Before we leave the module associated with crystal structures, I want to briefly introduce the concept of X-ray diffraction and how we use X-ray diffraction to gain the information about the crystal structure of materials. And in order for us to begin to start out, what we're going to do is we're going to look at a number of sieves, and I've illustrated what I mean by a sieve here in the figure. On the left, I have a sieve which is very coarse so that the largest diameter of particle that actually will pass through this sieve is pretty coarse, in comparison to the size of the particle that would pass through the mesh that I have to the right. So this is a way that ceramists, for example, or geologists can separate out particle size distributions of materials. And the characteristics that are important about these meshes are obviously the opening. And so the larger the opening, the larger will be the biggest particle that can pass through. And depending upon how coarse these meshes are, we'll have wires of different diameters. So meshes are then developed based upon how big the opening needs to be and therefore an associated wire diameter to accompany it. Now what we're going to do is to take two different meshes and note that the mesh size is on the order of microns. We have something that's called 125 Mesh. And that's a particular size. And it corresponds to an opening of 112 microns. If we look at Mesh 625, that's a much finer mesh. And what we will see there is that the opening is on the order of 20 microns. One of the things that I want to draw your attention to is the fact that we have a distribution of points, and those points are coming about as a result of passing a laser through the mesh. And what we have on the left is a coarser mesh, and the one on the right is a fine mesh. But they're made up of an array of woven in a square pattern wires. Now what becomes very important in addition to the fact that we develop this symmetry, what we see is that if we look at different mesh sizes starting out with a mesh opening of a 112, progressing all the way down to an opening of 20 microns, what we see is that the size of the spots that differentiate, one mesh size from another, winds up having a reciprocal relationship with the actual physical object that we're working with. So, the courser the mesh, or the larger the mesh size, the smaller will be the spacing between those scattering points associated with the laser beam passing through the mesh. So we develop then this concept of a reciprocal relationship between the real image and the associated diffracted image. Now let's turn our attention to how we actually study using X-rays to scatter using a crystal. Now, the atomic scale structures that we're looking at and we're trying to understand are typically between 0.5 and 50 angstroms. And in that particular case, we can use X-rays. Now, in addition to the fact that we're looking at spacings that are on the order of 0.5 to 50 angstroms, we're looking at periodic arrangements. And those periodic arrangement leads to what we refer to as constructive and destructive interference. Much like the patterns that we saw in the previous demonstrations. Now what also becomes important, not only can we describe the symmetry that's associated with the overall structure, we begin to understand not only the symmetry, but the spacing, and also the locations of the atoms that are within the crystal structure. And we know that, for example, that the scattering of these individual atoms will depend upon the particular electron density or the particular element. So, as we increase the atomic number, we increase the electron density. And so, by understanding not only the scattering phenomenon from the point of developing symmetry, we also will get an understanding of the locations of the atoms in the crystal. Now, using the concept of the duality between particles and waves, rather than using X-rays, we can often use electrons. And there are applications for looking at neutrons for a variety of different purposes. So, not only can we look at, use X-rays, we can look at, we can use electrons and we can use neutrons. Now, we know with respect to a looking at a wave, and here I've indicated two waves. And what I want to do is to take those two waves and add them together. And what will happen is I will get a wave pattern associated with the sum of those two wave patterns which is given in the figure to the right. Now what I can also do is I can describe a phenomenon that's referred to as constructive interference. So if I take two sine waves, and they are in phase with one another, I develop a constructive interference pattern which the amplitude is associated with the sum of each of those individual waves. And that leads to a constructive interference. On the other hand, if these waves are out of phase, then what I'd get is a destructive interference pattern where I get that flat line that's given on the right. So we're going to use this idea of constructive interference to describe how we can begin to understand the crystal structure of a material. What I'd like to introduce now is the simple geometry that's used in understanding the development of structure in a material. And we're going to use X-rays to probe this. And what's illustrated here is a diffraction set up where we have an incident beam of X-rays and a scattered beam of X-rays that go to a detector. And remember we want to have constructive interference, which means that those waves that are going to the detector must all be in phase with one another to have constructive interference. So now what I want to do is to come up with a little bit of geometry here. So I'm looking at the red arrow first and it tells me the distance from my source to my crystal surface and the corresponding distance between the surface and my detector. So what I have is this one particular ray, it's bouncing from the surface and going to my detector. And now what I'm going to do is to look at ray which is parallel to the first ray, and it's going to go down to a position inside of the crystal that leads to point A. And then I'm going to look at the ray that goes from point C to the detector. The reason that I'm looking at those particular lengths is that I want to make sure that the conditions are such that the incident and defracted rays, when I look at all of the rays, like ray one and ray two, those are in phase with one another and will lead to constructive interference. In order for that to occur, I must have a particular geometry that comes about. Namely the distance between A to B has got to be equal to the distance between B and C. And more importantly, what has to be true is that that distance, A plus B, or AB plus BC, has to be an integral number of wave lengths in order to have the X-rays in phase. And so I'm going to describe those path lengths. The path length AB and the path length BC are going to be equal to one another and they're going to be equal to that simple geometry that I have of d sin theta, where d is the spacing between the parallel lines of the crystal. And, as a result, I'm going to be able to add those two together so that I get AB + BC, and that's going to be equal to 2 times d sin theta. And, of course, we know that the distance of AB + BC has to be one whole wavelength. Now if we begin to look at other rays that are coming parallel to our original ray, that's probing deeper into the crystal, what we find is that the last set of vectors that we have added wind up creating a distance of AB + BC times the quantity 2. So that winds up giving us the fact that that path length must be now twice the number of wavelengths. So that set of beams, to be in phase with the other set will be equal to an integer, namely 2 times lambda. As a result what we find is, if we start looking at multiple layers below the surface, so we're talking about going in further and further, deeper into the crystal. And for all of these waves to be in phase, the requirement is that n lambda, where n represents the integer and that's going to represent what we refer to as the order of diffraction, is equal to 2 d sin theta. If we're only interested in the spacing we can talk about something we refer to as first order, where n's going to be equal to 1 and we can solve the geometry lambda is equal to 2 d sin theta. So what that means is, if I understand or I know what the value of lambda is, and I know where the diffraction is occurring for this particular distance of my grid, namely my d spacing, I would then be able to determine what that value of d is. I'd like to illustrate an example here where we consider a series of parallel 1, 1, 1 planes. And I've put up the figure where we describe the stacking sequence A, B, C. So each one of those then is a 1, 1, 1 plane. Now what I can do is I can come up with a relationship between the spacing between these planes and the edges of the unit cell. And what I'm going to do is to write that expression as d for the inner planer spacings of 1, 1, 1 is going to be equal to a0 onto the square root of 3. Now that's the body diagonal that goes through the cube, and because I'm only talking about the distances between one 1, 1, 1 plane and the next parallel, I then have to divide that expression by 3. So that would then give me the relationship between the d spacing, and therefore that d spacing, given a certain wavelength, that diffraction will occur at a specific angle theta. And then what I can do is I can describe the relationship, Bragg's Law, for first order diffraction then as lambda equals 2 d sin theta. So if I have lambda, I know what d is. I can tell you where the angle should be for the defraction. Or alternatively, if I actually measure the diffraction and I see at what angle I get diffraction for this particular set of planes, then I can determine the spacing, the despacing from this expression. There is a simple general relationship that exists between the despacing for any set of planes, and remember these planes are integers. And that relationship simply reduces to a0 divided by the sum of the squares of the indices, h squared, k squared, and l squared, in the form that's on the slide. So now what we've done is to talk a little bit about the concepts of the rudimentary geometry associated with X-ray diffraction. Thank you.