Okay. Let's. Let's get going. Welcome back. This is the ninth lecture of the condensed matter course.
A couple quick things. First of all, the videos from the lectures are starting to appear online.
You can get to them. They're linked from my Web page. If you want to try them out, watch them again.
If you find it useful, let me know if you have any trouble with the technology.
We're still troubleshooting some things. Also, the message board.
So far, people haven't used it all that much. Please don't be shy.
I promise you'll find it helpful. So at this point, we're about a third the way through the course,
so it's worth sort of stopping for a second and taking stock of where we are and where we've been and where we're going.
So far, we've really been looking at trying to build up some sort of microscopic understanding of the macroscopic properties of materials.
We started with a couple of rather crude pictures of materials, and then we started looking at more detailed materials, but only in one dimension.
So eventually we'd like to have some detailed pictures of materials, but in three dimensions.
So we're going to turn our attention to three dimensions now.
And this actually makes our life a little bit more difficult, a little bit more complicated.
But but this is the eventual objective to describe real materials at this point.
It's worth thinking back to one of the things that I said in the very first lecture
that the world of condensed matter physics is extremely broad and diverse.
And once we understand how it is that atoms bond to each other, you know, and ionic covalent, hydrogen,
metallic or then evolves bonds, you can start building up an enormous variety of different materials to study.
And so it seems appropriate that we do a little bit of an obligatory tour of some
of the types of condensed matter that we're not actually going to focus on,
but that are out there and are worth knowing that at least that they exist.
So here are just a few examples of condensed matter physics.
This is the one type of condensed matter physics.
We are going to study the crystal in solids and I'll have a whole lot more to say about crystal and solids later on.
The general idea of a crystal and solid is this some sort of periodic arrangement of atoms?
And very frequently the macroscopic morphology of the crystal reflects its underlying structure.
As you can sort of see in this in this picture, this particular picture of a crystal is is quartz, silicon dioxide in a beautiful crystal.
And you can see that it's a structure somehow is is reflecting how the atoms are put together.
One can also have slightly more complicated types of crystals,
whereas in most crystals in these crystals, atoms are held together in ionic or covalent bonds.
You can have situations where Van der Waals bonds are extremely important.
So in this picture, this is known as a molecular crystal. What happens here is you have a lot of atoms coming together to form a molecule.
They're strongly bonded in the molecule via covalent or ionic bonds.
But then the molecules come together to form a crystal where it's very weak bonding between the molecules, in this case, van der Waals bonds.
This particular picture is a depiction of C60 60 carbon atoms coming together to make a soccer ball shape.
Or if you're on this side of the ocean and it's a football shape, I guess. On the other side of football has a different shape.
Anyway, so they come together to make these C60 molecules in the C60 molecules, they're very inactive species,
but they can still bond together, forming to form a crystal via van der Waals bonding.
One can have situations where you have something where it's not so organised a little bit more disordered.
This is what's known as an amorphous solid. The word amorphous comes from Greek, meaning without form.
Obviously the object has form of some sort, but it's a disordered form.
This is another depiction of silicon dioxide, but it's amorphous silicon dioxide.
The microscopic structure here, the blue objects are supposed to be silicon.
And the and the red objects are oxygen.
Every silicon is bonded to four oxygen. Every oxygen is bonded to two silicon.
And generally we are familiar with this silicon dioxide, amorphous as window glass, has a lot of the same properties as its crystal and cousin Quartz.
But it is a slightly different material.
We can also have disordered structures such as liquids very here and in liquids there's attraction between atoms or molecules,
but it's not so strong to keep it held in a permanent configuration that the atoms can still
have a thermal energy to move around and that gives the liquid its property that it can flow.
You can have situations where you have partial order, where there's some amount of flowing in,
some amount of Christianity or some amount of order that remains.
So this picture is now is what's known as a liquid crystal.
The order that remains is that all of these molecules, these things that look like torpedoes or also are they all retain their orientation,
they're all angled the same direction and they all line up in stacks.
But within each stack, they're disordered and they can actually flow within the stack.
And that's what makes them a liquid crystal.
And there's a huge variety of liquid crystals, depending on what type of order you retain and what type of order gets thrown away.
Halfway between the disorder and disorder is also the idea of the quasicrystal.
This particular picture is what's known as a Penrose tile named after Roger Penrose, one of the most famous mathematicians here at Oxford.
One of the most famous mathematicians in the world, actually.
Anyway, this is made up of two different types of of tiles, these little blue blue objects and these little green objects.
