PROFESSOR: Since z, that's straight up.
And that is going to provide for you a cell that has the
shape of a square prism.
This would be a1.
This would be a2.
And this would be a3--
I'm sorry, this would be c.
And if you took the choice for the third translation as 1/2
half of a1 plus 1/2 of a2 plus some amount z up above the
base of the cell, and then redefined a T3 prime that
would be equal to--
sorry.
What am I doing here?
Yeah.
This is right.
1/2 of a1 1/2 of a2, and z straight up.
Define a T3 prime as 2T3 minus a1 minus a2.
And that will define for you, again, a cell in the shape of
a square prism with a1 and a2.
And now the translation that went up directly over the
center of the base of the net below.
Twice that minus a2 minus a1 brings you back to a third
translation T3 prime that's normal to the base.
So both cells have the same shape.
This is a primitive.
This is a body-centered tetragonal lattice.
And the symbol that's used to represent the body-centered
lattice is I. One of the few cases, again, where one has to
be bilingual.
The German word for body-centered is
innenzentriert.
So there's Schoenflies at work again.
So primitive body-centered represented by I.
And that brings us to one final case, and that is the
plane groups that have a hexagonal shape.
And here we have a funny situation in that the same
choices of the third translation do not work for
all of the plane groups.
If you have a base to the cell--
and again, the two edges of that net are
identical by symmetry--
if it has a three-fold axis in it and nothing else, than
there are two choices.
You can either have T3 be equal to 0a1 plus 0a2 and an
amount z straight up.
And that will give you a cell that is a primitive cell.
This would be a1.
This would be a2.
And just as we did for tetragonal, we'd call the
third axis c, This will work for a three-fold axis.
So there is a space group P3.
This will also work for all of the other groups.
So there's P3M1 and a P31M.
And a P6 will work.
And a P6mm will work.
So look it here.
We've got five additional space groups in addition to
the lattice type.
This is the primitive hexagonal lattice.
For a three-fold axis, we have another choice.
We could pick T3 so that it took the origin three-fold
axis and put it directly over this one.
Looks as though you could have a different distinct lattice
if you move the origin lattice point over the three-fold axis
that sits at a location 1/3, 2/3.
This one is at 2/3, 1/3.
Let me demonstrate, I think convincingly and clearly, that
those two choices are exactly the same thing.
And the way that I can show that is, once again, to draw
this net a1, a2.
And suppose I pick the part of the offset of T3 that is
within the plane of the net as going from this lattice point
to this one here.
Then if I go twice that translation T3, that's gonna
put me directly over this three-fold axis.
And if I go three translations T3, I'll be directly over the
lattice point on the ground floor.
So putting the original lattice point over this
three-fold axis or this three-fold axis is exactly the
same thing.
And I can change one into the other just by flipping this
thing 180 degrees.
So it's obviously the same choice.
This description of T3, if I define the direction of c now
as 3T3 defined in this fashion, minus a1 minus a2, is
gonna be a funny situation.
It's gonna be a peculiar sort of double body-centered cell.
Of course, along the long diagonal of the hexagonal
cell, I'll have one lattice point that's 1/3 of the way
along that long diagonal, and another interior lattice point
that's 2/3 of the way along that long diagonal.
OK.
This double body-centered cell, so to speak, actually
can be redefined in terms of a primitive cell, not a
primitive cell that is this particular description of a
primitive hexagonal lattice.
This is not a primitive hexagonal lattice.
But suppose I go from this interior lattice point along
the long diagonal of the cell to this location
and call that a1.
Remember that there is a three-fold axis sneaking down
through the center of this triangle, goes through the
lattice point, and comes out the center of
the triangle below.
If I go up to this lattice point and up to this lattice
point, I have three translations that are
straddling the three-fold axis.
And that three-fold axis rotates one into the other.
So this new definition of a cell that is in fact
primitive a1, a2--
and they're all equivalent by symmetry.
I always get in trouble when I try to draw it, but that
actually defines a cell with a shape that
we've not seen before.
That is a cell that has the shape of a rhombohedron.
And as I say, I always get in trouble when I try to draw it.
It's got three translations like this and three
translations skewed by 60 degree sitting on top.
Everybody can see that's a rhombohedron, can't you?
Say yes.
Be nice.
So this is another primitive cell that's a choice for this
triple cell.
