PROFESSOR: Any questions about where we left off-- up to
where we left off?
OK, what I'll do then is give you a few more examples of the
combinations in 22 to show which ones we have to retain
as frameworks for crystallographic point groups
and which ones exist as groups but which involve rotational
symmetries that are not permitted to a lattice.
So we've seen a combination of three orthogonal twofold axes
and then projection that would look like this.
And the international symbol for that point group is just a
running list of the different axes that are present, 222.
The next group in the sequence would be 3 2 2, where we took
a 120 degree rotation.
We combine that with a twofold axis perpendicular to it and
the new twofold axis comes out and reminds you again of
things that are quite clear but which are easy to forget--
that this angle here is 1/2 of 2 pi over 3.
Don't forget that 1/2.
So the neighboring twofold axis is 60 degrees away and
then if we allow these axes to operate on each other, the net
symmetry consistent set of axes looks like this.
But let us look at a solid that has this symmetry.
And such a solid would be a trigonal--
triangular prism.
And again we can use the corners of this polyhedron as
the reference locations of our motifs.
So let us put a first motif here, number 1.
Let's rotate it by 120 degrees to get a second motif here.
And then let us rotate that one down by a twofold axis
coming out of one of the edges.
That will give us number 3 that is down here or of the
same chirality.
And how do we get from 1 to 3 directly in one shot?
And the answer is about a twofold axis that comes out of
the face of the prism.
So again the prism that we've used as our reference is a
prism that looks like this.
We've got one twofold axis coming out of the face.
The next twofold axis is the one that is equivalent to a
threefold rotation followed by a twofold rotation.
Now here we hit a situation in deciding on the name for the
combination that is analogous to what we found in
two-dimensional plane groups for 3mm.
We saw that only one mirror plane was distinct.
We look at this arrangement of axes--
this was axis a, alpha, this was axis b, beta.
But if I repeat one of these twofold axes by 120 degree
rotations, this one is the opposite end of this one, this
one is the opposite end of this one, and this one is the
opposite end of this one.
So there are only three kinds of-- there are three, twofold
axes and they are related by the 120 degree rotation.
Another way of saying that is that if we look at a trigonal
prism, each of the twofold axes comes out of an edge and
out of the opposite face.
So there they all do the same thing in this regular prism--
twofold axes extend between corners of the opposite face.
So there are only--
there's only one kind of twofold axis present in terms
of being symmetry independent.
So just as we called 3mm, 3m, and we call this one 3 2
because all the twofold axes are symmetry equivalent.
The next one that is crystallographic would be a
combination of a 90 degree rotation with a pair of
twofold axes that are normal to it and separated by 1/2 of
pi over 2, the [? throw ?] of a fourfold axis.
And the symbol for this one would be a fourfold and now
there are two different kinds of twofold axes.
And if we look at a regular square prism, we can again
show that what we've been demonstrating for these other
prisms is true.
One twofold axis would come out of the face, the other
twofold axis would come out of the midpoint of an edge, and
the fourfold axis would come up here.
And a 90 degree rotation, from here to here, followed by a
twofold rotation about this axis, gives us as a net
effect a 1, 2, 3.
And that rotation from 1 to 3 about this twofold axis gives
us the combined mappings.
Notice that the order in which we do them is unimportant.
For example we could do the same thing but do two 180
degree rotations about the twofold axes.
Let's say we start by doing a rotation about
this twofold axis.
From here down to here.
And then we do a twofold axis about the--
twofold axis that comes out the face and that would take
number 2, and rotate it up to number 3.
And the way we get from 1 to 3 directly is by a rotation of C
pi over 2 about the square face of the prism.
So do them in any order you like--
two 180 degree rotations, or a 90 degree rotation, one of the
two full rotations with 90 is the other twofold--
the other type of twofold axis.
The 90 degree rotation plus the second type of the twofold
axis is the same as the first.
So the result can be permuted and turns out to be the same
combination.
The next one that we would hit if we proceed systematically
is non-crystallographic.
And this would be a fivefold axis.
And I'm foolhardy to even start trying to sketch this in
three dimensions.
Nevertheless, nothing ventured, nothing gained.
Start with a twofold axis out of one of the edges and rotate
from 1 down to 2.
Follow that by a twofold rotation about the twofold
axis that comes out of the face.
And that gives us one number 3 up here, and lo and behold,
the way you get from 1 to 3 directly is by a rotation
through 1/5 of 2 pi.
