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This is now the 10th lecture of the condensed matter, of course. When we left off last time, we were talking about lattices in three dimensions,
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and we covered the simple cubic lattice or primitive cubic lattice, and we've moved on to the body centred, body centred, cubic body centred.
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Cubic lattice or BTC or cubic dash I and we have a picture of that the conventional unit sell the body centred cubic lattice look something like this.
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There is one point shared among all of the eight corners, one eight times eight of them,
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and then one in the centre of the conventional cell making two lattice points.
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In the conventional cell you can look at the thing from above and in this plan view scheme where you label all of the heights appropriately.
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So this point here corresponds to this point here and it says unlabelled points are at height zero and a so this point is at height zero and height.
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A Now if you actually put atoms at the lattice points of the body centred cubic lattice,
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you notice that you pack the spheres fairly efficiently, you fill the hole that you would have had if it was just a simple cubic lattice.
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And because of that, this is a much more common arrangement for atoms, lots of elements sodium,
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lithium ion, potassium, all take body centre cubic lattice configurations.
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It's obvious from this picture that the coordination number, the number of nearest neighbours is eight.
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If you think about the sphere in the centre, it has obviously four neighbours on top and four neighbours on bottom,
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making a total coordination number of eight. It may not be obvious to you that every point in this lattice is equivalent to each other.
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That's one of our definitions of a lattice that every lattice point should have the same environment.
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But so we'll look at this picture here. When I see this, this sphere here is obviously in the centre of the yellow outline cube.
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But if we take one of the yellow, one of the corners of the yellow outline cube like that one, it is also in the centre of another cube.
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So each sphere thinks that they're inside at you.
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So every sphere, every point of the lattice is equivalent.
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That may not be a sufficiently convincing argument to convince you that the BC is actually a lattice.
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So what we should do is we should write down the lattice vectors, minus vectors,
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which we'll write as u v w times the lattice constant a with the following possibilities.
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Possibility one either with either.
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Possibility one all.
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What? She says lots of things that come out in the sound is falling out.
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Can you. Can you. Something's.
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Something's gone. Gone wrong. Can. Can someone back there fixing it?
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Is the other microphone. Or should I just be quite happy with this?
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No idea. Maybe. Maybe we kill the sound, and then I'll just yell.
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Is that better? Can someone work it?
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Yeah. No. I mean, there should be someone back there.
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Who's. Who's. There's not someone back there. No.
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I have no idea. It's just. Just keep going and see if we can suffer through it for a while.
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Someone that's the person's back is back there is fixing it.
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Okay, so hopefully this will get fixed. Stop me if it doesn't.
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Okay. Anyway. You can read, you can still read if even you can't hear.
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So we're going to we're going to write down the lattice vectors as you've RW times
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a with either all of the you VW integers and the integer case corresponds to.
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Well if you look in this and the conventional unit cell it's the corners of the unit cell.
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So this one has coordinate 100, this one has 110 and so forth and so on.
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The other possibility is you, BMW, you the W all half odd integers, half odd integers,
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integers, and by half odd I mean one half, three has five halfs and so forth.
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And that corresponds to the, the, the lattice points in the centre of the cube.
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For example, this point here would have coordinates one half, one half, one half.
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So this one would be one half, one half, one half. This one would be one half, three half, one half and so forth and so on.
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Now we should check to make one of our definitions of lattice vectors of,
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of a lattice that we should be able to add any two lattice vectors together and get another lattice vector.
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It should be a closed set under addition. So we should check. Well, if we add integers to integers, we get back to integers.
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So that's good. If we add integers to half odd integers, we get back half odd integers.
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So that's good. If we add half on integers to half on integers, we get back integers.
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So it's going to work either way. Add any of these to each other, you'll get back to something.
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It's either this kind or this time. So that's that's good. So that makes this a lattice.
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So let's see. Here we go. There we go.
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It's. It's useful to write down the primitive lattice vectors elves for this lattice,
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which we can take to be 100 times a010 times a and then one half, one half, one half times a.
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So let's see that on this on this plot here. So it's 100 times, 010 times and one half, one half, one half times a.
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And to convince yourself that these things are primitive lattice vectors,
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all we have to do is convince ourselves that you can get to any point of the lattice by adding these together in integer combinations.
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So for example, if I wanted to get to this point here, I would take this guy twice, one, two, and then subtract this one.
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And that would take me from here to here and I get to this lattice point and you can sort of work out that,
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that by adding these together in all integer combinations, you can get eventually to every every lattice point good.
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So far, so good. All right, now a warning.
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And it's a warning that that everyone in the world should take. Do not make the mistake of calling caesium chloride a body centred cubic lattice.
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The rule of a lattice is that every lattice point should look identical.
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So there is a point in the middle of the body here, but it's not identical to the other point.
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So it's not a body centred cubic lattice, it's a simple cubic lattice with a basis and the basis has two in equivalent atoms,
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one is 000, which is caesium, and the other is that one half, one half and half, which is chlorine.
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There's a difference. So it's not body centre cubic lattice. You will find books that make this mistake.
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This goes beyond difference in nomenclature. It's just wrong.
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It is incorrect to call it by central cubic lattice caesium, pure caesium where there is a caesium atom at all of the points here,
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even the one in the centre that is a body centred cubic lattice, but you're welcome to call it, you're also allowed to call it body central.
