PROFESSOR: Everyone seems to be here.
At least all the seats are filled up, so
why don't we begin.
One thing that I did finally, to be specific, I told you
that we would have three quizzes and that they would be
uniformly spaced one third, and 2/3 and 2.89 thirds of the
way through the term.
And I figured out where those dates would be.
So since I have them, let me give you the date.
So Quiz 1 is going to be on Thursday, October 6.
Quiz 2 will be on Tuesday, November 8, and Quiz 3 will be
on Thursday, December 8.
And this, as advertised, is exactly one third of the way
through the lectures.
This is exactly 2/3 of the way through the lectures, and this
is the next to the last meeting that we'll have.
This is the week before the final week of the terms. so I
deliberately kept it a little bit away from the usual
end-of-term crunch.
Being person who is sensitive to symmetry and order, I find
it highly regrettable that we cannot have the
quiz on October 8.
Then you would on the 8th of October, November, December,
you have a quiz.
Unfortunately, October 8, much as that would beautify the
schedule, is on a Saturday, so we can't do it so.
I could do it, but I don't think anybody would come.
I've been, believe it or not, going through these problem
sets in great detail.
I didn't want to hand them out until I had the list--
and you may or may not know-- that is made up for the
registrants in each class, And we get a nice array of
photographs with names and department numbers and years
numbers under them.
That something, by the way, that only the instructor in a
class requests, not even the secretary can do it.
So this is not widely broadcast.
I found to my dismay though when I finally got it that
about one third of the pictures are missing.
And I don't know whether you folks have been photographed
fairly late on the term, but a third of the
pictures are missing.
I already been putting some faces in my memory banks, but
to match up names and faces so that I can hand out the
problem sets individually, I need more photographs.
So I'll do it one way or another next time, and we'll
see some photographs, additional photographs, have
appeared in the interim.
I've given you three problem sets so far, and they were
really just for fun as well as to get you thinking about the
right sorts of things, but there were little puzzle.
I'm, unfortunately, going to have to tell you from now on
no more cutesy little puzzles.
You're going to get real problem sets,
long, tedious, drudgery.
They're optional though, so that's the saving grace.
So I'm going to pass it around the problem set 4.
And this asks you to demonstrate some of the things
that we talked about last time and convince yourself that
they really work and ask you to make some simple
applications of Miller-Bravais indices to
two-dimensional lattices.
And then finally, the last problem is to get you thinking
about patterns.
And I've asked you to identify the lattice points and the
symmetry elements in two patterns.
I'm almost willing to wager that not one of you will get
it completely right, that there's going to be some
little thing that you missed or did wrong.
At one time, I bet a bag of potato chips for everybody in
the class against the class wagering against me one bag of
potato chips.
Well, I'm going to watch my weight, and I've won that
invariably, and I really can't eat that greasy sort of stuff.
In any case, you'll see that in the case of symmetry and
patterns it is immensely simplified when you know
exactly what to look for, when you know the number of
possibilities was finite,.
And when we're through with this, you can ask just one
question about a pattern and then know exactly what the
symmetry of that pattern has to be and exactly where to
look for everything else.
I can guarantee you that by the time you finish this class
you will never sit in a bathroom and stare at the tile
floor and see it in exactly the same way again.
I think that's a good observation because I like to
think that that's what education is all about, not
sitting in the bathroom, but seeing things differently
after you've had the experience
than you did before.
That's what education is all about.
So have fun with the patterns.
And again, it will become--
one seat up here.
Today then we're about ready to embark on an adventure.
As I said last time, we'll develop two-dimensional
symmetries first because there are relatively few of them,
and one could do this rigorously
and in great detail.
We're going to have to touch on things more lightly when we
get to three-dimensional symmetries so we can finish at
a decent point in the term.
Let me remind you that we have so far identified two types of
symmetry that can exist in a lattice, onefold, twofold,
threefold, or sixfold rotation axes.
Any number of different rotational symmetries are
possible, but if you're going to want them to be compatible
with a lattice, you must restrict the rotation axis to
one of these five, including a onefold axis, which is no
symmetry it all.
And in two-dimensions, besides translation, we saw the
operation of reflection.