So this would be the non-crystallographic point
group, 5- well it's not going to be 5 2 2.
And that has a whole bunch of twofold axes separated
by 1/10 of 2 pi.
And just as in 3 2, twofold axes here all come out of the
face and out of the opposite edge.
So this would be called 522.
A nice symmetry but not crystallographic, so we can
promptly forget about.
So what comes out of this is a family of groups that are all
of the form n 2 2.
The crystallographic ones are 222, 32, 422 which we've
looked at in detail, and one that I won't draw because
there's so much symmetry it get's messy.
This is 622.
There is a Schoenflies notation.
You remember the language that we encountered for our two
dimensional point groups.
We used m for mirror in the international notation.
We used C standing for cyclic group, subscript s standing
for Spiegel in the Schoenflies notation.
The Schoenflies notation for all of this family of symmetry
is Dn, and the D stands for dihedral.
And the reason for that name is that the difference between
all of these groups, besides the n-fold axis, is this angle
between adjacent twofolds and this is a dihedral angle.
I have a set of planes passing through a common axis; the
angle between those planes is termed a dihedral angle.
So this is called D for dihedral and then a subscript
that gives the rank of the axis.
So Dn generically.
This is D2, this is D3, this is D4, and this is D6.
Comments or debate?
Yes, sir.
AUDIENCE: [INAUDIBLE]
PROFESSOR: [INAUDIBLE]
--out of this face.
So I got from here down to the diametrically opposed axis.
And I rotate it about the adjacent one which comes out
of an edge and--
what did I do here.
Here I did A pi over 2 from 1 to 2, and then I did B pi,
where this is B pi, and they turned out to be C pi, which
is this one here.
So going from here to here down to number 3 is the same
as going from 1 to 3 in one shot about a twofold axis
normal to the face.
And actually I'm courageous to try to do this in three
dimensions.
We could do it in projection and then things are used--
this for a point that's up, and use this for a point
that's down.
And then what we've done is to go from 1 that's up, to 2
that's up, and then we rotate it about this twofold axis.
That was 3, that's down.
So when we get into complicated symmetries where
you just can't do a proper job drawing them in an
orthographic drawing, we'll do them in projection and use a
solid dot for something that's up and an open circle for
something that's down.
Is there any other way we can combine things?
Well what you would have to do is use these three relations,
and plug and chug your way through all of the
combinations which were enumerated in the handout.
And I'll save ourselves a lot of work by saying that there
are only two more combinations.
And these are combinations of axes at angles that have
relevance to directions in a cube.
One of them is a twofold axis with a threefold axis with a
threefold rotation.
Again remember these are not
equations in symmetry elements.
This is really A pi, combined with B 2 pi over 3, combined
with C 2 pi over 3.
And the angles that fall out of this are all crazy things
like 109 point something degrees and they make no sense
whatsoever.
They're not nice things like some multiples of 2 pi, 90
degrees or 120 degrees.
And they make no sense at all until you refer them to
directions in a cube.
And this one, 2 3 3, consists of a twofold axis coming out
of the face of the cube, a 120 degree rotation, so this is B
2 pi over 3, this is A pi.
And the other one, C2 pi over 3, corresponds to a threefold
axis coming out of another body diagonal.
I don't know if I want to be gutsy enough to try to
illustrate that that really works.
But one thing that we should do is to let these axes go to
work on one another, and see what comes out.
First thing we can say is that this threefold axis--
if we extend it, it comes out of the bottom
diagonal of the cube.
This one, if we extend it, will come out of this diagonal
of the cube.
So there's a threefold here, and a threefold here.
The twofold axis gives us a threefold axis that will come
down this way.
So there's a threefold axis here and a threefold axis
coming out here.
And then the twofold axis will rotate this threefold axis
over to the remaining pair of corners.
So we have created this by looking at a twofold rotation,
combined with the threefold axis, combined with the
rotation of another threefold axis.
But in point of fact, if you let the twofold axis operate
on the threefold axis coming out of one body diagonal, you
get threefold axes automatically
out of all body diagonals.
So this is given the international symbol 2 3,
because if you start with one twofold axis out of a face
normal and one threefold axis out of--
along a body diagonal, the twofold axis, acting on that
threefold axis gives you one along every body diagonal.
And the threefold axis--
let me point out that a cube standing up on its body
diagonal, with a threefold axis coming out here and these
faces sloping down into the blackboard.
If I have a twofold axis coming out of one face, the
threefold axis puts a twofold axis on this face and rotates
again 120 degrees and puts a twofold axis on this face.
