PROFESSOR: All right.
The end of this lecture generated so much excitement
that we've continued on into the start of the next segment.
So I think we'd better begin.
What I told you, as several people were clever enough to
observe, that the number of planes between the origin and
intercept plane is equal to A times B times C, where the
intercept is A T1 out plus B T2 out equals C T3 out.
That is true only if the numbers A, B, and C are
mutually prime.
If it's something like 4, 2, 1, you won't get
that number of planes.
You'll get a submultiple.
And the reason for that is that some of the planes, when
you generate them by these three different translations
in different directions, some of them are mapped on top of
existing planes.
And I'm not going to attempt to prove this, but let me just
tell you the result, that if A and B contain some common
factor, p, and B and C contain some common factor, q, and in
the worst possible case, A and C contains some common factor,
r, then the number of intervals is equal to A times
B times C divided by p times q times r.
And probably the best way to show that is let you show, on
a problem set in the very near future, to actually map out
the planes in the two-dimensional situation, for
a and b that are mutually prime, and then do the same
thing for a and b that contain some common factor.
So this is the so-called Miller indices, the
Miller-Bravais indices, and everybody hears about them,
but it is not at all clear why one takes this very indirect
and non-intuitive way of defining the
orientation of a plane.
And the reason for this, ultimately, is that in x-ray
diffraction, you want to integrate up the scattering of
all atoms in a crystal.
So you want to integrate over a plane that contains the
atoms, and this locus has a convenient form only if you
choose these indices to find the orientation of the plane.
And what comes out of this, as I said to a couple of people
during intermission, is that the amplitude of the scattered
wave has some summation over all of the atoms.
And in that phase of that ray, that is e to the plus 2 pi i,
hx plus ky plus lz.
Very, very simple form.
And it has that form only because the locus, the way
this plane is defined was done very, very carefully.
So it's something beyond just simple geometry that forms a
reason for it.
Let me say a couple more things in general about the
Miller-Bravais indices.
They are very, very intuitive, counter-intuitive.
Let's suppose that this is a, and this is b, and this is c.
And I put down here, again, the equations of the entire
set of planes in the stack.
And let's look at the first planes and ask what the
intercepts are.
The intercepts on x, y, and z are going to be 1 over h on x,
1 over k on y, and 1 over l times z.
So the first plane from the origin is going to be 1 over h
of the a translation, 1 over k of the b translation, and 1
over l of the c translation.
So this gives us one way of determining, given a
particular rational plane, show it relative to the three
translations.
From the first plane from the origin, h is going to be 1
over the intercept on T1.
I'll call it a, which is what we usually do.
k is going to be 1 over the intercept on b, and l is going
to be 1 over the intercept on c for the first
plane from the origin.
So if you have a drawing of the set of planes relative to
the translation, such as the one that I obliterated from
just a moment ago, reciprocals of the intercepts of the first
plane from the origin give you h, k, and l very easily.
The thing that is counter-intuitive is that, if
you look out of context at the three indices, h, k, and l,
you have the feeling that as l gets larger, the intercept of
the plane on c should gradually creep up from here,
to here, to here, and so on, as l gets larger.
Actually, the reverse is true.
As l increases, the intercept on c gets smaller and smaller
and smaller and smaller, until finally, when l is infinity,
the intercept is 0.
So it's counter-intuitive.
The intercept on the axis gets larger as the Miller index
gets larger.
Let me give you a proprietary Uncle Bernie's secret method
for determining Miller indices without stress and strain.
This works every time.
Find the intercepts of any plane.
And we know from these equations what those
are going to be.
For the nth plane, the intercepts are going to be n
over h, n over k on b, and n over l on c.
Take the reciprocals.
And the reciprocals are going to be h over n, k over
n, and l over n.
And then find whatever integer, n, you must multiply
those reciprocals by in order to convert them to mutually
prime intercepts, and you've got the Miller indices in a
no-brainer, automatic fashion.
Another bit of jargon.
We will also use the basis vectors to define locations
and to define directions in a lattice.
To distinguish three integers that indicate a plane, and you
all know this from previous experience, the parentheses
says plane, an individual plane.
There are other times when you would like to use the indices
of one representative plane to indicate the entire set of
planes that are related by the symmetry of the crystal.