And they stack together, they fit perfectly together and in some very ordered arrangement.
But in fact, all the way out to infinity, the pattern never repeats.
It's a ordered but non-repeating pattern. And there are materials that have this property that are known as Quasicrystals.
So it was very controversial that they existed for quite some time.
It has actually been determined that, yes, indeed they do exist.
And the Nobel Prize was awarded for it just a couple of years ago.
There's also the idea of a polymer, which is sort of a one dimensional, long, very long molecule.
DNA is one that you're probably familiar with, but there's many polymers like polyethylene, for example.
And polymers have a lot of properties that are unlike other materials that we might be familiar with.
For one thing, in their fluid phase, they're sort of like a bowl of spaghetti.
A lot of these strands sort of all mixed in together, and they have a very non-Newtonian behaviour in terms of their fluid dynamics that
their viscosity depends very strongly on the timescales that you're considering.
So if you've never done this experiment with cornstarch and water, you can sort of mix up a group of cornstarch and water that will pour like a fluid.
You can put your hands in it like a fluid, but if you hit it with a hammer, it will crack.
It's really impressive, you know? The issue is that that the the long strands they take some time to untangle.
So if you try to move them very quickly, they'll feel like a solid. But if you give them time to relax, they'll flow past each other.
So that's just a very brief tour through some of the topics in condensed matter that we could study.
Most of them we're not going to study. Mainly we're going to. Yes. It is I.
Ice is ice is a crystal. A genuine crystal is an order crystal.
Most forms of ice, there's like 20 forms of ice.
Most of them are actually the ones we're familiar with is is generally a crystal and solid is what it's ordered.
But they actually mark this. You make a point,
ice is a little bit different because there's some slight rearrangement of the of the oxygen
hydrogen atoms on a microscopic level that makes it not perfectly periodic all the time.
But but it's mainly a crystal. We can we can discuss the this the the slight subtlety associated with ice that makes it slightly different.
Yes. Okay. Mainly what we're going to study is crystal structure, crystal in solids.
And here, before we get going, I have a couple of apologies. The first apology is that I'm going to have to use a lot of PowerPoint slides.
And I know PowerPoint slides are annoying. It's difficult to learn from PowerPoint slides, but unfortunately I can't draw this.
All the slides that I post are going to be on the web page. Most of them are in the book in one, in one form or another.
Hopefully that will be both and the lectures will be on online as well.
Hopefully that will give you enough information to be able to to learn the material.
So that's the first apology. The second apology is particularly in this lecture.
And the next lecture we're going to have to suffer through a lot of definitions, and that's a little bit tedious.
But unfortunately, we need to have a language in order to be able to describe the things we're talking about.
We need to be able to say, what is this picture we're looking at?
Where are the atoms sitting? We need some sort of vocabulary in order to communicate.
So that's we're going to have to just bear with that and suffer until we until we understand what all the words are.
And then we can start moving on with the physics again.
The second the third apology is that unfortunately, not all books agree on what the proper definitions of terms are.
So you will find some books that call a dog a cat and some that call a cat and a monkey.
So it's kind of a mess. The terms that I use are the ones that are generally accepted at Oxford.
They're actually probably the most commonly used terms for for certain things.
I will try to point out cases where there is some disagreement as to what the terms mean or what terms should be used.
But just be warned if you look in other books there, and there's also caveats within within my textbook as to which terms are controversial.
So just be warned of that, that that is one of the issues we have to deal with.
Okay. So we're going to start with the first definition of the day, a crystal.
Crystal is a periodic arrangement.
Periodic arrangement. Meant usually meaning of atoms.
So for example. Oops. That's a carousel.
So there's a simple crystal on the left, and then you have more complicated crystals over on the right.
They can have many different types of atoms. The atoms can be in very complicated arrangements.
But as long as this periodic is a crystal, a very important definition,
I'm actually going to give several definitions of this word, which are all equivalent a lattice.
So definition a maybe the most useful?
Is it a set of points? Defined as defined as integer sums.
Sums of. Here's another word.
Primitive lattice vectors.
Lattice vectors. Okay, now here's a case.
I'll explain what primitive large vectors are in a second. But here's a case where there's a word that people disagree on primitive.
Sometimes people drop the word altogether and they don't say it. Sometimes it's principle.
Principal lattice is sometimes replaced by translation and is sometimes replaced by the word basis.
That's a particularly inconvenient because we use the word basis for something completely different.
So that makes a real mess. Vectors is always vector.