And now we're going to begin to put things together and
make elaborate combinations.
The first thing I will ask is can we have more than one of
these symmetry elements present and operating about
the same locus at the same time.
The answer to that would be, why not.
Because if we look at a sixfold rotation axis, that's
really what a twofold rotation axis does combined with what a
threefold rotation axis does.
And we plot these on top of one another, and what we end
up with is the arrangement of motifs that is generated by
sixfold axis.
So in a sense, a sixfold axis is a twofold axis superimposed
on a threefold axis.
That kind of a naive way about thinking of this though.
What we're really saying is that what a sixfold axis has
is the set of operations 1 A 2 pi/ 6, A twice 2 pi /6 --
that would be 120 degree rotation--
and A pi, A 5 pi/6.
And there's 1, 2, 3, 4, 5.
And we're missing one.
60, 120, 240.
So that would be A twice 2 pi/3, 4 pi/6
So what we're really saying is a sixfold axis consists of
these six elements, and these elements constitute a group
because I can combine any two of these rotation operations
and find something that's already a member of the set.
By saying that a sixfold axis consists of a threefold axis
sitting on top of a twofold axis, what we're saying is
that there's one collection of elements here, namely
identity, or same thing as A 2 pi and A pi, that is a subset
of these, or we could say a subgroup.
So what we're doing in saying that these two sets of
rotation operations exist simultaneously, we're saying
that the twofold axis is a subgroup of the sixfold axis.
And similarly, we can separate out three other elements, a
onefold axis, same as A 2 pi; a 120-degree rotation, A 2
pi/3; and a 240-degree rotation, A 4 pi/3.
And these three operations, a group of rank 2, is what
constitutes another subgroup, a subset of elements which by
themselves satisfy all of the requirements of a group.
So this is one example of how we can have more than one
symmetry element, a set of operations existing about the
same locus and space.
This is not how one would go about the deriving these
higher symmetry however.
Because what we have to do is to add something to the set,
something that's called an extender, and then show that
all the requirements of a set being a group is satisfied;
namely, that a combination of any two elements in the group
is also a member the group, that for every operation an
inverse exists, and the identity operation is a member
of the group.
So you can build up more higher symmetries, more
complex symmetries by adding some operation called an
extender and then taking all of the products of these
elements and see what new operations arise.
So we can have more than one symmetry element operating
about the same locus and the set of individual operations
that it embodies.
And here sitting all by itself is a mirror plane.
Why don't we combine a mirror plane with the rotation axis?
And we've already seen examples of that.
For example, in the square pattern that these tiles make
up, there's a fourfold axis in the middle of each of the
squares that are also mirror planes that pass
through that location.
So let me look at a combination that is a little
simpler to handle.
And what I'm going to say is here sits a mirror plane with
a reflection operation that I'll call sigma 1 for the
individual operation.
And let me say that I combine now in that space a second
reflection operation about a line that
intersects the first one.
And the question I'm going to ask now is happens when I take
a first motif, number 1, reflected in the locus of the
first mirror plane to get number 2, and then reflect
that a second time in the locus of the second mirror
plane to get one that sits up here.
So the question is now what is the operation sigma 1 followed
by the operation sigma 2 equals to?
We can almost answer by the process of elimination.
If this one is left handed, reflection changes the
polarity, so number 2 is right hand.
And if we reflect a second time, the right-handed one
goes to a left-handed one.
So what we're asking is how do I get from one left-handed
motif to another left-handed motif?
Of the three operations that exist in a two-dimensional
space could be translation.
But, clearly, I can't take the first one and slide it
parallel to itself and make it coincide with the third.
So translation would not change chirality, but that
won't work.
What's left?
Rotation.
Yes.
You want to clarify something first?
AUDIENCE: That third one, it looks kind of left-handed.
[INAUDIBLE].
PROFESSOR: Oops.
It does.
It does indeed.
Sorry about that.
When you've got your nose poked right in these things,
it's easy to overlook it.
Yes.
Absolutely right.
That should be over canted over a bit more like I had it
the first way.
Very good thank you.
I don't want to proceed much.
Yes?
AUDIENCE: What did you [INAUDIBLE]
axis [INAUDIBLE]?
