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3.091 Introduction to Solid State Chemistry, Fall 2004

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3.091 Introduction to Solid State Chemistry, Fall 2004 Transcript – Lecture 15

First of all, welcome to the families. It's family weekend. So, welcome to the parents, sibs, and other members of extended family. I hope you have a lovely weekend on campus, and especially welcome to 3.091.

I have nothing special planned today, just a run-of-the-mill lecture so you get a sense of what it's like to be sitting in one of these seats that you've paid so dearly for. I hope we can convince you that you've made a smart decision in putting your son or daughter on this campus.

You will come to the quick conclusion it's not because of the facilities. Facilities here are nothing special. Classrooms, labs, and so on, it's the peer group. It's not the faculty. Faculty are OK, but I think it's the peer group.

I think it's the students are going to be the friends and colleagues that makes it a special experience here. I'm teaching freshman chemistry, but it's taught from the Department of Materials Science and Engineering.

And, I just want to make one little plug. If you walk up and down the infinite corridor, by and large it's ugly because it's decorated with administrative offices, with one exception. As you get to the east end, it looks like an engineering school because there are labs associated with the Department of Materials Science and Engineering.

And, if you ever wondered what's going on in those labs, it's your chance to find out. So, this afternoon from 3:30-5:00, it will be possible to tour the undergraduate laboratory, which is on the south side, and the nanomechanics lab on the north side.

So, if you're finding uncomfortable silence with your son or daughter, and things are kind of dragging on, walk down to the east end of the infinite corridor, and you can see some interesting things.

They're going to have some demos, and whatnot. And, the other announcement is that on Tuesday, we will have quiz six based on homework six, but there is a few questions on homework six touching upon x-rays.

And at the rate we're going, we will not have covered that material. So, ignore the questions on x-rays. But, the rest I think you should be familiar with. And, we'll test at the end of the period instead of the beginning.

That way, you will have some time to interact with the recitation instructor. So, what I want to do today is something really important. And, it's fitting that the parents are here. Up until now, by and large, perhaps with the exception of some of the band gap material, what we've been covering, you could've seen bits and pieces of that in high school. And, if you've taken a lot of high school chemistry you might be familiar with it. Well, starting today, say goodbye the high school. This is solid-state chemistry now. That's why we have the crystal models appear.

And, what we're going to do is we're going to begin with a little bit of review. And so, if you'll take a look at the handout that's going around, this will, of course, be posted on the web afterwards.

But, just to follow along, I know the parents are here. It gives them some comfort factor to hold a piece of paper in their hands. So, that's your Linus blanket for this morning. So, I want to give the big picture.

And the big picture is that the thesis of 3.091 is that electronic structure informs bonding, which then informs state of aggregation. And just to remind you, we started back at the beginning of September with the Bohr model of the atom, and saw that it was inadequate.

Sommerfeld put in a few patches, came quantum numbers, Aufbau Principle, multi- electron atoms. Then we could explain the patterns in the periodic table. And then we posited octet stability as something that atoms strive for.

And that gave rise to, first of all, just inert gases, explained them. But then we said, you know, things might engage in electron transfer. We could describe ionic compounds, then electron sharing, covalent compounds.

Then we talked about metals, which is 75% of the periodic table, and en route saw the Van der Waals bonds as a fourth type of primary bonding. And then, we looked at secondary bonding, which is prevalent in covalent compounds, and we saw these three types, dipole-dipole, London dispersion, and hydrogen bonding.

And, ultimately, that allowed us to decide whether something is a solid, or a liquid, or a gas at a particular temperature. And, we are particularly interested in 3.091, since this is solid-state chemistry, the circumstances under which something is a solid.

So, up until now, I think you're comfortable with classifying materials on the basis of their bonding type, in other words, is it a covalent solid, or ionic solid, or a metal. But, I want to show you a little more interesting way of classifying because we are going to classify things according to atomic arrangement.

And so, if you will flip over the sheet, we'll talk about atomic arrangement, and how it plays into classification of solids. By the way, solid, let's define solids. Solid is something that is dimensionally stable.

