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PROFESSOR: Examine every
possible means for combining

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the symmetry at once, but
there is a seemingly

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paradoxical trick that
we can yet pull.

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And let me indicate what
is true here for 2mm.

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OK, so this is a twofold axis.

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Has mirror planes perpendicular
to it.

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If one of these is a mirror
plane, the other one has to be

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a mirror plane as well.

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So there's no way we could make
one a mirror plane and

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one a glide plane.

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OK, that requires a net that
is exactly rectangular.

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So let's put in the
twofold axis.

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And I add one to the
corner of the cell.

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As we well know we have to have
twofold axes at all of

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these other locations.

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We want to put a mirror
plane in the cell.

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We could pass it through the
twofold axis, and that would

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be the same as P getting
to P2mn back again.

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But why do we have to put the
mirror plane through the

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twofold axis?

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We have to have the twofold axis
left unchanged when we

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add the mirror plane, because
if we created a new twofold

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axis we create a new lattice
and we'd wreck the lattice

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that we've constructed.

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But why, why, oh why couldn't
we put the mirror

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plane in like this?

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That's going to leave the
twofold axis alone.

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It's going to leave the
translations invariant.

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Why don't we do that?

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Why not?

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So here, trick number five,
or wherever we are now.

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You can add the symmetry
elements of a point group to a

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lattice, but not necessarily
at the same point.

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You can interweave them.

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But the constraint is that this
addition must leave the

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axis invariant.

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Let's leave the twofold
axis invariant.

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And vice versa.

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That is to say the mirror
planes can't create new

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twofold axes, the twofold
axis can't

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create new mirror planes.

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And let's make sure we
understand the reason why.

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If I've got a twofold axis here
and a twofold axis here,

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I have to have a translation
that's twice their separation.

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If this distance is delta,
then I have to have a

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translation that's
automatically

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created at twice delta.

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This is the same as saying
twofold axis with a

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translation gives you a twofold
axis halfway along.

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If I take the twofold axis, I
get the translation back as a

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consequence.

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So what I'm saying is if we take
the first twofold axis

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and repeat number one to number
two, and then add

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another twofold axis which
repeats number two to number

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three, you have a third one that
is related by translation

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of twice delta.

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So if you're going to interweave
the symmetry

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elements, you have to leave the
arrangement of symmetry

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elements at the same intervals,
otherwise you would

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have created a translation or
more that is incommensurate

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and incompatible with
the other ones.

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So this is a third trick.

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And we are going to need
a new theorem.

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Let's first of all look
at the possibilities.

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We can have the twofold
axis interweave

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with a mirror plane.

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How about putting
a glide plane in

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between the twofold axis.

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In other words, replace the
mirror plane by a glide plane.

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This twofold axis hangs off
here, reflect it across and

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slide it down to here, and you
get this twofold axis here.

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So that's also a self consistent
compatible

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arrangement.

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How about a centered net?

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And that's more difficult.

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A centered net has twofold axis
and all the places of C2,

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and then twofold axes at
locations like one quarter,

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one quarter as well.

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We can't put a mirror
plane in here.

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That's not going to work,
because it will generate a new

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twofold axis, and that changes
the translation

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T1 to half its value.

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We can't even put a glide plane
here because this would

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slide down to here, reflect
across, and that's

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not going to work.

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So the number of possibilities
is limited, but we'll see that

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there's one other case as well
for a fourfold axis.

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OK, we need a new combination
theorem that says if you have

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an operation a, pi, and combine
it with a reflection

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operation that's removed from
it by some distance delta,

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what is the result?

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That general theorem has worked
out for you on the

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notes, but I think what I'll do
it in the interest of time

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is simply look at how this
arrangement of symmetry

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elements would move
things around.

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And here I've taken the motif
and hung it at the lattice

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point, and repeated it
by the twofold axis.

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The mirror plane would take this
and reflected it across

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to here, then take this one and
reflect it across to here.

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And do the same thing down at
the bottom of the cell.

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Let me now ask you what sort of
plane has arisen that would

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be perpendicular to
the mirror plane?

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Could be interweaved or passed
through the twofold axis.

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Anybody want to hazard
an answer?

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OK.

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We have to have some sort of
correspondence theorem that

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says if you've got a twofold
axis, have one plane, you've

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got to have another plane
90 degrees away.

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Do I know how everything
is related?

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I know how this is
related to this.

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I know this is related
to this.