So let me give one specific example.
Suppose we had a cubic crystal, where these are the
three translations that form the edge of the unit cell.
And let's say that this is a1, this is a2, and this is a3.
I'm using a1, a2, and a3, not a, b, and c.
But let's not go there.
OK, these are lattice points.
And if I look at this plane that's perpendicular to a1 and
cuts it at one lattice point, l, this would
be the 1, 0, 0 plane.
And I would indicate that that is an individual plane that
we're talking about.
But if this crystal is cubic, all six faces of the set, 1,
0, 0; 0, 1, 0; 0, 0, 1; and then the plane that's in the
opposite direction, minus 1, 0, 0; 0, minus 1, 0;
and 0, 0, minus 1.
These six planes define the surfaces of the cube.
So there are times, since these are equivalent to planes
and will have the same properties, the same hardness,
the same [? edge ?] bits and everything else, there are
times when we may want to refer to the entire
symmetrical set.
And we do that.
We call that set a form.
It is a symmetry-related set.
And you can pick any one you like.
Generally, you pick one with simple indices, like the one
that is perpendicular to the translation a.
And then you indicate by braces, which is
the symbol for set.
This means the set 1, 0, 0, and that would correspond to
all six of these faces.
The number, if you do this in reverse, and I'm going to give
you the indices, what are the separate faces?
That depends on the symmetry.
So to give you an example, if I have a unit cell that has
three orthogonal translations, and this is a, and this is b,
and this is c, so that there is no cubic symmetry to it.
This face, which would be 1, 0, 0, and the one that cuts
the axes at minus a, minus 1, 0 0, those are going to be the
only faces with a single 1 and a pair of 0's which are
equivalent by symmetry.
So in this case, for this brick shaped unit cell, 1, 0,
0 would mean this pair of two faces, rather than six.
So to go from a representative face to the set of planes, you
have to know the symmetry.
If you can write the planes on the basis of symmetry, take
one representative and let that be the form.
So it leads one to observe that crystals undergoing
plastic deformation have bad form.
Yeah?
AUDIENCE: Is that different from when you
talk about a direction?
PROFESSOR: Yes, yes.
You will see in just a moment that we use the same system of
three integers, but in order to distinguish which one he's
talking about when one's thinking about integers with a
different set of numbers.
Again, if we use the lattice as the basis of a coordinate
system, another thing we might want to specify is the
orientation of a rational direction.
And let me go to a two-dimensional case.
Assume that c goes straight up.
Directions of easy plastic deformation will very often
involve integral number of translations.
For example, this direction here is obtained by going one
translation in the a direction plus one
translation in the b direction.
So we could use the pair of integers 1, 1, and if we are
normal to see we'd use a 0 for the third integer.
But how are we going to tell whether we're talking about a
direction or a plane?
And the way one does that is to use square braces to
indicate one particular direction.
You may want to also indicate the location of a particular
atom within the unit cell.
In order to do that, one could really use the location of a
vector, the components of a vector, xa plus yb plus zc,
the vector from the origin to that atom as coefficients
which correspond to the atom location.
So if that is the case, the numbers
are usually not integers.
And one simply uses the three integers in
an unadorned fashion.
So the lattice translations give you the basis of a
coordinate system, and we use those to specify features,
rational or not, of planes and directions and coordinates.
OK, any questions here?
OK, we are next going to take the first small step in a
process of synthesis.
We have found that there are, in two dimensions, five
distinct kinds of lattices.
The oblique, the rectangular, the centric rectangular, the
square, and the hexagonal.
Five kinds of lattices.
We've shown, to this point, that each of these lattices
can accommodate a certain rotational symmetry.
1 or 2 here.
m and m for the rectangular and centric rectangular.
4, the fourfold axis for a square.
3 or 6 for the hexagonal lattice.
So what I would like to do next, after we make a brief
direct diversion, is to combine with each of these
nets one of the two symmetries which can be
accommodated in that net.
And what we will have determined then are the
patterns' lattice and symmetry, which can exist in
two dimensions.
The sort of things we look at actually, like the pattern of
square tiles on the floor, or the two-dimensional pattern
that is on a fabric design.