It doesn't flow under its own weight. It has a volume of its own, whereas a fluid, either a liquid or gas, takes the volume of its container. The food goes to the walls of the container. The liquid in a gravity field sinks to the lowest point in the container.

But, the container determines the shape. A solid retains its shape. So, now I want to classify solids by atomic arrangement. So, I don't care if they are metals or ionic compounds or covalent compounds. There are two classes: ordered and disordered. Isn't it nice when you can take things and classify them, just bifurcate? It's one or the other, just move through that decision tree. So, let's look at ordered solids.

Ordered solids have a regular atomic arrangement, whereas disordered solids have a random arrangement. I put an asterisk next to random because it's not totally random. There is some local order, but there's no long-range order.

So, the order is short range. And, what do we call such solids? The ordered solids are called crystalline solids. And, we have a plain, everyday word for such an ordered solid. It's called a crystal, and I've been using that term a lot.

I talked about Davisson and Germer irradiating a nickel crystal. That's to distinguish the material for something that's disordered. If it's disordered, we say that the atomic arrangement is amorphous.

And, there is a simple Anglo-Saxon word for an amorphous solid: glass. Glass is an amorphous solid. You might think glass has to do with whether something is transparent or not. And, I want to disabuse you of that.

We've talked about transparency. And, what do we know about transparency? If we want to talk about transparency, if something is transparent, what do we do? We ask ourselves, is the band gap in that material greater than three electron volts? If the answer is yes, it's transparent.

It is less than three electron volts, it's an absorber. So then, what does glass mean? Glass means no long-range order. Let me give you some examples. We could say, for example, we've got diamond.

Diamond is crystalline. And, yet it's transparent. Why is it transparent? Band gap is about 5.4 electron volts. Obsidian, this is a glassy rock. It's got no long-range order. It's an amorphous material.

And, it is opaque. Obsidian is opaque. So, here's an example of an opaque glass and a transparent crystal. So, as of now, I want to make sure that nobody in 3.091 from this day forward ever, ever says glass is transparent.

I will show you before the end of the month metallic glass. How can it be a metal and glassy? Well, it must mean that there is metallic bonding operative. But, the atoms are not in a regular arrangement.

And, that material is not transparent to visible light, because you know that metals have no band gap whatsoever. So: something important here to retain. So, the parents will now learn what the students have learned.

When we go into a new unit, we begin with a history lesson. So, let's go in the way back machine. We'll go back to early efforts in trying to describe ordered solids. So, that's why crystallography is the study of ordered solids.

Crystal comes from the Greek, krystallos, which is one of the terms that might refer to ice. It's a certain form of ice, but anyways, that's the origin. So, let's go back to the first marking point in history. Robert Hooke, back in 1660, was studying cannonballs. They were doing military research already back then. What do you think? And, he posited that a crystal must owe its regular shape to the packing of spherical particles.

So, if you pack regularly, you'll get long-range order. And then, about a decade later, there was Niels Steensen, who was a Dane, spent most of his scientific career in Italy. And, he observed that quartz crystals had the same angles between corresponding faces regardless of their size.

And, see, what they were looking for is trying to make a connection between the macroscopic and the atomic world. In other words, here's a simple question. If I see something has a macroscopic shape very regular and cubic, can I infer from that that if I divide, divide, divide, divide, divide, if I get down to atomic dimensions, there will be some cubic repeat unit? Yes or no? That's the question they were wrestling with.

And then, around the end of the century, Christian Huygens in 1690 studying calcite crystals made drawings of atomic packing and bulk shape. And, here's a drawing from his book in 1690. And, Tom, if you could switch the video over to the document projector, I've got a calcite crystal here.

Here's the calcite crystal. All right, and now, auto focus is off, auto focus is on, OK, so now, you can see, look at the regularity of this. Imagine, you're looking at this. This is all you have to work with.

There's no scanning electron microscope. There's no nanoscope, nothing. That's what you've got to work with. And what he comes up with, Tom, if we could go back to the computer video please? So, starting with that crystal, this is what he posits.

