WEBVTT

00:00:12.320 --> 00:00:16.340
PROFESSOR: My watch says five
after 2:00, so why don't we

00:00:16.340 --> 00:00:18.710
get started?

00:00:18.710 --> 00:00:22.740
This is going to be
a strange lecture.

00:00:22.740 --> 00:00:25.260
I'm expecting some wise guy in
the back to say, all your

00:00:25.260 --> 00:00:27.020
lectures are rather peculiar!

00:00:27.020 --> 00:00:30.990
But I have a phone call from
Europe coming in sometime

00:00:30.990 --> 00:00:33.340
around 2:15, 2:30.

00:00:33.340 --> 00:00:36.950
Somebody's coming to visit us
next week, and we're going to

00:00:36.950 --> 00:00:38.340
set up a seminar.

00:00:38.340 --> 00:00:40.550
So I don't know exactly when
it's going to come in, but I

00:00:40.550 --> 00:00:41.730
have to be in my office.

00:00:41.730 --> 00:00:46.090
So what I will do is
adjourn at quarter

00:00:46.090 --> 00:00:47.970
after, 20 past the hour.

00:00:47.970 --> 00:00:50.730
And then so I don't keep you
here sitting restively

00:00:50.730 --> 00:00:52.580
wondering when I'm going
to come back, we'll

00:00:52.580 --> 00:00:54.460
adjourn until 3 o'clock.

00:00:54.460 --> 00:00:57.580
So you seem to have been taken
longer and longer breaks

00:00:57.580 --> 00:00:58.680
during intermission.

00:00:58.680 --> 00:01:04.010
So I'll give you a full 3/4 of
an hour so you can get it out

00:01:04.010 --> 00:01:04.849
of your systems.

00:01:04.849 --> 00:01:08.680
And then, we'll take shorter
breaks from here on in.

00:01:08.680 --> 00:01:11.910
OK this is going to
be pretty much our

00:01:11.910 --> 00:01:14.310
last lecture on symmetry.

00:01:14.310 --> 00:01:18.640
And we will begin, at least in
half of the next class on

00:01:18.640 --> 00:01:21.700
Tuesday, to begin talking about
physical properties, and

00:01:21.700 --> 00:01:24.190
in particular tensor
properties.

00:01:24.190 --> 00:01:28.650
But I'd like to say a little
bit about the derivation of

00:01:28.650 --> 00:01:34.030
space groups and take a look
at how this information is

00:01:34.030 --> 00:01:37.940
tabulated for you by the kind
folks who prepare the

00:01:37.940 --> 00:01:39.700
international tables.

00:01:39.700 --> 00:01:44.700
And there are really very strict
parallels between what

00:01:44.700 --> 00:01:45.970
we did in two dimensions.

00:01:45.970 --> 00:01:48.970
And that's the reason why we
did it so thoroughly and

00:01:48.970 --> 00:01:52.220
systematically, except that
there are three dimensions.

00:01:52.220 --> 00:01:56.920
And because we are taking 32
point groups and putting them

00:01:56.920 --> 00:02:01.150
into 14 lattices, there are a
lot more combinations to be

00:02:01.150 --> 00:02:01.760
considered.

00:02:01.760 --> 00:02:04.900
And there are a lot more of
them that are unique.

00:02:04.900 --> 00:02:09.080
Also, they're a little bit more
difficult to visualize

00:02:09.080 --> 00:02:09.850
and depict.

00:02:09.850 --> 00:02:13.440
So there are some new
conventions in preparing a

00:02:13.440 --> 00:02:16.720
representation of three
dimensional symmetries on a

00:02:16.720 --> 00:02:19.740
sheet of paper or page of a book
which, of necessity, has

00:02:19.740 --> 00:02:20.970
to be two dimensional.

00:02:20.970 --> 00:02:23.620
So we'll go over some of
those conventions.