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That's by a reflection plane.

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I know how the twofold
axes relate things.

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How is this pair related
to this pair?

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And the answer is that the way
they are related is by rights

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and lefts on here.

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This is right, this is right.

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Reflection changes handedness
to left, so I've got to have

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some way of getting from this
pair to this pair that

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involves a change
of handedness.

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And I see nobody is really
jumping out of their seat, but

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there is a glide
plane in here.

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Take this pair, slide it along
by half of T2, and flip it

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across by reflection.

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And there is a glide
plane right here.

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And therefore of necessity
there's a glide plane here.

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And this glide combined with the
perpendicular translation

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will put a glide plane
in here as well.

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And you look in your tables.

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This is plane group
number seven.

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And this one has
a twofold axis.

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It has a primitive rectangular
neck, so this is called P2m,

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and now the second plane at
right angles is not an mirror

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plane, it's a glide plane.

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So this is called P2mg, or
there's a shorthand form that

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leaves out the two.

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PMG is the shorthand
symbol for it.

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So there's a new plane group
that is based on orthogonal

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symmetry planes.

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One a mirror, one a glide,
and twofold axes.

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Let me ask you now what is the
point group of a crystal that

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would have this relation between
the atoms that are

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down in the guts
of the crystal?

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What would be the point
group of the crystal?

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We've got two different
point groups.

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We've got point M, we've
got point group two.

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What would be the point
group of the crystal?

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And this is a new problem,
a new concept.

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All of the plane groups that we
looked at until this point,

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the symmetry elements on an
atomic scale down inside the

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plane group were exactly the
same as the point group that

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we had added to the
lattice points.

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But here we've got two different
point groups that

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had been put into the lattice.

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So do we say that this
crystal has symmetry

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two or symmetry M?

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Or just point group 2m.

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That won't work because if you
have one mirror plane in a

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point group, you have
to have another.

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What this introduces is
a very subtle point.

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The point group is inherently
a macroscopic symmetry.

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When I say a crystal has a
particular point group, what I

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mean is that if I look at the
exterior faces of the crystal,

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I would say there's a
twofold axis here.

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And that's about all, if these
spaces are pair wise distinct.

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I have a crystal that looks
like this externally.

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I would say that that crystal
has symmetry 2mm

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based on the faces.

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If I found that the etch pits on
the surfaces were not quite

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the same on this face and this
face, I would have to throw

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out the twofold axis, perhaps,
which means this mirror plane

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would go out as well.

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But also I would do things
like look at the optical

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properties.

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I would measure the
conductivity as

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a function of direction.

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I would look at the slip systems
and yield stresses in

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different directions, and the
assignment of a point group is

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an experimental observation that
requires that you look at

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all possible properties.

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Not only shape, but properties
like conductivity and the

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magnetic susceptibility, and
also perhaps the way the

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crystal diffracts X-Rays.

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That's another physical
property.

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And then when you've done as
many measurements as you

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choose to make, you say as far
as I can tell, all behavior of

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this crystal is determined
inconsistent with this

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particular point group.

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But you're doing macroscopic
things.

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You're doing macroscopic
things.

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When you talk about the plane
group or space group, you're

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talking what goes on on
a local atomistic

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scale in the unit cell.

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So what would be the point
group of this crystal?

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It would have a twofold axis
that would be manifested

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externally.

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It would have a mirror plane,
which makes the left hand side

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00:12:59,570 --> 00:13:02,410
want to be the same as
the right hand side.

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00:13:02,410 --> 00:13:06,020
And if this were not a mirror
plane here, but a glide plane,

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what you would say is that this
face, if I reflect it and

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00:13:12,440 --> 00:13:17,160
slide it along by one angstrom
is going to become this face.

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00:13:17,160 --> 00:13:22,280
And this face here, if I reflect
it and slide it back

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00:13:22,280 --> 00:13:26,020
by one angstrom is going
to become this face.

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00:13:26,020 --> 00:13:28,860
But what will that do to
the external shape?

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00:13:28,860 --> 00:13:32,740
I just extend all these
surfaces, and it's going to be

206
00:13:32,740 --> 00:13:34,310
exactly what I've drawn here.

207
00:13:37,060 --> 00:13:38,810
How would you distinguish
a mirror plane

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00:13:38,810 --> 00:13:39,840
from a glide plane?

209
00:13:39,840 --> 00:13:43,070
If a crystal has a mirror plane,
a face that sits here

210
00:13:43,070 --> 00:13:45,490
passes through some atoms.

