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PROFESSOR: OK, OK, settle down.
It's time to start learning.
One announcement.
Tomorrow we'll have the weekly quiz.
In lieu of the Tuesday celebration, we're going to
move to Thursday, since the holiday moved everything,
compressed the week.
So last day, we were talking about doping of
semiconductors, and I want to finish up that unit.
Just to refresh your memory, we looked at how we could
change the behavior of the semiconductor by introducing
impurity atoms. And when the behavior of the semiconductor
is dominated by the impurity atoms, we term
that behavior extrinsic.
In other words, it's not the behavior of silicon itself,
but it's the behavior of silicon as determined by the
presence of some dopant atom.
And we looked at the special case of doping with a
supervalent impurity.
And here's the energy level diagram for that situation,
where we've got the valence band down here, we've got
conduction band up here, separated by an energy gap.
That's all characteristic of plain vanilla silicon.
But then when we dope with the supervalent impurity, there is
an extra electron.
And that is put into a donor level that lies just a tiny
bit below the bottom of the conduction band.
And then, thanks to thermal excitation, which gives us an
average energy of about 1/40 of an electron bolt, we get,
for all intents and purposes, 100% excitation of the
electronics that sit in the donor level, excitation up
into the conduction band.
And this is what electrical engineers term ionization.
And I also reminded you last day that this is the donor
level, and for each donor atom, there sits a donor level
at the same value of energy, but these are so far apart
point, owing to the dilution, the dilute concentration of
the impurity, that we don't violate the
Pauli Exclusion Principle.
So these are all sitting at the same level and that's why
they've given temperature promotes all of them up into
the conduction band, where they are mobile.
And then, just to complete the review, we call this n-type
because, thanks to doping, we generate electrons in the
conduction band, and electrons are negative.
Supervalent doping gives us n-type.
Then the total number of electrons in the conduction
band is the sum of those that would have been generated by
plain old thermal excitation.
Thermal excitation operates all the time.
So with thermal excitation, recall, we promote from the
valence band up into the conduction band.
So we break one of these bonds, shoot an electron up
here, and leave a hole behind.
So this generates the pair.
Hole in the valence band, and electron in
the conduction band.
That's always operative, but owing to the energies
involved, the band gap is on the order of one electron
volt, whereas the donor level sits only about 1/50 of an
electron volt below the conduction band.
So we get very, very little promotion at room temperature.
And in fact, it's something like 10 to the minus 19 is the
fraction of promotion, and if you say roughly 10 to the
twenty-third per cubic centimeter, that means you've
got about 10 to the fourth electrons per cubic centimeter
due to this thermal excitation from the valence band to the
conduction band.
Normally, when you dope a semiconductor, you dope it
around parts per million level.
And parts per million, that's 10 to the minus 6.
If you take 10 to the minus 6 times 10 to the 23, you get
about 10 of the 17.
Now you can see, 10 of the 17 is vastly larger than 10 of
the fourth.
So for all intents and purposes, once you've doped
something, this contribution is essentially 0, and so the
electron population, the conduction band, is that
dictated by the impurity atoms, which is why we say
that this is exhibiting extrinsic behavior.
The intrinsic properties are not
visible, they're not palpable.
They're immeasurable.
So intrinsic is substantially less than extrinsic.
Now, the last thing I wanted to do for you before we say
goodbye to this fascinating topic, which lays the
groundwork for everything that we know in the modern
electronic era, parenthetically, is to look at
the other type of doping.
I've showed you how to make an n-type semiconductor.
How do we make a p-type semiconductor?
So in that case, what we can do is dope
with a subvalent impurity.
What do I mean by subvalent?
Well, the valency of silicon is 4, so I want
something less than 4.
So a good example is boron into silicon.
So boron is group 13, or 3, all right, if you look on your
periodic table.
And silicon is group 14, according
to the IUPAC notation.
So let's go back and take a look at how that might appear.
So I'm going to draw the silicon crystal.
Remember, this is going into a single crystal of silicon.
So silicon is sp 3 hybridized, and we have silicons
everywhere in the structure.
So we're going to put silicons.
And I'm trying to show sp 3 hybridization.
So this is 3 legs each in the plane, then you've got 109
degrees here.
So that's silicon, and so on.
I get the three of them just to complete
enough of the picture.
And now what we're going to do, is introduced boron.
And boron goes into the silicon lattice
and covalently bonds.
It doesn't go sit in some void space in between the silicons.
It actually sits on a silicon site and
substitutes for the silicon.
Now, boron is group 13.
It's got 3 valence electrons.
And so it forms bonds with 3 silicons.
And now there's a fourth silicon here, and that silicon
has an electron, but the boron doesn't
have a fourth electron.
Here's where it gets interesting.
The driving force to complete the picture here is so great
that the system will actually pull an electron out of a
silicon-silicon bond--
and so I'm going to use a different color chalk, and I'm
going to indicate, just for argument's sake, that we're
going to pull an electron out of this bond here, break this
bond, and shoot that electron over to here.