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Cubic is the same thing as simple cubic with a basis with basis where the basis
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that we're talking about here includes a point at 000 and a point at one half,
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one half, one half. Those are equivalent statements.
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I suppose the the simpler of the statements is to say it's a body centred cubic lattice,
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but you can write a body central cubic lattice is just saying a simple cubic
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with a basis including one lattice point at 000 and one last point at one half,
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one half, one half good. So far so good. All right.
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Now, you might wonder, why is it we're suffering through using a conventional unit cell that has one more one that has two lattice points in it,
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instead of using a primitive unit cell that only has one lattice point in it.
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Wouldn't that be easier? Is the reason we don't use a primitive, primitive unit cell because the primitive unit cell is kind of ugly looking.
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So this is the inner sites, primitive unit cell for the basic lattice and it's a truncated octahedron.
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I think it has 14 sides and you can kind of see what's going on here.
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This is in the centre of this cube. There's one of the lattice points and here are the eight neighbours which you can see.
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All the eight neighbours and the hexagonal faces are the perpendicularly bisecting plains between
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the lattice point in the centre of the cube and the last point in the corner of the cube.
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The square faces are the perpendicularly bisecting plains.
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Between the lattice point in the centre of the cube and the lattice point in the neighbour centre in cube.
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So if you go over one cube and you have a lot of points in the centre of that cube
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and you make a perpendicular bisecting plane that gives you these square faces,
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remember the way you construct the bigger site cells by making perpendicular by sectors.
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So you know that everything in space that's closest to the one lattice point in the centre is on the inside of that object.
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So it's an ugly shaped object. That's why we don't use primitive unit cells for the BC lattice.
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But if you do take these ugly shaped objects, these truncated octahedron,
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they will stack together very nicely and tile all the space, filling all the space appropriately as they're supposed to.
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There's one more type of lattice that we need to discuss, which is the phase centred cubic lattice.
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Phase centred. Centred.
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Cubic or FCC or cubic dash f.
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You may have noticed I wrote a centred the American way, not the British way.
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And the reason I do this is because I did the other way. My brain would explode.
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It's just, you know, you get used to doing something one way and that you just can't change.
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Either way is correct. So depending on what side of the ocean you're on. So let's see what the face under cubic lattice looks like.
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So you start with the simple cubic lattice, the points in the corner of the conventional unit cell.
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And then you add one point in the centre of every face, like this one in the centre of this place,
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this one of the centre, this face is one of the centre of this face and so forth. How many lattice points are there in the.
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So they are going to have to make these harder.
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Yes, they are four points in the convention itself. There is to see where the four points are.
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Is eight corners, each of which counts one eighth.
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So it's one eighth inside the convention itself.
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Then there are six bases and the point in the centre of the face is half inside the red cell and half outside the hotel.
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So it's six times one half equals four lattice points within the convention itself.
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You can also draw the convention itself with this plan view scheme.
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Which you know.
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So here are the points in the corners, as it says unlabelled points out height zero on a so this point here corresponds to this point here,
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which is that height is zero on a you can see the points on this on the halfway up the faces at height
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labelled a over two so that's this point this point this point in this point that's this one this one.
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This one and this one. And then the one in the centre here is also at height zero on a this one and this one here.
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Okay. Now, if you imagine arranging atoms together or spheres together in the FCC lattice configuration,
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it is actually the most efficient way that you can possibly pack spheres together.
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So if you have a bunch of tennis balls, you're trying to stick them into a box.
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The highest density packing of those tennis balls into the box is an FCC lattice.
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There are other packages which are equivalently dense, but you're never going to do better than the FCC packing.
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We don't study the other types of backings because they don't have orthogonal axes, but they are equivalent to FCC.
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In fact, the statement that you can't get a more dense packing of atoms than FCC was conjectured by Kepler in 1611, and it was only proven in 1998.
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So it's a very long standing standing theorem, but it turns out to be true.
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You can't do any better.
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You can't pack more things into a small space if they're spheres, then using FCC lattice because you get so many spheres in a small space.
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If you think about atoms trying to track each other, it's a very common it's a very appealing configuration for atoms and so many elements.
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Take FCC configurations copper, silver, gold, calcium.
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Many others are FCC. Okay, so since there are four atoms per unit cell, we should be able to view the FCC lattice.
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As for entropy penetrating simple cubic lattices.
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So let's see if we can see that this is an FCC lattice of spheres and this is the red.
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I have marked out a simple cubic lattice. This is a simple cube.
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It's a little hard to see, but then you can actually pick out if you look very carefully.
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There are three other enter penetrating simple cubic lattices which are mixed in here.
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And although it may not be completely obvious that the environment of every single sphere is identical to the environment of every other sphere.
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Now, since that's not completely obvious, we should do the exercise of actually writing down the lattice vectors.
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Lattice vectors? For the FCC.
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Gladys and I claim they take the form again. U v w u v w times a where either 1uvw all integer.
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And again, that would correspond. Backing up one more that would correspond to the the points in the corners all integers.
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So this one is is 100 this is 110 and so forth and so on.
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And the other possibility. So you VW where either either possibility one they're all integers or two.
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Two of them are half odd.
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Integer and one.
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Integer. Okay.
