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,
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.
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.
There is one point shared among all of the eight corners, one eight times eight of them,
and then one in the centre of the conventional cell making two lattice points.
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.
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.
A Now if you actually put atoms at the lattice points of the body centred cubic lattice,
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.
And because of that, this is a much more common arrangement for atoms, lots of elements sodium,
lithium ion, potassium, all take body centre cubic lattice configurations.
It's obvious from this picture that the coordination number, the number of nearest neighbours is eight.
If you think about the sphere in the centre, it has obviously four neighbours on top and four neighbours on bottom,
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.
That's one of our definitions of a lattice that every lattice point should have the same environment.
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.
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.
So each sphere thinks that they're inside at you.
So every sphere, every point of the lattice is equivalent.
That may not be a sufficiently convincing argument to convince you that the BC is actually a lattice.
So what we should do is we should write down the lattice vectors, minus vectors,
which we'll write as u v w times the lattice constant a with the following possibilities.
Possibility one either with either.
Possibility one all.
What? She says lots of things that come out in the sound is falling out.
Can you. Can you. Something's.
Something's gone. Gone wrong. Can. Can someone back there fixing it?
Is the other microphone. Or should I just be quite happy with this?
No idea. Maybe. Maybe we kill the sound, and then I'll just yell.
Is that better? Can someone work it?
Yeah. No. I mean, there should be someone back there.
Who's. Who's. There's not someone back there. No.
I have no idea. It's just. Just keep going and see if we can suffer through it for a while.
Someone that's the person's back is back there is fixing it.
Okay, so hopefully this will get fixed. Stop me if it doesn't.
Okay. Anyway. You can read, you can still read if even you can't hear.
So we're going to we're going to write down the lattice vectors as you've RW times
a with either all of the you VW integers and the integer case corresponds to.
Well if you look in this and the conventional unit cell it's the corners of the unit cell.
So this one has coordinate 100, this one has 110 and so forth and so on.
The other possibility is you, BMW, you the W all half odd integers, half odd integers,
integers, and by half odd I mean one half, three has five halfs and so forth.
And that corresponds to the, the, the lattice points in the centre of the cube.
For example, this point here would have coordinates one half, one half, one half.
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.
Now we should check to make one of our definitions of lattice vectors of,
of a lattice that we should be able to add any two lattice vectors together and get another lattice vector.
It should be a closed set under addition. So we should check. Well, if we add integers to integers, we get back to integers.
So that's good. If we add integers to half odd integers, we get back half odd integers.
So that's good. If we add half on integers to half on integers, we get back integers.
So it's going to work either way. Add any of these to each other, you'll get back to something.
It's either this kind or this time. So that's that's good. So that makes this a lattice.
So let's see. Here we go. There we go.
It's. It's useful to write down the primitive lattice vectors elves for this lattice,
which we can take to be 100 times a010 times a and then one half, one half, one half times a.
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.
And to convince yourself that these things are primitive lattice vectors,
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.
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.
And that would take me from here to here and I get to this lattice point and you can sort of work out that,
that by adding these together in all integer combinations, you can get eventually to every every lattice point good.
So far, so good. All right, now a warning.
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.
The rule of a lattice is that every lattice point should look identical.
So there is a point in the middle of the body here, but it's not identical to the other point.
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,
one is 000, which is caesium, and the other is that one half, one half and half, which is chlorine.
There's a difference. So it's not body centre cubic lattice. You will find books that make this mistake.
This goes beyond difference in nomenclature. It's just wrong.
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,
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.
Cubic is the same thing as simple cubic with a basis with basis where the basis
that we're talking about here includes a point at 000 and a point at one half,
one half, one half. Those are equivalent statements.
I suppose the the simpler of the statements is to say it's a body centred cubic lattice,
but you can write a body central cubic lattice is just saying a simple cubic
with a basis including one lattice point at 000 and one last point at one half,
one half, one half good. So far so good. All right.
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,
instead of using a primitive unit cell that only has one lattice point in it.
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.
So this is the inner sites, primitive unit cell for the basic lattice and it's a truncated octahedron.
I think it has 14 sides and you can kind of see what's going on here.
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.
All the eight neighbours and the hexagonal faces are the perpendicularly bisecting plains between
the lattice point in the centre of the cube and the last point in the corner of the cube.