That way, we get r months around the boron.
It says p3 hybridized.
It just doesn't have that extra electron.
But now it rips the electron out of this
silicon-silicon bond.
And what's the consequence of breaking this bod?
What's the electrical feature that we've created here?
A hole.
So now we've created a hole somewhere else in the crystal
in order to satisfy the desire of boron to
get that fourth bond.
And now can you see that for every boron that I introduced
into the crystal, I'm going to make a hole
somewhere in the crystal.
And on the energy level diagram, where
do those holes live?
Those holes live in the valence band.
So let's go make the energy level diagram.
So up here we have the conduction band, and
downstairs, we have the valence band, as before.
And I'm going to assume that we got the thermal promotion.
But it's so tiny in comparison to the amount of boron we're
going to put in.
I'm not going to muddy the water here and show that
you've got pair formation.
Because the extent of it is so small, it doesn't make any
difference.
So now what happens?
Now, I know I'm going to generate holes here, and I've
got to show this energy level.
And this bonding level is different from
this bonding level.
Agree?
Silicon-silicon bond has different bond energy from
boron-silicon bond.
So that means the top of the valence band
is different from--
because this valence band is silicon-silicon-silicon.
So it turns out that, and this is not to scale, that the
energy level of this bond is just a little bit higher.
And this is an energy level that involves the accepting of
electrons, whereas in the other case, we
were donating electrons.
Hence, that's called the donor level.
This is called the acceptor level.
And it's about 1/50 of an electron volt above the top of
the valence band.
And so what happens is that in order to make this bond, we
move something out of the valence band up into the
acceptor level and generate a hole.
And so for every boron that we put in, we have
another broken bond.
And all of these lie at the same level, but the dilution
is so high, that they're so far apart, that they don't
violate the polyexplosion principle.
They're thermally promoted up, and away we go.
So now, under these circumstances, the number of
negative charge carriers no longer equals the number of
positive charge carriers.
Because I've got all these positive charge carriers.
Thanks to the introduction of boron, p is greater than n.
So we've made a p-type semiconductor, by the
introduction of a subvalent impurity.
And we can redo the entire central part of last day's
lesson, which was the Bohr model.
Now, this is the sexiest part of the Bohr
model I've ever seen.
This is really, really cool.
The hole is mobile!
And the boron is a little bit shy of protons, right?
It's 3 in a land of 4.
So it's negative.
So I've got a stationary negative center, and I've got
something positive revolving around a negative center.
This is an inverse Bohr model!
Because the Bohr model has a honking big positive center
with a dinky little electron revolving around.
We've got the immobile negative center, and the
mobile hole revolving around.
Go through, you get the same set of quantum--
I mean, there's a whole bunch of quantum levels in here.
Fantastic!
How far that Bohr model can take us.
So now you know how to make a p-type semiconductor, you know
how to make an n-type semiconductor.
And then what you can do is put them together.
And you can put, say, boron-doped silicon opposite
phosphorus-doped silicon.
Be careful here.
The fonts are really critical.
This is uppercase P, for phosphorus.
This is lowercase p, for positive doping.
So don't go, oh, P, phosphorus,
that means it's p-type.
No, P gives you extra electrons.
It's n-type.
So that's what uppercase P.
This is a lowercase p.
This is p-type.
So now I've got p-type, n-type, and so
what do I have here?
I've got a p-n junction.
And this is the beginning of solid state devices,
rectification of A/C current, diodes, you name it.
And it all starts with this, the chemistry.
This is the birth of the transistor, and all the modern
electronics that we have based on that.
Later on, we'll show you how you get the boron in, and how
you get the phosphorus in.
You don't just come in and sprinkle it on top, and hope
that it goes to the lattice site.
There's a lot of processing involved.
OK.
So this is where I want to hold for the unit on
semiconductors, and how it fits into the
grand scheme of things.
So in the books, you'll see--
this is a very crude drawing, because this is showing n-type
semiconductor with grossly exaggerated numbers of donor
electronics up here.
There's no mention of the donor level.
So this is sort of
semiconductivity, sort of pre-3091.
This is extrinsic semiconduction for idiots, I
guess you'd call it.
And then, same thing here.
You see all of the holes in the valence band.
We know better, because we know there's an acceptor level
and there's a donor level.
OK.
So now we're going to switch gears.
Now here's the big transition, right now.
If the last two lectures didn't leave high school
behind, today we leave high school behind.
So those of you who have been coasting up until now, thanks
to your very good high school background,
start paying attention.
Because it's high school no more.
So just to remind you where we've come.
We started over here, at the beginning of the semester.
And the big theme is going to be, electronic structure
informs bonding, which informs state of aggregation.
State of aggregation is solid, liquid, or gas.
So I want to show you how we get to where we are.
So we did all of these topics, right?
Bohr, or Sommerfeld quantum numbers, multielectron.
And then we came to octet stability.
And with octet stability, we went a long way.
We got into ionic bonding.
We got into completing the valence shell.
And in fact, all of this is a consequence
of the octet stability.