PROFESSOR: We could do that.
And let's do it, and we're going to encounter
this sooner or later.
So we have a first mirror plane reflect from a first
one, which is right handed, to second one that's left handed.
And then define a second locus, and that one would take
the second one and move it over to here, the same
distance on the other side of this mirror plane.
And I'll let you answer the question since you asked it.
How is the first one related to the third one?
AUDIENCE: Translation.
PROFESSOR: Translation.
Yes.
AUDIENCE: It's translated.
PROFESSOR: Exactly So if this distance is delta, the first
and the third are related be a translation
which is twice delta.
So that would be in addition where we'll eventually want to
consider when we have patterns that are based on a lattice.
But right now, I want to consider symmetry operations
that leave at least a point in space and move perhaps a line
in the case of the mirror plane.
But back to the original question, what is this first
reflection followed by a second reflection?
We've just thrown out translation.
That won't work unless the mirror planes are parallel to
one of those.
The only thing left is rotation.
And in point of fact, even looking at this diagram, you
can get from the first one to the third one by a rotation.
So the answer to the question is that a combination of two
reflection operations is a rotation operation about the
point of intersection of the mirror planes.
But we can go further and say exactly how large this
rotation is.
Let's draw an extra line in here.
And let's say that this angle is alpha.
And if I repeat this by reflection, that
angle is also alpha.
Let's label this angle beta.
Repeat that by reflection, and that angle must also be beta.
The angle between the two mirror planes is the angle mu.
And clearly mu is equal to alpha plus beta.
So the answer to the question quite generally is have two
mirror planes intersect at an angle mu, and their successive
operation is going to be equivalent to the net result
of rotating through twice of the angle
between them mirror line.
So we don't have to think about that anymore now.
We know if we combine to mirror planes
we get a net rotation.
So that is a basic result, and that's a first example of
something that I'm going to call by my own pet term.
I'm going to call them combination theorem, the
theorem that tells you what you get when you combine two
operations.
And what this is going to give us is a way of filling in one
box in the group multiplication table.
So knowing what two mirror planes combined at an angle
should be, I can now combine the mirror planes at angles
which gives me a rotation operation which is one of the
ones that's allowed for a lattice.
In general, if I'm not yet putting these symmetry
elements in a lattice, I could combine them in such a way
such that I got a 17-fold rotation as the angle.
And then the angle between the mirror planes would
be half of 2 pi/17.
Could be a lovely symmetry, nice for a pendant or a ring,
but not for a lattice and not for a crystal.
So I'm going to want to combine mirror planes at
angles that correspond to rotations that are compatible
with the lattice.
So what I could do is take a first reflection
locus, sigma 1.
And if I want the rotation angle to be 180 degrees for a
twofold axis, I would want the second mirror plane with the
operation sigma 2 to be at one half of pi with
respect to the first.
And for this specific combination then, reflection
in the first mirror planes followed by reflection in the
second mirror planes to give me a third one, is going to be
equivalent to--
Oops.
Back here--
it's going to be equivalent to a twofold axis.
So sigma 1 followed by sigma 2 at an angle one half of pi is
going to be equivalent to the rotation operation A pi.
This is not yet a complete pattern because here's a
mirror planes that wants to reflect this motifs as well.
So let's just reflect number 3 across the mirror planes 1.
And then we're going to get a fourth object.
Now let's show that this constitutes a group.
We've got the identity operation doing nothing.
We've got the operation sigma 1.
We've got the operation sigma 2, and we've got
the operation a pi.
I'll put the same operations again in the vertical column.
Here's operation 1 doing nothing, sigma 1,
sigma 2, and A pi.
So doing nothing and following it by one of these four
operations just gives you the same thing back again.
And doing these operations followed by 1 will give me the
same operation back again.
If I reflect twice across sigma 1 from left to right and
right to left, I'm back to where I started from.
So sigma 1 followed by sigma 1 is the identity operation.
Sigma 1 followed by sigma 2 is what we just did.
That turns out to be the operation A pi.
And sigma 1 reflection and following that by A pi is the
way in which I get to the fourth one.
So that is going to be equal to sigma 2.
Well, let me rips through the last ones fairly quickly.