So the threefold axis relates all twofold axes coming out
normal to the cubed face.
And they were really just two independent axes in this
combination.
So 2 3 is the international symbol-- one kind of twofold
axis, one kind of threefold axis--
inclined at these crazy angles that are in a cube.
The Schoenflies notation for this is T, and that stands for
tetrahedron.
And let me try to convince you that if I look at a
tetrahedron, that that is the arrangement of pure rotation
axes in a tetrahedron.
And the way to show a tetrahedron is to
inscribe it in a cube.
So if I connect these two faces together and these two
faces together, that will define for me a solid that has
four triangular faces.
Easier to recognize it when we put it up on one face.
So this is a tetrahedron.
Schoenflies symbol is T. And I love to get to this part of
the semester and be at this point at the end of the hour,
because if I draw a stereographic projection of
the twofold axes, in this symmetry, I can take a couple
of more twofold axes and add it to this combination which
destroys the group.
But lets me wish everybody a happy Halloween and exit to a
stunned silence at the end of the hour.
So this is a nice point group for October.
There is one more and that is the highest symmetry of all.
And this is a combination of a rotation--
A pi over 2, a 90 degree rotation, with a rotation B 2
pi over 3, rotation through one third of the circle, and
the rotation C pi.
And the directions between the axes that come out there don't
come out some multiples of 2 pi.
Again they are crazy angles that make no sense at all
unless you refer them to directions
that occur in a cube.
The fourfold axis is in the direction that corresponds to
the normal to a face.
The twofold axis corresponds to a direction out
of one of the edges.
And the threefold axis corresponds to a direction
that is a body diagonal.
So again this is a mess.
If we let those axes work on one another however, we will
produce, more readily appreciated in a stereographic
projection, fourfold axes along the directions that
correspond to face normal.
So the cube, threefold axes coming out of the body
diagonals and twofold axes in between all of the fourfold
axes in directions that correspond to the lines from
the center of the cube out through the edges.
So this is a group which would be called 4 3 2.
There's one kind of fourfold axis related to all the others
by the other rotation axes that are present.
One kind of threefold axis that is related to all of the
other threefold axes along the body diagonal by other
rotation axes that are present.
And twofold axes, one kind, all coming out of the edges.
So this is the group that, in international tables, is
called 4 3 2.
And the Schoenflies notation, this is called O. And that is
the general reaction when one sees this lovely combination.
You go ooh and O is what it's called.
But the O doesn't stand for a gasp, it stands for an
octahedral.
This is the symmetry of an octahedron--
rotational symmetry of an octahedron.
So that's it.
That is the bestiary of ways in which you can combine
crystallographic rotation axes in space about a fixed point
of intersection.
And there are eleven of them.
There are the axes by themselves--
1, 2, 3, 4, and 6.
There are the dihedral groups, 222, 32, 422, and 622..
And then the two cubic groups, T and O. In the international
notation, I'm mixing metaphors.
These are 23 and 432.
I call to your attention the insidious similarity of the
two combinations of axes, 32 and 23.
When the 3 comes first, this is a group of the form n22.
When the 2 comes first, that is the tetrahedral group.
So if you count them all up, there are 4, 2 is 6, and 5.
There are eleven axial combinations.
Quite a few more than the situation in two dimensions
where we just had single rotation axes 1, 2, 3, 4, 6.
Now we have those as in two dimensions but the dihedral
groups and the two cubic arrangements of axes as well.
So we don't have to stretch our vocabulary too much more
to be all inclusive here.
OK, comments?
Takes your breath away, doesn't it?
All right, let me indicate the next step in outline.
What we will next do is introduce our remaining two
symmetry operations into the picture.
We have the eleven axial combinations.
As I said, we can regard these as a framework that we can
decorate with mirror planes and or the inversion center,
which has not appeared until this point because inversion
is inherently a three dimensional transformation.
So what we're going to do is to take these axial
combinations and add another symmetry
operation to the group.
And this as--
we used the term earlier, this is an extender.
We have something that constitutes a group by itself
then we muck things up by adding another operation.
Remember in all of this we are combining operations, not
symmetry elements.
So we'll take an axial combination and add the
reflection sigma, a reflection operation sigma.
Or take a rotation and combine it with the operation of
inversion as an extender.
So let's itemize the sorts of extenders we should consider.
And the ground rules are that the extender should leave the
arrangement of rotation axes invariant.