And this consequently gives its name to this lattice.
This is called a rhombohedral lattice, regardless of whether
you pick the primitive cell in the shape of the rhombohedron
or this peculiar double body-centered cell.
So this is a rhombohedral lattice.
And this is represented in a space group symbol by the
letter R, standing for rhombohedral.
So just the three-fold axis with this choice of
translations, that would be space group R3.
AUDIENCE: [INAUDIBLE]?
PROFESSOR: Yeah.
a1 like this. a2 like this. a3, I'm taking a right-handed
system, comes from the centered lattice point.
So let me draw it in projection.
It's easier to see.
Here's the outline of the hexagonal net.
Here's the three-fold axis.
And I've taken a1 coming up like this.
I've taken a2 coming up out of the board like this.
And I've taken a3 coming out like this.
OK.
So these are the three edges of the rhombohedron.
Notice that I cannot have that lattice with a lot of the
hexagonal plane groups.
I cannot do it for six-fold axis.
Because a six-fold axis--
P6 and P6MM have a six-fold axis here and three-fold axes
in the middle of the cell.
So I cannot have a rhombohedral lattice for P6 or
P6<M. The primitive hexagonal lattice is the
only thing that works.
And then we have this curious situation with the two
alternative settings of point group 3M in a hexagonal net,
P3M1 and P31M.
For P3M1, I have symmetry 3M at both the origin lattice
point and in the center of these two triangles.
So for P3M1, in addition to a three-fold axis, I could put
in 3M1, stack up plane group P3M1.
But I cannot do this for 31M.
That's impossible.
Because if you look at P31M, there is symmetry 3M at the
origin and only symmetry 3 in the center of the triangles.
So if I try to pick this as a third translation for plane
group P31M, I'm taking symmetry 3M and plopping it
down on top of symmetry 3, and that wrecks the plane group.
So for the rhombohedral lattice, I can
only have 3M1 or P3.
P6 has a six-fold axis at the origin lattice point.
The two centers of the triangles are three-fold axes.
So primitive for the six-fold axis, either P6 or P6MM, is
the only choice.
So if anybody's counting-- and I wasn't--
we've got 11 lattices so far.
And the only ones that we haven't been able to get by
simple stacking of the plane groups are the systems that
you all know as your favorites.
The cubic or isometric stems for the fact that the
translations are identical in all three directions.
Let me quickly dispose of those by starting with a
four-fold axis coming out of one face of a tetragonal cell
and saying now if this is a cubic symmetry, I have a
three-fold access that comes out, sort of, in the direction
of a body diagonal.
But this three-fold axis is gonna rotate that four-fold
access to this direction and to this direction.
We've seen that if I look down along the diagonal of a
equally axed tetragonal prism, the three-fold axis takes this
face, rotates it into this face, and rotates
it into this face.
So I have to have a four-fold axis coming out of every one
of the faces.
And having that arrangement of four-fold axes means that not
only will the two translations be identical in the base of
the cell, as is the case for tetragonal, but if I put in
that three-fold axis as an extender, all three edges of
the cell have to be identical by symmetry.
So I can do this for a primitive tetragonal lattice.
And that's gonna give me a primitive isometric lattice.
I can do this for a
body-centered tetragonal lattice.
So that would be indicated by the symbol I. So it's a
body-centered cubic lattice.
And if you add up everything we've done to this point,
there are 13 space lattices, a very unlucky number.
And as you all know, there are 14.
We've missed one.
So what have we not done?
We stacked up all the plane groups in all possible ways.
We've taken the four-fold axes out of the face normals, all
three face normals of a tetragonal cell.
And that forced it to become a cubic lattice.
What have we done wrong?
Actually, we can get the fourteenth space lattice by a
little bit of sleight of hand that is not obvious.
Let's say that this is the base of a tetragonal lattice
that we used as the parent for our cubic lattice.
Let me change that primitive square in the base of the
tetragonal cell and change it into a centered square.
There'd be no reason for doing that for a plane group.
But let me notice now that if this is the direction of a
four-fold axis, rather than having the three-fold axis
come out of this direction relative to this super double
square, I can make the three-fold axis come out in
this direction.
Ah-ha.
So what I have now is a cell that is centered, has a center
in the base and has translations that are
perpendicular to the base like this.