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00:13:45,490 --> 00:13:49,860
Those atoms are repeated
by reflection.

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00:13:49,860 --> 00:13:53,810
And being slit up by an amount
tau which is on the scale of

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00:13:53,810 --> 00:13:56,060
atomic dimensions, and
that would give

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00:13:56,060 --> 00:13:57,990
rise to a face here.

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00:13:57,990 --> 00:14:01,700
But can you macroscopically
assign any difference to the

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00:14:01,700 --> 00:14:05,150
fact that two faces
atomistically don't meet, but

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00:14:05,150 --> 00:14:08,680
are separated by
3.2 angstroms?

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00:14:08,680 --> 00:14:09,200
No.

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00:14:09,200 --> 00:14:12,130
What you see is one face like
this, and one face inclined to

220
00:14:12,130 --> 00:14:16,090
it with a slope that is the same
for either a glide plane

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00:14:16,090 --> 00:14:17,360
or a mirror plane.

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00:14:17,360 --> 00:14:21,450
So another truth about
crystals is that

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00:14:21,450 --> 00:14:34,670
macroscopically a glide line
plane manifests itself as a

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00:14:34,670 --> 00:14:35,920
mirror plane.

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00:14:46,440 --> 00:14:50,480
So paradoxically this crystal,
which only has a twofold axis

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00:14:50,480 --> 00:14:53,550
and one mirror plane down in its
guts is going to look as

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00:14:53,550 --> 00:14:59,830
though it has point group two in
it, even though there is no

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00:14:59,830 --> 00:15:03,650
sight, atomistically within this
arrangement of atoms that

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00:15:03,650 --> 00:15:04,900
has symmetry 2mm.

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00:15:11,428 --> 00:15:12,390
AUDIENCE: I have a question.

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00:15:12,390 --> 00:15:12,740
PROFESSOR: Yeah?

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00:15:12,740 --> 00:15:15,115
Sorry.

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00:15:15,115 --> 00:15:16,365
AUDIENCE: [INAUDIBLE]

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00:15:21,290 --> 00:15:23,230
PROFESSOR: That's a very
good question.

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00:15:23,230 --> 00:15:29,390
And actually, to answer it
properly requires knowing

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00:15:29,390 --> 00:15:32,880
something about diffraction.

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00:15:32,880 --> 00:15:35,840
The symmetry of the diffraction
pattern that you

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00:15:35,840 --> 00:15:39,520
observed would look as though
it had a mirror plane in it.

239
00:15:39,520 --> 00:15:45,270
Which is to say if we put a beam
of white radiation down

240
00:15:45,270 --> 00:15:48,950
along this direction of a
crystal, imagine these things

241
00:15:48,950 --> 00:15:52,440
all extending out in a third
dimension, that what we would

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00:15:52,440 --> 00:15:57,700
see among the arrangement of
spots is a twofold axis in the

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00:15:57,700 --> 00:15:58,560
center of [INAUDIBLE]

244
00:15:58,560 --> 00:16:02,150
photograph we'd see a mirror
plane running this way, and a

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00:16:02,150 --> 00:16:03,510
mirror plane running this way.

246
00:16:03,510 --> 00:16:07,455
So we would see spots on the
[INAUDIBLE] pattern, which

247
00:16:07,455 --> 00:16:08,720
would look like this.

248
00:16:11,510 --> 00:16:16,650
So the glide plane would
also behave it

249
00:16:16,650 --> 00:16:19,020
were a mirrored plane.

250
00:16:19,020 --> 00:16:21,350
And, maybe, if you
think of what

251
00:16:21,350 --> 00:16:24,470
goes on with the fraction.

252
00:16:24,470 --> 00:16:27,590
If you ask yourself, if I
brought in x-rays this way and

253
00:16:27,590 --> 00:16:30,700
diffracted them off a layer of
atoms related by this glide

254
00:16:30,700 --> 00:16:36,300
plane, and then I can't do this
actually to the crystal.

255
00:16:36,300 --> 00:16:41,190
But, suppose I then inverted the
direction of the incoming

256
00:16:41,190 --> 00:16:48,320
beam, and slitted along
by five angstroms.

257
00:16:48,320 --> 00:16:50,440
Doesn't make any sense.

258
00:16:50,440 --> 00:16:52,830
The diffraction from these
planes is going to look the

259
00:16:52,830 --> 00:16:56,100
same whether I bring in from
one side or the other.

