is simply given by the difference between the lengths B and A marked in the figure, since these are the only two segments that differ for the two photons. The incident ray vector is along a direction u0 unit vector that defines the direction of the incident light. The magnitude of the incident ray vector depends on the index of refraction of that particular solution. Now in an analogous way, we can define the scattering vector. The scattering vector is along a unit vector us, which is along the scattered direction. Now let's assume that we can fix a detector that is held at a fixed angle data relative to the source that is the incoming radiation. Using these definitions, the two lengths B and A are simply given by the dot product of the distant vectors along the scattering and the incoming unit vectors, respectively. Now the path difference is simply given by the difference between these two lengths. The path difference itself results in a phase difference given by the following equation. Now the electric field of light scattered by the ith particle relative to the phase of light scattered by the kth particle is given by a sinusoid with the phase information contained in the argument of the sinusoid. Now note that in this equation, E0 is the amplitude of the incident electric field, and C incorporates a number of experiment specific parameters, for example, distance to the detector, wavelength, polarizability and so on. Now the total intensity of the scattered light is actually proportional to the square of the total electric field, hence the average scattering intensity is given by the intensity averaged over a complete oscillation period. The square of the cosign can be expanded using standard trigonometric identities. After using a little bit of algebra and trigonometric manipulations, we can show that the scattering intensity is simply dependent on the phase difference. Now we still need to determine the unknown experiment dependent constant C. How can we do this? Well we can do this by doing the following experiment. We can measure the scattering intensity at zero scattering angle, that is at the same angle at which the incoming photons are the incident. This gives us knowledge of all the unknown parameters in this equation. Now let's define a quantity called the structure factor, S. Now it's immediately obvious from the definition that the real part of the structure factor is the desired scattering intensity. The structure factor poses rotational invariance, that is S(k) = S(-k). Hence, this leads to the following simplification. Hence, knowledge of the scattering intensity now leads us to a knowledge of the structure factor. Now how can we relate the structure factor to the pair distribution function? Now, let's expand the structure factor into two parts, one part where the index i is equal to j, and the second part, where the index i is not equal to j. Now when i equals j the summation simply yields the total number N. Now when i is not equal to j, invoking the condition that we do not care about the index of variables, we only need to evaluate the average between two particles, say 1 and 2. So all of the other parts of the average from 3 to N cancels out between the numerator and the denominator, leading to the following simplification. Now it can be shown that this can be easily simplified to. Now the structure factor contains information about the factor termed as g tilde of k, this is simply the Fourier transform of g(k). This demonstrates that the results of a scattering experiment gives a direct measure of the pair distribution function of the particles in a liquid. Now where does this come in useful? Where is it handy? Now, we've discussed in the water lecture that water consists of two phases, a low density liquid and a high density liquid. Now how can we identify these two phases? Well these two phases have a distinctly different pair distribution function that allows us to fingerprint these two different phases. The high density liquid does not have characteristic solvation shells, and has a broad distribution in its first and second solvation shell. On the other hand, the low density liquid has a distinct first and second solvation shell. These signatures allow us to distinguish liquids and determine its solvation properties, which are crucial to their function in a wide variety of applications. Now to summarize, in this lecture we learned of a model called the Lennard-Jones potential to describe the interaction between two nuclei. We went on to describe the classical partition function and showed the emergence of an important quantity called the configurational partition function, which contains all the information of a classical partition function. We proceeded to try and characterize liquids and learned that the pair distribution function is an extremely useful quantity to fingerprint a liquid. We worked through the use of x-ray diffraction as an experimental tool to determine the pair distribution function. The knowledge of the pair distribution function allows us to identify the liquids and their properties, and their functionality.