The square faces are the perpendicularly bisecting plains.
Between the lattice point in the centre of the cube and the lattice point in the neighbour centre in cube.
So if you go over one cube and you have a lot of points in the centre of that cube
and you make a perpendicular bisecting plane that gives you these square faces,
remember the way you construct the bigger site cells by making perpendicular by sectors.
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.
So it's an ugly shaped object. That's why we don't use primitive unit cells for the BC lattice.
But if you do take these ugly shaped objects, these truncated octahedron,
they will stack together very nicely and tile all the space, filling all the space appropriately as they're supposed to.
There's one more type of lattice that we need to discuss, which is the phase centred cubic lattice.
Phase centred. Centred.
Cubic or FCC or cubic dash f.
You may have noticed I wrote a centred the American way, not the British way.
And the reason I do this is because I did the other way. My brain would explode.
It's just, you know, you get used to doing something one way and that you just can't change.
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.
So you start with the simple cubic lattice, the points in the corner of the conventional unit cell.
And then you add one point in the centre of every face, like this one in the centre of this place,
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.
So they are going to have to make these harder.
Yes, they are four points in the convention itself. There is to see where the four points are.
Is eight corners, each of which counts one eighth.
So it's one eighth inside the convention itself.
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.
So it's six times one half equals four lattice points within the convention itself.
You can also draw the convention itself with this plan view scheme.
Which you know.
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,
which is that height is zero on a you can see the points on this on the halfway up the faces at height
labelled a over two so that's this point this point this point in this point that's this one this one.
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.
Okay. Now, if you imagine arranging atoms together or spheres together in the FCC lattice configuration,
it is actually the most efficient way that you can possibly pack spheres together.
So if you have a bunch of tennis balls, you're trying to stick them into a box.
The highest density packing of those tennis balls into the box is an FCC lattice.
There are other packages which are equivalently dense, but you're never going to do better than the FCC packing.
We don't study the other types of backings because they don't have orthogonal axes, but they are equivalent to FCC.
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.
So it's a very long standing standing theorem, but it turns out to be true.
You can't do any better.
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.
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.
Take FCC configurations copper, silver, gold, calcium.
Many others are FCC. Okay, so since there are four atoms per unit cell, we should be able to view the FCC lattice.
As for entropy penetrating simple cubic lattices.
So let's see if we can see that this is an FCC lattice of spheres and this is the red.
I have marked out a simple cubic lattice. This is a simple cube.
It's a little hard to see, but then you can actually pick out if you look very carefully.
There are three other enter penetrating simple cubic lattices which are mixed in here.
And although it may not be completely obvious that the environment of every single sphere is identical to the environment of every other sphere.
Now, since that's not completely obvious, we should do the exercise of actually writing down the lattice vectors.
Lattice vectors? For the FCC.
Gladys and I claim they take the form again. U v w u v w times a where either 1uvw all integer.
And again, that would correspond. Backing up one more that would correspond to the the points in the corners all integers.
So this one is is 100 this is 110 and so forth and so on.
And the other possibility. So you VW where either either possibility one they're all integers or two.
Two of them are half odd.
Integer and one.
Integer. Okay.
So we're going to show that this actually works. But before going on.
So for the one one more chocolate, what is the coordination of the coordination of 12?
Let's actually see on this on this on this plot, why it is the coordination number is is 12.
I'm going to stop asking questions going around that chocolate this way. I'm going have to ask harder questions.
Okay. So if you see the coordination of 12 of the take this sphere here,
the one that's cut in half, it has four neighbours one, two, three, four at the same height.
It has four neighbours at a slightly lower height. Here you can see two of them that are touching it also.
And then if you went to a slightly higher height, the same distance up that these are down, you would have four more.
So it's four at the same height, four slightly lower and four slightly higher, which may not look all that convincing.
But we can show it using this rule.
So this rule would give that to us for free.
The reason we would know this is because if you if you look at this point here.
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.
What does that correspond to? That corresponds to somewhere in the middle of the face.
For example, this guy here, he's over one half.
He's back one half, but he's up an integer height. Whereas this guy over here, he's over one half.
He's up one half. But his height is an integer, happens to be his depth is an integer.
This one here is over a half, upper half and is an integer back.
So if it's on a face, it has to have out integers and one integer.