260
00:16:56,100 --> 00:17:01,400
So a glide plane, in diffraction
symmetry,

261
00:17:01,400 --> 00:17:04,140
manifests itself as
a mirror plane.

262
00:17:04,140 --> 00:17:07,569
So how can you determine the
presence of glide planes using

263
00:17:07,569 --> 00:17:08,050
diffraction?

264
00:17:08,050 --> 00:17:09,020
And you can.

265
00:17:09,020 --> 00:17:15,640
And the answer is, that the
glide plane causes the

266
00:17:15,640 --> 00:17:20,249
intensity diffracted it from
planes with certain indices to

267
00:17:20,249 --> 00:17:21,900
be identically zero.

268
00:17:21,900 --> 00:17:26,569
You've probably heard of these
magical extinction rules

269
00:17:26,569 --> 00:17:28,650
they're called.

270
00:17:28,650 --> 00:17:35,780
And it says that if h plus k,
plus three times L is two pi

271
00:17:35,780 --> 00:17:39,450
plus four, and the crystal is
green, then you've got such

272
00:17:39,450 --> 00:17:42,350
and such, the reflection
being absent.

273
00:17:42,350 --> 00:17:45,800
Actually it is--

274
00:17:45,800 --> 00:17:49,680
I'm getting distracted-- but, it
turns out that glide planes

275
00:17:49,680 --> 00:17:53,230
affect just a sheet
of reflections

276
00:17:53,230 --> 00:17:54,930
in reciprocal space.

277
00:17:54,930 --> 00:17:58,450
And the reason some reflections
are absent is that

278
00:17:58,450 --> 00:18:02,710
that sheet of reflections
corresponds to what the

279
00:18:02,710 --> 00:18:06,540
projection of the structure
onto that plane would do.

280
00:18:06,540 --> 00:18:09,820
And, if you project a structure
that has a glide

281
00:18:09,820 --> 00:18:14,160
operation onto the plane of the
glide plane, the structure

282
00:18:14,160 --> 00:18:18,190
looks like the lattice
translation is half as big.

283
00:18:18,190 --> 00:18:22,860
And, if that's the case, the
diffraction spots are twice as

284
00:18:22,860 --> 00:18:25,190
widely separated.

285
00:18:25,190 --> 00:18:29,690
And it turns out that all of the
reflections that have an

286
00:18:29,690 --> 00:18:34,280
index corresponding to the
direction of tau are absent if

287
00:18:34,280 --> 00:18:37,710
that index is odd.

288
00:18:37,710 --> 00:18:40,220
So, if you knew a little bit
about it to begin with, that

289
00:18:40,220 --> 00:18:41,690
perhaps answers the question.

290
00:18:41,690 --> 00:18:44,680
So yes, you can determine the
presence of glide planes.

291
00:18:44,680 --> 00:18:47,440
And later on, we'll talk
about screw axes.

292
00:18:47,440 --> 00:18:50,510
You can determine their presence
unambiguously, but

293
00:18:50,510 --> 00:18:53,300
you do that not the symmetry
of the diffraction effects,

294
00:18:53,300 --> 00:18:55,066
but from systematic absences.

295
00:19:01,460 --> 00:19:07,240
OK, another thing I suggested we
could do is to put a glide

296
00:19:07,240 --> 00:19:09,380
plane in between the
two-fold axes

297
00:19:09,380 --> 00:19:11,310
instead of a mirror plane.

298
00:19:11,310 --> 00:19:15,480
And this says that I have a pair
of atoms repeated by the

299
00:19:15,480 --> 00:19:19,730
two-fold axis, the glide plane
would take that pair, shift

300
00:19:19,730 --> 00:19:22,880
them down by half of the cell,
and reflect them over.

301
00:19:22,880 --> 00:19:28,616
So this would be the
pattern of objects.

302
00:19:32,650 --> 00:19:36,320
This is not a lattice point,
because if these are right

303
00:19:36,320 --> 00:19:38,390
handed, that's a right-handed
pair, this is

304
00:19:38,390 --> 00:19:41,890
a left-handed pair.

305
00:19:41,890 --> 00:19:44,870
Now let me try you again.

306
00:19:44,870 --> 00:19:48,920
I know how these guys left
and right to this

307
00:19:48,920 --> 00:19:52,080
glide plane are related.