That's not bad. Remember, this Huygens is the same one that gave us optics. You see, these people were polymaths. They weren't narrow specialists. They were generalists. And, they made substantial contributions in more than one field.

Tom, let's go back to the document camera. Let's go back to the document camera. Here's a piece of tin. So, I mean, if I had to draw this piece of tin, I might draw a different drawing from what Huygens drew because you can see the angles are different here.

Look, these are right angles. Well, here's a piece of beryl. This is beryllium aluminum silicate. Look at that. That's a hexagon on en,d almost. So, what's going on? Here's a piece of cryolite.

Looks like somebody borrowed it and dropped it. Look at that. See, bad data. Can you imagine drawing that? See, that's what happens. They give you lousy samples, expect you to do cutting-edge research.

[LAUGHTER] OK, Tom, let's go back to the computer before I digress. All right, so here we are. Here's another example. I couldn't bring this one in. These are big chunks of basalt off the coast of Iceland.

And, you can see, they have other forms. All right, so far so good. Now, let's go into about a hundred years later over to France. This is Rene-Just Hauy, and he's studying at Sorbonne. And, he's studying the cleavage of calcite. Basically, what he does is he comes in to work everyday and he breaks things. And, he breaks the calcite, and he looks up the little pieces of calcite. And what he observes is that all shards are rhombohedral.

It doesn't matter how small he breaks the calcite crystal down. He always gets rhombohedral shards. So then, he says, I'm going to posit that that's the base unit of calcite. And furthermore, because he's a Frenchman and they are steeped in mathematical tradition, he decides to model this mathematically.

And he says, OK, we all know that if we wanted to fill three space with the same unit, we could choose a number of shapes, right? We could start with a cube, and I could fill this room with cubes of identical size.

I could fill this room with sort of orthorhombic boxes, sort of rectangular but square cross-section. I could have something that's rectangular with a rectangular cross section. I could do something that's rhombohedral.

How do I know? Look at the crystal. I don't see any holes here. So, what Hauy did is he said mathematically, I want to determine, what is the maximum number of distinguishable shapes that will fill three space? I call this the milk carton problem.

How many different shapes of milk cartons could you make that you would be able to fill the truck perfectly? And, he came up with seven of them. And, I'm getting ahead of myself there. The seven of them are shown here.

This is in your notes, and these are the labels for them. And I said here's the basic unit. So, you have three lengths, A, B, C, and you have three angles, alpha, beta, and gamma. And, depending on which mix you choose, for example the cube is here, A equals B equals C.

So, three edges, all normal to one another, and then here's the orthorhombic. They're not equal to each other, and so on. So: seven basic shapes. And, here they are drawn for you. And, this is taken out of the lecture notes, the archival lecture notes.

So, this is analogous to the tiling problem, just to drive home the point. So, if you want to do the tiling problem, how many distinct shapes of tile are there in two space? So, I think we can agree, we can go with square tiles.

Or we could go with rectangular tiles. What other tiles do we have? Triangle, well, triangle is actually, it's not a basic unit because I can't, it doesn't observe translational symmetry, but if I close the triangle like this, and have a 60∞, so this will become the basic repeat unit, OK? Then there's a regular parallelepiped where this unit, or angle, rather, is not equal to 60∞.

And this, in fact, is a basic unit that Frank Gehry uses in building his - if you go over to the Stata Center, you will see that the basic unit that's used to make those irregular shapes is this. This is space filling.

And all of these are labeled, and so on. This is the clothing problem. How do you cut a two-dimensional piece of cloth to fit over a three-dimensional shape? It's not a trivial problem. That's why most clothes don't fit. It's very simple: fashion designers don't understand the simple geometry. And, how do we describe size? For men, at least, they take a chest size and the sleeve size. For women they just go, oh, she's an eight.

An eight? One number? This is why things don't fit. You have to get down to the unit cell. Here's one that won't work. The pentagon won't work. And, I'm not trying to make a political statement; I'm simply saying - so, here's what you can do.