And that led us into the various
types of primary bonding.
Ionic, covalent, metallic.
And in some instances, van der Waals.
Solid argon is van der Waals bond.
It's the only kind of bonding, so it's the primary bonding.
But in some cases, like HCl, how do you get liquid HCl?
How do you get solid methane?
So we had invoke secondary bonding for some of these
covalent molecules, and that invoked dipole-dipole
interaction, the London dispersion force, which is the
same as the van der Walls bond, but I decided to mix
things up, so that both men get a little bit of attention,
none of them feel slighted.
And finally, hydrogen bonding, in those special instances.
And that allowed us to then determine whether something
was a gas, a liquid, or a solid.
And when it's a solid, we're happy.
Because this is 3091, and we're interested
in the solid state.
By the way, what is a solid?
Well, here's an operative definition.
That which is dimensionally stable, has a
volume of its own.
It doesn't distort in a gravity field.
All right?
Liquids will distort.
Fluids, gases will distort.
Now, there's two ways of classifying solids.
One is bonding type.
That's what we did here.
But the second way of classifying solids is based on
atomic arrangements.
All right?
So let's look at that.
There's only two types of atomic arrangement.
Ordered and disordered.
I love these taxonomies that split everything
into a choice of 2.
That way, you just sort of go through life and decide, is it
left or is it right?
Is it up or is it down?
Is it ordered or is it disordered?
So what are the characteristics
or an ordered solid?
It has a regular atomic arrangement.
That means it's got a long-range order.
And I'll show you that in an ordered solid, I know not only
where my next nearest neighbor is, where my tenth nearest
neighbor is.
I know where my hundred and fifty second nearest neighbor
is, because it's an ordered solid.
And we have a simple Anglo-Saxon word for an
ordered solid.
It's called crystal.
So when I say something is a metallic crystal, I don't mean
it's something that's got magical properties, and if I
put it over my head, I get powerful or something.
It just means I've got atoms in an atomic regular
arrangement.
Now contrasting to that is a disordered solid.
And in that case, we have a random arrangement.
But I put an asterisk here because, as they say in
California, it's not totally random.
It's only random up to a point.
No long-range order.
There may be short-range order.
In other words, I know what my next nearest neighbor is, but
I probably don't know where my tenth nearest neighbor is
Because the coordination shell is established, but the next
coordination shell, and the next coordination shell,
they're not established.
And I'm going to show you, with examples, what that means
So we call those kinds of disordered solids amorphous
solids, as opposed to the regular solids, which are
crystalline solids.
And there's a plain Anglo-Saxon word for a
disordered solid.
It's glass.
Now you might think, well, gee, doesn't glass mean that
it's transparent in visible light?
No, no.
Maybe up until now, but I want everybody who's ever taken
3091 from me to be disabused of the notion that glass means
transparent.
How do we think about the question of transparency?
And here I mean transparency vis-a-vis visible light.
I'm not going to say, what's transparent to x-rays?
That's pedantry.
So how do we think about whether something is
transparent to visible light?
We just ask ourselves, is the band gap greater than 3
electron volts?
The band gap is greater than 3 electron volts, visible light
zooms right through.
It doesn't have enough energy to excite the electrons, be
absorbed, and re-emit.
So that's what transparency means.
So I can have the kind of material that's used in things
like eyeglasses.
It's transparent to visible light because of
its high band gap.
Diamond, diamond is transparent to visible light,
and you wouldn't dare insult diamond by saying
that diamond is glass!
In fact, watch the old gangster movies from the
1930s, that was the derogatory term for fake diamonds.
You call them glass.
So how is it that diamond, which is crystalline, it's
transparent to visible light, and yet the window glass that
we have is transparent to visible light?
It has nothing to do with state of order.
Has everything to do with this.
In fact, David, if we could shoot to the document camera,
I'll show you a piece of glass that's not transparent to
visible light.
How do we zoom in on this thing?
OK.
Auto focus.
So what I'm showing you is a piece of obsidian.
It's a naturally occurring mineral.
It's rich in silica, and it's turned dark because it's got
some iron impurity, and the iron absorbs in the visible.
And this is clearly not transparent to visible light,
and yet, it is a disordered solid.
This is a glassy solid.
In fact, the white speckles there
are devitrified obsidian.
It's actually started to crystallize.
So the crystalline form is the part that's transparent.
If you could peel off this crystalline form, you'd
actually end up with something that's
transparent to visible light.
So here's an example of where the crystalline form is
transparent to visible light, and the glassy form is opaque.
So glass has nothing to do with transparency.
This is how you think about the question of transparency.
OK.
So let's cut back to the slides, David, please.
So we're now going to start by looking at ordered solids.
So we're going to take next several lessons and talk about
ordered solids, and then after that, we're going to take a
few more lessons and talk about distorted solids.
Now first of all, what's the term crystal?
Where does it come from?
It comes from the Greek word, crystallus, which is the Greek
word for ice.
That's where we get the term crystal.
So we're going to start with a history lesson.
That's how we start everything in 3091.