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So we're going to show that this actually works. But before going on.
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So for the one one more chocolate, what is the coordination of the coordination of 12?
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Let's actually see on this on this on this plot, why it is the coordination number is is 12.
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I'm going to stop asking questions going around that chocolate this way. I'm going have to ask harder questions.
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Okay. So if you see the coordination of 12 of the take this sphere here,
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the one that's cut in half, it has four neighbours one, two, three, four at the same height.
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It has four neighbours at a slightly lower height. Here you can see two of them that are touching it also.
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And then if you went to a slightly higher height, the same distance up that these are down, you would have four more.
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So it's four at the same height, four slightly lower and four slightly higher, which may not look all that convincing.
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But we can show it using this rule.
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So this rule would give that to us for free.
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The reason we would know this is because if you if you look at this point here.
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Well, first of all, let's take a look at this this item two here, two of them half odd integer, one of them integer.
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What does that correspond to? That corresponds to somewhere in the middle of the face.
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For example, this guy here, he's over one half.
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He's back one half, but he's up an integer height. Whereas this guy over here, he's over one half.
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He's up one half. But his height is an integer, happens to be his depth is an integer.
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This one here is over a half, upper half and is an integer back.
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So if it's on a face, it has to have out integers and one integer.
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If it's on a corner, it's all integers. Now we can check.
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We can check here that again.
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We should have the rule that if you add any two things from these sets to each other, you should get back something from this from one of these sets.
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So, for example, if you add integers to integers, you get back integers.
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If you add integers to half odd integers and one integer, you'll get back in.
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The fact that you're adding integers to something doesn't change where there's a half odd integer or an integer,
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so you get back something in the set to again.
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But what's not obvious is you take two things which have to have all integers in one integer and you add them together.
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What do you get? So let's try. So if you have, for example, one half, one half, one, and you add it to three half, one half zero.
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So both of these are two of them, half odd integers, one of them integers.
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So both of these would be on the face. You get two comma, one, comma, one, which is all integers.
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And that's good. But I could have done it differently. You could have had one half, one half, one added to say one half, zero three half, for example.
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Then what you get is one comma, one half comma, three halves, and still two of them are half odd integers, and one of them is an integer.
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So if you add together two things from set B,
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you'll get either something from set that B set to you get either something from set one or something from set two.
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So it forms a close set under addition. Now, given that we know that these are the coordinates of the of the vectors in the FCC lattice,
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let's see if we can figure out that that coordination number 12 a little more easily.
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So let's start with this point here. That's .000 and look for the closest things to 000 closest to 000.
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Okay. Well, okay, this guy here, it looks pretty close.
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His coordinate is one half, one half, comma, zero. So one half, one half comma zero.
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And if that fits the definition of one of the points in the SCC lattice, two of them half identities and one of them integer and it's pretty close.
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Can't get any closer. But we could have made this plus or minus and we could have made this plus or minus as well.
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Right. Those all fit the definition as well. And we could have put the zero in any of the three spots.
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Three possibilities. So we have four possibilities of these plus or minus signs and three possibilities of where we put the zero.
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So we get four times three equals 12 is the coordination number.
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Okay, good. Okay. All right.
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So it's worth also writing down a set of primitive lattice vectors for the FCC lattice.
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So, please, for the FCC lattice, a really good example are just the closest vectors are make pretty good POVs.
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So one half one half zero times a one half zero one half times a and zero one half one half times a make pretty good primitive lattice vectors.
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There they are. And again, to convince yourself that they really are primitive large vectors,
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what we have to do is convince ourselves that we can get to every point on the FCC lattice by adding integer combinations of them.
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So for example, if we wanted to get to this point here, we would have to add one of each of them.
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So with this one, this one plus this one would take us to here. And if we wanted to get to here, we would have to add.
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Well, I'm not sure. Yeah, okay. How do you get there?
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Two of these and then minus one of these or something. And anyway, so you can play around with it.
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It works. You can get there. How do you how do you get here.
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Yeah. So we two of these would get you here then you need.
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Well, uh. Mm hmm.
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Maybe minus one. Yeah, minus one of these, then. Plus one of these. There's something that that will get you there.
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Okay. Anyway, all right.
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So since the FCC lattice has FCC lattice has for.
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Four lattice points per conventional unit cell. Just like we did with the book, we could write it as simple cubic simple cubic times a basis.
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Times a basis where the basis now includes basis equals.
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The point at 000. The point at one half one half zero.
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The point at one half zero one half.
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And the point at zero one half. One half.
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That would give me the four lattice points per unit cell.
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And if I translate those four, that points in my conventional cell around to every lattice point.
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So that is point is that 000 and these are all displaced from the original lattice point that
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will reconstruct the entire phase centred cubic lattice or another way of thinking about it.
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These are the displacements of the for inter penetrating simple cubic lattices that make up the FCC lattice.
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As with the BC lattice, there's a reason we don't use the primitive units l for we use a conventionally.
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It's not the primary itself because proving itself is extremely ugly.
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Here it is. It's a truncated dodecahedron that has 12 sides. Those 12 sides correspond to the 12 nearest neighbours.
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So what we have in the centre is one of the lattice points and then we have the 12 nearest neighbours,
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four of them slightly above four at the same height and four in the slightly below.
