Chemistry: Crystal Packing

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So, I've shown you x-ray defraction and, in principal, you believe that we can know where the atoms are in a crystalline lattice. And I've shown you the seven crystal systems so, at least in principal, you accept that there are ways to arrange the atoms. Well, it turns out those are primitive cells, meaning that there are only atoms in the corners, but not all crystals are limited to that. You can actually have additional atoms elsewhere in the unit cell as well. Let's focus just on the cubic system plus one other that I'll bring up later on. Here's simple cubic, so go back to the cubic primitive cell, and if we put one atom in each one of the corners, that gives us simple cubic. But we could imagine something related to this in which we put an additional atom in the middle of the cube. Suppose we just add one more atom. Well that's going to give us something called body centered cubic, where there's an additional atom in the middle of the cube. This is "bcc" for "body centered cubic". And then there's another one where, suppose we add an atom in the middle face of each of the faces of the cube, and then gives us this, where you can see that not only do we have atoms in the corners, but we also have an additional atom in each one of the six faces of the cube and this is known as "face centered cubic."
Well, let's go back ...we mentioned earlier that for the simple cubic, there's one atom per unit cell. Remember this represents more than one unit cell, because only one-eighth of each of these cubes is inside the unit cell. And if you take a look at the graphic you'll see what I mean. Well, in the body centered cubic, the atom that's in the middle of the body center, that's entirely in one unit cell, and then one-eighth of each of the corners is also in the unit cell and so that means there are two atoms per unit cell in the body centered cubic in contrast to the simple cubic structure. And then when we look at the face centered cubic structure, the face centered cubic structure has one atom from one-eighth of each of the corners, plus there are six atoms in the faces of each of which each face atom belongs to a total of two unit cells. Or another way to say that is, that half of each of the atoms that's in the face belongs inside the unit cell, and so there are actually four atoms inside the face centered cubic unit cell.
Now, one interesting aspect about the face centered cubic lattice is that this is considered a close packed unit cell. In other words, this is as close as you can pack spherical objects. If you pack a bunch of oranges, for instance, in a box, what are you going to do? Well, you're going to arrange them in something that looks like a triangle and then you're going to put another atom on top of those three and you're going to build it up. And, what I'm here to tell you is that the face centered cubic lattice is just such a structure. In other words, this is as high a density of atoms as you can create. The spaces between the balls are as small as they can be. And you might believe that, for the simple cubic, there's a lot of space on the inside between the four balls, and similarly for the body centered cubic, you might believe that they are not as close as they could be, but in the face centered cubic, in fact, there are.
Now, there are two ways to pack identical spheres in order to get them as close packed as possible and one of them is the face centered cubic lattice, and the other is the hexagonal close packed. And I'll show you how you get between those two with another model here.
So, we have a layer of six atoms, so this is as close as you can pack six objects and then the next layer is going to fit, again, sort of, here's three, and then the next one is going to fit in the indentation between those three. So, if we do that, we get that layer, and this is going to be really clear in the graphic box, so you might want to take a look at that. And then, on the third layer, the next layer up, we have a choice. We can either put the next layer so that it is right on top of the original green layer, and, once again, they are fitting into the gaps just like one on top of three, and that's the hexagonal close packed structure. And another way that I might illustrate that is to take out the extra yellows here, and I'll show you that it looks like a hexagon. From on top, it looks like a hexagon. So that's the hexagonal close packed structure.
Now, to get to the face centered cubic lattice, what we do is, we start just as we did before, but now, instead of placing the third layer right on top of the second layer, we place it over the tetrahedral holes that have not been covered yet and it gets to something that looks like that, where the next layer, then, is going to look go... Now, it's the third layer that's going to be exactly on top of the green layer. And so, this green layer is a repetition of the green layer below and the balls are not exactly right, not exactly perfectly lined up, but if you look at the graphic, you'll see what I mean by how you can build the face centered cubic structure from a close packed structure. And it has the same density of atoms as the hexagonal close packed does.