00:02:23.620 --> 00:02:26.970
But the main reason for going
a little bit further is that

00:02:26.970 --> 00:02:31.070
there is another surprise
lurking in there for us as

00:02:31.070 --> 00:02:36.250
soon as we begin to combine
translation in three

00:02:36.250 --> 00:02:38.930
dimensions with a symmetry
element.

00:02:38.930 --> 00:02:44.880
But let me go through
in some detail the

00:02:44.880 --> 00:02:46.650
monoclinic space groups.

00:02:52.710 --> 00:02:57.550
So we have at our disposal
to decorate with symmetry

00:02:57.550 --> 00:02:59.305
elements two lattices.

00:03:02.940 --> 00:03:14.190
And they are a primitive
monoclinic lattice with two

00:03:14.190 --> 00:03:18.445
translations, a and b, that
define an oblique net, and a

00:03:18.445 --> 00:03:21.560
third translation, c, at right
angles to that net.

00:03:21.560 --> 00:03:25.990
And then, either two choices
for a double cell--

00:03:30.790 --> 00:03:33.720
and I'll draw these in
projection because I can do

00:03:33.720 --> 00:03:36.040
both of them with one diagram.

00:03:36.040 --> 00:03:44.720
Either this is a and b and the
extra lattice point is in the

00:03:44.720 --> 00:03:47.460
center of the cell halfway
up from the base.

00:03:47.460 --> 00:03:48.775
This would be body centered.

00:03:54.800 --> 00:04:00.640
And that's represented by
I for innenzentriert.

00:04:00.640 --> 00:04:03.660
In German, that's German
for body centered.

00:04:03.660 --> 00:04:09.350
Or the alternative would be to
pick a cell like this, in

00:04:09.350 --> 00:04:12.970
which case the extra lattice
point gets caught in the

00:04:12.970 --> 00:04:14.800
center of one of the faces.

00:04:14.800 --> 00:04:19.120
So this depiction here would
be a side centered lattice.

00:04:24.420 --> 00:04:27.800
And it could have the extra
lattice point in the middle of

00:04:27.800 --> 00:04:30.370
the face out of which b comes.

00:04:30.370 --> 00:04:34.450
And in that case, the symbol
for the lattice is B. Or

00:04:34.450 --> 00:04:37.590
alternatively, without changing
any of the nature of

00:04:37.590 --> 00:04:40.100
the specialness of the lattice,
the extra lattice

00:04:40.100 --> 00:04:41.580
point could be in the
middle of the face

00:04:41.580 --> 00:04:43.250
out of which a comes.

00:04:43.250 --> 00:04:46.770
And that would be
an A lattice.

00:04:46.770 --> 00:04:51.960
With the first, setting with the
c axis unique, there is no

00:04:51.960 --> 00:04:56.310
c lattice because the oblique
base of the cell would not

00:04:56.310 --> 00:04:59.890
give us anything new if
it were centered.

00:04:59.890 --> 00:05:03.530
All right, so there are
those two lattices.

00:05:03.530 --> 00:05:05.815
And then, there are three
point groups.

00:05:09.460 --> 00:05:15.180
And they were 2, m, and then a
combination of a twofold axis

00:05:15.180 --> 00:05:18.200
perpendicular to
a mirror plane.

00:05:18.200 --> 00:05:22.060
So it looks as though in
principle there are going to

00:05:22.060 --> 00:05:24.286
be 6 combinations.

00:05:24.286 --> 00:05:26.600
In point of fact, there are
a lot more because of the

00:05:26.600 --> 00:05:28.790
surprise that we have
in store for us.

00:05:31.680 --> 00:05:37.260
Before proceeding further,
though, let me ask

00:05:37.260 --> 00:05:38.930
rhetorically the question.

00:05:38.930 --> 00:05:45.340
Which lattice type would you
pick as the standard one?

00:05:45.340 --> 00:05:49.550
And here, there's a major
decision that has to be made.

00:05:54.760 --> 00:06:04.750
Should the labels on an
axes give a unique

00:06:04.750 --> 00:06:06.000
symbol for the lattice?