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And the faces of the truncated dodecahedron here are the perpendicular by
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sectors of the segment between zero and the and the and the nearest neighbours.
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Okay. All right. That's all we need to know about the SCC lattice.
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In fact, that's all the three dimensional lattices we're going to study.
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It is worth knowing that there are 14 types of lattices in.
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In all the ones we've studied are the cubic primitive, the cubic body centre, the cubic phase centre.
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We also talked about the tetrad, all simple diagonal and simple earth Arabic.
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And then you have a whole bunch of other ones here.
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You may notice that there's, there's analogues that we don't need to know, analogues of the body centre,
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tetrad body centre, or they're iambic phase centred or thrown back and so forth.
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You may notice there's no face centre tracking. All the reason for that is because in fact by turning the lattice sideways,
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it would be in one of the other classes as well, so you'd be over counting.
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So anyway, these are sometimes known as the Bravo lattice types. After Bravo, who is the first person to write them down correctly?
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And so we give Brave a credit for doing them. So these are the only possible lattice types you can have.
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It's kind of a deep mathematical statement that any periodic structure in three dimensions,
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anything you can have in three dimensions is one of these lattices times some basis.
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So if you can write down all these lattices and you can have any basis you choose,
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no matter how complicated you can make anything, any periodic structure is one of these lattices.
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Times, some bases.
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The only ones we're actually going to have to know this year are the three cube X and and the three simple cubic tetrad along with ceramic.
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Very rarely do we need to know tetrad and fourth Romick. I think they only rarely come up on exam.
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It's really these three across the top that actually show up on exams.
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Okay. So this is an example of what you can put together when you when you have a lattice and a basis.
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This is a sodium chloride structure, a very typical salt structure, sodium chloride, very ionic sodium gives up the electron.
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Chlorine takes the electron. And if you look at it carefully, you can see that sodium is forming an FCC lattice.
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It's even labelled cubic F. So maybe it's easiest to see in this picture.
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These are all depictions of the same lattice. None of them are actually great, but.
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We may have to make do. So here you can see the green sphere is here in this picture form, an FCC lattice.
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So on the corners and in the centre of the faces as well.
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But in addition to the sodium atoms which are on the FCC ladder lattice, there are chlorine atoms as well.
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And you can describe the position of the chlorine atoms by saying for every sodium atom there is a chlorine atom displaced by one half,
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one half, one half from the sodium atom. That will that will get you the position of all the chlorine atoms.
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Here's a plan for you. The blues would be the sodium and the chlorine.
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So, for example, if I started the sodium and I go over a half back a half and then up a half to the next layer, I get to a chlorine atom.
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When you describe a basis, you should always describe the basis of the primitive unit cell.
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So here I don't tell you where all the four. I don't talk about the conventionally, the cell, which would have for sodium and for chlorine.
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I just tell you, there's it's an FCC lattice and immediately, you know that the conventional your cell will have four lattice points in it.
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And on the position of the lattice at 000, there's a sodium blue and then displace from that by a half, a half a half output put chlorine.
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So I only have to describe to you one sodium and one chlorine to tell you everything.
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Once I've told you it's a simple Cuba. It's a it's a F.C.C. lattice.
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Notice also that it's not unique how I describe the position of the chlorine with respect to the position of the sodium.
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I could have told you that the chlorine was displaced one half zero zero.
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That would give you the equivalent structure. Or I could have decided that chlorine was my 000 and sodium was displaced from it.
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That would also give you an equivalent structure. So there's various different ways to describe the same the same lattice.
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Here's another structure that we run into very frequently. The diamond structure, carbon.
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It also is the structure of silicon and germanium. It is also based on the FCC lattice.
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So you can see, for example, in the top face here, there's some guy, there's one in all the corners and there's one in the centre of the face.
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The side face is one in all the corners, there's one in the centre of the face and so forth.
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So it's definitely has an FCC lattice in it. But then that that doesn't tell you where all the positions of the carbons are.
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They're additional carbons.
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And you can describe the position of the carbons by just as being displaced by one quarter, one quarter on quarter from every lattice point.
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So for example, in the plan view, we start with this carbon at 000 and then we displaced by a quarter,
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one quarter on quarter and we find another carbon at that position as well.
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So again, I don't have to describe the position of all of the eight carbons in the conventional units cell.
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I only have to tell you that it's an FCC lattice and and it has a basis of two atoms, 12000 and one at one quarter, one quarter, one quarter.
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And that's sufficient to describe everything. Okay. All right.
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What's the lattice? What's.
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What is the lattice? What is the lattice? What?
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It is a lattice. It's a lattice with a basis.
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What? What lattice type? Who said FCC.
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Okay, good. Good. Wow. See this advantage to sitting up close?
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You get more chocolate that way. Incidentally, I switched from chocolate types because I tried some of the chocolate yesterday.
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It wasn't all that good, so hopefully this works better. I'm chocolate connoisseur.
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Okay, so it's FCC. It's. It's. So this is gallium arsenide structure, also known as the zinc blend structure.
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It's exactly the same as the diamond structure, except that you've taken the second carbon and you turn it into a different type of atom.
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So here, let me actually to show you. So the yellow at are form an FCC.
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So you can see that the the yellow atoms in the corners and the centre of the faces.