A history lesson.
So we go way, way back to Isaac Newton's time.
The Reformation, Charles, Oliver Cromwell, all those
exciting times.
In merry old England, there was Robert Hook.
You know him from Hook's Law, in mechanics.
So he was doing military research in 1660.
He was trying to understand what was the optimal way to
stack cannon balls.
So when you're on a battle front, you want to figure out
how to keep your material compactly arranged.
You don't lie them on the ground.
So from that, he concluded that crystal that is a regular
array must owe its shape to the packing
of spherical particles.
So that's what he was thinking,
or trying to imagine.
Remember, Democritus said that we have these indivisible
particles, but no one had ever seen them before.
Then in 1669, a Dane by the name of Steenson who was
working in Italy was looking at quartz crystals.
And he was studying various quartz crystals.
By that I mean, quartz crystals of different sizes.
And what he observed was, didn't matter what the size of
the crystal was.
They always had the same angles between
corresponding faces.
So if I give you a big crystal and a smaller crystal of the
same material, and a smaller crystal, and they all have the
same shape, Can you imagine that you might conclude that
that is reflective of something down at the atomic
level, and if you could get down to the atomic level, that
spacial arrangement would hold right now?
Because that's how it grew, right?
It started from a small number of atoms and grew, grew, grew.
Why would it start with one shape, and all of a sudden
change to another shape?
It doesn't make any sense.
At least, not to me.
And certainly not to Steenson.
And then we go to Holland, 1690.
Christiaan Huygens.
And he was studying calcite crystals.
And he drew drawings of atomic packing and bulk shape.
This is 1960.
I'm going ot show you an image from his book.
This was drawn in 1690.
See, this is stacking of cannonballs, but he was
looking at a calcite crystal.
Actually, Dave, could we cut back to the document camera?
We're going to do a fair bit of this.
I'll show you calcite crystal.
This is what he had to work with.
It looks like this.
To give you a sense of scale--
what do I have?
Do I have anything you'd recognize?
OK, here's a soda can to give you a sense of scale, OK?
This is one honking big calcite crystal.
I got it as a gift from someone who watched 3091
lectures on OpenCourseWare.
And he said, if I ever come to town, can
I attend your lecture?
I said, sure.
And he showed up, he attended the lecture.
He was a man probably in his early 40s, and as a pastime he
collects gemstones.
And he gave me this giant calcite crystal, because the
one I was using before was so pathetic.
He said, I've got to give you one.
And then, I thought, jeez, if I could ever come upon
somebody who's in the gold business--
and you know, he might give me a big honking crystal of
face-centered cubic metal called gold--
which broke $1,000 an ounce yesterday, but--
anyways, if you're out there, and you're watching these
lectures, I'm here.
It's 8201, and you can see any anytime.
My assistant's name is Hillary, she'll take the call.
All right.
So let's go back to the slides.
Dave, could we cut to the slides, please?
Thanks.
All right.
So now--
oh, I'm going to show you a few others.
Remember, here's the idea, that we're looking down to the
elemental level.
So if you look on your periodic table, it says 10,
you see, 10, it's supposed to be tetragonal.
So here's a crystal of 10.
Dave, cut back, please.
We're going to go back and forth today a lot.
There's a piece of ten 10.
So you might look at it and say, gee, that
looks kind of cubic.
But if you look carefully, the vertical dimension is greater
than the horizontal.
In fact, it's got a square base.
And then, this is the tall one.
Which, if we cut back to the slides, let's see.
And that's what tetragonal looks like.
And we'll study all of these in a few minutes.
These are some giant crystals I didn't dare drag in.
This is sort of human-sized scale.
This is balsalt, which is a magnesium iron silicate.
It's on a coast of Iceland.
Dave, let's show them a few more.
Here's one more, if we go to the document camera.
This was beryllium aluminum silicate.
This is a barrel.
And you can see, this has got a hexagonal habit.
So something's going on at the atomic level that's different
as we go from one crystal structure to the other.
What else have I got here?
Here's some calcite, here's some fluorite.
Look at this one.
This is square pyramidal but it's ionic.
So don't give me any of that sp3 d2 stuff.
Because it's not going to work here.
All right.
Let's go back to the slides, please.
So then we jump to 1781, to France.
Rene Juste-Hauy was at the Sorbonne, and he was studying
the cleavage of calcite crystals.
In other words, he was taking these things
and breaking them.
So he'd go to work every day and break things.
And he studied the shards, and he found that the shards were
all rhombahedral.
the same shape as the parent crystal.
No matter how small the shards got, they
always looked like this.
He didn't end up with cubic shards.
He didn't end up with long slivers.
He always ended up with rhombahedral slices.
So the other thing is that he didn't end up with any voids.
No matter how he cut this thing, he never got voids.
So he reasoned that this must be the way things are packing
in three space perfectly.
So then he said, gee.
If they pack perfectly rhombahedrally--
we all know that we can stack boxes together perfectly,
cubes together.
So he, being a Frenchman, said, what's the mathematical
formulation?