Here's what happens if you try to fill two-space with a pentagon. You see, it won't butt up. There is these white gaps. The reds touch the reds, but it won't fill. So, this is how Hauy worked in three-space and determined that certain units will not butt up against one another and fill three space perfectly.

OK, so we are clicking. Things are going well. But we still haven't got atomic arrangements. All we've got this milk cartons so far. So now let's think about the next step. The next step came about 50 years later, 1848, which was an auspicious year in Europe.

It was another Frenchman, August Bravais. And, he proved mathematically that there are 14 distinct ways to arrange points in space. And, what do I mean by that? Here's a cartoon that shows the cube.

We've agreed that the cube is a cookie-cutter that will fill three space. But when I start putting atoms in the cube, I have three distinguishable arrangements. So, here, on the left, I have something where I put atoms only on the corners.

In the second one, I put atoms on the corners, and one in the center. Now, think, look at the one in the center. That one in the center sees eight nearest neighbors. Each one of these see only six nearest neighbors.

So, these atoms have a different atom arrangement, a different atom environment from the atoms in this arrangement. So, the left one is called simple cubic. The center one is called body centered cubic because there is an atom in the center.

And it doesn't matter. I can arbitrarily shift the center up to this corner. And in this corner atom sees the same nearest neighbors in the same arrangement relative to the center. Here's body centered cubic.

Choose any one of those. If I sit here in the center, I've got one, two, three, four, five, six, seven, eight, or I can go to this one over here, and it sees one, two, three, four. And you just keep going.

It's going to see the same arrangement. And yet, the cookie-cutter that fills space is the cube. And yet, there is one other possibility, and that's face centered cubic. Face centered cubic is here.

Well, you have the square face, and an atom in the center of each face. And, that's distinguishable. That's distinguishable. So, Bravais, again, Frenchman, steeped in math, goes into the mathematics and asks how many different ways can I put atoms in two these seven crystal systems that Hauy has specified, and get distinguishable point environments? And, the answer is 14. The answer is 14, 14 ways to arrange points in space. So, now we're getting somewhere. And, by the way, if you look at this, here are the 14 different Bravais lattices. And, they are shown, this is also from the lecture notes.

And, all we are putting here is one dot at each lattice point. But it doesn't have to be a dot. What I'm going to show you in a second is I can put sets of atoms, groups of atoms. In other words, if I wanted to describe the positions of every apple in an orchard, what I could do is say that the Bravais lattice is telling me the planting of the trees.

And then I can hang different types of apple arrangements at each point. So, this is what we are getting to. We are trying to break this down. Otherwise, we're going to have a gazillion different atomic arrangements.

And what we're able to do is confine everything to simple set of 14. So, what I want to do now is show that we can look at different sets of elements at these points. So, let's see what happens if we look at face centered cubic and put different atomic groupings at each point.

So, ultimately what I'm in search of is the crystal structure. And, the crystal structure is the atomic arrangement in three space. And, the crystal structure is the sum of two components. First is the Bravais lattice.

And, what the Bravais lattice is, it's really just a point environment. It's a point environment. And I can put anything I want at the point, including nothing. I mean, I can make this the equivalent of a John Cage piano piece with just a bunch of rests of different time signature.

Do you know what I'm talking about? Listen. That's rests in 4/4 time. See that? See this? That's a simple cubic lattice. It's a set of points in space. OK, so it's a point environment, but let's hang something on the points.

And, what we hang on the points is called the basis. And, this is the atom grouping at each lattice point. This will all become clear, I assure you, with a few examples. So, let's look at some examples.

Let's look at examples, say, for face centered cubic. Let's see what we can do with face centered cubic. So, I'm going to choose, in all cases, the Bravais lattice is FCC. So, that's shown up there on the right side.

Only, what you're seeing here is the drawing with just a single atom at each lattice point. So, if the basis is something, just a single atom, so that could be a metal. So, an example for that might be gold.

See, this is gold. This is a FCC Bravais lattice with one atom at each lattice point. How do I know it's gold? Look at the color. It's obviously gold. That's how I know. That's FCC gold, but there's a whole bunch of metals that are faced centered cubic including aluminum, copper, platinum; these are all FCC metals.

