In this lesson we're going to be describing point defects or what we've already established as vacancies. So let's look for example about what happens when we introduce a vacancy inside of a pure substance. So here is our picture. And right in the center of the visual you can see where the vacant side is. All around that vacant side we see a certain amount of distortion of the atoms in the vicinity of where that atom has been taken away. And as a result of that we create distortions of the spheres in the vicinity of the vacancy. It turns out that in addition to having vacancies as point defects, we can also have interstitial atoms as defects. In this particular case we're going to be talking about the same materials. So this for example pure aluminum or pure copper, and occasionally one of those atoms is sitting in the wrong location. It's sitting at an interstitial position. And as a result of that atom sitting in that interstitial position it winds up creating distortions around it as well. So the point defects then come in two types. One of the types is the vacancy and the other type is the interstitial position. And so again we have the distortion here. Now the question becomes, how do these vacancies actually arise. So, I've illustrated up here in two dimensions, a array of spheres that represent my atoms in a two dimensional arrays, and the lines that connect them represent the bonds. Now what I'm going to do is to take that structure, and I'm going to randomly break some of these bonds. And when I do that, I'm going to start moving those atoms that resulted as a consequence of breaking those bonds, and I'm going to move them to the surface. And then when I do that I leave behind a vacant lattice site. And I going to go ahead and then take that atom and move it to the surface of the material. So, as I go through here, what you're going to begin to see is, I am creating these vacancies. They're happening randomly throughout the structure. And I've indicated where those vacant sites are by the red square that comes in. And this process goes on and on until I have a certain concentration of vacancies that exists in my material at a given temperature. Now, when I begin to think about what's happening during this process, I am actually putting energy into the system. So, I'm putting the energy into the system and the reason for doing that is I am actually breaking the bonds, and that requires a certain amount of energy. But at the same time if I start randomizing these vacancies, or missing atoms, throughout the structure, I'm going to wind up compensating for that energy. And as a result I get back some of the energy from the randomization process, and we can look at that from the point of view of some of the dynamics by considering I can express the free energy function G in terms of two parameters. The enthalpy, and the entropy of the system. Now, for a solid state process, I can let the enthalpy be related directly to the internal energy of the system. So when I look at enthalpy, I know that enthalpy is divided into energy plus this DeltaPV term, or pressure volume term. In the solid state processes and condensed matter, we're going to make the assumption that the change in energy associated with neglecting that term is actually fairly small. So I can then describe the change in energy associated with the energy that I am putting into the system and the energy that I get back, or the Gibbs free energy that I get back as a result of the randomization processed through the entropy term. Now, there's another way that we can think about a structure which contains vacancies. We can think about them as a solution. So if we have a pure substance, like aluminum or like copper, and we have vacancies distributed through it. We can consider the atoms in the structure as being the solvent, and we can then consider the vacancies as the solute. So now what we have is a dilute solution of vacancies. Now, when we look at the behavior of the concentration of vacancies as a function of temperature, we're going to see a very important point, and that point tells me that as I increase the temperature, I am increasing the energy that I'm putting in to the system, there by able to break more bonds, and as a consequence, I'm able to increase the concentration of vacancies. And we see it written in the form of an exponential equation. If we make that equation linear. Which I have done in the bottom line. What I can do is at two different temperatures, I can measure the concentration of vacancies by for example, making precise measurements of the densities of these different temperatures. Then what I can do is I can then calculate based upon that reasoning, I can calculate using this exponential function. I can calculate the activation energy that's associated with it and it comes about by looking at the slope of that line so I'm then able to make an estimate on the activation energy of the process. The term that I have in there, R is the normal gas constant, that in the cases that we're going to be dealing with it, at least initially, we're using units of joules per mole. So that gives us then a picture of the development of vacancies and the important fact that as temperature goes up, the vacancy concentration increases. Now what we would like to do in our subsequent discussions is to consider other types of materials as well. Thank you.