And the other thing is that some some books will call lattice brave lattice instead of lattice reserving lattice for real meaning crystal.
So that makes a complete mess. Some people call a crystal, a lattice and a lattice.
A bravo lattice. Please ignore that. That's just.
I mean, it makes everything incredibly confusing. Anyway, this is a good set of definitions that we will use here at Oxford.
The exam committee agrees with these definitions, so hopefully it won't get too complicated.
We just have to deal with it because that's the way the world is.
Okay, so let me actually define give examples of what these crystals, what these lattices are.
So we did this in one day. You have a bunch of points lined up in one D with a distance, eh, between them.
So then we can write.
The positions are of the points in the lattice as an integer a times a, a is the A here is the primitive lattice factor prim lat vec.
Actually, I'll frequently call that p lv primitive lattice vector and then n is an integer.
So this is something we've seen before. We have integer times, primitive lattice factor gives us the points in the lattice.
So I wanted to do a little bit more complicated.
So the lattice points, lattice vectors here are integer times,
a primitive lattice vector A1 plus an integer and two times another primitive lattice vector A1.
The A's are the primitive lattice factors.
The Elves, the A's should be A's should be non linear, non linear,
so that we have something that's really two dimensional and the ends are integers and our integers element of integers.
Okay, so let's show an example of this. So here we have a lattice and we can A1 and A2 are the primitive lives vectors.
If we think of this point in the centre as being the point zero,
every point in this lattice is an integer, it's an integer times A1 plus an integer times A2 for example.
This point here is to A1 plus one A2 OC and frequently we will notate r equals n1 comma N2 in square brackets
to indicate N1 times A1 plus and two times A2 like it's indicated up here to comma one to indicate this point,
if this point here is taken to be the origin.
Now, I should also comment that this notation is frequently abused, that sometimes people will say the point one half, comma,
zero or something like that to mean a point, which is not a lot is point, the point which is in between lattice points.
So if you want to specify this point here, you would say go a half the distance of the primitive last factor.
A1 The more important thing is that the square brackets mean a point in space and for the
integers coefficients here then you mean a lattice point of is not integer coefficient.
That is not a lattice point. Okay, so far so good. All right, so far, pretty easy.
We can do the same thing in three D. Three.
We would write R equals R and one A1 plus end to a two plus and 3a3.
And the A's should be non complainer. A non complainer.
So we're talking about something that's three dimensional and we can notate it as R equals brackets and one and two and three like that.
One thing that you should be aware of is that the choice of primitive lattice factors of pelvis, primitive lattice vectors is not unique.
So you can choose different sets of primitives, large vectors, and they're perfectly okay.
For example, you could choose these two things as primitive lattice vectors.
You could use these two things as primitives, large vectors. You could choose these two things experience large vectors.
The important thing is at any point on the lattice, should the express of all as an integer some of the primitive lattice vectors.
So far, so good. All right. First chance to win fabulous prizes.
Are these primitive ladies factors? Okay, who was the first person I heard?
Okay, now I know you were in someone somewhere. Someone else over here. It was you because of such liars.
So. All right, so there was some voice over here.
I heard it over here, but I don't know who get it. All right, one more chance, okay?
Yes. These are primitive latch factors,
is it's lattice vectors because you can get to any point in the lattice by some sort of integer, some of those large factors.
Okay. How about these are these primitive, large vectors?
No, they are not. So they are not.
Because if you add them together in all integer combinations, they miss every other lattice point.
Okay. They would be primitive lattice factors for this lattice if I took away half the lightest points.
Then you have primitives, large vectors. All right.
Everyone always misses that one. Okay. Every year. Okay.
So my second definition of a lattice.
A lattice? Sort of equivalent definition of a lattice.
This is B is that it is a okay addition.
Of the name change. It is a set of vectors.
Set of vectors such that the addition of two.
Of two gives a third.
Okay.
And this actually, it's pretty easy to see that this is equivalent to the prior definition if you have N1, comma, N2, both of those being integers.
So it's an element of the lattice and you add it to N3 come and for both of those being integers, so it's an element of a lattice,
you get five and six and those are then integers also and therefore an element of the lattice as well.
A third definition C is that a lattice looks the same.
Looks the same. From viewed from viewed from any lattice point.
Okay. So for example, if I set myself here, so this thing is now is a lattice,
if I set myself here and I view the environment around me, I say, okay, there's a there's a point to my left this distance.
There's a point to the right the same distance as a point up right, up, left, down, right, down, left.