OK, and so in this case, we would call the crystal structure, the crystal structure is also FCC because you've only got one atom per lattice point. So, the lattice point, the Bravais lattice and the crystal structure are the same: face centered cubic. I could put a molecule at each lattice point. For example, methane, I could put the five atoms, carbon and the four hydrogens at each lattice point. And, if one of you is the first human to go to Jupiter, when you get to Jupiter, and you see solid methane, you will be heartened to know that it will be in the FCC crystal structure.

It has to be. Even proteins, even DNA, if it partially crystallizes, is going to crystallize in one of those 14 Bravais lattices. It has to. So, solid methane, and this will also be FCC. We could put an ion pair, for example, a cation and an anion at each lattice point.

And that would give us sodium chloride. And sodium chloride gives us something that's called the rock salt crystal structure. And, this model here is sodium chloride where the greens are chlorine, and the golds here are sodium.

What you do, is you might say, well, look, they describe a square so why don't we just call this simple cubic? I have different atoms of different lattice points. So, that won't work. Instead, if you could just train your eye to look at the greens, look at, one, two, three, four, five.

There's a fifth green in the center of a square. Then, train your eye to look at the golds: one, two, three, four, make a square, five in the center. So, instead, say let's look at the pairs. Green and gold: one.

Green and gold: two. Green and gold: three. Green and gold: four. And green and gold: five in the center make a face centered cubic lattice with a rock salt crystal structure. That's shown here in this cartoon.

So, whether you count the greens or you count the blues, it doesn't matter. But it's the ion pair. So, that means this is a derivative from the FCC Bravais lattice. And then, there's one other one.

There's one other one. I can put an atom pair. I can put an atom pair. Here I put five atoms, a molecule. Here I'm going to put an atom pair. The example I'm going to give you is two carbons at 109∞.

And what will that give me? That's an example of diamond or silicon or germanium. And, here's diamond right here. This is diamond. If you train your eye, the atom that I'm holding has four struts coming out of it.

That's the sp3 hybridized, four struts coming out of a single carbon. Every carbon has four struts coming out of it. But, watch this. When I lie this down in a single plane, and I bring it up, you see where it got the post its? Look at how the post its describe it.

One, two, three, four, five. It's face centered cubic, and with each lattice point, I take two carbons. Two carbons here, two carbons here, two carbons here, and I go all the way through the lattice.

And, I end up with diamond cubic crystal structure as a Bravais lattice with a basis of the atom pair. And, this is diamond cubic. So, diamond cubic is the crystal structure, but it's derived from an FCC lattice. So, again, the overarching theme here is the integral of the Bravais lattice. So, this is the set of points, right? This is the point set in three space, the Bravais lattice. And then, what you put at each point gives you the crystal structure.

And, you could go on, and on, and on, and you can have a ball with this stuff. Oh, here, this is, to show you in two space, this is interesting one. This is an Escher print. This is the dogs. So, what's the crystal structure here? So, let's start looking.

Well, you might go one, two, three, four, five. You might say, well, this looks like simple face centered cubic. Actually, face centered and body centered are the same in two space, right? The body, how do you describe a sponge in two space? Think about it.

OK, the trouble is, this dog is facing in the other direction. So, that won't work. So, instead, all these dogs are facing the same way. And now, if you look carefully, take as the basis, see, those could be lattice points.

The Bravais lattice is the red dots. The basis is the set of four dogs. The two facing the right, and these two facing the left, now, two right, two left, two right, two left. So, it's the four dog set, you know? Those four dogs are the basis.

So, bingo. So, now, what do we have? We have simple cubic. This is SCP, simple cubic puppy. That's a two-space, all right? Look at this one. I put the dot here on the abdomen of this creature. So, what have you got? This is a rhombus.

Or some of you might have initially called this the triangle. But, you see that when you put the two of them together, this has translational symmetry. And, in fact, look at how many of these entities belong to that lattice point.

All of this, all of this, all of this, because the tail to this is up here. So, the repeat doesn't start until down here. You could cut it this way. I don't care. But the point is this is a concept of the basis.

