 Okay, let's go ahead and draw each of these cubic cells, and then we're going to talk about the number of lattice points, each one of these halfs, okay? And I'll describe to you what lattice points are as well. So if everybody could draw with me, right? So what you kind of got to do is twist your mind from two-dimensional to three-dimensional, just kind of like what these pictures are up here. So, I'm going to carry it away. Okay, so a simple cubic is going to look something like this with atoms at each one of the corners of the cube there. Hopefully you guys can see that I've made eight dots. Okay, and what you'll find is the lattice points for a unit cell. So that's z, so the number of lattice points in that cell is going to be the effective number of atoms that are contained within the unit cell. Is everybody okay with that? So in the case of a simple cubic or a primitive cubic, I believe your book uses, the number of lattice points for a simple cubic is going to be those eight atoms, okay? So eight times the amount of each of those atoms that's within that cubic cell. So there's one eighth of each one of those atoms within that cubic cell. Is everybody okay with that? And if you're not okay with it right now, think about it. I'll show you a picture or I'll show you an actual model when we get to the lab and you can see for yourself. Okay, so one eighth of each one of those is within that unit cell. So you go eight times one eighth, one eighth, and that equals one. So that's the number of lattice points. So effectively there's one atom within that unit cell. Is everybody okay with that? So let's do a body-centered cubic now. Okay, so the body-centered cubic, there's the cube that we're going to use. Remember it's got atoms on each one of those corners as well. Okay, so let's do that. But it's also got one directly in the center, right? Body-centered. Is everybody okay with that? So like that. So kind of try to twist your mind to make yourself see that one in the center. So let's figure out the number of lattice points for a body-centered cubic. So it's going to be very similar to this one, right? Because there's those eight atoms on the corners that are going to have one eighth of each one of them inside of that unit cell. Is everybody okay with me saying that? So what would we do? You can help me out if you want. Eight times. What is it? One eighth. Good job. Okay. Plus, what do you think it would be? Plus what? Plus one. Plus one. Very good. Why one? Because of the one that's in the middle. Because there's one in the middle. Okay? So there's an eighth of eight of them, plus one whole one in the middle. Very impressive. That's awesome. So what is that? Eight times one eighth is one, plus one. It's two. It's two. I don't even need my calculator. It's not even me. Everybody's okay with that? No. It's not even me. Everybody's okay with that? Makes sense? And let's do the last one, the face-centered qubit. So I expect if I were to draw one of these on the test that you guys could identify all of the stuff that we're talking about right now, and of course more when we talk about some others. Okay? Face-centered qubit also has atoms on each one of those corners of the cube. Remember? And if you don't, it's right up there. But what else does it have, guys? One on each of the faces, right? So how many faces are there on a cube? Six. Very good. So let's draw one on each of the center of those faces. And again, this is going to, hopefully you guys can see it, but maybe after I see that, I didn't want to use different colors because I didn't want you to think they were different types of atoms because we're talking about metallic solids here. So face-centered qubit, is it going to be similar to these ones in any way? Yes, it was the eight. Yeah. Because of that eight one. Yeah, very good. So Z of FCC is going to have, as its first term, eight times one eight. But it's going to have something added to it, right? What would that be? Six. Six. Very good. Awesome. Six times one half. And why would it be six? Because of course there's six on each one of the, or one on each one of the six faces. And there's half of that atom that's within the unit set. Is everybody okay with that? So if we do this, of course, eight times one eight is one. Six times one half is three. So one plus three is four. So the number of lattice points for the face-centered qubit is going to be four. Is everybody okay with this? Yes. Why is there one half? Why is there one half? Because when you look at these ones on the face here, only half of them is within the unit set. Does that make sense? Because you're kind of bisecting that thing, right? Any other questions on this? Okay, it's very important for what we're going to learn in a few seconds.