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So, for example, here here's a face and there's a yellow in the centre of it.
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Then the blues are just displaced one quarter, one quarter on quarter from every yellow atom.
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Now, some of the you know, it may look a little puzzling.
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It looks like we have more yellow atoms than we have blue atoms, but we don't, because some of the yellow atoms are only half inside the unit.
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So the FCC, there's four lattice points within the unit cell.
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And so there should be four yellow atoms if you count them, including, you know, half an eighth of it's on the corner and a half is on the side.
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You'll count for yellow atoms and then they're obviously for blue atoms completely inside the inner cell as well.
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Okay, one more comment. This is a subtlety.
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If Mike Glazer happens to be your tutor, [INAUDIBLE] be very happy that I tell you this.
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He's a real class crystallographer and he always holds my feet to the fire to force me to tell things the way they really are and not tell any lies.
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So here I'm going to tell you the subtlety to not tell you any lies. So suppose we have a material like this.
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It appears to be a simple cubic system with a basis of three.
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There's one on the corners and then two somewhere in the centre.
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They're not right in the centre so it's not BC but there's two somewhere in the centre and there's one shared by all the eight corners.
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So we would say it's a simple cubic with a basis including three, three, three atoms.
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Now if I measure the edges of this cube and I say they're all the same and, and all the three directions I would say it,
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you would be tempted to say it's simple cubic crystal, but a crystallographer would say it's not.
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And the reason they would say it's not is because it doesn't have the symmetry of a simple cubic object, a simple cubic object.
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You should be able to turn it in all six directions and it should look the same whichever direction you look at it from.
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And this thing doesn't look the same if you look at it from the top. As far as if you look at it from the side, why is that important?
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Well, the Crystallographer knows that if it doesn't have the symmetry, there's no good reason that its height should be the same as its width.
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They're not the same. They're not equivalent in any sense. So the height doesn't have to be the same as the width.
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And so if I tell you whether the the edges have the same length, he would say, well,
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go measure it more carefully and you'll discover they're not the same length.
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So the Crystallographer knows if there's not a good symmetry, reason for the edge likes to be the same, then they're not the same.
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And in crystallography, if I wrote this down, the crystallographer would come back to me and say, No, they're actually not the same length.
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You just think they're the same length. So go measure it again and you'll discover that they're not the same length.
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It would be an unbelievable coincidence if there was not a good symmetry reason
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for the edges to be the same length and they end up being the same length.
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Okay, so that's to make the real crystallographer happy.
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And that is all we have to say about crystal structure for today.
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Okay, good. So we've now learned everything we need to know about crystal structure in three dimensions.
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All the lattices we're going to have to discuss we know about is the basis.
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But at the end of the day, we really want to describe physical phenomena in these crystals.
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And more often than not, what we're interested in is waves of some sort,
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whether the vibrational waves or phonons, whether they're electron waves, whether they're electromagnetic waves.
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And the world of waves is the world of reciprocal space.
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So we have to back up and understand some things about reciprocal space.
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When we have complicated crystals. So let's remind ourselves some things we learned about reciprocal space in one dimension.
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Well,
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in one dimension we had a direct lattice lattice of the form r sub n equals a times n and was the integer a was the primitive lattice vector P.O.V.
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And then from direct space we have the reciprocal space.
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Reciprocal lattice which we can write as GM equals two pi over a times m,
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and now here to pi over a is the primitive lattice vector in reciprocal space.
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Primitive lattice vector for the reciprocal lattice. Now, why was it that we chose this to be the reciprocal lattice?
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Well, we chose it because if you take K to K plus any element GM in the reciprocal lattice, we get back the same wave gets same wave.
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Why is that? Well, let's do it carefully. You do the i k dot r k that are get shifted to e d d i k plus g m r and and that becomes equals.
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Either the i k dot r and either the i gmm r and and this factor here is just one because it's this
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is one because it is either the i two pi over a times m times a times n and that's just one.
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So we're going to generalise that into more dimensions in any dimension.
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In any dimension we define define the reciprocal lattice define.
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Resupplied us in Sep lat via R points as points g.
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Give actor such that. E to the i g dot r and oc g.
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Let's not put an index on it yet. G r n. Equals one for all.
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For all are an indirect lattice.
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Indirect let us. So this is our general definition of the reciprocal lattice vectors g in the reciprocal lattice.
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Now, this is a nice definition. It's a very useful definition, but I have not in any way proven to you that this definition defines a lattice.
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I've claimed it defines a lattice, but I haven't shown you it defines a lattice.
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And the proof that the set of points g in reciprocal space that satisfy this equation equals one for all are end in the direct lattice.
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The proof that that defines a lattice in reciprocal space is a little subtle and a little tricky.
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So we're going to go through it and see if we can make it convincing that it's true.
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So first of all, we'll take the direct lattice vectors. We have to define the direct lattice vectors.
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Direct lattice vectors x, we'll write them in terms of the primitive lattice vectors.
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So R equals N1, A1, plus N2, a two plus n three, a three where the A's are the primitive lattice factors.
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A's are PVS. A's are please.
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But the last fact is, okay,
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so we define the direct lattice now and then we are going to guess I'm going to make a guess of what this is a very good guess.
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It's correct guess. And we have to prove it in a moment. Guess the pelvis is the pelvis of the reciprocal lattice principle that.