They were steeped in mathematics there.
That wasn't a slur against the French.
It's a compliment.
He said, what are the mathematically distinct
shapes, that if we stack them together in three space they
will fill three space with no void?
And the answer is seven.
Seven space-filling volume elements.
By the way, the gabled milk carton isn't one of them.
And so he called the crystal system, so I call seven
distinct shades of milk cartons.
And these are the seven crystal systems, and they're
described geometrically, and you'll get a chance to go
through them.
The cube is one of them.
Tetragonal.
Here's the calcite, rhombahedral.
There's barrel, hexagonal, and so on.
The seven different ways you can feel three space.
And this is from the archival readings of my predecessor,
Professor Witt.
Actually, just to give your a sense, here's the
two-dimensional analogy.
If I said to you, we're going to open a company, and we're
going to make bathroom tile, and we want to make it sort of
an upscale company, you know, not everybody wants boring old
square tiles.
But definitely, I can cover two space with square tiles by
putting them side by side, one on top of the other.
What's another way I can fill?
I can go with rectangular tiles.
So this dimension is different from that dimension.
What's another way?
I can use a rhombus, right, at 60 degrees.
Or I can use just a plain old parallelepipede at some
arbitrary angle.
And if you go over to the Stata Center, to the Gehry
building, take a look at the shape that they
chose as the unit cell.
This is the repeat group, if you like, or the unit cell,
because that's the one that's going to be cookie cutter
replicated, side by side.
When you're putting the skin on a building, you have to
make everything fit with no holidays, right?
Because you don't want gaps.
what they chose was this area element.
And all the pieces of stainless steel are cut to
this shape.
And that's how you make things fit.
How you take something and make it fit around a three
dimensional object is tricky.
It's the same thing as the clothing problem, right?
How do you take a flat piece of cloth and make it fit the
contour of the human body?
At least, you know, in men's clothes, you know, we've got
dimensions.
We've got sleeve length, got waist, chest. We've got all
these numbers.
For a woman, 8.
One number?
Fashion designers don't know crystallograpy!
If they knew crystallography, they'd know how
to specify the shapes!
If you understand this, you can start your own business
and make clothes that fit.
And I guarantee you, there's a market out there for
clothes that fit.
What about this one?
Pentagon.
We're going to make pentagonal bathroom tile.
Well, when we install these, we're going to have to use two
different colors.
You can see this up here, right?
This is pentagonal bathroom tile.
It doesn't fit.
You've got all these white spots.
So this is not a unit cell.
This is not one of the crystal systems for two dimensional.
So this one's no good.
We better put an x through it, in case someone fell asleep,
and wake up, and then tell me that that's OK.
All right.
So then we move about half a century forward, to the
turbulent year 1848, also in France.
Auguste Bravais.
And what Bravais did, is he said, all right.
This is just filling space.
But where are the atoms?
I haven't told you where the atoms are, right?
You know, as far as you're concerned, maybe there's one
atom in each of these boxes.
Or maybe there's atoms at the corners of the box.
Or maybe there's atoms at the corner and the
center of the box.
Where are the atoms?
I've simply told you that I can fill three space with
boxes, or with these rectangular boxes, and so on.
So he set out mathematically to prove how many different
arrangements of points there are in space.
It's sort of like saying, if I want to tell you where all the
apples are in the orchard, I'm going to tell you where the
trees are, and I'm going to tell you where the
apples are on a tree.
So where do the atoms go inside these boxes?
Turns out there's 14 different arrangements.
And what do I mean by that?
This is a good example.
So we've already established that the cube is a
space-filling volume element.
It's one of the seven.
But look.
If I'm one of the atoms at the corners here, I've got six
nearest neighbors.
This is also a cube, but it's got atoms at the corners and
an atom in the center.
You might say, well, what's the big deal?
If you're in the atom in the center, and you look to the
nearest neighbors, you've got eight nearest neighbors.
Here you only have six nearest neighbors.
So this is spatially differentiated from this.
So what why don't we keep going?
And by the way, the same environment of that central
atom that's coded green is the same environment of any of the
atoms on the corners.
They're all symmetric.
So both are cubes, but they have different space point
environments.
And here's a third one, where we put atoms at the corners
and atoms on each of the faces.
So this atom on the face, you can see, it's got one, two,
three, four atoms in the plane, three
behind, three in front.
It's got 12 nearest neighbors.
This has got eight nearest neighbors, this has got six
nearest neighbors.
So Bravais went through and he said, let's take those seven
crystal systems. Let's put atoms in distinguishable ways.
How many distinguishable ways can we come up with atom
arrangements?
And the answer is 14.
Now, what we're going to do in 3091, is we're just going to
confine our conversation to the cubics.
Because you use orthogonal vectors, they're all the same
length, the mathematics are simple, and the
principles are the same.
And by the way, boatloads of materials are
in the cubic system.
So it's not as though we just chose something that's
convenient but irrelevant.
So here are the different Bravais lattices.
So you can see the cubic, there's simple cubic,
body-centered cubic, face-centered cubic.