And I can measure the distances, I can measure the angles. And if I sit here another lattice point over here,
it has exactly the same environment to point to the right point to the left point to the upright up left, down, right, down, left.
Exactly the same distance is exactly the same lattice, exactly same angles.
If if this is true for every one of these points, then what you have is a lattice.
So for example, is this a us? I don't know who is the who.
I heard this over here. Who is? Who is? Who is it? Is it you?
Are you okay? All right. Five, four.
All right. Okay. So I have to get out some of these. I have, like, six of them here.
So, okay, this is not a lattice. And there are several ways to see why it's not a lattice.
So, first of all, from the first definition, are there primitive lattice factors which we can add up in all integer combinations which would
give exactly these points and only these points when sum together with integer coefficients?
Well, there aren't. But it may not be so obvious to see that that's true, that you can try coming up with two primitive lattice factors.
They want to take two and see if we can come up with these points and only these points.
You can't do it, but it's much easier to use some of the other definitions like this one we should have.
If we think of this point here as being zero, then we have this lattice factor here in this lattice factor here.
When you sum them together, they should give a third lattice vector.
But if you sum these two together, you get a point in the middle of this hexagon and it's not there.
That's not one of the points in our set. So this is not a lattice because you can't sum two things together and always get a third.
The third definition is that the environment of R is the environment of all points should be the same
and the environment R is not the same as the environment a p r has a has another point to it south,
whereas p has a point not to its south but to its north.
So P and are in equivalent points being Q are equivalent.
Okay. So even among the professional literature, people will refer to this thing as the hexagonal lattice.
And that's just wrong. It's not a lattice. You shouldn't call it the hexagonal lattice, although people always do.
So be warned about that.
I know this is horrible to be giving you all these caveats that people say these things wrong all the time, but it's just true they do.
Okay, so let's go on. Fact.
Any periodic structure. Is a lattice of repeating.
Motifs. Okay.
Any periodic. This is just a fact. Any periodic structure is a lattice of repeating motifs.
So here's my this is same picture in the book. I just like these these things.
They're armadillos, I guess. So this this periodic structure of armadillos.
And in fact, if you take a lattice here, it's a triangular lattice made up of a bunch of triangles.
It's a nice lattice. And you replace each of these points with an armadillo.
You get the whole periodic structure. So I've drawn it as lattice times, the repeating structure, which is the armadillo.
Well, what about this? This is certainly a periodic structure. So can we see how this thing is actually a lattice of repeating motifs?
Okay. I claim that it's a triangular lattice, this triangular lattice of blue dots times this repeating motif,
and that might not be so obvious to see, but let's see if we can make it obvious.
So first, what I'm going to do is I'm going to put a big red dot in the middle of each vertical edge.
Okay. So here's a big red dot here, a big red dots. And now it's looking a little bit more likely that, first of all,
these big red dots are clearly forming a triangular lattice just like this triangular lattice down here.
Now, make it a little bit more obvious by putting a green diamond around this repeating motif.
And you can see that, in fact, if I stack together these green diamonds,
each one of them centred on these big red dots, it will reconstruct the entire honeycomb.
This thing that people call the hexagonal lattice that isn't a lattice.
So it should be called the honeycomb, I guess. All right.
Is that clear? Okay, good, good. So far, so good.
So that brings us to another very useful definition, which is a unit cell.
Unit cell equals a region of space.
And basically it's the repeating motif. It's the region of space of a space which.
When repeated. And stacked and stacked together.
And sometimes we use the word child, meaning repeated child means repeated and stuck together reconstructs.
The Fall fall periodic structure.
We use this term, units sell previously in one dimension.
Now we can just use it in general dimensions.
So this green diamond would be the unit cell for this repeating structure.
Okay. And let's see if we can do another one. Here's our armadillos again.
So we have a unit cell which contains one armadillo.
We can stack them together. What's important about unit cells is that they have to stack together with no space.
They should be like bathroom tiles. They should fit together exactly with no space between them.
Another definition which is useful is a primitive unit cell.
In itself equals a unit cell containing.
Exactly one lattice point. Exactly one lattice point.
So, for example, in this picture, this would be a primitive unit cell because it contains one big red dot and that's the lattice point.
And okay, so we can't see the lattice points in here, but if you remember where they were in those blue pictures,
the the armadillo is sitting on top of the latest point. So each in cell contains exactly one lattice point.
I guess here's here's another case. Here's a nice lattice. There's a primitive unit cell because it's if you can tile all space with it,
you put them together, tile all space, and each unit cell contains one lattice point.