If it's on a corner, it's all integers. Now we can check.
We can check here that again.
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.
So, for example, if you add integers to integers, you get back integers.
If you add integers to half odd integers and one integer, you'll get back in.
The fact that you're adding integers to something doesn't change where there's a half odd integer or an integer,
so you get back something in the set to again.
But what's not obvious is you take two things which have to have all integers in one integer and you add them together.
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.
So both of these are two of them, half odd integers, one of them integers.
So both of these would be on the face. You get two comma, one, comma, one, which is all integers.
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.
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.
So if you add together two things from set B,
you'll get either something from set that B set to you get either something from set one or something from set two.
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,
let's see if we can figure out that that coordination number 12 a little more easily.
So let's start with this point here. That's .000 and look for the closest things to 000 closest to 000.
Okay. Well, okay, this guy here, it looks pretty close.
His coordinate is one half, one half, comma, zero. So one half, one half comma zero.
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.
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.
Right. Those all fit the definition as well. And we could have put the zero in any of the three spots.
Three possibilities. So we have four possibilities of these plus or minus signs and three possibilities of where we put the zero.
So we get four times three equals 12 is the coordination number.
Okay, good. Okay. All right.
So it's worth also writing down a set of primitive lattice vectors for the FCC lattice.
So, please, for the FCC lattice, a really good example are just the closest vectors are make pretty good POVs.
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.
There they are. And again, to convince yourself that they really are primitive large vectors,
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.
So for example, if we wanted to get to this point here, we would have to add one of each of them.
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.
Well, I'm not sure. Yeah, okay. How do you get there?
Two of these and then minus one of these or something. And anyway, so you can play around with it.
It works. You can get there. How do you how do you get here.
Yeah. So we two of these would get you here then you need.
Well, uh. Mm hmm.
Maybe minus one. Yeah, minus one of these, then. Plus one of these. There's something that that will get you there.
Okay. Anyway, all right.
So since the FCC lattice has FCC lattice has for.
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.
Times a basis where the basis now includes basis equals.
The point at 000. The point at one half one half zero.
The point at one half zero one half.
And the point at zero one half. One half.
That would give me the four lattice points per unit cell.
And if I translate those four, that points in my conventional cell around to every lattice point.
So that is point is that 000 and these are all displaced from the original lattice point that
will reconstruct the entire phase centred cubic lattice or another way of thinking about it.
These are the displacements of the for inter penetrating simple cubic lattices that make up the FCC lattice.
As with the BC lattice, there's a reason we don't use the primitive units l for we use a conventionally.
It's not the primary itself because proving itself is extremely ugly.
Here it is. It's a truncated dodecahedron that has 12 sides. Those 12 sides correspond to the 12 nearest neighbours.
So what we have in the centre is one of the lattice points and then we have the 12 nearest neighbours,
four of them slightly above four at the same height and four in the slightly below.
And the faces of the truncated dodecahedron here are the perpendicular by
sectors of the segment between zero and the and the and the nearest neighbours.
Okay. All right. That's all we need to know about the SCC lattice.
In fact, that's all the three dimensional lattices we're going to study.
It is worth knowing that there are 14 types of lattices in.
In all the ones we've studied are the cubic primitive, the cubic body centre, the cubic phase centre.
We also talked about the tetrad, all simple diagonal and simple earth Arabic.
And then you have a whole bunch of other ones here.
You may notice that there's, there's analogues that we don't need to know, analogues of the body centre,
tetrad body centre, or they're iambic phase centred or thrown back and so forth.
You may notice there's no face centre tracking. All the reason for that is because in fact by turning the lattice sideways,
it would be in one of the other classes as well, so you'd be over counting.
So anyway, these are sometimes known as the Bravo lattice types. After Bravo, who is the first person to write them down correctly?
And so we give Brave a credit for doing them. So these are the only possible lattice types you can have.
It's kind of a deep mathematical statement that any periodic structure in three dimensions,
anything you can have in three dimensions is one of these lattices times some basis.
So if you can write down all these lattices and you can have any basis you choose,
no matter how complicated you can make anything, any periodic structure is one of these lattices.
Times, some bases.
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.
Very rarely do we need to know tetrad and fourth Romick. I think they only rarely come up on exam.
It's really these three across the top that actually show up on exams.