Sigma 2 followed by signal 1.
Sigma 2 followed by sigma 1 is the same as A pi.
Sigma 2 followed by sigma 2 gets us back to where we
started from.
That's the identity operation.
Sigma 2 followed by A pi reflect from here to here and
then rotate.
That's the same as sigma 1.
And the final sequence, rotation followed by
reflection would give me sigma 2.
Rotation followed by sigma 2 will give me sigma 1.
And doing the rotation operation twice is the same as
the identity operation.
So are the group postulates satisfied?
Yes.
The combination of any two elements is a
member of the group.
For every operation, an inverse exists.
And we can answer that question very easily by merely
looking at the column under a particular element, and
somewhere we find the identity operation.
So sigma 1 is its own inverse.
So, yes, an inverse exists for every operation.
And then finally, identity is a member of the group.
So everything's lights up.
Bells ring.
This set of four elements is entitled to
call itself a group.
And in particular, it is a group of rank 4 because there
are four elements in the group.
I show you a cool thing.
I noticed that the number of objects in the pattern, the
number of motifs in the pattern, is exactly the same
as the order of the group.
Why?
Because these four operations tell you how to get from any
operation to the remaining three and tell you how to get
it into itself.
So identity relates it to itself.
Sigma 1 reflects it to here.
A pi rotates it down to here.
Sigma 2 reflects it down to here.
So there's always a one-to-one correspondence between the
elements that are in the group and the rank of a group and
the number of objects that are in the pattern.
Yes, sir?
AUDIENCE: How do you show if the inverse exists again?
PROFESSOR: The inverse exists if I can find something that
combined with an element in my basic set gives me the
identity operation.
So it's the operation followed by its inverse has to be the
same as doing nothing.
So if I look under each element and find the identity
operation, then sigma 2 is its own inverse.
If I look under A pi, A pi is its own inverse.
This is generally not the case.
This is a very simple symmetry.
So really we can work with commas and little figures, and
that's one way of getting these symmetries.
But group theory I think you can see is a very nice, very
elegant language for describing some
characteristics of these patterns.
Now little bit of jargon.
As if this material were not confusing enough, there are
two different languages that are used to denote these
unique combinations.
This notation where we simply have a number for a rotation
axis or an m for mirror plane is something that was first
proposed by two
mathematicians, Hermann and Mauguin.
And this was adopted as the preferred notation in the
international tables, that hefty tome that I brought it
on the first day of classes.
So this is also referred to synonymously as the
international notation.
Let me put these things out horizontally.
For rotation axes by themselves, the symbol in
Hermann notation is simply n.
And for lattice, n--
as we've seen many times--
is restricted to what in Hermann-Mauguin notation would
be either 1, 2, 3, 4, and 6.
There's another type of notation that is use quite
frequently by condensed matter physicist, and this is called
the Schoenflies notation after one of the mathematicians who
was the first to derive the three-dimensional symmetries.
This was a curious thing.
There were three people in different parts of the world
about the turn of the century-- and this was before
email or even air mail--
who were trying to derive the three-dimensional symmetries.
Schoenflies was one of them.
And all three of them got to the same place at about the
same time, and the results were obtained.
But each one used his own method and had
his different notation.
Schoenflies was one of the people who did this.
Schoenflies was a mathematician and based his
notation on group theory.
Something called a cyclic group is what
the C stands for.
And a cyclic group is one in which all elements are power,
so to speak, of some basic operation.
For example, a fourfold axis consists of the set of
operations A pi/2; a 90 degree rotation; A pi, which can be
written as A pi over 2 squared; A pi over 2 squared;
[? then we need ?]
A pi/2; and A pi/ 2 again; A 3 pi/2, which can be viewed as
doing A pi/2 three times; and finally, the identity
operation, which is the same as A 2 pi, which is equivalent
to doing the basic operation A pi/2 four times.
So the group of rank 4, there are four elements in the
group, each one is a power of the basic operation A pi/2.
So a rotation axis by itself as a cyclic group.
Schoenflies indicates these generically by C subscript n,
where n is the rank of the axis.
Hence, this is C1.
This is C2.
This is C3.
This is C4.
And this is C6.
A mirror plane by itself is indicated m in the
international notation.