However, one should be warned that the primitive unit cell.
Primitive even cell. Cell is not unique.
You have different choices as to what you want to use for your premium itself.
Self. For example, here's a lattice written, you know, for different times and you can choose plenty of unit cells to be any shape.
So you can choose from this shape the shape to shape the shape as long as they fit together and tile all of space.
You have a very nice primitive cell. Some of them are more convenient than others.
One should be warned that something like this is also a good primitive unit cell.
And that's a little less obvious that that contains exactly one lattice point.
But you have to think about the the things that are partially enclosed in the unit
cell like this guy has only counting is theta of this the of the full circle.
And this one counts as theta of the full circle. This one is primary minus of the full circle.
This is primaries full theta of the full circle.
And if you collect all those pieces of the the lattice point you put them together, you'll get the full two pi.
Okay, so it's one lattice point, but it's broken up into four pieces. Okay?
It's one lattice point. Broken into four pieces. Maybe this one. This is also a good primitive unit cell.
This is a lattice point more obviously broken into two pieces.
You assemble them together, you get the one lattice point back.
And this one is a is a simpler unit cell, which is maybe easier to see that that contains exactly one one lattice point another definition.
I know we have a lot of these conventional units cell.
Units l equals conventional as convenient a unit cell a non primitive you to cell non primitive unit cell that is convenient,
which usually means usually meaning orthogonal axes or orthogonal.
Orthogonal axes. Right. Think so?
So, for example, here we have a triangular lattice and a conventional unit cell here with orthogonal axes rather than primitive unit cell.
So how many lattice points are in that? Can someone up there who has it, someone way back there.
All right. I believe you because you look honest. Okay.
So, yes. Okay.
Another another definition. So over here, a beginner site sell, beginner sites sell.
Vigna was a very important physicist if you haven't run into his name already.
Site was less important, but also important. The region.
The region. Around a lattice point closer.
To that lattice point. That point, then to any other.
Okay. So you have a whole bunch of lattice points you would like to find the region around.
One last point that is closer to that lattice point than to any other lattice point.
And there's a way to do this, which is the beginner sites, construction beginner sites,
how you find that region, construction, construction, which I mean, I will explain this construction.
It's easier to to describe it than to write it out. But I'll write down use perpendicular perpendicular by sectors.
So let's see how we do this. Okay, so here's a lattice, and I've labelled one of the points of the lattice black.
We'd like to find the big site cell of that lattice point.
In other words, what is the region of space which is closer to that black lattice point than is to any of the the blue lattice points?
Okay. How do we do that? Well, you pick a neighbouring point like this one point up here and you put down a perpendicular bisected this red line.
Now everything on the lower side of that red line is closer to the black point than it is to the blue point labelled one.
Right. Definition of perpendicular by sector. Now, let's do that with another point over here, this point two,
and now we have this vertical perpendicular by sector and everything on the left hand side of the
of the of the vertical line is closer to the black point than it is to the blue point label to.
And you do the same thing for three and four and five and six and all the way around until you wall off a region.
And everything in that region which I've now call a green, is closer to the black point than it is to any of the other blue points.
Is that clear how that works? Okay, good. So then you have this nice region, which is a vector site cell,
and it turns out that the thing there site cell is primitive is primitive, even cell in itself.
Well, first of all, it obviously contains only one lattice point.
So that's good at tile space because you can put a finger size cell around every point in space.
And obviously every point in space is the biggest is in the being site cell of some lattice point, whichever last point is closest to you,
you're in that vector site cell and so at tile space and encloses one lattice point and each one of these has the same shape.
So it's a nice primitive in itself. Okay, good.
Everyone happy so far? Okay, now I'm going to this.
I think. I think this is not on the syllabus, so you don't really need to go it.
But it's pretty cool. Known as a boring cell to spell it right.
And I will annoy several annoy var on a cell which is a beginner side cell.
For a non lattice. For a non lattice.
Collection points of points.
So imagine you just have a bunch of points, say, in a plane and you wall off the region,
which is closer to one of the points than to any of the other points.
This concept of avoiding lysell shows up not only in physics, it shows up in chemistry, biology, epidemiology.
It shows up very heavily in computer graphics. It shows up in just all over the place, including pizza delivery.
So I'm going to explain to you why it's important for pizza delivery.
And you will you'll realise why the born cell is such an incredible thing.
So okay, so this is this program you can click the link to on my website.