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We'll call them Bs-vi as compared to ACB for the direct lattice and will define them by the following equation.
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B sub I got it with a sub j equals to pi delta i j.
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It's a rather important equation. Now, first thing you might wonder is how do I know, given a set of primitive lattice vectors in direct space?
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A How do I know that there's a set of vectors? B Which satisfies this?
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Are this equation sort of a dual space, dual vector space basis?
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Well, there's a fairly easy way to show you that you can find these given some A's, which is by writing them down.
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So let's write them down. So b i equals to pi aj cross a k over to A1 dotted into a to cross a three.
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Okay. And this is for our i j k equals two, one, two, three or three.
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One, two or two, three, one.
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So in the cyclical way. So I claim that this expression for be will satisfy this definition of B and to check it.
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Let's let's just find out if it's true. So let's take for example, b one dotted with A1.
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So we'll write out B one. B one is to py a to cross a3 over A1 dotted with A to cross three.
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And then we want to dot that into A1 and we see that the numerator in the denominator are actually identical.
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So we just get to pi as we're supposed to.
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However, if we took B1 dotted into A2, we would get the same same expression here on A2, cross a three of top,
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and then this whole expression downstairs, right, and then dotted into A2 and that thing equals zero because A2 Cross A3 is orthogonal to A2.
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When you turn it into a two, you get zero.
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So indeed the B1 and A1 gives you two pie B one, two, A2 gives you zero and you can check that it works for all the other i j combinations as well.
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Okay. So now we have a guess for what our reciprocal lattice vectors are.
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So we would guess that we would write down reciprocal lattice vectors g as m1, b1 plus m2, b2 plus m3 b3.
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Okay. That's going to be our guest right now.
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But if we wanted to try to prove that these GS are reciprocal lattice factors, are are we only if want to prove the GZ lattice.
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What we need to show is that the M's can only be integers.
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Okay. Now, if I want to just pick any point in reciprocal space, I can choose to write that point as G with M's arbitrary.
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So let me start with M's arbitrary. M's arbitrary, in other words, real numbers arbitrary.
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And that allows i.e. consider any G consider any vector any vector g not necessarily on a lattice.
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So I'm going to consider any possible g to begin with.
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Now then what we're going to do is we're going to impose the definition of the I got R equals one.
384
00:38:26,110 --> 00:38:31,950
So we're going to force on it one equals either the i g r.
385
00:38:32,260 --> 00:38:41,739
Okay, what does that mean? Either the i g is m1 b one plus m2 b two plus m3 b three.
386
00:38:41,740 --> 00:38:50,350
I'm about to run out of room and one a1, a1 plus and to A2 plus, m3 A3.
387
00:38:50,710 --> 00:39:00,370
Okay. And I need this to, if I'm going to impose the condition e I got r equals one, I have to impose the condition that this thing equals one.
388
00:39:00,730 --> 00:39:05,410
Well, using our orthogonal B condition B by that AJ equals to pi delta.
389
00:39:05,560 --> 00:39:15,220
J That is one equals either the two pi i M1 and one plus m2 and two plus m3 and three.
390
00:39:16,470 --> 00:39:24,630
Now, if I want this thing to equal one for every possible direct lattice vector for every possible end, the only way this will be true.
391
00:39:25,020 --> 00:39:28,139
This is true for all. N for all.
392
00:39:28,140 --> 00:39:33,930
And and only if m are integers.
393
00:39:34,950 --> 00:39:46,359
Our integers. Okay. So that means that in order to satisfy our definition of the reciprocal lattice that each of the I got are equals,
394
00:39:46,360 --> 00:39:52,690
one for all are in the direct lattice. The only way we can do that is if we choose g of that form with the integers.
395
00:39:52,960 --> 00:39:58,340
Therefore G is a lattice. And not only do we know that G is lattice, we know what its primitive lattice factors are.
396
00:39:58,360 --> 00:40:02,890
So we just derived the fact that G is a lattice with those primitive lattice factors.
397
00:40:03,400 --> 00:40:09,790
So far, so good. Happy with that? Okay, good. A couple of interesting facts fact.
398
00:40:10,120 --> 00:40:13,240
Actually, I think this is a homework problem, maybe on the revision homework or something.
399
00:40:13,570 --> 00:40:23,740
The recipient is SEP flat of FCC is B.B.C. PCC and vice versa.
400
00:40:25,630 --> 00:40:31,450
Kind of an interesting statement. You can check it and I think you probably will for one of your homeworks.
401
00:40:32,140 --> 00:40:43,690
Another interesting comment is it seems that in 2D, in 2D, same rules apply, rules apply.
402
00:40:44,620 --> 00:40:49,920
But you might wonder, how do you handle this formula in 2D?
403
00:40:49,930 --> 00:40:56,920
Because I only have two vectors, not three. The way you do you handle that formula is just choose.
404
00:40:59,290 --> 00:41:03,939
Choose A three equals Z had a point coming out of the plane.
405
00:41:03,940 --> 00:41:06,220
So in other words, if you live in 2D,
406
00:41:06,430 --> 00:41:12,190
you imagine there's a normal to a plane and you treat that as your third primitive lattice factor and you go from there and you're in business.