So, now we have all these different structures and where are we going to go from there? Well, let's think about something like sodium chloride. Sodium chloride has a one-to-one anion to cation ratio, and how are we going to build that up? Well, it turns out that sodium chloride has a face centered cubic lattice of chloride ions with sodium cations in the octahedral hole. So what's an octahedral hole? Well, so here's a layer and we both agreed that this is a piece of the layer of the face centered cubic lattice and then there's another layer that comes along and if we set them up so that they are sort of staggered 180 degrees with respect to each other and we think about what's in the middle, what's in the hole? Well that hole is going to be octahedral and that's really obvious when we look at this way, as opposed to looking at it from the three-fold angle. And so, what we can think of as - there's our sodium and it's sitting in an octahedral hole just like that. Well, where is there an octahedral hole inside the face centered cubic lattice? And the answer is - right in the very center of the cube. There's actually a hole there that's an octahedral hole. It consists of the atoms - the hole is created by the atoms that are in the middles of the faces of each one of the cubes. And then there are actually more octahedral holes. There's an octahedral hole, there are three more sort of net octahedral holes, and they are on the edges of the cube. So there's one here, one here, one here, one here, and then, here, here, here and here, and also on the bottom. And each one of the atoms that might fit there; so in other words, squeezed in there is also going to be an octahedral hole. Of course the rest of the atoms that make up the hole are part of the next unit cell. But, if you count, you have four octahedral holes in the unit cell and four atoms in the unit cell, because there are four chlorides total in the unit cell and so that's where we get our one to one structure. So, if you look at the graphic, you'll see that sodium chloride is a face centered cubic lattice of chloride with sodiums in the octahedral holes.
So, if we think about cesium chloride, in contrast; in cesium chloride, the chlorides are in a simple cubic unit cell with the hole, the eight-fold hole, so there are eight chlorides filled by a cesium. So, cesium chloride is a simple cubic structure of chlorides with a cesium atom in the middle. Once again, we have one total chloride in the unit cell for a simple cubic lattice, and one cesium that's going to be entirely contained inside the unit cell and so our ration is one to one. Now, for zinc sulfide, there are two zinc sulfite structures; there's sphalerite or zinc blend and that consists of sulfides in a face centered cubic lattice and then, if you think about it, there are not only octahedral holes in a face centered cubic lattice, there are tetrahedral holes as well. And what do they consist of? Well, imagine that these three atoms are the atoms in three adjacent face centers. So, one of these is in the face center of this face, one of these is in the face center of the bottom face, and then one of these is in the face center facing me. Well, then if we take the corner as well, and bring it up like that, what have we got? We've got a tetrahedron. And so the space in there, right in there, of the tetrahedron, that's what we call a tetrahedral hole. And there is one octahedral hole for every atom in the unit cell of a face centered cubic lattice, and there are two tetrahedral holes in every unit cell of a face centered cubic lattice.
And again, take a look at the graphic and you'll see what I mean. It will be more clear. Anyway, so if you fill up half of those holes, then what you have is a one to one structure. And so zinc sulfide and also diamond and also silicon carbide, has half the tetrahedral holes filled with a cation or with another atom, and that gives rise to the sphalerite or zinc blend structure. And then, if we go to the hexagonal close packed structure, and put sulfurs into half of the tetrahedral holes, that's what gives you the wurtzite structure. So zinc sulfite exists with sulfurs in both of the cubic close packed and the hexagonal close packed structure with zincs in tetrahedral holes. Again, you get the one to one stoichiometry that you expect - by filling the appropriate number of tetrahedral holes. So every zinc is going to be tetrahedral. It's going to have a tetrahedron of sulfurs around it and you can either convince yourself or take it on faith, that every sulfur has a tetrahedron of zincs around it. Bottom line is we actually can characterize and categorize the structures of all kinds of salts, all kinds of metals - I forgot to mention that silver, for instance, crystallizes in a face centered cubic lattice where each of these is a silver atom - so we can go through and we can describe the crystal structure of all of these different compounds and what we can do is we can learn something about how big the atoms are or how big the ions are and how they pack and why they should be stable or unstable, and that gives rise to an entire field of structural chemistry.
Condensed Phases: Liquids and Solids
Solid State: Structure and Bonding
Crystal Packing Page [2 of 2]