The basis is many, many units that are repeatable, and can be positioned at each lattice point. So, now, what I want to do is talk little but more about the cubic system because we're going to focus on a cubic system in 3.091 largely because it has a simple geometry.

Most of the periodic table, you'll find, is in cubic system, FCC or BCC. And then, the math is simple because you're in Cartesian coordinates. So, let's take a look at what we've got going for us here.

I want to look at the properties of cubic crystals. So, here we are. This is a table out of your lecture notes, and I'm going to work through this table. And, I urge you to get friendly and comfortable with this table so that you understand the properties.

And, why am I making you learn this stuff? Because I think it stands to reason that if you look in this crystal, you see that atoms touch along certain directions. And, atoms are far apart along other directions.

Well, we've already seen in 3.091 that there's a correlation between atomic contact and bonding. And, bonding is related to a whole host of other properties. For example, mechanical strength, there are people in this room that a thinking about being mechanical engineers, aero-astro engineers, nuclear engineers.

You need to understand the relationship between the crystal structure and the strength of the material. Obviously, if I look down an atom direction, if I look down an atom direction where I have a high density of atoms, that's going to be a direction of strength.

If I look down a direction that has a low population of atoms, that's going to be a direction of weakness. And then subsequently, if I want to cleave a crystal, then where my going to cleave it? I have to understand the crystallography.

The electrical properties, the optical properties. What's the index of refraction? It's the bending of light in response to the field set up by the bonds. So, you want to go into optical electronics, you've got to know the stuff.

Everybody has to know this stuff because chemistry is central to everything. It's all chemistry. This conversation is brought to you by secondary bonding. It's all. That's it. So, let's look at this one in some detail.

And what I want to do first of all is the unit cell; let's look at the unit cell. So, here's FCC, face centered cubic, so there's the lattice. And, now what I'm going to use is a hard sphere model.

So, I'm going to make the spheres, the atoms, so big, that they actually touch along their principal axis of contact. So, face centered cubic, the atoms touch along that face diagonal. And, the unit cell, this is called the unit cell.

This is the repeat unit. The unit cell is that which repeats in three space and fills three space. It's the basic unit of the crystal system. And, the edge of the unit cell is given the dimension, a.

So, the edge of the unit cell, and this is a cube so it's all the same edge, so the volume of the unit cell is equal to a cubed. That's trivial. So, we got that. Oh, lattice points per unit cell, lattice points per unit cell, let's look at that.

Let's look at this red one. How many lattice points? Well, we can count lattice points by counting atoms. If you see, there's eight corner atoms, and how much of each of those corner atoms lies within the bounds of this unit cell? It's one eighth.

So, lattice points per unit cell is going to equal eight times one eighth. This is the corners. This is the corners. And then, we've got these face atoms. Eight times 1/8 is one. And, you see these blue ones? These blue ones are half inside this unit cell and half inside the adjacent unit cell.

And I've got one, two, three, four, five, six of them, two in this direction, three principal directions, and two times, so that's six times one half, which is - this is the face atoms. OK, six times a half is three.

So, three plus one is four. So I have, effectively, four lattice points per unit cell. I'm not saying atoms because I could replace each of these single atoms with a methane molecule. So then I'd have five atoms per point. So that's why I'm being a little bit fastidious here, lattice points per unit cell. What else are we looking at? Oh, nearest neighbor distance. Well, nearest neighbor distance: that's kind of obvious.

Here's a drawing. This is a, this is a, and that distance is root two times a. And this is half of root two, so that's given here, a over root two, nearest neighbors, well, you can see that. Nearest neighbors: let's count from this one.

It's got four at the corners, right, and it's got three in this plane, three in the other plane. So, that's going to add up four plus four plus four is nearest neighbors is equal to 12. Some people call this the coordination number.

And, this has got the highest number of nearest neighbors when you have hard spheres of equal dimension of any crystal system. You can see here in body centered cubic, it has eight nearest neighbors whereas simple cubic only has six nearest neighbors.

