In this lesson, we're going to be talking about qualitative discussions of slip in hexagonal close pack systems. If you recall, one of the ways that we can describe packing in the hexagonal close pack is to compare it to packing that we see using the hard sphere approximation for FCC. Remember the way the FCC structure worked was we had ABCABCABC packing, where as in the case of the hexagonal clothes pack as shown on the right, the stacking sequence is ABAB. So if we look along the C axis, that is the direction of the arrow with respect to the hexagonal close pack structure, that arrow direction, the magnitude of that is two thirds of the magnitude of the vector that takes you from the A through the B and the C back to the A again, which is in the face centered cubic structure. So what we've done in the past is to relate those two directions, the 1,1,1 direction in the FCC structure, and the C direction in the hexagonal closed packed. Now, what we were able to do then is to relate in the C axis how long the direction is or repeat direction is in the HCP. So we start out and recognize that, that arrow that's given the unit of one. The magnitude of that is a zero under the square root of three, so the distance of c is two-thirds of that. Then what we can do is to use the close pack directions in the playing of the one, one, one in the face center cubic structure. And that's going to be equivalent to the packing distances that we see in the HCP structure as well. And what we're then able to do is to recognize that a in the hexagonal system is going to be related to A0 under the square root of 2, divided by 2. So that's the distance in the FCC structure. So now we have both c and a expressed in terms of the dimensions of the unit cells with respect to the face center cubic system. And when we take a ratio, what we get is the ratio of C over A is the square root of eight over the square root of three. Now we've talked about this previously and the module that had to do with crystallography but I wanted to bring this to your attention. And remember, what this C over A Is in this particular case, it happens to be the ideal packing arrangement for the ATP structure, meaning that the spheres are perfectly or the structures are perfectly spherical for the atoms. And as a consequence, we have this ideal c over a ratio. When we look at actual materials, materials like cadmium through beryllium as illustrated on this table, what we see is that when we compare the c/a ration in this particular system. systems to the ideal hexagonal closed pack structure we see deviation with respect to that c over a ratio. When we look at zinc, for example, and we're going to be looking at zinc a little bit further along in this lesson. But when we look at zinc, it has a c/a ratio which is greater. This would suggest that we're stretching a bit in the c direction. It may also indicate that, rather than having perfect spheres, we have structures in terms of the atoms as being deviating from that, maybe an ellipse, so that the deviations lead to this increase in the c axis between those Densely packed plains in the HCP system. And then again when we start looking at elements like magnesium through barilium, we are now below what happened in terms of the packing sequence as compared to the ideal case. So let's take a look at the ideal system with respect to the HCP structure. We're looking at the ABA packing sequence. And you can see how those planes are related to one another and we refer to these planes as the basal planes. Those are the planes in the hexagonal. Packing plane. And they are the densest packed planes in the Hexagonal System. And as a consequence, what we would expect to see is, we would expect to see only slip on those densely packed planes. And they would be in the directions of the vectors indicated on the diagram. As into the direction of A1, A2, and A3 which described the HCP system. Now, it turns out if we deviate the C over A ratio, we begin to look at the process on other planes and of on the case that we have up here on the diagram. What we're actually looking at is the possibility of having slip on the prism plane. So, what we would have to do is to look at the atoms and how they are distributed on this plane and calculate the relative planer density. We come along and another possibility is that we can have slip on a pyramidal plane which is given in the diagram on the left. And another pyramidal plane that takes you halfway up from a to b, and that's another possible plane. So we can have a variety of different planes with respect to the deformation in HCP. But I want to point out that when you look at the close pack directions and the close pack planes, what you see is when you're talking about the number of potential slip systems, what you have is effectively slip on one plane, which is that densely packed plane and in the three directions a1, a2 and a3. So we're limited into the amount of slip that we can have in this system. What we're going to do is to use this to an advantage to show the possibility of slip in the systems and then to describe the fact that the critical reserve shear stress is a property of the material and not only that. We obey Schmid's law, which is the law that we described in a previous lesson. So when we take a look at deformation in the hexagonal system, what we can do is we can look at the critical resolve sheer stress that we have and now what we want to think about is how that close packed basal plane of the HCP structure works. One of the things that we know from an earlier lesson is that the plane of maximum shear that we can have in a cylinder of material Is it 45 degrees? So what that's going to help us do is if we were to take our single crystal of zinc and orient it, so that the basal plane was at 45 degrees. We will then be on the plane of maximum share and in this particular case, this is where the deformation is going to occur. Now if we take data by recognizing and determining what the critical resolve shear stress is at 45 degrees, what we can then do is to go back and use the equation that's at the top of the slide that describes Schmidt's Law. And then what we're able to do, by keeping the angle cosine theta, which is the slip direction and the deformation direction, keeping that constant. And then what we can do is to calculate what the variation and the critical resolve sheer stress would be. And that's the solid line. So the solid line, then, represents the values that we would calculate using Schmidt's Law. Now if we look at the red dots, the red dots actually represent the real data. So what this is telling us is that if we measure the critical resolve shear stress, or we measure the stress necessary to determine the deformation at different orientations, what we find is that we are in effect satisfying the condition of Schmidt's Law. And so, here is our diagram again. Here is our slip plane, we're looking at variations with respect to the slip plane. That is oriented at different orientation with respect to the stress axis and consequently were able to reproduce the curve on the left hand side by recognizing that we can determine at 45 degrees. What the critical resolve sheer stress is by looking at the minimum in this curve So that's our angle fined that we'd be interested in, and it is related to the angular relationship between the slip plane, normal, and the deformation access, and theta then represents the angle associated with the Force and the slip direction. In this lesson we talked a little bit about the process of deformation in a hexagonal close-packed system. We recognized that there is one densely packed set of planes, namely the basal planes, in the HCP system. And that packing density is going to be similar to the packing density that we have in FCC. But because of the limitation in the number of slip systems, we begin or the system begins to look for other possible slip systems where the planes are not quite as densely packed. Then we consider the possibility of looking at the Schmitt's law to in fact show that the critical result of sheer stress that we measure in a material where we can restrict slip like the zinc single crystal. Then what we're able to do is to make some calculations and in fact show that the slip process does follow Schmidt's rule. Thank you.