So we'll derive those exhaustively.
The thing that results is something that is called a
plane group.
And I'm going to take this through in detail because
there are not that many combinations that can be made.
And the number is enough to count on
your fingers and toes.
You've got to use both, so they're a fair number, but not
a staggering number.
If we were to do this exercise in three dimensions, there
would be 230 distinct combinations.
And that's too much for anybody to go through.
I know of very few people who work professionally in this
area who've actually sat down and derived every single one.
If you know the principles of the derivation and know how
you could go about doing it if you were forced to, that's
sufficient.
The results are tabulated in an admirable compilation.
We mentioned that at the beginning.
This is the International Tables for X-ray
Crystallography, Volume 1.
So what we should end up with is a way for you to go to this
reference data and know how to interpret it
and make use of it.
But this is a nice microcosm.
There are not many plane groups, and to do this
exhaustively, will let us see the sort of reasoning that
goes into it.
I have to tell you that this is not in Buerger's book.
We're going to do two-dimensional
crystallography first, and then extend
it in a third direction.
In fact, I know of no book that does this.
And so you're going to get a unique, handwritten document
by none other than yours truly that will do this in detail
every step of the way.
And I don't know of any other place where this is done.
Now, what is the origin of this name?
Plane, well, these are symmetries
confined to a plane.
And then there's this mysterious term, group.
And the corresponding thing in 3D is something
called a space group.
This is a language derived from a branch of mathematics
that is called group theory, and it is exactly the language
that describes things that we've mentioned
conversationally, such as if you take two transformations
and perform them in succession, they're going to
give you a third that has to be present.
If you've got two present, there must be a third.
We talk about symmetries being self consistent.
If you have a rotation operation that is not an
integral submultiple of 2 pi, it's not going to be self
consistent because you go around, and around, and
around, and you will never come exactly full circle.
So group theory is a branch of mathematics that deals with
concepts of this sort.
So it's not very hard to develop, and it gives its
ideas to the notation that is used in crystallography.
So let me say a few words about group theory.
What is a group?
A group is a set of elements--
And what are elements?
Elements are things--
For which a law of combination is defined.
And the nature of the elements can be very,
very different things.
They could be numbers.
They could be complex numbers, and the law of combination
here might be multiplication.
They could be matrices, for which the law of combination
is defined as matrix multiplication.
Or they could be pots of different color paint, for
which the law of combination is defined as mixing the two
colors 50-50.
I don't know any practical application of that sort of
group, but nevertheless, it follows the definition, a set
of things for which a law of combination is defined.
And in addition, which satisfies three so-called
group postulates.
And the group postulates are the combination of any two
elements is also a member of the group.
The second group postulate is that the identity--
I'm going to call it just identity--
and let me define, identity is doing nothing--
Is also a member of the group.
And if we define the identity operation as I, this means
that for every element, I followed by a or a followed by
I, is the same as a by itself.
If the law of multiplication were just arithmetic
multiplication, then number 1 would be the identity
operation because 1 times any number gives you the number.
Any number times 1 gives you the same number back again.
And the third postulate is that, for every element, an
inverse exists--
let's call the elements a and a minus 1--
such that a followed by a minus 1 is equal to the
identity operation, and the inverse element followed by
the original element, a, is also the identity operation.
So if the law of combination were defined as
multiplication, you would have to find, for every number,
another number, which multiplied by it, gives you
the number 1.
OK, let me now illustrate with a couple of simple examples.
Let's consider the set of numbers 1, n minus 1.
The number of elements in the group is said to be the rank
of the group, or some people like to use the term order.
And I don't like that term because second order or fourth
order is a term applied to quantities which are not
terribly important.
So we expand this except for higher order terms, which are
negligible.
And I don't like that connotation.
So I will not use order, although it's sometimes used
to talk about the rank of a group.
The rank or order of the group is simply the number of
elements contained in a set.
So let's show that the numbers 1 and minus 1 constitute a
group of rank two.
Is the combination of any pair of elements also a
member of the group?
Well, to do this, it's convenient to set up an array
called the group multiplication table.
And what you do is you simply put the elements of the group
along a row.
And in this case, it's 1 and minus 1.