Because if it doesn't, we are going to create a rotation
operation that does not conform to the constraints
that we used in Euler's construction.
So for example, if we take 222, which contains the
operations A pi, B pi, C pi and identity,
that's the group 222.
If we would add to the arrangement 222 a mirror plane
that snaked through some arbitrary fashion like this,
that mirror plane is going to reproduce the twofold axis
over to here.
And that is either going to not constitute a group because
this twofold, this twofold, and this twofold don't conform
to Euler's construction.
Or alternatively, if we put it in carefully at 45 degrees
with this twofold axis, we're going to get twofold axes that
are 45 degrees apart and that's going to change this
into a fourfold axis.
So if the addition of a mirror plane does not leave the
arrangement of rotation axes--
rotation operations--
invariant, we're either going to get something that's
impossible and does not constitute a group.
Or else we're going to get a combination of rotation
operations of higher symmetry which we've already found
because we went through that process of combination using
Euler's construction in an exhaustive fashion.
So the rule then is that if we add the reflection operation
sigma, then the arrangement of rotation operations must be
left invariant.
OK let's look at the single axes.
If there's an n-fold axis, the ways we can add a reflection
operation to that axis is to pass the reflection operation
through the axis.
And this is going to look very much like the two dimensional
point groups of the form nmm except that rather than having
a mirror line, imagine the whole works as extending
upwards along the rotation axis and space.
So this extender is called a vertical mirror plane.
The other way we could add a mirror plane to an n-fold
rotation axis is to put the mirror plane in an orientation
that's perpendicular to the rotation axis.
That didn't exist in two dimensions because that plane
that's perpendicular to the axis is the
plane of our paper.
And unless we wanted to have a two-sided group that was on
both the top of the paper and the bottom of the paper, and
we make up the rules since it's our ball game.
And that could be a group and these would be the two-sided
plane groups, a plane point groups, but we
didn't do that here.
Adding the mirror plane, the reflection operation sigma, in
a fashion that is normal to the rotation operation, A 2 pi
over n, is another distinct combination.
And this is called, very descriptively a horizontal
mirror plane.
That looks like about all you can do except for the cases
where we have more than one kind of rotation axis present.
So let me use 422 as an example.
We could add a horizontal mirror plane perpendicular to
the principal axis of symmetry, and that would be
the horizontal sigma.
We could add a vertical mirror plane.
And now there are two ways we can do it.
We could put the mirror plane, the operation sigma, through
the fourfold axis and in a fashion that was perpendicular
to the twofold axis.
And we will retain the term of vertical
sigma for that addition.
But the other thing that we could do would be to put the
reflection operation interleaved between the
twofold axes.
That's going to take this one and flip it into this one,
this one flip it into this one, flip these back and
forth, and that doesn't create any new reflection.
And this is referred to as a diagonal reflection plane.
Diagonal to what?
Diagonally interleaved between the twofold axes.
And these are distinct additions and they will lead
to different groups.
In as far as addition of reflection operations is
concerned, that's about all we can do that's distinct.
And notice that this is for the groups Dn, and
tetrahedral, and octahedral only.
The distinction here is not defined for
just a single axis.
So there are three possible extenders here-- a vertical
mirror plane, a horizontal mirror plane, a diagonal
mirror plane, added to each of the eleven arrangements of
rotation axes.
And then the final extender that we could
add is to add inversion.
And the symbol for the inversion operation is 1 bar.
And obviously if one point in space is going to be left
invariant, you either add this on a single axis, and that is
what you'd have to do for the groups Cn, or at the point of
intersection.
And that would be the case if more than one axis, and that's
the groups of the form n22, T,
and O. That's it.
That's the job.
So we should consider each of these possible additions of an
extender systematically.
I don't propose to do every single one independently.
If we do a couple, you'll get the general idea.
And I think because I have an honest face, and you've come
to trust me, I can just describe the remaining results
to you and we won't grind through every single one.
Now the enormity of what I've proposed becomes apparent when
I say that we now are going to have need of a number of
different--
what I call combination theorems, that let us complete
the group multiplication table.
And deduce, as a consequence, which symmetry operations must
come into being because of these additions.
So we'll want to know what happens when you add a
vertical sigma to a rotation operation, A 2 pi over n.
We'll want to know what happens when you add a
horizontal sigma to a rotation operation, A 2 pi over n.
And we're going to want to know what happens when you add
a diagonal mirror plane, this really is a special case of
the vertical mirror plane.