A mirror plane is also a cyclic group.
There are two operations, reflection and reflection back
to where you came from.
Doing the reflection operation twice is the
same as doing nothing.
So Schoenflies also call this one C. And the one thing that
is without meaning in English is his subscript S. And
Schoenflies was German, and the S stands for spiegel,
which is the German word for mirror.
We have in many cities a paper that's
called the Daily Mirror.
In Germany, there's a paper called Der Spiegel.
So they use it the same way in everyday life
as well as in notation.
So C sub S in the Schoenflies notation is a mirror plane.
Cyclic group the S stand for spiegel.
For the combinations of which we have seen only one, in the
Schoenflies notation these are all of the form Cnv, a
rotation axis Cn with a vertical mirror plane passing
through it.
So this one, for example-- we've yet
to look at the others--
would be C2v.
So we'll have to draw the first, and then I'll give you
the complete set.
So this is a rotation axis with a mirror plane passing
vertically through it.
Let's do a few more.
And I think having done a couple in great detail, we'll
see what the others will look like quite readily.
Let's take a fourfold axis and add to the rotation operations
A pi/2, a first mirror plane that I'll label sigma 1.
I can permute the order of operations and say that sigma
1 followed by the rotation operation A pi/2 is going to
be equal to a second reflection operation that is
equal to one half of pi/2 away from the first.
So I'm claiming that if I reflect in the first mirror
plane and then rotate by 90 degrees that should be another
mirror plane at 45 degrees to the first.
So let me do exactly what I advertised.
Here's the first, one right handed.
Reflect across to get a second one, which is left handed.
Reflect, then rotate, and that would bring my left-handed one
up to this location here.
So here's 3, and it's left handed.
And lo and behold, just as advertised, I get from the
first one to the third one by reflecting across a mirror
plane, which is one half of pi/2.
If let these symmetry elements operate on each other, what
I'll end up with is a set of mirror planes
at 45 degree intervals.
And what I'll have is a pair of objects hung in the same
fashion at every mirror plane.
Is that too fast?
You want to go through that a little slower?
Oh, go ahead say, go through it more slowly.
Everybody's afraid to say, yeah, yeah, and been seem like
a class dummy.
Do you want me to do it more slowly?
I see people still writing, so I think what you'd rather have
is me be quiet for a bit while you catch up to where I am.
In the Schoenflies notation, this lovely thing here is C4.
It's a rotation axis of rank 4 with a vertical mirror plane
added to it.
And the international notation, now the
Hermann-Mauguin notation, is just a running list of the
individual symmetry elements that are present.
And now I really are going to have a mouthful.
This is a fourfold axis, so the symbol for that is 4.
This is a mirror plane.
That's a mirror plane, and that's a mirror plane.
So it looks as though I should call this 4mmmmmmmm, which
comes 4mmmmmmmm.
Well, that is a typical reaction.
It's a lovely symmetry.
But that is a mouthful.
So it isn't really necessary to give all these m's So the
international notation is a running list of the
independent symmetry elements, and that's the new wrinkle
that I'm introducing.
It's a running list of the independent symmetry elements.
And all these mirror planes are just different sigma that
exist as operations in the group.
But how many different kinds of mirror planes are they?
Well, there are two different kinds of mirror planes, both
in terms of the way in which they function in the pattern.
The two motifs related by reflection hang close to this
mirror plane, but they're widely separated for this
mirror plane.
So the motifs do different things relative to those two
kinds of mirror planes.
Another way of asking what's independent is if I start with
this 1 mirror plane and repeat it by 90-degree rotations,
I'll get these 4 mirror planes 90 degrees away.
So they are not independent in that these mirror planes are
all related by the rotational symmetry that's present.
You don't get this mirror plane in any fashion other
than saying, if I combine the rotation operation with this
reflection operation sigma, the net result is this
reflection plane.
So there are two mirror planes that are distinct in this
arrangement of symmetry elements, distinct in the
sense that they function in different ways in the pattern;
distinct in the sense that no other operation that is
present will throw these two operations into one another.
Another example, and this is in fact symmetry 4mm, is the
square tile.
If you look at the mirror planes there, they are 45
degrees apart.