With tetragonal, there is simple tetragonal,
body-centered tetragonal.
There's no face-center tetragonal.
If you try to make face-center tetragonal it's degenerate,
reverts to one of the other systems.
So these are the distinguishable systems. And
everything, including crystalline proteins--
if the protein crystallizes, it forms atomic arrangements
in one of these Bravais lattices.
Everything conforms to one of these Bravais lattices.
And if it doesn't, it means it lacks long-range order, in
which case it is a glass.
That's it.
So here's face-centered cubic.
You can see the placement.
But there's something that's been simplified here.
Right now what you see is a single atom, designated by a
hard sphere.
But at each of these lattice points, I can put
more than one atom.
So let me show you what I mean by that.
What I want to do, is put different numbers of atom
arrangements at the lattice point.
So those positions, the distinguishable positions, are
called lattice points.
And now I want to define the crystal structure--
the crystal structure is the complete accounting of atomic
arrangement.
It's the complete description of atomic arrangement.
And we don't have to go too far, because we just have to
get the base unit, the unit cell, and then just
repeat the unit cell.
Complete description of atomic arrangement.
And so how we going to go about this?
We're going to start with a Bravais lattice.
And what is the Bravais lattice?
The Bravais lattice is simply a point environment.
And this will mean something to you in about three minutes,
when I give you the rest of the description.
So it's a set of points in space and the basis.
And what is the basis?
The basis is the atom group at each lattice point.
So what do we mean by that?
Well, let's look at what's up on the board.
What's up on the board is face-centered cubic.
So let's do all of these.
So we're going to say that the Bravais lattice, in this case,
in face-centered cubic, FCC.
That's what this one is, up on the board.
And we've got dots indicate the points.
Because really, the points are nothing.
It's a concept.
See, it's French.
It's a concept, right?
What's this?
See what I'm holding?
I'm holding a Bravais lattice.
It's just a set of points.
It's like a John Cage piece, you know, a bunch of rests.
So if I go to the piano and I play rests in
five four time, listen.
Isn't that great?
Man, it's fantastic.
All the different rests, and different--
So this is a set of lattice points.
Now I'm going to put something on them.
You think I'm kidding.
I'm not.
All right.
Now what's the basis?
If I put a single atom, we have what's up on the slide.
And so that's the representation of metal.
So for example, gold is FCC, with a single gold atom at
each lattice point.
Aluminum, copper, platinum.
These are all FCC metals.
So there's a single atom at Bravais lattice point, and so
the crystal structure is known as FCC.
It's the same as the Bravais lattice.
It's called FCC.
OK.
Now, we can also put a molecule at each lattice site.
So I'm going to put up the FCC here.
Here's the FCC.
There's the cube, the set of lattice points, and the
lattice points are--
I'm going to just illustrate on the front face.
One, two, three, four, and a fifth atom at the front face.
Now these are the points.
I could either put simple atoms there, and the central
image there shows what happens when you have the close-packed
arrangement, the atoms are so big that they touch.
Or what I could do, at each lattice point,
I could put a molecule.
So for example, I could put methane.
I could put CH4.
A methane molecule.
Because when one of you gets to Jupiter, and you see the
methane ice, and you go, oh.
That's FCC.
Because everybody else on the mission hasn't taken 3091.
They don't have a clue what they're looking at.
But you go, that's face-centered cubic.
And they go, huh?
And this is at each lattice site.
You see, my point is, now look, here's the difference.
All five of these, the carbon and the four hydrogens, sit at
this lattice site.
All five atoms sit at this lattice site.
It's not that the carbon goes here, one hydrogen goes over
at one lattice site, one hydrogen over here.
You don't have the methane straddling the unit cell.
You have all five atoms here, five atoms here.
At every lattice point, I have all five atoms at
each lattice point.
So in that case, we also have, it's FCC.
So methane ice is face-centered cubic.
Because it's spherical.
They pack just like gold atoms. You know, from a
distance, we can model this as a sphere, as a point.
Or we could put an ion pair.
An example of that is sodium and chloride, and this is
called rock salt crystal structure.
Rock salt.
Let's look at that one.
So at each last lattice point, I'm going to put two ions.
Not one, two.
This is taken from your text.
So you might look at this and say, well, here's the front.
You might look at this and say, well, isn't this kind of
like simple cubic?
Because I've got one, two, three, four, but I've got
different atoms if I try it that way.
See, this is taken from a different book.
I did some highlighting on it.
So this is sodium chloride again.
One, two, three, four, five.
With this lattice site, I put the chloride atom and the
sodium atom.
The pair of atoms are associated with
this lattice site.
You might say, yeah, but this is way over here.
Doesn't matter.
I'm going to anchor all of those to this lattice site.
To this lattice site, I'm going to anchor this and the
sodium ion.
So this is not simple cubic.
Because if I try to call this simple cubic, I have different
atoms at different lattice sites, and that's a no-no.
Has to be the same.
So if I associate the pair, chloride ion and sodium ion,
chloride ion and a sodium ion
not depicted.