407
00:41:12,700 --> 00:41:19,370
All right. A couple of minutes left. We're going to try to in the next couple of lectures, we're going to try to do some interpretation.
408
00:41:19,420 --> 00:41:25,149
Some uses of this reciprocal lattice is a rather important statement that people frequently make.
409
00:41:25,150 --> 00:41:31,900
The reciprocal lattice is the 48 transform for you transform.
410
00:41:34,130 --> 00:41:43,040
Transform of direct direct this may be isn't surprising because the reciprocal
411
00:41:43,040 --> 00:41:46,819
lattice lives in k space and the direct lattice lives in real our space.
412
00:41:46,820 --> 00:41:52,040
And we know to get from our space, the k space, you frequently have to do things like Fourier transform.
413
00:41:52,790 --> 00:41:57,650
But let me see if we can make this a little bit more rigorous. Let's do it in one D again.
414
00:41:57,950 --> 00:42:01,580
So our lattice vector is our n equals eight times MN.
415
00:42:02,060 --> 00:42:05,290
And how are we going to Fourier transform that lattice?
416
00:42:05,300 --> 00:42:15,200
Well, let's make a function row of x, which is a sum over all lattice points of a delta function at the position of each lattice point.
417
00:42:16,640 --> 00:42:21,080
This is what's known as a delta function comb. Delta function comb.
418
00:42:21,530 --> 00:42:25,720
Have you seen this before in some in quantum mechanics or something called.
419
00:42:26,330 --> 00:42:31,310
Okay, it looks kind of like this. Here's the x axis and then here's zero.
420
00:42:31,790 --> 00:42:41,480
Here's a, here's to a A, and it has these great big delta function peaks at the position of each of these lattice points.
421
00:42:43,070 --> 00:42:54,500
Okay. Now try 40 transforming us. So 40 transform of all of x equals integral the x into the ik x.
422
00:42:54,920 --> 00:42:59,210
Then we have this row of x function and we'll plug in the row of x function.
423
00:42:59,720 --> 00:43:09,830
We'll pull out the sum so we get some over an integral of the x to the i k x and delta function of x minus r and.
424
00:43:11,420 --> 00:43:23,240
And we let the Delta Function Act so we get some over n e to the i k r and which we could also write as some over and into the i.
425
00:43:23,270 --> 00:43:26,389
K. And okay, so what is this?
426
00:43:26,390 --> 00:43:34,250
We have this for a transform, a row of x and it's a sum over all of these, these phases in the expression, well,
427
00:43:34,790 --> 00:43:45,320
if K is an element of the reciprocal lattice of GM, then every term here is of the form into the eigen r and GM.
428
00:43:45,590 --> 00:43:52,070
And, and so every term is one. So this then becomes sum over N of the number one, which is infinite.
429
00:43:53,150 --> 00:44:00,140
If you have an infinitely big system if K is not an element of the reciprocal lattice.
430
00:44:01,710 --> 00:44:08,580
Then what we have, then what we have is you'll have a situation where this complex phase here is not equal to one.
431
00:44:08,790 --> 00:44:16,290
So it has some complex phase, some arbitrary complex phase. And then if you go out to a lattice point, which is twice as big double PN,
432
00:44:16,530 --> 00:44:19,589
you'll get twice the complex phase and you go out to something is three times as big.
433
00:44:19,590 --> 00:44:23,729
You get three times the complex phase and these complex phases keep rotating around and around and around.
434
00:44:23,730 --> 00:44:26,910
So you get some over oscillating phases.
435
00:44:29,410 --> 00:44:32,710
Which goes to zero. They all cancel out and you get zero.
436
00:44:33,400 --> 00:44:37,870
So at the end of the day, maybe I'll move over to hear what we have.
437
00:44:40,890 --> 00:44:50,430
Is that the 48 transform of this delta function comb is a sum over all possible reciprocal lattice
438
00:44:50,430 --> 00:44:58,530
vectors k minus GM of a peak and infinitely big peak at the position of the reciprocal lattice vector.
439
00:44:58,530 --> 00:45:06,000
And if you do it carefully, you get a factor of two pi over a out out front a build.
440
00:45:06,230 --> 00:45:11,790
The last factor is two pi over a is it's the same two pi that shows up whenever you do for a transforms.
441
00:45:11,790 --> 00:45:14,369
There's always two PI's floating around, they're always there.
442
00:45:14,370 --> 00:45:18,599
You're not probably going to be held responsible for ever getting this pre factor right?
443
00:45:18,600 --> 00:45:27,209
I suspect so. The principle is that if we have a delta function common in real space, if you Fourier transform that the result is infinite.
444
00:45:27,210 --> 00:45:29,700
If you're sitting on a reciprocal lattice site,
445
00:45:29,700 --> 00:45:36,030
if your K is a reciprocal lattice vector and it's zero otherwise and that becomes a delta function, come in k space.
446
00:45:36,480 --> 00:45:42,210
Okay, now the same thing more or less holds in three dimensions.
447
00:45:43,710 --> 00:45:58,740
Three D so in three d, in three d we'll write row of x vector equals sum over lattice points are of a delta function x minus r.
448
00:45:59,130 --> 00:46:04,260
Again, if you for a transform it you'll get four a transform a row of x,
449
00:46:04,260 --> 00:46:08,430
you let the Delta Function Act and now a three dimensional delta function Delta three.