But one of those mirror planes comes out normal to the edge
of the tile.
The other mirror plane comes out of the vertex of the tile.
So they are different in the way they function in the space
which has this symmetry.
So, mercifully, we don't have to call this 4mmmmmmmm.
We drop the last 6 m's, and this one is called simply for
4mm, 2 kinds of mirror planes with a fourfold axis.
If you're familiar with the way in which these were
derived, the symbol tells you exactly what you've got.
So it's a very useful notation for these symmetries.
Let me pause here, give you a chance to catch up.
Yes, sir?
AUDIENCE: So wait, all you did here is take two separate
mirror planes at an angle of 45 degrees between, and you
ended up with a fourfold symmetry.
PROFESSOR: Exactly.
AUDIENCE: But you didn't put on the fourfold symmetry, it
was just what fell out of it?
PROFESSOR: No.
I Got this mirror plane to begin with by combining this
reflection operation sigma with the operation A pi/2, and
that's where this operation came from.
But a general theorem that says if I have a rotation
operation A alpha and combine it with a reflection operation
sigma, the combined effect is a reflection operation sigma
2, which is alpha/2 away from the first.
So, again, showing you for this now rather messy diagram,
if I start with the reflection operation that takes 1 of them
throws of the 2 and then rotates 2 up to location 3,
the way I get from 1 to 3 in one shot is to reflect in this
locus sigma 2.
Yes, sir?
AUDIENCE: You could have started with any one of those
two and got the third.
PROFESSOR: Absolutely.
Absolutely.
The operations that are present operate
everything in the space.
So when I say that there's a mirror plane here that relates
this one to this one, it also relates this one to this one,
this one to this one, this one to this one, that mirror
planes operates on everything.
You can't say that a symmetry operation grab this little
packet of space and moves it to another packet
removed from it.
To say it's a symmetry of a pattern or of a crystal, it
has to leave everything invariant.
OK.
But, again, in terms of the language of groups theory, if
you combine this pair of operations, then when you
combine everything that you get pairwise, you will get in
this case a total of, how many operations?
How many operations are in this pattern?
I said a moment ago the rank of the group is the number of
operations that are present.
It's the number of objects that are in the group because
each operation in a group tells you how to get from
anyone to all of the other.
So, 2, 4, 6, 8, this is a group of rank 8, and there are
8 operation that are present.
And I can rattle them off quickly.
Four operations, identity, 90, 180, 270 for
the fourfold axis.
That's 4.
This mirror plane.
The mirror plane.
And this mirror plane.
That's 8.
Four for rotation.
Four for reflection.
Other questions?
All right.
Let me then wrap up this quickly and get
onto something new.
Did I hear the hiccup or a question?
No.
It was a hiccup.
Not a yawn I hope.
Let me do the highest symmetry of all in two dimensions, and
this is if I take a sixfold axis and combine it with a
mirror plane.
So here's the first operation sigma.
This one is something of a bear to draw.
This is sigma 1.
Sigma 1 followed by the operation A 2 pi/6 should be
equal to a reflection that is an angle one half of 2 pi/6 30
degrees away from the first.
So this will be sigma 2.
So if I reflect from 1 to 2 and then rotate by 60 degrees,
I'm going to get one that sets up here.
And the way I get from number 1 to number 3 in one shot is
by a reflection sigma 2.
Now if I draw in all of those are mirror planes when they
are repeated by the sixfold axis, I shouldn't have given
this my 6mmmmmm treatment because there
are 12 mirror planes.
Two objects hanging on this one in one fashion spaced in a
different way that are belly to belly.
They're back to back here, so it does something different in
the pattern.
A pair hanging here.
A pair hanging in here.
A pair hanging and here.
A pair hanging in here.
And finally, a pair hanging in here.
So there are a total 2, 4, 6, 8, 10, 12
motifs in this pattern.
Pairs hanging on mirror planes that are 30 degrees apart and
hanging disposed and pointing in different ways on the
adjacent mirror planes.
So this is one that we would call in international notation
6mmm and in Schoenflies notation C6v.
So we're making extraordinary progress here.
The one that I did for you initially was C2v, and that's
2mn in international notation.