308
00:19:52,080 --> 00:19:55,440
How about these ones that are
up, to the ones that are down

309
00:19:55,440 --> 00:19:57,430
here in the middle
of the cell?

310
00:19:57,430 --> 00:20:00,650
There's got to be, from our
correspondence principle, some

311
00:20:00,650 --> 00:20:03,010
plane going in this direction.

312
00:20:03,010 --> 00:20:05,593
Anybody want to hazard a
guess at what it is?

313
00:20:08,431 --> 00:20:11,410
The answer to that question
resoundingly is, no, not at

314
00:20:11,410 --> 00:20:13,306
this hour the afternoon,
thank you.

315
00:20:13,306 --> 00:20:14,000
Yeah?

316
00:20:14,000 --> 00:20:14,860
AUDIENCE: Left one.

317
00:20:14,860 --> 00:20:16,910
PROFESSOR: Yeah, very good.

318
00:20:16,910 --> 00:20:20,850
Right ones have to go left ones,
and they're going to go

319
00:20:20,850 --> 00:20:24,440
left and right about
these planes here.

320
00:20:24,440 --> 00:20:28,080
And, if I combine this glide
plane with t2, there has to be

321
00:20:28,080 --> 00:20:30,660
another glide plane in here.

322
00:20:30,660 --> 00:20:34,530
If I combine this glide plane
with t1, there'll have to be

323
00:20:34,530 --> 00:20:37,650
another glide plane in here.

324
00:20:37,650 --> 00:20:42,890
OK, and that turns out to be
plane group number eight,

325
00:20:42,890 --> 00:20:49,106
P2gg, not P2mm, both mirror
planes have become glide.

326
00:20:57,385 --> 00:20:58,960
All right, we're almost there.

327
00:20:58,960 --> 00:21:05,650
If you look at the hexagonal
symmetries, there is just no

328
00:21:05,650 --> 00:21:10,680
way that you can interweave
things, and an attempt is made

329
00:21:10,680 --> 00:21:12,530
in the notes that
I handed out.

330
00:21:12,530 --> 00:21:14,400
You just cannot do it.

331
00:21:14,400 --> 00:21:21,820
But with P4M, P4, there is
one final possibility.

332
00:21:21,820 --> 00:21:25,890
And I'll just set that up
and not carry them out.

333
00:21:35,610 --> 00:21:45,865
OK, I'm going to put on just P4
with twofold axes in here.

334
00:21:48,680 --> 00:21:55,120
Now we have planes that go in
orientations like this.

335
00:21:55,120 --> 00:22:01,220
And then there is another plane
45 degrees away that is

336
00:22:01,220 --> 00:22:04,950
a symmetry independent plane.

337
00:22:04,950 --> 00:22:08,210
And the question now, I'll let
you have a crack at it, is

338
00:22:08,210 --> 00:22:13,750
there any way in which I could
take planes in the diagonal

339
00:22:13,750 --> 00:22:17,860
orientation or parallel to the
cell edge and interweave them

340
00:22:17,860 --> 00:22:20,770
in between the rotation axes
in such a fashion that the

341
00:22:20,770 --> 00:22:24,200
rotation axes were
left invariant?

342
00:22:24,200 --> 00:22:26,800
Again, conversely if they're
not left invariant, we will

343
00:22:26,800 --> 00:22:31,630
have created new translations
and wreck the lattice.

344
00:22:31,630 --> 00:22:35,320
So how could we slip in planes,
either reflection or

345
00:22:35,320 --> 00:22:37,560
glide, and leave the
axes invariant?

346
00:22:44,298 --> 00:22:48,186
AUDIENCE: One quarter
and three quarters?

347
00:22:48,186 --> 00:22:49,160
PROFESSOR: Yeah.

348
00:22:49,160 --> 00:22:51,916
Do you want parallel
to the cell edge?

349
00:22:51,916 --> 00:22:52,360
AUDIENCE: Horizontal.

350
00:22:52,360 --> 00:22:52,600
PROFESSOR: Horizontal?

351
00:22:52,600 --> 00:22:55,060
Yeah.

352
00:22:55,060 --> 00:22:56,330
This [INAUDIBLE]

353
00:22:56,330 --> 00:23:00,540
here has a fourfold axis up,
a fourfold axis down, a

354
00:23:00,540 --> 00:23:02,510
fourfold axis up.

355
00:23:02,510 --> 00:23:05,960
Twofold axis down, twofold axis
up, twofold axis down.