And then you put the same elements along a column, and
then you combine them.
1 times 1 is 1.
Minus 1 times 1 is minus 1.
1 times minus 1 is minus 1.
Minus 1 times minus 1 is plus 1.
Sometimes, as I'm giving this part of the lecture, somebody
passes down the hallway, and they hear 1 times 1 is 1.
1 times minus 1--
and they go in reverse to see what in the world could be
going on in this high-powered mathematical lecture.
That's why I close the door.
OK, so any combination of elements is also a
member of the group.
Is the identity operation present?
Yes because minus 1 times plus 1 or plus 1 times minus 1
gives you the same element back again.
So for every element in the group, the combination of any
two elements is present.
For each operation, we've shown that an inverse exists
and for every element there's an identity operation, and the
inverse exists.
So the pair of numbers, not terribly exciting, 1 and minus
1 where the law of combination is defined is a
group of rank 2.
I'll give you another example.
I won't bother to carry out all the terms, but the numbers
1, minus 1, i, and minus i, where i is equal to the square
root of minus 1, and where the law of combination is defined
as multiplication, constitute a group of rank 4.
With this simple introduction, I think one can see that
symmetry transformations can be regarded as operations that
are elements in a group.
And let me give you a simple example.
Let's look at a combination.
We still haven't considered the mind-boggling possibility
of having more than one symmetry element present in a
space at the same time.
But let's suppose we have two mirror planes that are
completely orthogonal, and see what sort of pattern they
would generate.
Here's a first motif.
If I call this mirror plane m1, and this mirror plane m2.
m1 is going to take this object and reflect
it across to here.
m2 will take this object and reflect it down to here.
Take this one, and reflect it down to here.
And now I know how this is related to this, how this is
related to this, how this is related to this.
But this one and this one are exactly the same thing.
And those two objects are not related by reflection.
If this is a right-handed one, this is a left-handed one,
this is a left-handed one, and this is a right-handed one.
So if something relates these two, it has to be something
that does not produce an enantiomorph.
The only thing that's possible then would be translation.
And these two guys are not parallel to one another.
You can't get this one by just sliding
this parallel to itself.
The only thing that's left is rotation.
And, lo and behold, we can get from this one to this one by a
180-degree rotation.
And we can get from this one to this one by
a 180-degree rotation.
So here's an example of a way in which we've combined
without showing why this is possible, or how I got there.
Here's a combination of symmetry elements that gives
us a possible arrangement of objects in space.
It's going to be convenient to have a notation to indicate
such combinations.
And we call them Harry, George, and Sam, or some
popular, affectionate name.
Or call them number 1, number 2, number 3.
But a nice notation is going to be a descriptive one which
tells you what you have.
And what is done in crystallography is to indicate
these possible combinations by a running list of the
different kinds of symmetry elements that are present.
And here, we've got a twofold axis, for which the symbol is
2, and two mirror planes that are different and function in
different ways in the pattern.
So this combination is called 2mm.
OK, that's getting ahead of our story.
But what I wanted to do now is to show you that this set of
symmetry elements contains four operations that
constitute a group.
There's the operation of a one-fold axis.
That's the identity operation.
There is a reflection across the first
type of mirror plane.
And notice, now, the utility of having a symbol that
indicates a specific operation rather
than a symmetry element.
There's a second reflection operation, sigma 2, and there
is a rotation operation through a 180 degrees, A pi.
So I'd like to show that these four operations constitute a
group of rank 4.
So how do we show this?
We first set up the group multiplication table.
And then we simply combine these
four operations pairwise.
Put the same 1, sigma 1, sigma 2, and A pi down
this side of the array.
Doing nothing, doing nothing, is the same as a one-fold axis
doing nothing.
Doing a reflection sigma 1 followed by doing nothing
gives you sigma 1 back again.
Doing a reflection sigma 2 followed by doing nothing
gives you sigma 2 back again.
And similarly, rotating A pi followed by doing
nothing is the same.
If I go along the columns, doing nothing followed by
sigma 1 is sigma 1.
Sigma 1 and 1 combined is sigma 2.
A pi combined with sigma 1 is A pi.
So far so good, but not surprising.
Doing sigma 1 and following it by sigma 1 is reflecting left
to right across mirror plane 1, and then reflecting right
back again.