So imagine this blank sheet. Here is a map of Oxford.
You know, nothing's written on it yet and you own two pizza shops and one of them is in North Oxford.
We'll put it here and one of them's out in Cowley. It's over here. So that blue line is the perpendicular bisected between those two.
Okay, now if someone calls you up and says, I want a pizza, if they're in this region, they're closer to this red point.
So you have the guy in North Oxford deliver, whereas if it's in if it's down in this region, well, it's closer to this point.
So you have that guy deliver. Now you're making a lot of money, so you buy a couple more pizza shops.
So let's put in a couple more. One, two, three. Okay.
You have a bunch of pizza shops and now these are the voluntary sales of each of these red points.
So this region here, everything inside this region here is closest to this red point.
That is one I sell. So if someone calls within this region, it's serviced by this this guy who's closest to him and so forth and so on.
And you can keep going on and on and on and adding points.
And these cells indicate the cells always wall off the area, which is closest to that red point.
And you can kind of guess that there's going to be millions of applications of the idea of the voting.
I sell in lots of different areas of science, including pizza delivery, which should be a science by now.
Okay. All right. So that's all we have to say about Vaughn Neisloss.
So next, the word basis. This should not be confused with the word basis that we had when we talked about primitive lattice vectors.
And some people call them primitive basis vectors. That's not this. This is why we do not use the word basis.
When we talk about primitive lattice vectors, we don't call them primitive basic vectors because we use a bird basis here.
Sorry about that confusion. Okay. So basis is a description of objects in the unit cell.
In unit cell. With respect.
With respect. Our key ASP is key. With respect to our reference.
Lattice point. Okay.
So for example. Oops. Okay.
There. So here we have a periodic structure. I put the lattice, the reference lattice point I've drawn in blue in each of these unit cells.
And then you'd have to describe everything that's in the unit cell reference to that one lattice point.
So you would say there is an armadillo sitting on top of your blue dot. Okay, maybe that's not so, so useful.
Let me do a more a more physics oriented one.
So here's a crystal here, made up of two different types of atoms the large light grey atom and the small, dark grey atom.
The black points of reference points that give me the corners of the unit cell.
So here's a unit cell in here. It's just been blown up a little bit, so you can see it a little bit better.
So here we have a unit cell next year in a cell, and if you stack the unit cells together, you reconstruct the whole the whole crystal.
Now, a basis is that we should list off all of the objects in the unit cell reference to the reference lattice point,
which we call coordinate zero zero. So we would say, for example, there's a large light grey atom at position.
A over two a over two here. With respect to this point, zero zero.
Then there are small, dark grey atoms at positions.
And for April 4th, we have a for a fourth we have a four, three and four and eight for a four so far.
So once I've given you all of this list of where all the objects are in the unit cell, what the objects are and what their relative vector is.
With respect to the reference lattice point, I then give you a basis for the crystal clear.
So if I tell you the lattice and I tell you the basis, I've told you the whole crystal.
So this is the language we need in order to describe the whole crystal.
Now sometimes it is useful. Okay, so here it's useful to take the reference lattice to be coincident with some atoms.
So here's. Here's a crystal made up of red and blue objects, red and blue atoms.
Now you can take a unit cell that looks like this. It contains one red atom, one blue atom.
But what people do more often is they put the unit cell with corners on the red atom, or they could have chosen blue.
In this case, we're going to choose choose red. Then we would say our basis is the red atom is at the zero zero position of the unit cell,
and the blue atom is a distance one half in the A1 direction from the red atom.
So it's a position one half comma zero. Now note it's really important not to over count the red atom.
So you don't say that there's a red atom at zero zero and another red atom at one zero another red atom at one one another red atom at at zero one.
That would be counting four red atoms. And there's only one red atom in that unit.
Cells divide up into four pieces, but you only have to list it once it's zero zero.
And then it's implied that if it's at zero zero, it's also at all the other pieces.
It's other, you know, it's quarters the way it's divided up, if it's one piece of it is here, the other pieces of it are going to be there.
So you don't over list it. Is that clear listed just once, something people frequently make mistakes on.
All right. So now we can move on to what we're finally building up to three D lattices.
Three lattices. So this is an important statement, maybe the most important statement I will make all day.
You will not need to know. To know.
By the decrees of the exam committee. You will not need to know about lattices in 3D.
In 3D where axes. Axes are not not orthogonal.
Why? Well, because then you won't have anything to learn next year.
No. I mean, the reason is it's just because it gets a lot algebraically more complicated.