And if this is the direction of the four-fold axis and this
is the direction of the three-fold axis, that
three-fold axis is going to put a centered lattice point
on the remaining faces of the cube.
So it's only if I notice I can make a double square base to
the cell again, put the three-fold axis of a cubic
symmetry along the direction of the body diagonal, I
generate lattice points in the middle of the other faces as
well, and this gives me the lattice known as F, the
face-centered cubic lattice.
And the edges of the cell here are all identical, equivalent
by symmetry.
So they're labeled a1, a2, a3.
Have you ever seen a lattice constant for a
cubic crystal a0?
Has anybody ever seen that?
A lot of people do it.
And if you see that notation in a publication, you will be
entitled to sneer at it knowingly.
It's logical, three, two, one, zero.
By extrapolation, if you've got an a3, an a2, a1, you
gotta have an a0, right?
Well, not necessarily.
That sort of algebra doesn't apply to
labeling of cell edges.
It comes from a curious historic precedent.
Come back, again, to the fact that for literally more than
100 years, people developed symmetry theory and had a
pretty good idea that crystals were based on packing of
units, molecules, whatever they were, that were on the
nodes of a lattice.
But they couldn't measure the size and shape of the lattice.
And that was the brilliance of the famous experiment by Max
von Laue because that, in one fell swoop, show that the
mysterious radiation x-rays were just electromagnetic
radiation of a very short wavelength.
And that's what he really gained notoriety for.
But at the same time, he proved for the first time,
unequivocally, that crystals were based on a lattice.
And that was the other side of his double-edged
accomplishment.
Up until that time, people learned about crystals by
studying the morphology and measuring the angles between
crystal faces and recording different faces on a crystal.
I mean, I can just see one of these old guys working a
little bit before his wife made him come down and sit for
supper because it was getting cold.
And he'd let out a, whoop, huh?
A 13 27 face.
What the hey.
It's a new face on potassium tartrate, something like that.
This was the way these guys got their jollies.
And on the basis of the angles between faces, as I said on
several occasions, they could determine a ratio of axes, a
ratio of a to b to c, if their assignment of indices to faces
were correct.
And how did you know?
Well, along came x-rays, a and you could measure for sure.
So what people started doing to distinguish a to b to c
ratios determined from crystal morphology, they put the zeros
on it to indicate real values from diffraction.
And when little people came afterwards and they didn't
realize the reason for doing that-- nobody's done this for
over 100 years because everybody determines lattice
constants using diffraction--
but some people have seen these zeros.
So they say, I'm not doing proper notation.
And you know how furious crystallographers get if you
don't use proper notation.
So let me write a0 as the lattice
constant of my cubic crystal.
You see this commonly.
And there's been no need for it for 100 years.
So anyway, that's the origin of the subscript zero.
There are a lot of dumb things that take place among normally
rational people.
I said a moment ago-- and I crossed my
fingers when I said it--
that c is the label that is assigned to the unique axis,
the axis that is unique because of symmetry.
Sometimes you see monoclinic crystals, which have a
two-fold axis and/or a mirror plane perpendicular to the
two-fold axis, labelled with this as b, this is a, and this
as c, and the angle between them-- that's the general
obtuse angle--
as beta.
There is a story behind this lapse of
standardness in notation.
How do people decide on what proper notation is?
Well, there is an organization called the International Union
of Crystallography, which is the equivalent of the
International Union of Pure and Applied Physics and the
International Union of Pure and Applied Chemistry.
And they meet every three years.
And there are special commissions for people who
like to fight and argue out conventions and nomenclature.
And this is such a silly abomination that there was a
move made when they came out with their new edition of
international tables to change this and do it the way you do
for all of the other crystal systems, namely a, b with
gamma between them, and c coming out this way as the
direction of the two-fold axis.
And this would be the mirror plane.
And that would put this in accord with tetragonal and
hexagonal and everything else.
And it was logical and seemed the right thing to do.
So how do people decide on nomenclature?
This is something that people fight with with more passion
than anything else in science.
So you make a proposal.
The proposal goes to the national committees of
crystallography in countries over the world.
They discuss it.
They fight about it.
They vote on it.
Then it comes back to the International Union.
And finally, if everybody seems agreed, they present it
at the next meeting of the International Union Of
crystallography for a vote.
And that's exactly what happened with the proposal to
standardized notation.