The only one that I left out to this point is a threefold
axis compared with a vertical mirror planes.
And let's start with the operation A 2 pi/3, combined
with that a first reflection operation sigma 1 that passes
through it.
And for reference, I'll drawn in some lines that are
separated by intervals of 60 degrees.
So this reflection from 1 to 2 followed by a rotation 120
degrees should give me this one as number 3.
The first is right handed.
The second is left handed, and the third stays left handed.
And lo and behold, the way I get from 1 to 3 directly is by
a reflection sigma 2 across a mirror line that is--
I'm sorry.
This is number 1 down here-- across a mirror line that is
30 degrees away from the first.
If I would complete the pattern, reflect this one
across to here, take this pair and reflect it or rotate it up
here, and I have six objects.
So this is a group of rank 6.
And it has characteristics that are quite analogous to
those we did for the other rotation axes in all respect
except one.
The international symbol for this combination is C3v.
We've got a threefold axis.
We've got the mirror plane that we added.
And then we've got a mirror plane that is 30 degrees away
from the first--
I'm sorry--
60 degrees away from the first,
one half of 120 degrees.
I claim that this is not the proper symbol because the
mirror planes that are listed should be the
symmetry-independent, distinct sort of mirror planes.
And is that the case with this one?
The answer is no.
It isn't.
Here is one mirror plane.
It's got a pair of motifs hanging on it.
This mirror plane here is a mirror plane that has a pair
of motifs hanging it.
Same is true of this mirror plane.
So all six of these mirror planes are doing the same
thing in the pattern.
Each one has a pair of a motifs on either side of it in
the same fashion.
And I can get one mirror plane and the motifs hanging on it
by a rotation of 120 degrees.
So there's only one kind of mirror planes, so we don't
need that m.
So all of the other rotational symmetries, m, a onefold axis
with a mirror plane; 2mm, 4mm, 6mm have two
kinds of mirror planes.
C3v has only one kind of mirror planes.
And another way of showing that is if I look at a
trigonal prism.
It's got a mirror plane coming out of a corner and out of the
opposite face.
Mirror plane coming out of the corner and out
the opposite face.
Mirror plane coming out of a corner and out
the opposite face.
And that's exactly the rearrangement of symmetry
elements here.
So each mirror planes when drawn in relative to a
trigonal prism, which is a body that has this symmetry,
each mirror planes does exactly the same thing, as
opposed to the square tile or square prism that has one kind
of mirror plane that comes out of faces and one kind of
mirror plane that comes out of corners.
So there are two different types of mirror planes.
Let's add them all up, summarize, and we've got the
cast of characters that we can be use in deriving
two-dimensional symmetries.
We've got C1, which is a onefold axis; C2, twofold
axis; C3, that's a threefold axis; C4, and that's a
fourfold axis; C6, that's a sixfold axis.
Then we've got the additions that involve adding a mirror
plane to a rotational axis.
So this is simply m and CS in Schoenflies notation.
Then we added a mirror plane to a twofold axis.
We've got C2 with a vertical mirror plane, 2mm; C3v; and
not 3mm, but only 3m; C4v, and that's 4mm; and C6v, and
that's 6mm.
Add them all up, and there are 10 unique possibilities, no
more, no less.
Yes, sir?
AUDIENCE: I'm missing a point.
Why is there the mirror plane sigma 2 rather than rotating
sigma 1 is not different?
How do you?
PROFESSOR: The same thing is going on, on this mirror plane
as on this mirror planes, on this mirror plane.
Or if you like to see it in terms of a geometric solid,
the mirror planes here in a trigonal prism all of them do
the same thing.
They come out of one corner and out of the opposite edge.
Each of these has one pair of objects hanging on it.
And that is true of all three of them.
And if you like, this end of the mirror plane here is
nothing more than one that's related to the first one by
one of the rotations of the threefold axis extended back
in the opposite direction.
So the mirror plane doesn't just work up here.
It works down here as well.
It works all through the space.
So any way you want to look at it and whatever terms work for
you, each of these mirror planes is the same, but they
have a different type of end to them.
There's something different hanging at one end than
at the other end.
Think of it in terms of actual physical object.
AUDIENCE: [? What about the ?] case of the fourfold
[? there? ?]