Okay. So this is an example of what you can put together when you when you have a lattice and a basis.
This is a sodium chloride structure, a very typical salt structure, sodium chloride, very ionic sodium gives up the electron.
Chlorine takes the electron. And if you look at it carefully, you can see that sodium is forming an FCC lattice.
It's even labelled cubic F. So maybe it's easiest to see in this picture.
These are all depictions of the same lattice. None of them are actually great, but.
We may have to make do. So here you can see the green sphere is here in this picture form, an FCC lattice.
So on the corners and in the centre of the faces as well.
But in addition to the sodium atoms which are on the FCC ladder lattice, there are chlorine atoms as well.
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,
one half, one half from the sodium atom. That will that will get you the position of all the chlorine atoms.
Here's a plan for you. The blues would be the sodium and the chlorine.
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.
When you describe a basis, you should always describe the basis of the primitive unit cell.
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.
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.
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.
So I only have to describe to you one sodium and one chlorine to tell you everything.
Once I've told you it's a simple Cuba. It's a it's a F.C.C. lattice.
Notice also that it's not unique how I describe the position of the chlorine with respect to the position of the sodium.
I could have told you that the chlorine was displaced one half zero zero.
That would give you the equivalent structure. Or I could have decided that chlorine was my 000 and sodium was displaced from it.
That would also give you an equivalent structure. So there's various different ways to describe the same the same lattice.
Here's another structure that we run into very frequently. The diamond structure, carbon.
It also is the structure of silicon and germanium. It is also based on the FCC lattice.
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.
The side face is one in all the corners, there's one in the centre of the face and so forth.
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.
They're additional carbons.
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.
So for example, in the plan view, we start with this carbon at 000 and then we displaced by a quarter,
one quarter on quarter and we find another carbon at that position as well.
So again, I don't have to describe the position of all of the eight carbons in the conventional units cell.
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.
And that's sufficient to describe everything. Okay. All right.
What's the lattice? What's.
What is the lattice? What is the lattice? What?
It is a lattice. It's a lattice with a basis.
What? What lattice type? Who said FCC.
Okay, good. Good. Wow. See this advantage to sitting up close?
You get more chocolate that way. Incidentally, I switched from chocolate types because I tried some of the chocolate yesterday.
It wasn't all that good, so hopefully this works better. I'm chocolate connoisseur.
Okay, so it's FCC. It's. It's. So this is gallium arsenide structure, also known as the zinc blend structure.
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.
So here, let me actually to show you. So the yellow at are form an FCC.
So you can see that the the yellow atoms in the corners and the centre of the faces.
So, for example, here here's a face and there's a yellow in the centre of it.
Then the blues are just displaced one quarter, one quarter on quarter from every yellow atom.
Now, some of the you know, it may look a little puzzling.
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.
So the FCC, there's four lattice points within the unit cell.
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.
You'll count for yellow atoms and then they're obviously for blue atoms completely inside the inner cell as well.
Okay, one more comment. This is a subtlety.
If Mike Glazer happens to be your tutor, [INAUDIBLE] be very happy that I tell you this.
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.
So here I'm going to tell you the subtlety to not tell you any lies. So suppose we have a material like this.
It appears to be a simple cubic system with a basis of three.
There's one on the corners and then two somewhere in the centre.
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.
So we would say it's a simple cubic with a basis including three, three, three atoms.
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,
you would be tempted to say it's simple cubic crystal, but a crystallographer would say it's not.
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.
You should be able to turn it in all six directions and it should look the same whichever direction you look at it from.
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?
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.
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.
And so if I tell you whether the the edges have the same length, he would say, well,
go measure it more carefully and you'll discover they're not the same length.
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.
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.
You just think they're the same length. So go measure it again and you'll discover that they're not the same length.
It would be an unbelievable coincidence if there was not a good symmetry reason
for the edges to be the same length and they end up being the same length.
Okay, so that's to make the real crystallographer happy.
And that is all we have to say about crystal structure for today.
Okay, good. So we've now learned everything we need to know about crystal structure in three dimensions.
All the lattices we're going to have to discuss we know about is the basis.
But at the end of the day, we really want to describe physical phenomena in these crystals.
And more often than not, what we're interested in is waves of some sort,
whether the vibrational waves or phonons, whether they're electron waves, whether they're electromagnetic waves.
