Now what we'll do is to apply the concepts that we learned in the last lesson to describe how we can take a single crystal that's oriented in a specific direction and come up with the stress necessary to initiate deformation on a particular plane. Before we do that, let's go back and review some of the concepts that we learned earlier. First of all, we know that slip occurs on the densely packed plane and in the face center cubic. Those densely packed planes are the Type 111, which is given on the slide. In addition to that, the slip directions are in the directions of the closest approach. That is, the atoms that are touching one another, and we know from this diagram that we see we have a0/2 onto the vector (1,1,0). We now have described what we refer to as the slip system, the slip plane, and the slip direction. Now, if we go back and we think about a face centered cubic crystal, what we're looking at now is onto the plane of densest packing. Remember that, on the plane of densest packing, we're looking at a vector coming out of that plane that would be the 111 direction, and what I've done is to indicate the arrows for the distances of closest approach for the atoms that are in that plane. I've defined a particular axis, defined a direction of my vectors and I've applied the vectors to these various atoms that are sitting adjacent to one another in this close packed structure. So if we look at all of the possible slip systems in a cubic material that stays center cubic, we can see that they form a tetrahedron. With respect to that tetrahedron, every face is a 111 type face and every edge of the tetrahedron Are directions that are parallel to 1, 1, 0's. Now what we can do is figure out exactly how we can put all of these concepts together and review some of the basic ideas of vectors and planes and how we determine The vectors to the planes. So what I have, I've illustrated here a one one type plane so you can see I intersect x, y, and z each at one, take the reciprocals and I will then get the one one one plane. Then what I did was to take The corners. So the first corner is the position 0, 0, 1. And then I look down here and I see the position 1, 0, 0. And if I do tip, you subtract the tail of the arrow and the tip, then I would get the vector A which lies along The direction [101]. And remember what I've done is I've used square brackets to describe that particular direction. Now when we look at the B Vector, once again, its the tip of the arrow minus the tail and then what we get is the [011] vector. Okay, so those are our two vectors that lie in this plane. And what I can then do is to determine the third vector, which is going to be along the direction from the corner position at 1,0,0, to the other corner position at 0,1,0. Now one of that things that I know is I've been able to define the slip plane as being the plane where I have my three intercepts. This was introduced in. A previous module where we were introducing the notions of crystallography. Then, what we're looking at is, in a cubic crystal, the one thing that we know is that the vector that's normal to the plane in the cubic material has the same indices as the plane. In other words, if I know the one one one plane then i immediately know that the normal to that is going to be the one one one direction, so that's one way that I can easily find out what the particular plane is. Then an alternative way is I can take Two vectors on the plane and I can cross those two vectors and if I set them up as a right handed system what I see is that a cross b is going give me the normal. And what happens in that one is I use my normal vector analysis to come up with the vector i plus j plus k or Looking at it from the point of the way we described vectors in crystallography. That would be simply the 1-1-1 direction. So these are different ways that I can establish what we mean by the particular slip plane, and I think it's instructive for you to go through and make sure you understand each of these procedures, so you can easily identify directions and planes. Thank you.