It's off the diagram here.
And I put these dots at the center of each chloride ion.
Can you see that I'm at the face of FCC?
So that's an FCC Bravais lattice, but it's got two
atoms, in this case, both ions, associated with each
lattice point.
I will come to the dogs in a second.
And now there's a last one I can do.
And that's got two atoms, but not an ion pair.
We're going to put an atom pair.
Got an atom pair, and that looks like this.
I'm going to put a carbon atom, one carbon atom, and a
second carbon atom.
And this angle here is 109 degrees.
And I'm going to put this pair of carbon atoms together, and
when I do that, and I replicate that through FCC, I
end up with diamond cubic.
And I'll show you a three-dimensional example of
that next day.
So just bear in mind that if you have an ion pair, you end
up with rock salt.
If I show you this pair, this is due to the sp3
hybridization.
And so examples of this one are diamond, not graphite,
just diamond, silicon--
so everything that I've shown you about single crystals, and
intrinsic and extrinsic semiconduction, is diamond
cubic crystal structure.
But it's not one of the Bravais lattices.
It's an FCC Bravais lattice with two
atoms per lattice site.
That's diamond cubic crystal structure.
But it extends to two dimensions.
So here's an Escher print.
It's got all these dogs.
In two dimensions, what's the crystal structure here?
Well, there's no such thing as body-centered cubic.
There's no third dimension in two dimensions.
So you've only got simple cubic or face-centered cubic.
So what do we have here?
Well, you see, you've got a problem.
Because the black dogs are facing the opposite direction
of the white dogs.
So you can't go one, two, three, four, and make
anything that way.
So instead, I put dots on the, I don't know what you call
this part of the dog.
I don't know.
This part of the dog.
The leg?
Whatever you call this thing.
Yeah.
You put it on the same spot of the dog--
maybe I should have used his eye, I know how to call it.
Anyway.
So I put these things up, right?
And that means that if I associate the four dogs with
this point, the two dogs facing the right, and these
two dogs facing the left.
So these four dogs are associated with
this lattice site.
Then I move up here, and I've got the same thing.
One, two, three four.
Two dogs facing right, two dogs facing left.
So that means I've got a simple cubic bravais lattice,
and the basis is four dogs.
Four dogs as my basis.
And with that combination of a simple cubic lattice and four
dogs is the basis, I can make the entire structure.
That's what this is all about.
That's my unit cell, that's my repeat unit.
So there's simple cubic, right there.
So instead of a single atom at each corner, or these are
spheres that are touching, we got the four dogs.
Here's another one.
Once I got the Escher, I kept flipping through the book.
I don't know what this thing is, but it's creepy.
Anyway.
It's wallpaper.
If it's wallpaper, it's ordered.
If it's ordered, it's a two-dimensional crystal.
If it's a two-dimensional crystal, it has to conform to
one of the 14 Bravais lattices.
So which one is it?
So I looked at this for a while, and I took this happy
entity here, and I put a dot in its belly.
So wherever I found one of these, I put a dot.
and then I connected the dots, and what do I get?
I got the rhombus.
Now, what's the basis?
That's the Bravais lattice.
What is the basis?
So I went over here and I said, well, if I start from
this thing and I go all the way up, I got some
tail, blue tail here.
And this thing way down here looks like
it's missing a tail.
So if I start from this, go all the way down to here, and
capture all of this material in between and associate it
with that lattice point, that can be my basis.
And then I take that set of imagery, and I put it on top
of this, where it's centered at about this thing's navel,
and put it around here.
You can see, the translation is obvious there, and the
translation is obvious here.
There's the wallpaper.
So it's part of a Bravais lattice, only in two
dimensions, you see?
This is crystallography.
That's all it is.
Crystallography.
All right.
So the next thing that we have to do, it's kind of a little
bit boring, but you need to know this stuff, because it's
formative for the next unit.
It's the characteristics of cubic lattices.
Now what do I mean, the characteristics?
I mean the geometric characteristics.
So there's a table here that I've taken out of your
archival reading.
If you go to the website, there's a set of notes that
were written by my predecessor, Professor Witt.
And if you go and look this up, this unit on
crystallography, there's this table.
And it shows such things as the unit cell volume.
Well, what's the unit cell?
The unit cell is this cookie cutter cube that's the repeat
unit, and it has a dimension here called a, which is the
lattice constant.
It's the size of the box.
So what's the volume of the unit cell
Well, duh, it's a cubed.
That's obvious, all right?
Doesn't matter whether it's simple cubic, face-centered
cubic, body centered cubic.
How many lattice points per unit cell?
Well, that's-- how many atoms do I get per unit cell?
If you look at this, you say, well gee, it's--
you know, if you've got the simple cubic, you've got them
at the eight corners, so it's eight, right?
Well, no, because each atom on the corner is inside this
cell, but it's also inside the neighboring cell, and it's
inside each of the eight neighboring cells.
So I have eight times one eighth.
So the total amount of atom within the cube is only one.