00:06:13.210 --> 00:06:15.860
In other words, if you have
the double cell for a

00:06:15.860 --> 00:06:20.340
monoclinic crystal, should you
define the cell to make it

00:06:20.340 --> 00:06:23.160
body centered always, or to
make it A centered or B

00:06:23.160 --> 00:06:24.410
centered always?

00:06:26.950 --> 00:06:33.350
Or should the labels be defined
by the relative

00:06:33.350 --> 00:06:35.030
lengths of the axes?

00:06:54.490 --> 00:06:56.460
So those are the two
possibilities.

00:06:56.460 --> 00:06:59.710
And there's no right
or wrong way.

00:06:59.710 --> 00:07:02.970
You pays your money and
you makes your choice.

00:07:02.970 --> 00:07:09.230
And on pragmatic grounds,
this is the

00:07:09.230 --> 00:07:12.000
convention that's followed.

00:07:12.000 --> 00:07:15.790
And it's purely a pragmatic
decision because lattice

00:07:15.790 --> 00:07:19.530
constants have been determined
for literally tens of

00:07:19.530 --> 00:07:21.340
thousands of materials.

00:07:21.340 --> 00:07:23.530
Sometimes, the lattice constants
are known and the

00:07:23.530 --> 00:07:25.160
space group is not.

00:07:25.160 --> 00:07:30.470
But it would be terrible if the
label that you applied to

00:07:30.470 --> 00:07:38.710
the axis could be C attached
to the longest one, the

00:07:38.710 --> 00:07:40.500
shortest one, or the
intermediate one.

00:07:40.500 --> 00:07:44.710
It would be impossible to
calculate any database for

00:07:44.710 --> 00:07:48.910
lattice constants displayed
by particular materials.

00:07:48.910 --> 00:07:53.790
So there's all this-- this is
what the decision was made.

00:07:53.790 --> 00:07:56.110
This is the decision that
was made many years ago.

00:07:56.110 --> 00:08:02.490
And for monoclinic crystals,
the labeling of the axes is

00:08:02.490 --> 00:08:07.050
determined by the c axis
being the unique axis.

00:08:07.050 --> 00:08:10.630
That is, the axis that is along
the twofold axis or

00:08:10.630 --> 00:08:13.210
perpendicular to the
mirror plane.

00:08:13.210 --> 00:08:20.190
And then, the labels b and c are
defined by magnitude of b

00:08:20.190 --> 00:08:22.570
greater than the
magnitude of a.

00:08:22.570 --> 00:08:27.690
So I inherently drew this is
in the proper fashion.

00:08:27.690 --> 00:08:30.110
So the labels are applied
to monoclinic

00:08:30.110 --> 00:08:31.360
crystals in that fashion.

00:08:34.270 --> 00:08:37.799
The price you pay for that is
that the symbol for the

00:08:37.799 --> 00:08:41.039
lattice for a monoclinic crystal
that has the double

00:08:41.039 --> 00:08:46.630
cell could either be A, B, or
I depending on the relative

00:08:46.630 --> 00:08:47.690
lengths of the axes.

00:08:47.690 --> 00:08:50.610
If these were the two shortest,
it would be a body

00:08:50.610 --> 00:08:51.890
centered lattice.

00:08:51.890 --> 00:08:54.910
On the other hand, if this and
this were the shortest pair,

00:08:54.910 --> 00:08:57.030
then it would be a side
centered lattice.

00:08:57.030 --> 00:08:59.730
So the symbol for the lattice
type will bounce around

00:08:59.730 --> 00:09:01.710
depending on the dimensions
of the translations.

00:09:04.210 --> 00:09:08.080
So that's a complication
in the space groups.

00:09:08.080 --> 00:09:10.960
In the two dimensional
plane group, that

00:09:10.960 --> 00:09:12.240
issue never came up.

00:09:12.240 --> 00:09:16.730
We just took B greater than A
for the rectangular and the

00:09:16.730 --> 00:09:18.775
oblique nets.

