Today we're going to talk about the unit cell and a very important feature is the Miller indices. We're going to use Miller indices to define crystallographic planes. You know an individual plane or 2D surface is going to be defined by two vectors in the plane and a normal vector. But we'll also define them using the Miller Indices. So if we have two vectors in the plane, V_1 cross V_2, that's going to give us the normal vector. Hence we can define that plane by h, k, and l of the normal vector or the two vectors in the plane, or by this bracket called the plane, this represents an individual plane. What are those brackets? We've already utilized the square brackets and the carat brackets for directions and now we're going to use a smooth brackets to identify individual planes. Why are we interested in planes? When we talk about mechanical properties, especially like metals, they will deform along specific crystallographic planes along specific direction. If I look at how many things are made, in this example, a laser, it has to be grown layer by layer on a specific point to get the ideal properties and we know that if a gem such as diamond, we need that diamond to be cut along a specific crystallographic plane. That gives us its brilliance and a symmetry. Because remember the whole objective with our fancy diamond we want the light to come in and bounce off. That can only occur by minimizing the number of defects by specifically cutting it along specific crystallographic planes. How do we index the plane? First thing we do, we have to define our origin. Typically, we're going to use a right-handed coordinate system and when we say a right-handed coordinate system, x cross y should always give me a positive z. Make sure when you're drawing planes, you stay with a right-handed coordinate system. The first thing we do is define the origin O. Typically it's going to be at the point 0, 0, 0 where the unit vectors are orthogonal. Then what we're going to do is record where the intercepts are. In this particular example, the plane here, this blue 2D surface, intersects the x-axis at 1 and it intercepts the y-axis at 1, is parallel to the z-axis and I know the mathematicians are going to say that I'm lying, but we're going to say it intercepts at infinity. Now, what we're going to do, we take the reciprocals of the intercepts, 1 over 1 is 1, 1 over 1 is 1, 1 over infinity is 0. Now we have 1, 1, 0. If we have negatives, we put a bar on top and finally, we put it in, oh, and we have to only have integers. Miller indices can only be integers, so we multiply whatever factor to remove the any fraction and now we've put it in smooth brackets to give us specific plane. This plane is referred to as the 1, 1, 0 plane. Now, in the cubic system, we have the luxury that the normal vector here, h, k, and l is equal to the Miller indices in square brackets, so here I have the 1, 1, 0 plane. Its normal vector is going to be 1, 1, 0, one the luxuries of utilizing the cubic system.