This took place maybe, I don't know, in the 1950s.
Everybody was in favor.
The discussion proceeded rationally.
Then this little guy jumps up in the front row, short of
stature, balding of head, with a very piercing nasal voice.
And he gets up and he enters into this diatribe.
b-axis unique was good enough for Curie.
b-axis unique was good enough for Mauguin.
B-axis unique was-- and all the names were French.
Actually, he wasn't French.
He was Belgian.
So he went on and on and on.
And he so cowed this group of 300 rational delegates, that
they didn't cave in entirely.
What they did was they entered every entry for monoclinic
crystals in the international tables twice, once with this
set of labels, once with this set of labels.
And to indicate that this was the preferred notation, this
is labeled the first setting.
And to indicate second-class citizenship no matter what
Curie, Mauguin, Bravais, and all of these other guys felt,
this was called the second setting.
And I'm not kidding.
I'll bring in a copy of the international tables for x-ray
crystallography.
Every entry for a monoclinic crystals is in there twice.
AUDIENCE: And no one in the past 50 years has tried to get
rid of the second set?
PROFESSOR: No.
Actually, the reason is it is expensive to
revise these tables.
And I brought in on the first day, you don't remember, my
copy of international tables for x-ray crystallography,
which is about this big and cost $100.
And then the new version which is twice as large
dimensionally, four times as heavy in terms of weight, and
I think about eight times as expensive in
terms of actual cost.
So you don't do this casually.
If there's something actually new comes out, you make a
little supplement that people can stick
in the pages somewhere.
AUDIENCE: Don't they still have
PDFs nowadays or something?
PROFESSOR: The last volume came out before PDF, before
very intensive use of computers for such data.
The journals now are all done online.
You submit papers online and they're archived online as
well in addition to hard copy.
But that's something that was before the time of the
publication of the last set of volumes.
Well, I am almost done with talking about lattices.
But I have a thought question.
You've all heard of this castle in Ireland that has
this famous rock in the side of the castle called the
blarney stone?
And you, to get the gift of gab, have to lean over this
rather dangerous wall and somebody holds your ankles and
you go down and you kiss the blarney stone.
And that gives you supposedly the gift of gab.
But kissing is a dangerous occupation.
You've all heard about kissing disease mononucleosis?
And with all those people smooching the blarney stone,
I've always been afraid the blarney stone might
come down with mono.
And if the blarney stone became ill in this session,
would it have to be
transferred to the Mono Clinic?
I don't know.
All right.
Here in all their glory are the 14 space lattices, or
Bravais lattices as they're finally called.
The labels a, b, c, et cetera, are not on them, but the c
axis is vertical.
You'll see that the hexagonal rhombohedral lattice is shown
referred to hexagonal axes.
In the international tables, you will find all the
information both referred to rhombohedral lattices and to
the triple hexagonal cell as well.
Now, into these lattices one should drop those of the 32
point groups that the lattices can accommodate.
That's gonna be a lot of additions.
Besides that, there is the option of having the symmetry
elements of the point group not intersect at a common
point, but interleave them as we did, for example, with the
glide planes in P2GG, and in some of the other
two-dimensional plane groups as well.
Then we would do what we also did in two dimensions, take
the mirror planes and replace them by glide
planes, one or the both.
And you might think that we really have an enormous amount
of work to do.
We haven't considered any symmetry combinations that
involve a horizontal mirror plane.
For example, we look at what symmetry 2 requires and we
found the primitive monoclinic lattice, and the side-centered
monoclinic lattice, and the
body-centered monoclinic lattice.
So there's more than one lattice type into which we can
drop a given point group.
But what would you have if you had a mirror plane
perpendicular to the two-fold axis?
What kind of lattice will that require?
OK.
Let me, in one masterful swoop, convince you that we
have done almost the better part of the work.
Let us ask what does inversion require of a lattice?
What does the symmetry of inversion, which didn't appear
in any of the plane groups at all because that's inherently
a three-dimensional point group operation?
Well, think about it.
All inversion says is that, if, hey, there's a translation
that goes up this way, there should be another translation
minus T in the opposite direction.
And any lattice whatsoever can make that claim and satisfy
that requirement.