00:09:22.210 --> 00:09:24.940
OK, so let me do quickly
a couple of the

00:09:24.940 --> 00:09:26.620
monoclinic space groups.

00:09:26.620 --> 00:09:31.720
Let's take a twofold axis and
put it into a primitive

00:09:31.720 --> 00:09:34.210
monoclinic net.

00:09:34.210 --> 00:09:41.870
And this is exactly the same as
our plane group, P2, with

00:09:41.870 --> 00:09:49.720
the twofold axes extended
parallel to the c translation.

00:09:49.720 --> 00:09:56.130
So we would have twofold axes in
all of these orientations.

00:09:56.130 --> 00:10:03.090
And that's simply P2, little p
in two dimensions, with the

00:10:03.090 --> 00:10:05.200
twofold axes extended
normal to the plane

00:10:05.200 --> 00:10:05.880
of the plane group.

00:10:05.880 --> 00:10:07.270
So this is p2.

00:10:07.270 --> 00:10:09.560
This is capital P2.

00:10:09.560 --> 00:10:11.550
And that is way we
distinguish plane

00:10:11.550 --> 00:10:14.710
groups from space groups.

00:10:14.710 --> 00:10:18.510
So the symbol for the lattice
is a capital symbol for the

00:10:18.510 --> 00:10:19.760
space groups.

00:10:23.410 --> 00:10:29.866
So I didn't have to use any
new combinations theorems.

00:10:29.866 --> 00:10:32.140
I simply say there's
a third translation

00:10:32.140 --> 00:10:33.370
perpendicular to the net.

00:10:33.370 --> 00:10:35.470
So everything protrudes
out of the plane group

00:10:35.470 --> 00:10:36.720
into a third dimension.

00:10:42.549 --> 00:10:57.400
Let me now combine a twofold
axis this a and this b and

00:10:57.400 --> 00:11:02.020
this c with a lattice that is
not the primitive lattice but

00:11:02.020 --> 00:11:03.250
one of the double cells.

00:11:03.250 --> 00:11:05.720
And I'll take it, for
convenience, to be the body

00:11:05.720 --> 00:11:08.530
centered flavor because
it's easier to draw.

00:11:11.330 --> 00:11:17.760
And the operation that we've
added to the lattice, if this

00:11:17.760 --> 00:11:23.230
is a combination of a twofold
axis, is the symbol A pi.

00:11:23.230 --> 00:11:26.590
And I know what happens if
I combine A pi with a

00:11:26.590 --> 00:11:28.760
translation that is
perpendicular to

00:11:28.760 --> 00:11:30.420
the rotation axis.

00:11:30.420 --> 00:11:34.740
I get a new rotation axis, B pi,
that's halfway along the

00:11:34.740 --> 00:11:37.030
perpendicular part of
the translation.

00:11:37.030 --> 00:11:38.780
And that's where all
of these additional

00:11:38.780 --> 00:11:40.800
twofold axes came in.

00:11:40.800 --> 00:11:46.150
But if a lattice is a body
centered lattice, there is now

00:11:46.150 --> 00:11:49.750
a translation that goes up to
the centered lattice point.

00:11:52.300 --> 00:11:53.460
So what's that going to be?

00:11:53.460 --> 00:12:01.080
What is T followed by A pi
followed by T, where T is 1/2

00:12:01.080 --> 00:12:04.340
A plus 1/2 of B?

00:12:04.340 --> 00:12:09.180
And that is a part that is
perpendicular to the axis, and

00:12:09.180 --> 00:12:15.820
then a third component, c, which
is parallel to the axis.

00:12:15.820 --> 00:12:18.470
As with all these theorems, what
you do is you draw it out

00:12:18.470 --> 00:12:20.790
once and for all and see what
it turns out to be.

00:12:28.070 --> 00:12:31.110
So let's do that.

00:12:31.110 --> 00:12:34.820
Let's say that this is the
translation that goes up to

00:12:34.820 --> 00:12:38.360
the point 1/2, 1/2, 1/2.