And it was sort of just agreed upon by the exam committee that in third year we can all have a
3D lattices just to prevent us from getting just knee deep in all sorts of algebra of angles.
We're just going to stick with orthogonal axes in three dimensions, and I think that that rule has never been broken as far as I know.
So I've been told this is a hard and fast rule, so you can stick with it.
I hope it stays true.
So the simplest lattice in 3D cubic lattice, we colloquially call it a cubic lattice, but it's probably more properly called simple cubic.
Simple cubic. Or a primitive cubic or cubic dash P for primitive.
And you probably know what this looks like. It looks kind of like this. This unit cell for this unit cell equals a cube.
Unit cell equals a cube.
And it looks like this. How many atoms are in that?
How many last points are in that unit? Sell one. Okay. Everyone's yelling out.
So I don't know how to give this challenge I have even myself. All right.
So, yes, it's one one last point in that in that unit cell.
So that's a primitive units cell. And it's one eighth times all of the eight corners, each corner is one eighth inside the unit.
So we can write down the lattice points in more proper way that are equals u v w
times the lattice the edge line to a where you v and w u v w are all integers.
Now. We were told at the beginning that we should be able to write lattice points as
integer sums of primitive lattice vectors so we can choose a one equals a times 100,
a two equals a010 and a three equals a001.
And these are the primitive last vectors p lives. Okay.
Good. There is yet another word it's useful to know is the idea of a coordination number.
Number. The number which is usually called Z, which is the number of nearest neighbours.
So what's the coordination number of this cubic lattice? Someone said six.
Who is the first person to say six? So over here it was okay.
I trust you. I believe you. One of these be. Yeah.
It still tastes good. It's good chocolate. So I see.
You know, the woman who lived the oldest recorded human lifespan Gene come actually 124 and a half years old.
She ate a kilogram of chocolate a week. It's true. It's on Wikipedia.
So. All right. So actually the simple cubic symbol, cubic lattice, although it's, um,
this is what it would look like if you put atoms on the position of each of the lattice points.
It's a, a cube, obviously, although this is a very simple lattice to describe.
It's actually extremely unusual in nature of all the materials, all the atoms on the periodic table,
only one only one element actually takes this simple cubic arrangement.
The element is polonium. And for one more, who discovered polonium inside someone over here?
It was I believe it was. It was Marie Curie.
And Polonium was named after her home country of Poland.
I have to mention that because Marie Curie is one of my heroes. All right.
So the reason the reason why this is so unusual,
you have to sort of think about stacking to that second sphere is if you ever try to put like tennis balls in a box or something,
you'll realise they really don't like to stay in this position. They really want to fill that hole in the middle.
There's a lot of extra space in that hole in the middle and someone will fall into that extra hall.
Another way to think about it is that, you know,
there's a bonding force that wants to get these atoms really close to each other, are fairly close to each other.
If you sort of think of it as hard spheres that are trying to attract each other,
that it's not really going to like to leave that great big hole in the middle.
Something's going to want to fill that hole. So that's why the simple cubic lattice is is very unusual among pure elements.
So there are some variance on the simple cubic lattice that we need to know to track it all.
Let us. Which means.
Axes perpendicular. Axes perpendicular.
To edges. Equal. Equal.
So draw a picture of that.
So here's simple cubic hair said triangle two the edges are equal, one edge is different, and the other we need to know is orthodontic.
Which is actually perpendicular, no edges equal.
Okay. So there you go. Simple with Arabic, all edge length.
Different, but at least all orthogonal.
Now, although simple cubic is a very unusual in nature, there are things that are based on simple cubic that are quite common.
So you do need to know about the simple cubic lattice. So let's go back to this picture here.
Remember, it's very convenient if you have more than one type of atom to put the reference lattice and the reference atom at the lattice point,
and then another atom, you can reference it. You can give a basis by giving its position with respect to the reference, the reference lattice point.
So a very important structure is the so called caesium chloride structure.
Name that way because the material caesium chloride takes that structure.
Obviously, caesium chloride, extremely ionic material, caesium gives up.
It's an electron very nicely and chlorine is extremely electronegative.
So it wants to take that electron. And so you get nice ionic bonding.
So the definition of caesium collide structure is cubic lattice with a basis simple cubic lattice or p cubic primitive cubic
lattice with basis basis where we put the caesium at position 000 on the lattice point and the chlorine at position one half,
one half, one half. That's in the middle of the hall that we had in this picture here.