And the world of waves is the world of reciprocal space.
So we have to back up and understand some things about reciprocal space.
When we have complicated crystals. So let's remind ourselves some things we learned about reciprocal space in one dimension.
Well,
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.
And then from direct space we have the reciprocal space.
Reciprocal lattice which we can write as GM equals two pi over a times m,
and now here to pi over a is the primitive lattice vector in reciprocal space.
Primitive lattice vector for the reciprocal lattice. Now, why was it that we chose this to be the reciprocal lattice?
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.
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.
Either the i k dot r and either the i gmm r and and this factor here is just one because it's this
is one because it is either the i two pi over a times m times a times n and that's just one.
So we're going to generalise that into more dimensions in any dimension.
In any dimension we define define the reciprocal lattice define.
Resupplied us in Sep lat via R points as points g.
Give actor such that. E to the i g dot r and oc g.
Let's not put an index on it yet. G r n. Equals one for all.
For all are an indirect lattice.
Indirect let us. So this is our general definition of the reciprocal lattice vectors g in the reciprocal lattice.
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.
I've claimed it defines a lattice, but I haven't shown you it defines a lattice.
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.
The proof that that defines a lattice in reciprocal space is a little subtle and a little tricky.
So we're going to go through it and see if we can make it convincing that it's true.
So first of all, we'll take the direct lattice vectors. We have to define the direct lattice vectors.
Direct lattice vectors x, we'll write them in terms of the primitive lattice vectors.
So R equals N1, A1, plus N2, a two plus n three, a three where the A's are the primitive lattice factors.
A's are PVS. A's are please.
But the last fact is, okay,
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.
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.
We'll call them Bs-vi as compared to ACB for the direct lattice and will define them by the following equation.
B sub I got it with a sub j equals to pi delta i j.
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?
A How do I know that there's a set of vectors? B Which satisfies this?
Are this equation sort of a dual space, dual vector space basis?
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.
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.
Okay. And this is for our i j k equals two, one, two, three or three.
One, two or two, three, one.
So in the cyclical way. So I claim that this expression for be will satisfy this definition of B and to check it.
Let's let's just find out if it's true. So let's take for example, b one dotted with A1.
So we'll write out B one. B one is to py a to cross a3 over A1 dotted with A to cross three.
And then we want to dot that into A1 and we see that the numerator in the denominator are actually identical.
So we just get to pi as we're supposed to.
However, if we took B1 dotted into A2, we would get the same same expression here on A2, cross a three of top,
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.
When you turn it into a two, you get zero.
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.
Okay. So now we have a guess for what our reciprocal lattice vectors are.
So we would guess that we would write down reciprocal lattice vectors g as m1, b1 plus m2, b2 plus m3 b3.
Okay. That's going to be our guest right now.
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.
What we need to show is that the M's can only be integers.
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.
So let me start with M's arbitrary. M's arbitrary, in other words, real numbers arbitrary.
And that allows i.e. consider any G consider any vector any vector g not necessarily on a lattice.
So I'm going to consider any possible g to begin with.
Now then what we're going to do is we're going to impose the definition of the I got R equals one.
So we're going to force on it one equals either the i g r.
Okay, what does that mean? Either the i g is m1 b one plus m2 b two plus m3 b three.
I'm about to run out of room and one a1, a1 plus and to A2 plus, m3 A3.
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.
Well, using our orthogonal B condition B by that AJ equals to pi delta.
J That is one equals either the two pi i M1 and one plus m2 and two plus m3 and three.
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.
This is true for all. N for all.
And and only if m are integers.
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,
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.
Therefore G is a lattice. And not only do we know that G is lattice, we know what its primitive lattice factors are.
So we just derived the fact that G is a lattice with those primitive lattice factors.
So far, so good. Happy with that? Okay, good. A couple of interesting facts fact.
Actually, I think this is a homework problem, maybe on the revision homework or something.
The recipient is SEP flat of FCC is B.B.C. PCC and vice versa.
Kind of an interesting statement. You can check it and I think you probably will for one of your homeworks.
Another interesting comment is it seems that in 2D, in 2D, same rules apply, rules apply.
But you might wonder, how do you handle this formula in 2D?
Because I only have two vectors, not three. The way you do you handle that formula is just choose.
Choose A three equals Z had a point coming out of the plane.