00:12:38.360 --> 00:12:41.560
Here is the parallel part--

00:12:41.560 --> 00:12:42.870
the perpendicular
part, rather.

00:12:42.870 --> 00:12:49.420
This is 1/2 of A
plus 1/2 of B.

00:12:49.420 --> 00:12:53.390
And then, we have a part
that's parallel to the

00:12:53.390 --> 00:12:55.910
rotation operation, A pi.

00:12:55.910 --> 00:12:59.330
And that is 1/2 of c.

00:12:59.330 --> 00:13:01.190
So let's just do it.

00:13:01.190 --> 00:13:03.030
Here's my first object
number one.

00:13:03.030 --> 00:13:04.260
It's right handed.

00:13:04.260 --> 00:13:09.520
I'll rotate 180 degrees
to get a second one

00:13:09.520 --> 00:13:10.720
that's right handed.

00:13:10.720 --> 00:13:15.220
And then, I will translate
it up to here.

00:13:15.220 --> 00:13:17.030
And here sits the third one.

00:13:17.030 --> 00:13:18.280
It's also right handed.

00:13:22.120 --> 00:13:24.580
How do I get from
one to three?

00:13:29.530 --> 00:13:32.180
It's not really clear
how I do that.

00:13:32.180 --> 00:13:34.640
And again, I'll draw it in
projection because this is

00:13:34.640 --> 00:13:37.000
looking pretty messy in
three dimensions.

00:13:37.000 --> 00:13:40.150
So here's my first one.

00:13:40.150 --> 00:13:43.615
I rotate 180 degrees to
get a second one.

00:13:43.615 --> 00:13:45.460
The chirality is all the same.

00:13:45.460 --> 00:13:48.190
And then, I translate
up to this point.

00:13:48.190 --> 00:13:51.610
So if this one is at z and this
one is at z, this one

00:13:51.610 --> 00:13:59.800
here will sit at z plus 1/2.

00:13:59.800 --> 00:14:02.860
And this is the third one.

00:14:02.860 --> 00:14:04.110
Anybody got any idea?

00:14:07.760 --> 00:14:08.800
Can't be reflection.

00:14:08.800 --> 00:14:10.410
They're of the same chirality.

00:14:10.410 --> 00:14:12.640
It can't be translation
because the

00:14:12.640 --> 00:14:16.325
orientation is different.

00:14:16.325 --> 00:14:16.814
Yes, sir?

00:14:16.814 --> 00:14:19.750
AUDIENCE: Then it has to be
some form of rotation.

00:14:19.750 --> 00:14:21.680
PROFESSOR: Right, has to be.

00:14:21.680 --> 00:14:28.040
But how do we get the object up
to a different elevation?

00:14:28.040 --> 00:14:35.220
Well, we've stumbled headlong
over a new type of operation,

00:14:35.220 --> 00:14:37.780
just as we discovered the glide
plane when we started

00:14:37.780 --> 00:14:40.330
putting mirror planes into a
lattice in combination with

00:14:40.330 --> 00:14:41.670
translation.

00:14:41.670 --> 00:14:45.950
The only way I can get from the
first to the third is to

00:14:45.950 --> 00:14:50.990
rotate 180 degrees about exactly
the same point that I

00:14:50.990 --> 00:14:54.050
would have before, namely at
1/2 of the part of the

00:14:54.050 --> 00:14:55.820
translation that is

00:14:55.820 --> 00:14:59.200
perpendicular to the operation.

00:14:59.200 --> 00:15:01.680
But then before putting it
down, I've got to take a

00:15:01.680 --> 00:15:02.910
second step.

00:15:02.910 --> 00:15:08.470
I've got to slide it up by a
translation component that is

00:15:08.470 --> 00:15:11.550
parallel to the axis about
which I've rotated.