That's why you've got to go through this derivation.
So here's an example for face-centered cubic.
You see the corners, one, two, three, four?
There's eight of those.
Eight times one eighth is one.
And then you've got these atoms on the face, and there's
one, two, three, four, five, six, right?
Six faces.
But each of these atoms on the face is only
half inside the cube.
The other half is in the other cube.
So I got six times one half is three, and that gives me four.
So that's how you go through, and figure out
what's going on.
And then you can look at things like nearest neighbor
distance, and what's the relationship between the
radius, if you're looking at hard sphere packing.
So here's a good example.
Here's the hard sphere packing model.
So I've got atoms all the same size, and
they're packed to touch.
And they're packed to touch so that in the end, I end up with
a face centered cubic arrangement.
And so what's the relationship between the radius and the
lattice constant?
And it's just simple trigonometry, right?
4r equals 2a squared, and you go through it, and that's how
you end up with all these formulas.
So that's what going on.
All right.
So I think we're going to move to the last five minutes here.
So just a couple of comments.
Today we played music when you came in.
It was Burning Down the House.
An old piece by Talking Heads.
Why was I playing Burning Down the House?
Well, I want you to study this painting.
This is a painting by Georges Braques, from 1908, And it's
called The Houses at L'estac, or if I speak Canadian, The
Houses at L'estac.
And you see the thing about these houses?
You see the ones in the background have better
definitions than the ones in the foreground?
And the ones in the foreground are all distorted, and kind of
falling over?
Well, when this was exhibited at the Salon, on the critics
lambasted it.
They said, this thing is absolutely childish.
And one of the critics said, this isn't
a painting of houses.
This is nothing but a stack of cubes.
And from this painting, and that negative comment, came
the term cubism.
And so the entire movement of cubism comes from this.
And since we're studying the properties of cubic crystals,
I thought the connection was pretty tight.
And that's why I chose that piece.
Don't go anywhere yet.
We still have a couple of minutes.
It's not better out there than it is in
here, I guarantee you.
All right.
So now I want to show you one of the properties.
Why do we care about atomic arrangement Suppose some of
you are interested in fiberoptics.
One of the things you have to do in fiberoptics is make
junctures, because you can't run a cable 100 miles without
making some junctures.
Well, atomic arrangement reflects microscopic
properties.
And one of the properties that you can have in a crystal is
something called birefringence, depending on
the atomic arrangement.
So what happens is if you have a ray that comes in from the
left here, it splits into two.
And that's no good, in certain instances.
In other instances, if you want a beam splitter, you want
something birefringent.
So look at all these crystals.
Quartz is birefringent.
Ice is.
You've seen it.
Calcite is birefringent.
What do I mean by that?
Well, Dave, can we cut to the document camera?
Right.
So here's a here's a crystal of--
well, this is my handwriting.
It's beautiful.
Let's write-- here's more.
Isn't that beautiful?
OK, enough about me.
Now let's--
All right.
So this is birefringent.
But what happens if we rotate it?
We can change the degree.
And if we're really clever, and we get it onto the optical
axis, we can actually get these things to superimpose.
Here's a piece of cryolite.
This is the stuff they make aluminum out of.
They dissolve aluminum in it.
And you can see that, see, I loaned it to somebody, and
they dropped it.
And when they dropped it, they damaged it.
That's what happens.
Look, it's all crunched up there.
But anyway, I don't want to get into that.
But so you can see, if we turn this and we get it right on
the optical axis, we can actually make these things
line up sometimes.
Anyway.
So you can see the property of birefringence.
OK.
Let's go back to documents, Dave. One more thing.
Last thing I want to show you.
You know gold.
Gold looks like this.
It's yellow, right?
Well, depending on how you, you know, those of you have
seen certain wedding bands know that if you want to make
a white gold, you add nickel to it.
And once the nickel content gets above a certain level,
the gold looks white.
In other words, just metallic reflective.
In Eastern Europe, it was common to add high levels of
copper, in which case, you get a beautiful rose-colored gold,
or pink gold.
I'm going to show you some blue gold.
This isn't a joke.
This is 14 karat.
It's 40 plus percent by weight gold, and
the balance is indium.
And when you add indium to gold, you get blue.
Document camera, please.
Just so that you remember.
OK.
That's gold.
This is gold foil.
Rutherford, et cetera, et cetera.
You know, this is yellow gold.
Ho-hum.
This is white gold.
Same deal, but with some nickel.
Now it's reflecting off of the yellow gold.
Boy, there's a lot of yellow in here.
Where's that coming from?
It's hard to tell.
All right.
Now Let's look at some blue gold.
This is blue.
So, but it's brittle.
So you know, what are you going to do with something
that's brittle?
You can't shape it.
So I have an idea.
So here's my idea.
See, what we do is--
see this?
We do, instead of this, we put this.
We drill a hole in it, we put hands on it, we sell it for a
lot of money.
It's blue gold.
It's blue karat gold.
What you can do if you understand crystallography.
All right, get out of here.
I'll see you on Friday.