450
00:46:08,820 --> 00:46:23,010
This becomes some over lattice vectors are of the i k dot r and then this thing becomes exactly the same way that if K is a reciprocal lattice vector,
451
00:46:23,310 --> 00:46:28,590
it's the sum over an infinite number of last vectors of the number one which diverges and you get infinity.
452
00:46:28,830 --> 00:46:32,309
If K is not a reciprocal lattice vector, then you don't get the number one.
453
00:46:32,310 --> 00:46:37,050
You get some complex phase and that complex phase rotates around and around and around and we add them up, you get zero.
454
00:46:37,320 --> 00:46:44,520
So you end up getting to pi cubed over the volume of the cell volume itself.
455
00:46:45,570 --> 00:46:55,680
Again, you're probably not going to be held responsible for the pre factor sum of reciprocal lattice vectors of k minus reciprocal that Inspector G.
456
00:46:57,160 --> 00:47:01,310
Okay. So far, so good. Now we can do a little better even.
457
00:47:02,680 --> 00:47:07,180
We can consider things that are more complicated than just a delta function comb.
458
00:47:07,180 --> 00:47:13,060
Because we're very frequently, we're very infrequently actually presented with a real delta function comb.
459
00:47:13,570 --> 00:47:19,180
So instead, we're going to consider any periodic function, periodic function.
460
00:47:23,290 --> 00:47:34,000
Role of X. And what I mean by periodic function is that role of X should equal role of x plus R where R is a lattice factor lat vec.
461
00:47:34,780 --> 00:47:39,579
So this function row of x equals row of x plus are where are the lattice factor?
462
00:47:39,580 --> 00:47:42,600
It has the periodicity of the lattice. Okay.
463
00:47:42,700 --> 00:47:50,920
Does that make sense? Yeah. Okay. Okay. So now let's take the for a transform of this function, for I transform a row of x equals.
464
00:47:52,220 --> 00:48:00,980
Integral d3x to i k dot x and this is the integral overall space row of x.
465
00:48:01,880 --> 00:48:07,460
Now I'm going to do a little bit of a trick. This is a useful trick that will probably come back to haunt you at some point.
466
00:48:07,730 --> 00:48:12,170
I'm going to take that integral over all space and I'm going to write it as an
467
00:48:12,170 --> 00:48:19,670
integral over all lattice vectors are then the integral d3x over the unit cell.
468
00:48:20,840 --> 00:48:24,800
Maybe the bigger sites units all around are around R.
469
00:48:25,400 --> 00:48:30,020
So I'm just breaking up that integral over all space into pieces, each piece being over the vector site.
470
00:48:30,020 --> 00:48:36,470
So around the position are you the i k not x x?
471
00:48:37,850 --> 00:48:53,190
Then what we'll do is we'll define a parameter why x equals r plus y that and we'll rewrite that integral then as integral oops some over r some
472
00:48:53,270 --> 00:49:14,209
of our r integral d3y of over the units cell of e to the i k dot I guess r plus y and then row of r plus y now row of our plus y row is periodic.
473
00:49:14,210 --> 00:49:18,290
So row of our plus y is the same as row of y. So that makes that easy.
474
00:49:18,800 --> 00:49:29,900
And then I can factor out the ix our term from this here and I get some over r e to the i k dot r.
475
00:49:31,240 --> 00:49:45,220
But that in parentheses times this in parentheses integral d3y over the unit cell of the i k dot y of rove y.
476
00:49:47,550 --> 00:49:51,900
Now, this this first time should look familiar since we just calculated a moment ago
477
00:49:52,650 --> 00:49:57,930
this thing here is the sum of all lattice vectors of each of the i k that are.
478
00:49:57,930 --> 00:50:03,450
If K is a reciprocal lattice factor, this is infinite. If K is not a reciprocal aspect of this thing is zero.
479
00:50:03,750 --> 00:50:06,720
So this thing is just as we had before.
480
00:50:07,020 --> 00:50:16,290
It's two pi cubed over the volume of the cell, some overall reciprocal lattice factors, g of delta of R of k minus g.
481
00:50:16,560 --> 00:50:21,360
So it gives you a delta function peak at the position of each reciprocal lattice vector.
482
00:50:21,870 --> 00:50:27,990
This term here is known as the structure factor s of K structure factor.
483
00:50:30,920 --> 00:50:39,860
Structure factor, and it's just the Fourier transform of the function we're considering within a single unit cell.
484
00:50:40,370 --> 00:50:45,620
So this is actually a rather interesting statement and rather important statement we're going to use many times later on.
485
00:50:45,740 --> 00:50:47,660
You take any periodic function whatsoever,
486
00:50:47,660 --> 00:50:57,049
yuphoria transform it and will only have that non-zero values at the position of reciprocal lattice vectors and it's value at those positions.
487
00:50:57,050 --> 00:51:01,430
Typical lattice vectors is weighted by the Fourier transform of the function within a
488
00:51:01,430 --> 00:51:05,299
single unit cell is going to be extremely important in the next two or three lectures.
489
00:51:05,300 --> 00:51:09,050
So I will see you on Monday. Monday, I think.
490
00:51:09,050 --> 00:51:11,810
Monday. I think I'll see you Monday. All right. Have a good weekend.