So that is going to be a one-fold axis doing nothing.
Doing sigma 1 following up by sigma 2 would reflect across,
and then follow up by reflection in the second
mirror plane.
I get from the first one to the final one by
the rotation A pi.
So this is A pi.
And if I do the first reflection, sigma 1, and
follow that by A pi, that gets me from number 1 to number 2
to number 3.
And I get from number 1 to number 3 directly by the
reflection operation, sigma 2.
OK, you notice what's happening?
I get the same four elements back again in
every column or row.
Sigma 1, 1, A pi, sigma 2.
And if I rattle these off quickly, this as A pi.
This is identity, and this is sigma 1.
And this will be sigma 2.
This will be sigma 1.
And this will be the identity operation.
So the first group postulate is satisfied.
These four elements combined pairwise always give you
nothing but one of these elements back.
An identity operation is present because the one-fold
rotation axis is the same as leaving the thing alone.
And we've seen that for every operation, an inverse exists
because 1 occurs once in each row and column.
So sigma 2 is its own inverse, and for sigma 1, sigma 1 is
its own inverse.
For A pi, A pi is its own inverse.
So an inverse exists.
So we've got a group.
And this is another way of saying, in general terms, if
we put two mirror planes and a twofold axis together in this
specific fashion, these operations, when they go to
work on an initial motif, reproduce a finite set of
objects and not a clutter that just fills space and never
closes upon itself.
Yes, sir?
AUDIENCE: I see why you have sigma 1, sigma 2, A pi in that
group, but why do you have the 1, which is a conversion?
PROFESSOR: OK.
If I didn't put it in-- no, that's the identity operation.
This is a one-fold axis.
Just this 2 was a twofold axis.
So one-fold axis is a nice symbol for identity because it
doesn't do anything.
It picks something up and puts it right back down
where it came from.
OK.
Yeah.
That's deceptive because 1, when we're talking about
multiplication of numbers, also functions as
the identity operation.
Yeah?
AUDIENCE: If all these symmetries are commutative,
[? that is, ?]
if they all commute, do you really [INAUDIBLE] bottom
diagonal [INAUDIBLE]?
PROFESSOR: In general, yes.
AUDIENCE: Generally?
PROFESSOR: Yeah.
Actually, this is a special group because if I have two
elements, a, and take b and follow it by a, that is the
same result as a followed by b.
And a group that has this character
is said to be abelian.
There's a standing rotten joke in mathematics that asks
rhetorically, what is purple and commutes?
The answer is an abelian grape.
I don't think it's terribly funny either, but if you had a
gaggle of mathematicians here, they would chortle and double
over in laughter.
Ha, ha, ha, abelian grape.
Well, I don't feel badly.
You react about the same way to some of my
funny things, too.
But you get used to that sort of thing.
All right.
The feeling I want to leave you with at this point is the
idea that group theory and some of its concepts are
exactly what we're meaning when we grope for a definition
of the fact that a certain collection of symmetry
operation should be finite.
That is to say, it's finite.
The operations combined on each other have to give a
finite number, must constitute a group.
And the main reason for being familiar with this is that
group theory forms the basis of a lot of the words that are
used to describe the
combinations of symmetry elements.
For example, what we have derived here, the symmetry
that's called 2mm, is something that's called a
point group.
Point because there is at least one point in space
that's left unchanged, and that's this point of
intersection of all three symmetry elements.
And it's called a group because the individual
operations of these symmetry elements, when combined
according to group theory, can be shown to
constitute a group.
The patterns that we get when we add symmetry elements to a
lattice are groups in the sense, if you regard the
pattern as extending infinitely in all directions,
that the combination of any two elements gives you
something that's also in the group.
But the number of elements is infinite.
For example, in the lattice, here's T1.
Here's T2.
Every other lattice point can be described as some
combination of T1 followed by T2.
But the numbers in front of those
translations can be infinite.
If we put a symmetry element in this lattice, like a
twofold axis, for example, the translations reproduce this
twofold axis to an infinite number of locations.
But yet, all of the group postulates are satisfied.
The number of elements in the group does, however, not have
to be finite.