So in other words, if you live in 2D,
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.
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.
Some uses of this reciprocal lattice is a rather important statement that people frequently make.
The reciprocal lattice is the 48 transform for you transform.
Transform of direct direct this may be isn't surprising because the reciprocal
lattice lives in k space and the direct lattice lives in real our space.
And we know to get from our space, the k space, you frequently have to do things like Fourier transform.
But let me see if we can make this a little bit more rigorous. Let's do it in one D again.
So our lattice vector is our n equals eight times MN.
And how are we going to Fourier transform that lattice?
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.
This is what's known as a delta function comb. Delta function comb.
Have you seen this before in some in quantum mechanics or something called.
Okay, it looks kind of like this. Here's the x axis and then here's zero.
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.
Okay. Now try 40 transforming us. So 40 transform of all of x equals integral the x into the ik x.
Then we have this row of x function and we'll plug in the row of x function.
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.
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.
K. And okay, so what is this?
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,
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.
And, and so every term is one. So this then becomes sum over N of the number one, which is infinite.
If you have an infinitely big system if K is not an element of the reciprocal lattice.
Then what we have, then what we have is you'll have a situation where this complex phase here is not equal to one.
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,
you'll get twice the complex phase and you go out to something is three times as big.
You get three times the complex phase and these complex phases keep rotating around and around and around.
So you get some over oscillating phases.
Which goes to zero. They all cancel out and you get zero.
So at the end of the day, maybe I'll move over to hear what we have.
Is that the 48 transform of this delta function comb is a sum over all possible reciprocal lattice
vectors k minus GM of a peak and infinitely big peak at the position of the reciprocal lattice vector.
And if you do it carefully, you get a factor of two pi over a out out front a build.
The last factor is two pi over a is it's the same two pi that shows up whenever you do for a transforms.
There's always two PI's floating around, they're always there.
You're not probably going to be held responsible for ever getting this pre factor right?
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.
If you're sitting on a reciprocal lattice site,
if your K is a reciprocal lattice vector and it's zero otherwise and that becomes a delta function, come in k space.
Okay, now the same thing more or less holds in three dimensions.
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.
Again, if you for a transform it you'll get four a transform a row of x,
you let the Delta Function Act and now a three dimensional delta function Delta three.
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,
it's the sum over an infinite number of last vectors of the number one which diverges and you get infinity.
If K is not a reciprocal lattice vector, then you don't get the number one.
You get some complex phase and that complex phase rotates around and around and around and we add them up, you get zero.
So you end up getting to pi cubed over the volume of the cell volume itself.
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.
Okay. So far, so good. Now we can do a little better even.
We can consider things that are more complicated than just a delta function comb.
Because we're very frequently, we're very infrequently actually presented with a real delta function comb.
So instead, we're going to consider any periodic function, periodic function.
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.
So this function row of x equals row of x plus are where are the lattice factor?
It has the periodicity of the lattice. Okay.
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.
Integral d3x to i k dot x and this is the integral overall space row of x.
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.
I'm going to take that integral over all space and I'm going to write it as an
integral over all lattice vectors are then the integral d3x over the unit cell.
Maybe the bigger sites units all around are around R.
So I'm just breaking up that integral over all space into pieces, each piece being over the vector site.
So around the position are you the i k not x x?
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
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.
So row of our plus y is the same as row of y. So that makes that easy.
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.
But that in parentheses times this in parentheses integral d3y over the unit cell of the i k dot y of rove y.
Now, this this first time should look familiar since we just calculated a moment ago
this thing here is the sum of all lattice vectors of each of the i k that are.
If K is a reciprocal lattice factor, this is infinite. If K is not a reciprocal aspect of this thing is zero.
So this thing is just as we had before.
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.
So it gives you a delta function peak at the position of each reciprocal lattice vector.
This term here is known as the structure factor s of K structure factor.
Structure factor, and it's just the Fourier transform of the function we're considering within a single unit cell.
So this is actually a rather interesting statement and rather important statement we're going to use many times later on.
You take any periodic function whatsoever,
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.
Typical lattice vectors is weighted by the Fourier transform of the function within a
single unit cell is going to be extremely important in the next two or three lectures.
So I will see you on Monday. Monday, I think.
Monday. I think I'll see you Monday. All right. Have a good weekend.