So let me now just quickly say that if we have derived the
lattice types, and by implication the space groups
as well, for a two-fold axis, for a four-fold axis, for a
six-fold axis, and so on, we have automatically satisfied
the requirements for any point group that results when you
add inversion to a two-fold axis, when you add inversion
to a four-fold axis, or add inversion to a six-fold axis.
So it turns out that the requirements of a six-fold
axis, which gave us the primitive hexagonal lattice,
are also met by 6 over M. The requirements of a four-fold
axis, which gave us the lattices primitive tetragonal
and body-centered tetragonal, are also satisfied for 4 over
M.
So, any one of the point groups that
differs from the 11--
from the 10 two-dimensional symmetries--
namely 1, 2, 3, 4, 6, M, 2MM, 3M, 4MM, and 6MM--
any symmetry that you get by adding inversion to those
two-dimensional point groups is automatically going to be
happy in the space lattices that we've
derived to this point.
So we have done an enormous fraction of the work that's
required to derive space lattices, and the
space groups as well.
So there is not too much more to be done.
And what I will do for probably the next meeting is
to derive a few of the space groups for the lower
symmetries which are not too complicated, just to show you
how the game plays out.
And then I will mention a little bit about the special
idiosyncrasies of the orthorhombic brick-shaped
cells where there is nothing more distinct about one
direction and the other.
All the inter-axial angles are 90 degrees.
The translations are general.
So there are peculiarities in notation there that are unique
to the orthorhombic system.
And then I want to spend, perhaps, one class talking a
little bit about crystal chemistry.
Have any of you had a class in solid state chemistry or
crystal chemistry?
OK.
One.
One of the main uses of space group theory is the
description of crystal structures.
So I think I might, as a minor digression, take an hour to
talk about packing and some of the concepts used to describe
the close-packed structures and ionic structures and give
you a problem that ask you, as I already did, to generate a
structure, simple-minded two-dimensional structure, and
do some interpretation of it.
So, I think this will be worth saying a little bit about just
so that one of the main applications of symmetry
theory does not escape without any discussion at all.
We will, after about two more lectures, have a complete
change of direction and start talking about properties of
crystals and how these properties are impacted by the
fact that crystals have symmetry.
So we're gonna use some of our results in symmetry, but not
all that much, but develop remarkable mileage in
predicting anisotropy in crystals just on the basis of
the symmetry which that crystal possesses, that
certain physical properties as a function of direction are
weird surfaces that have lumps and noses and bulges in them.
And certain properties have this anisotropy universally,
regardless of the chemistry or composition of the crystal.
Given the symmetry of the crystal, the anisotropy has to
be in this fashion.
We'll be able to show that some properties simply cannot
exist, period, regardless of composition and structure in
materials that have certain symmetries.
So that'll be the direction we take off in.
I mentioned at the beginning of the term-- you've probably
forgotten--
there is a lovely book on tensors and physical
properties by a gentleman named J. F. Nye, N-Y-E. And
the title of the book is Physical
Properties of Crystals.
This is a nice place to go for further reading.
We will go through roughly half of Nye's book with a
little more emphasis on the consequences of symmetry than
what he does.
But if you want to do some extra reading, I asked the
Coop to stock this book.
But I suggest that you not rush out and buy it.
It's available in paperback, which means it's cheap, but
only relatively so.
There's no such thing as a cheap technical book,
paperbacked or not, these days.
It's also on reserve in the library.
So, see if you need the extra help.
See if you would like to do some reading in Nye.
And make sure you want to before you go out and put your
money down at the Coop.
The last part of Nye is an elegant,
beautiful part of the book.
It deals with the thermodynamic relations
between different properties.
And it's a unique treatment that is developed by Nye.
We won't do any of that, but it's a nice thing to have.
So don't rush out and spend your money.
But if you want additional reading and just going to the
reserved book collection is not enough, then Nye is not at
all a bad copy of a book to have.
I think most of you have probably seen Nye, not the
book but the man.
How many of you have seen the famous Bragg bubble raft movie
on disk locations, where he floats sheets of bundles and
sheers them and you see his dislocation go zipping across?
At one stage in that movie, there's an assistant who
tiptoes, trying to be unobtrusive, across the
background.
But the camera catches him scurrying behind Bragg.
That's Nye.
So if you've seen the movie, you've seen
what Nye looks like.
OK.
I'll call it quits there for today.
And we'll see you next week on Tuesday.