So that is another distinction that's worth making, that
there are infinite groups and there are finite groups.
And the number of elements can be infinite.
Collection of symmetry elements that leave one point
in space unmoved is called a point group.
If we do something like this, put a twofold axis in
combination with a pair of translations, that is an
infinite set of operations that acts on all of space.
This is something that's called a space group.
Another designation, distinction, that's sometimes
useful to make are crystallographic point groups,
such as this one, 2mm, as opposed to combinations of
symmetry elements, which are perfectly lovely and which are
valid groups, such as a combination of rotations of 2
pi over 5, or a combination of a fivefold axis
with a mirror plane.
Perfectly lovely symbols, constitute groups, but these
are non-crystallographic
These would be
non-crystallographic point groups.
Satisfy all the requirements of the group, but if it's to
be in a crystal, the fivefold axis must be combined with
translation, and that's impossible.
So you can have
non-crystallographic point groups.
There are no non-crystallographic space
groups because they are, by definition, something that
involve translation yet constitute groups.
All right.
Set the stage for next time, so you all come back excited.
We have shown that there are a limited number of symmetry
elements that are possible in a lattice, rotation 1, 2, 3,
4, and 6 in a mirror plane.
And these are now examples of what we would call
crystallographic point groups.
These would be the sort of symmetry elements that are
candidates for two-dimensional patterns.
But are these the only symmetries that can be added
to lattices?
Could we not combine a mirror plane with a twofold axis?
The answer I say to that is, sure.
You bet you can.
You just did it for us.
But could you combine a mirror plane with a threefold axis, a
mirror plane with a fourfold axis, a mirror plane with a
sixfold axis?
The answer to all of those questions are yes.
So there are going to be spaces here, where there will
be additional groups.
And when we're done, we will have the two-dimensional
crystallographic point groups.
But now I have to be able to complete something.
If I take A pi and combine it with a mirror operation sigma
1, what do I get?
If I take A 2 pi over 3, a threefold rotation, combine
that with a mirror plane that passes through
it, what do I get?
Again, a characteristic of symmetry theory is that
whenever I take a motif and repeat it by operation number
1 to get a second motif, and then I repeat that by
operation number 2, I have three
things that are identical.
And there must be some way, some operation, that
automatically arises that tells me how number 1 is
related to number 2.
Number 1 is related to number 3.
Tied in with group theory, this says that if you combine
operation 1 in a space, if it's to be a group, if
operation number 2 is present, you must combine those two
steps, and whatever pops up must be a member of the set of
operations that are present.
So we're going to have to come up with something that I call
combination theorems.
And this is simply expressing the result of a combination of
two elements in a group multiplication table.
It goes with a caveat.
If you're so excited about this stuff, you're going to
run home and start reading Buerger's book as
soon as you get there.
You can regard these symmetry transformations as operators.
And you're familiar with other sorts of operators like d by
dx, d by dy.
And when you apply them in succession, you actually write
it as a product.
d by dy followed by d by dx you write as d squared dxdy.
Or you write it as d squared dx squared when you
differentiate twice.
You don't mean you take the differential of the function
with respect to x and then square it.
You mean you do this operation twice.
So these are all operators.
And what we usually understand is that if you do d by dy
first and then follow that with d by dx, the sequence of
operation goes from right to left.
And I think that will come right quite naturally to you.
You do that with differentials and other sort of operators.
That is what we'll do.
Not Martin Buerger.
A very strong-minded person, and he did what
he thought was sensible.
And he said we read things from left to right.
We read everything from left to right.
So I will be damned if I'm going to write d by dx, d by
dy to be a sequence of operations that goes from
right to left.
So all throughout Buerger's book-- and it's easy to get
adjusted to, if he writes A dot B, he means B followed by
A. Whereas normally, in the calculus of operations, we
will use, as the rest of the world does, this is A followed
by B. Trivial point, but one that can throw you for a loop
the first time you see it in Buerger's book
if you follow it.
Distinction does not matter if the group is abelian.
So that the order of operations doesn't make any
difference.
OK, that's enough for one day.
Beginning next time, know we will start to begin this
process of synthesis.
And the first thing we'll ask is, how do you combine a
rotation operation with a reflection operation?
See you next week.