And we'll want to know what happens when you add an
inversion center to a rotation axis.
So let me do a few of these and then next time we can
start off and start driving the three dimensional
symmetries.
So let's just repeat the ones that we've already done.
We said that if we have a rotation operation A alpha,
and we put a reflection plane through it, that A alpha
followed by a reflection plane passing through it-- let me
call this sigma V-- because this is the so-called
vertical, up orientation.
We've already seen that in two dimensions.
This is a vertical mirror plane, sigma prime, that is
going to be alpha over 2 away from the first.
So that is something that we've already seen in it's
entirety in the two dimensional point groups.
There was m sigma, there was 2mm, and that was C2V, 3m,
that's C3V, 4mm, and that was C6V, and 6mm, and that's C4V,
and that's C6V.
So that is the result that we obtained for two dimensions.
And you can see now the reason for distinguishing the mirror
plane by saying it is a vertical mirror plane because
this is in the three dimensional sense.
It's vertical parallel to and passing through the rotation
axis-- no longer a rotation point but a rotation axis.
So we've got those theorems.
What happens if we take a rotation operation, A pi, and
put a horizontal reflection operation through it
normal to that axis?
So I'll call this sigma h, a horizontal operation.
OK what we have to do is draw it out once and for all.
Here's the first one, let's say it's right-handed.
We'll rotate by A pi to get a second one which stays
right-handed.
And then we'll reflect it down in the horizontal mirror plane
to get a third one which is left-handed.
And now the question is, how is number one related to
number three?
Anybody want to hazard a guess?
We've got to go from a right-handed one to a
left-handed one.
But these two guys are oriented
anti-parallel to one another.
So how do we relate the first one to the third one?
I heard somebody mumble softly enough to remain anonymous.
Inversion.
Right.
So as we go along making these combinations, if we had not
been bright enough to think of the operation of inversion as
a general transformation where the sense of all three
coordinates is changed, we would have stumbled over
headlong right here.
Combine a rotation operation, A pi, with a mirror reflection
that is perpendicular to the axis.
The way you get from one to three in one shot is by
inversion through a point that is at the intersection between
the rotation axis and the mirror plane.
So rotation followed by rotation in a vertical mirror
plane that's perpendicular to the axis is the operation of
inversion at the point of intersection.
So again, if we had not been clever enough to invent it or
tell you about in advance, there it is.
When we start forming the group multiplication table, we
would have had to have defined this operation to describe
this relation.
AUDIENCE: Is that sigma sub h?
PROFESSOR: Ah yeah.
Let me write that, sigma h.
I thought that V didn't look like a V, so I changed it so
it looked like a V but it shouldn't be a V. That's a
horizontal mirror plane.
OK this is a new combination and if we see what operations
are going to be present in the group, we've got the two
operations of the twofold axis, 1 and A pi.
And what we have added is sigma h as an extender.
So 1 and A pi, this little box here is the subgroup that we
know and love as the twofold axis.
And then we'll write sigma h here and now let's fill in the
group multiplication table.
Doing the identity operation twice is identity.
Doing the identity operation followed by A pi is A pi.
Identity followed by sigma h is sigma h.
Identity followed by A pi, sigma h, lets me
fill in those boxes.
Do a rotation twice, that's the identity operation.
Do a rotation of A pi and follow that by a reflection, A
pi followed by reflection.
This is the inversion operation.
And so I should add inversion to my list of operations since
it's come up.
So 1 followed by inversion is inversion.
A pi followed by inversion is the horizontal reflection.
Horizontal reflection followed by A pi
is the same as inversion.
Horizontal reflection followed by horizontal reflection is
the identity operation--
brings me back to where I started from.
And a horizontal reflection followed by inversion is the
same as A pi.
So I'll have another object down here to complete the
three dimensional arrangement.
And this is the group for operations--
A pi, inversion, horizontal reflection, and
the identity operation.
So what do we call this one?
The operation A pi plus a horizontal reflection
operation gives rise to a group that is a twofold axis
of its operations,
perpendicular to a mirror plane.
And this written as a fraction means that the 2 is
perpendicular to the mirror plane rather than being
parallel and in the plane of the mirror plane.
So two codes for writing symbols.
A 2 followed on the same line by the m means that the m is
parallel to 2.
We know that when we write them as a fraction, that will
be our way of designating that this mirror plane is
perpendicular to a twofold axis.
OK it's the witching hour, 4 o'clock exactly.
That's when we ought to quit, so let's stop there.