Whoops, we don't want to look at that anymore. All right, and then the packing density, I want to show you the packing density. That's an interesting calculation. What the packing density indicates is how much, going back to Democritus, if we model the FCC crystal as hard spheres touching, how much of this is matter and how much of it is free space? That's given by the packing density.

And, that's equal to the volume of the atoms divided by total volume. So, if we choose the unit cell as the basis so we've got something to count on, let's go back. Well, let's go to the other one.

I like this one. It's a cuter drawing. OK, so, the volume of the atoms, well, I know I've got four atoms, the equivalent of four atoms. And, what's the volume of an atom? It's going to be four thirds times pi times the radius of the atom cubed.

And, what's the point of that unit cell? That unit cell is a-cubed. And now, I need to know the relationship between a and r so I can convert one to the other, and then I'll have a dimensionless group here.

So there, you can see the relationship between a and r. A-squared plus a-squared is the 16 r squared. So, if you go through all that, you can convince yourself that a equals two roots of two r in FCC.

So, plug that into there, and you will end up with the packing density will be equal to pi divided by three times the square root of two, which is a value of 0.74. So, that means in face centered cubic, in a face centered cubic lattice, you have 74% matter.

This is assuming hard sphere model. Sphere is so big that they touch. So, that's 26% void. And, you can go through, and you can see that as you look at this table that the packing decreases. In the case of body centered cubic, there's a little more open space.

It's 68%, and in the case of simple cubic, it's only about half packed. So, this table here is something that you want to get good and friendly with. The other thing I need to show you is crystallographic notation. And, what I think I'm going to do is invite you to look at the website. This will give you the nature of crystallographic notation. Position is basically as it is in your math classes. The only difference is we don't put enclosures.

There's no parentheses around a point. So, if the origin is 0,0,0, but don't put parentheses around it. So, a point is simply written as follows: so origin for a point, 0,0,0. That's all there is to it.

In the case of directions, this is going to take a little more time than I have set aside because I've got a few items of business that I need to take care of. So, I think I'm going to hold at this point, and we'll jump ahead to, see, I always have a few things up my sleeve.

OK, so let's go back to the beginning. When people came in, as is the custom, we have music playing. And what was the music today? Yeah, The Talking Heads, and the song was? "Burning Down the House."

So, why would I be playing Burning Down the House in a lecture about crystallography? Well, what inspired me to choose that piece of music was this painting, the Houses at L'Estaque. This was painted by Georges Braque in 1908.

L'Estaque is a community down in the southern part of France. And, when Braque painted this, he outraged the critics of the salon in Paris who condemned this painting. They said, look, it doesn't have a common perspective.

The houses in the back, it looks as though the lighting is coming from here, but the lighting over here is coming from another angle. He broke all sorts of artistic conventions. And, one critic ultimately condemned this painting of houses as nothing but a stack of cubes.

And from this came the term Cubism. And since we've been talking about the properties of cubic crystals, I thought the connection was pretty obvious. OK, I did. All right, we have a contest every year in 3.091.

The contest involves developing a mnemonic device for the certain groups in the periodic table, the lesser-known actinides and lanthanides. And so, I'm going to announce the winners today, and we'll see if they come to class.

But in their defense, there is a second lecture. So, if they are not here, we assume that they are at the second lecture, right? So, anyway, here's one. This was for the lanthanides. This was an Elizabethan sonnet that was written by Allison Burke.

Is Allison here? She obviously goes to the second lecture. OK, so, well, you'll see in a minute what the prizes are. OK, next one, Eddie Fagin, is Eddie here? You have to come down and claim your prize.

Come on down. [APPLAUSE] OK, so he has a choice of two items. One is the, congratulations, but I haven't shown you everything. First, here is his mnemonic verse. But, this is really hot. Look, one of the things that we learned is the Born Exponent, and but what's really cute is, I mean, I didn't notice this the first time.

I just caught it last night. So, this, of course, is a pun on the Bourne Identity, but down here, it's the 3.091 pictures is in the corner here, very sweet, very sweet. OK, so you have a choice of the hot necktie from the American Chemical Society, or, this is a ladies scarf.