PROFESSOR: In the fourfold, one comes out of the face.
One comes out at the edge.
The motifs are oriented and spaced at a different distance
from each of those neighboring mirror planes.
So these two are different.
The fourfold axis never rotates this
one into this one.
These two are not different because the threefold axis
rotates, if you do it twice, this end into this end of what
is one in the same mirror plane as what's down on here.
So what's confusing you I think is that there's a
polarity to the mirror planes.
Both ends of the mirror planes do not have the same
disposition of motifs on them, but there's nothing that says
that this has to be the case.
The motifs could just be at one end.
And that, if you think about it a little bit if it help,
figure out--
not on company time though, but on your own time-- what a
fivefold axis does.
And a fivefold axis is a non crystallographic symmetry.
But it turns out that C5v, which is a 5 with an m, has
only one kind of mirror plane in it as well and for exactly
the same reason.
A regular figure that has the symmetry is a pentagon.
A mirror plane comes out a corner and out of
the opposite face.
And that's true for all of the mirror planes that are in
there and separated by one half of 2 pi/5.
AUDIENCE: That's just whether or not the [INAUDIBLE] axis is
[INAUDIBLE]?
PROFESSOR: Yeah.
That's what I said.
Yeah.
So before I go through all of the rotation axes with
vertical mirror planes, the international notation would
be 2mm, 3m, 4mm, 5m, and 6mm, 7m.
For the odd symmetries, there's only one kind of
mirror plane in a pattern, in the tile, or
whatever you want to have.
So there are 10 distinct possibilities, and these are
called the Point Groups.
Why?
Because they are clusters of symmetry elements about at
least one point that's fixed and embedded
immovably in space.
They're called groups because, as we've seen for one simple
example, the collection of operations follows the
postulates for the set of elements which we have defined
as a group.
More specifically, we could call these the 10
two-dimensional crystallographic graphic Point
Groups, which is more of a mouthful.
But it emphasizes the fact that there are lots of
two-dimensional Point Groups, but the two-dimensional
crystallographic Point Groups are those that involve
rotational symmetry that are compatible with a lattice.
Hence, they are crystallographic.
But the number of Point Groups is actually infinite if you
include the ones that are not compatible with translation.
All right.
I think that's probably a good place to quit.
And guess what?
My internal clock has told me that this is the
time for our break.
This is one crossroads in our development.
And what we'll do next for the faint hearted who may not want
to have more of this stuff for one day than what we've just
done, we're now going to do the final penultimate
combination, and say we have also shown that there are 5
two-dimensional lattices.
And the final step will be to say if we have a pattern that
has symmetry and is based on translation, we can obtain
these by taking each of the 10 crystallographic Point Groups
in turn and dropping them into each of the five lattices that
can accommodate them.
We would not try, for example, to take a Point Group like 4mm
and try to drop it into the hexagonal lattice.
It's not going to fit.
But there will be two, maybe three ways in which we can add
a given Point Group to one of these lattice types.
And what we've done when we finished is we have
exhaustively derived the symmetries' translational
lattices and symmetry operations of reflection and
rotation that are possible for two-dimensional crystal.
You might say, well two-dimensional crystal, I've
heard that people who do thin film work to make monolayers
and really make a two-dimensional crystal.
That's not something you could say only a few years ago.
But why do we worry about two-dimensional symmetries if
we're not to be wallpaper designers?
Well, actually, I'll give you one example.
One of the difficulties you had in conveying the nature of
a crystal structure is taking something that's fairly
complicated in three dimensions and getting it onto
a two-dimensional sheet of paper.
And what you invariably do is you project the contents of
the unit cell down along one of the cell edges.
And what you have then is a two-dimensional crystal
structure with a lattice and symmetry.
So you'll see two-dimensional representations of
translationally periodic arrangements of atoms quite
frequently when you look at projections of actual crystal
structures, which is the only reasonable way to convey the
information of any structure once you get on beyond the
baby stuff of sodium chlorides, zinc, sulfide and
body-centered iron.
OK.
I'm going to pause, suck in air.
I am happy to turn you loose for 10 minutes and will meet
here again, let's say, at 10 after 3:00.