00:15:11.550 --> 00:15:16.320
So to complete my theorem,
A alpha followed by a

00:15:16.320 --> 00:15:22.070
translation that has a
perpendicular part plus a

00:15:22.070 --> 00:15:26.210
parallel part is a new type
of operation that involves

00:15:26.210 --> 00:15:30.020
rotating through 180 degrees.

00:15:30.020 --> 00:15:36.450
And this is located at 1/2
of T perpendicular.

00:15:36.450 --> 00:15:39.360
But it has a translation
component which I'll call

00:15:39.360 --> 00:15:40.970
generically tau.

00:15:40.970 --> 00:15:45.540
And tau is equal to the
parallel part of the

00:15:45.540 --> 00:15:48.120
translation.

00:15:48.120 --> 00:15:53.060
So if I separate out this new
sort of operation in all of

00:15:53.060 --> 00:15:59.180
its grandeur, what it does is
it takes a first object,

00:15:59.180 --> 00:16:02.920
rotates us 180 degrees, does
not yet put it down, first

00:16:02.920 --> 00:16:06.160
slides up by the amount tau.

00:16:06.160 --> 00:16:09.200
Doing the operation, again
rotates it 180 degrees,

00:16:09.200 --> 00:16:13.320
doesn't yet put it down,
slides it up by tau.

00:16:13.320 --> 00:16:18.700
Doing it again moves us to
another one up here.

00:16:18.700 --> 00:16:24.950
So what we've generated is a
helical spiral of objects

00:16:24.950 --> 00:16:29.450
about the central axis about
which we're rotating.

00:16:29.450 --> 00:16:32.140
And this is a new two step
operation that's called a

00:16:32.140 --> 00:16:34.310
screw axis.

00:16:34.310 --> 00:16:38.730
There is a persistent rumor that
it derives its name from

00:16:38.730 --> 00:16:41.860
the effect that it has
on 360 quizzes.

00:16:41.860 --> 00:16:43.460
But that is just
an ugly rumor.

00:16:43.460 --> 00:16:45.310
There's nothing to
that at all.

00:16:45.310 --> 00:16:48.930
So this is a new two step
operation that we'll call, in

00:16:48.930 --> 00:16:53.910
general, A alpha tau.

00:16:53.910 --> 00:16:57.900
And what it will do is to
generate a screw like

00:16:57.900 --> 00:16:59.680
distribution of objects.

00:16:59.680 --> 00:17:04.700
For those of you who have come
in late, since you've been

00:17:04.700 --> 00:17:07.290
demonstrating a predilection
towards longer and longer

00:17:07.290 --> 00:17:11.890
breaks, we're going to adjourn
now and have a 40 minute break

00:17:11.890 --> 00:17:13.849
so you can get it out
of your system.

00:17:13.849 --> 00:17:18.980
And we will resume at exactly
3 o'clock sharp.

00:17:18.980 --> 00:17:27.849
And we will launch into an
elucidation of the nature of

00:17:27.849 --> 00:17:29.099
screw axes.

00:17:40.050 --> 00:17:43.960
And again, for those of you who
were not here at the very

00:17:43.960 --> 00:17:48.190
start of class, the reason is
that I have a phone call that

00:17:48.190 --> 00:17:52.110
had to come in about this time
of day and no other time.

00:17:52.110 --> 00:17:55.300
So rather than keep you
shifting, wondering whether

00:17:55.300 --> 00:17:57.950
I'm coming back, we'll start at
three when I should be done

00:17:57.950 --> 00:17:59.050
with my business.

00:17:59.050 --> 00:18:00.810
OK, so that's it.

00:18:00.810 --> 00:18:02.910
You missed all the important
discussion of what's going to

00:18:02.910 --> 00:18:06.320
be on the next quiz, and nobody
will tell you because

00:18:06.320 --> 00:18:09.420
they want to keep class
average down.

00:18:09.420 --> 00:18:12.710
All right, so sorry for the
longer than usual interlude,

00:18:12.710 --> 00:18:14.310
but we'll start again
promptly at 3:00.