You don't know what's going on in his personal life. What are you talking about? [LAUGHTER] [APPLAUSE] Come on. But you know what I'm going to do? You're not going to make the choice right here in front of all these people.

You're going to talk to me afterwards and they are not going to know what you chose. Perfect. [APPLAUSE] OK, so the one that I'm choosing as the winner, I had all sorts of things given to me over the years.

The Elizabethan sonnet is one. But this is the one that takes the cake. This is a musical departure, a riff on the Nat King Cole song Love. And, this is an image of the CD. She turned in the CD to me minutes before 5 o'clock on the deadline date.

So, this is the image. And, there's the words. Listen to this phrase here. It imitates very, very in the song. It's beautiful. So, that's really good. [APPLAUSE] Is Ms. Gabriel here? Please come down.

She obviously is either here, or she is shy, or she goes to the 1 o'clock class. Anyways, I wanted you to see, and I want you to understand that this is part and parcel of what makes this student body unique.

I mean, I submit that this kind of thing would not happen at Harvard. [LAUGHTER] [APPLAUSE] OK, a couple of more things. Tom, can I get back to the document camera? I want to show some other crystals and make the point about symmetry and so on.

So, this crystal I'm going to show you is cryolite. And, cryolite is birefringent. Sorry, I'm going to have to ask you to go back to the thing. I needed to show this one thing, please, Tom? Can we go back to the computer? Thanks.

OK, so what I'm going to show you is something that's birefringent. The crystal symmetry has atoms just in parallel, a little bit off axis from one another. And, what that does is it gives rise to two indices of refraction, and so if you have a light ray entering the crystal off the alignment of the optical axis, the light will split into two.

So, you will get an ordinary ray and an extraordinary ray. And the separation of these two rays is related to the difference in the indices of refraction. Now, if you are in optoelectronics, and you're making a splice, and you've got a material that's birefringent, you'd better know this because all of a sudden you've got your light going off to the side, or maybe this is what you want.

So, what I'm going to show you is - now we can go to this. This is a piece of cryolite. And, it's birefringent. And, what I want to do is show you how it will act, if we can go to this, where am I, OK, I'm here, good.

Let's see, can I zoom in anymore? Good. And now, what we'll see is as I turn this, you see, now I've lined up. So, do you see this line here? Appears as a line, everything's as it should be. But then, as I turn this, cool. See that? Separation. And then I can bring it back. So, that's birefringence. And, this is another example of what you get with a crystal that has a particular crystal structure, one of the rhombohedral types.

Back to the computer, Tom, I've got to set up one other thing here. OK, so this gives you an indication of some of the materials that you may know that are - obviously calcite. Ice is birefringent, not terribly so.

Rutile, very much, titanium dioxide, one of the forms of titanium dioxide. Quartz: just mildly so. OK, now I want to show some very strange FCC. This is colored gold. You know what gold looks like.

It's very malleable. It's an FCC crystal, and yellow. By alloying with different elements, you can change color, right? But these are all FCC. You see, nickel is FCC. Aluminum is FCC. And, indium, so on, what I'm going to show you is the alloying of indium with yellow gold to give something that is 12 karat gold.

So, Tom, may we switch to the document camera? OK, let's get the cryolite out of the way. This is gold. This is the gold foil that Rutherford would've used in his famous experiment. OK, and now, this is indium gold.

It's a very soft, sort of a dove's egg blue color, very lovely color. So, you might ask yourself, well, what could you do with something like this, because it's somewhat brittle. You can see, here's the machine.

That gives you a better sense of it, just a little bit off. All right, so what I was thinking as what if we were to do something like this? We could take a watch like so, and then what we could do is put something like that here, see, and then we put quartz movement underneath, and we put some numbers on the front here, and some hands.

And, this thing's solid 12 karat gold. We could sell this thing for about $10,000 apiece. [LAUGHTER] And then, there's all those other colored golds. We can do all of this as well. So, I think this gives you a sense of what happens when you understand crystallography? And, I think with that, we are going to go for an early dismissal and wish you a good weekend.