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.
91. 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.
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.
91 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,
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 60í, 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
60í.
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 109í.
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.
91 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.
4. 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, 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.