So the chlorine is going to fill that hole. So what we get is something that looks like this.
We get the caesium is on the corner of the unit cell and the chlorine sitting in the centre of of the unit cell.
Okay. Now you might think of it this way as having caesium is on the corners of the unit cell in the chlorine, the centre of the unit cell.
But that's kind of arbitrary that we chose it that way because so here's another picture of the same structure.
The caesium are white in there and the chlorine is a green here.
So here's the unit cell we were just discussing with caesium on the floor, on the corners and chlorine in the centre,
but one could just as well have chosen this as the unit cell with the chlorine is on the corner and the caesium in the centre.
So what we really have is two interlocking simple cubic lattices.
One of them is made up of the caesium is one of them is made up of the chlorine.
Okay. And here's a, here's an animation that's supposed to show that.
So that's caesium chloride. There's the caesium here, caesium chloride again and there's, there's the chlorine.
So each of them is a simple cubic lattice and they're sort of stuck together.
So this type of structure, the caesium chloride structure, is actually quite a common structure in nature.
Now we can do something else, which is to consider pure caesium, pure caesium, which is a very similar structure, a pure caesium structure.
So we have so it's a cubic P lattice, p lattice,
but we maybe going to put this in parentheses because we're going to say that's a line moment caesium.
We're going to put the caesium at 000 and then another caesium at one half, one half, one half.
Okay. So all we're going to do is I'm going to take that caesium chloride and we're going
to get rid of the chlorine in the middle and replace it with another caesium.
Okay. So we can think of it as simple cubic with a basis where the basis is a caesium is 000 and a caesium at one half, one half, one half.
Now this is actually itself a different type of lattice.
This is an important type of lattice that we're going to be studying, which, well, it's known as the body centred cubic lattice.
Make sure I don't miss anything. Body centred. Cubic lattice lattice also known as cubic dash by.
So the word body centre is not it sounds like something out of yoga or something, but it's actually, you know,
the idea is that is it you've put a another point in the centre of the body of that
cube I is from German enters entry which must mean something in a German scholar.
It's something like ends in the centre of the body or something.
Anyway, anyway, so I'm going to claim that all the environment of all of these points is exactly the same.
So this is actually itself a lattice. So let's see how this how this works.
This is the unit cell of the body centred cubic lattice.
How many lattice points are in the body centre? Cubic lattice just.
All right. I didn't look at it. It's too.
This is a conventional unit cell of the body centre cubic lattice.
We take the conventional in itself to be the cube again, just because it's more convenient to do so.
There's one in the centre obviously, and then we have the one which is divided into eight pieces in the corners.
Now there's a convenient way to describe these three dimensional structures, which is to use what's known as a plan view.
A plan view is coming from a sort of architectural term where you look down on the thing from the top and you draw the heights,
you indicate the heights of all the things that you see.
So if you're up in a, you know, a helicopter and you're looking down on this conventionally itself,
you would see four things in the corners and you would see one in the centre and then you label the heights.
So this one is labelled with height eight over two and then down here it says unlabelled points or a height zero and a so this one's
a zero and a this one's at zero and a this one's a zero and a this one sits here on A And this one here is at height a over two.
Okay. If you put put atoms together in this body centred cubic structure, they would look more like this.
And it's actually it's much it's clearly it's filling up much more of the holes of the lattice.
This way is much more efficient sphere packing and there's quite a few materials that actually take this body centred form sodium,
lithium ion potassium is many, many, many more.
So what is the coordination number? Ed said it was over there.
Okay. Yeah. Okay. Eight. It's eight. So if you think of if you think, pretend you're the guy in the middle here, then you have eight neighbours.
Each one of those eight neighbours. What's less obvious is that if you happen to be one of these guys in the corners, you also have eight neighbours.
Okay. Every point here is identical. Maybe.
Let's show it here. So here's the conventional sell like this.
And here's the point in the centre.
Now, you could just as well have chosen this point to be the centre of a cube, and then we would have this cube with this point in the centre.
Okay. So again, remember that this is really two interlocking simple cubic lattices.
When we were thinking about it as being caesium chloride, there are two interlocking simple cubic lattices.
So if we choose this point as the centre, then the yellow ones make the cube around it.
If we choose this one as the centre, then the red ones make the cube around it.
Either way, it always has eight nearest neighbours. Okay.
The last thing. I think we're out of time.
We'll pick up next time and will prove that this thing is actually a lattice by the definitions of lattice that we gave at the beginning of the day.
All right. I'll see you tomorrow.