356
00:23:05,960 --> 00:23:11,440
So this would be a lovely
location for a glide plane.

357
00:23:11,440 --> 00:23:12,690
So that's one.

358
00:23:16,595 --> 00:23:24,890
From the fact that I am placing
the same diagram on

359
00:23:24,890 --> 00:23:29,700
the board yet again suggests
that there's another way.

360
00:23:29,700 --> 00:23:32,070
That's a perfectly good
suggestion, and that's one of

361
00:23:32,070 --> 00:23:34,100
the things we would
have to examine.

362
00:23:34,100 --> 00:23:36,860
Is there any other way I could
put in a mirror plane or a

363
00:23:36,860 --> 00:23:41,730
glide plane and leave
the axes invariant?

364
00:23:44,580 --> 00:23:45,390
Yeah?

365
00:23:45,390 --> 00:23:46,803
AUDIENCE: [INAUDIBLE]

366
00:23:46,803 --> 00:23:48,220
45 degrees?

367
00:23:48,220 --> 00:23:49,470
PROFESSOR: 45 degrees.

368
00:23:52,710 --> 00:23:53,140
Through here.

369
00:23:53,140 --> 00:23:55,770
I don't want to do it through
here because the fourfold axes

370
00:23:55,770 --> 00:24:00,175
sit on the same side, but you
want to put it in like this?

371
00:24:05,700 --> 00:24:07,205
I don't think you can do that.

372
00:24:21,970 --> 00:24:23,560
Glide plane won't work here.

373
00:24:26,080 --> 00:24:28,350
How about a mirror plane?

374
00:24:28,350 --> 00:24:31,300
There's no mirror plane that
goes through these twofold

375
00:24:31,300 --> 00:24:35,630
axes for P4mm.

376
00:24:35,630 --> 00:24:38,910
What we had there were mirror
planes here, and then going

377
00:24:38,910 --> 00:24:41,090
this way were glide planes.

378
00:24:41,090 --> 00:24:42,095
We've already got that.

379
00:24:42,095 --> 00:24:47,090
How about putting in a mirror
plane like that?

380
00:24:47,090 --> 00:24:49,510
This goes into this.

381
00:24:49,510 --> 00:24:52,120
Halfway along there would have
to be another mirror plane in

382
00:24:52,120 --> 00:24:52,920
that orientation.

383
00:24:52,920 --> 00:24:56,730
Here's a twofold axis that
demands that there has to be

384
00:24:56,730 --> 00:25:00,410
another one if it has a mirror
plane passing through it.

385
00:25:00,410 --> 00:25:01,660
So there's another way.

386
00:25:05,420 --> 00:25:13,350
One final one, and I will not
keep you guessing anymore,

387
00:25:13,350 --> 00:25:17,680
sneaking a peak at the answer.

388
00:25:17,680 --> 00:25:21,000
There is another way you could
do it, and that would be to

389
00:25:21,000 --> 00:25:24,720
put the glide plane down
diagonally through the cell.

390
00:25:35,940 --> 00:25:37,820
That is put the glide
plane here.

391
00:25:37,820 --> 00:25:41,400
That and take this twofold axis,
reflect it across to

392
00:25:41,400 --> 00:25:43,260
here, and slide it
down to here.

393
00:25:43,260 --> 00:25:48,230
So this twofold axis would be
related to this one by glide.

394
00:25:48,230 --> 00:25:51,950
So those three possibilities are
available for point group

395
00:25:51,950 --> 00:25:55,400
4mm and a square net.

396
00:25:58,430 --> 00:26:01,500
If you try them, and the simple
way of doing it without

397
00:26:01,500 --> 00:26:04,330
any general theorem would be
to draw in the way the

398
00:26:04,330 --> 00:26:10,510
symmetry elements repeat the
motifs, and then ask what they

399
00:26:10,510 --> 00:26:13,572
require for the remaining
symmetry elements.

400
00:26:13,572 --> 00:26:16,680
All of these give the
final result.

401
00:26:19,720 --> 00:26:21,815
So they yield the same result.

402
00:26:27,610 --> 00:26:34,540
You can develop a general
theorem for what you obtain if

403
00:26:34,540 --> 00:26:40,750
you have an operation a, alpha
combined with a plane, and

404
00:26:40,750 --> 00:26:44,680
make it a general plan, a glide
plane that is removed

405
00:26:44,680 --> 00:26:50,130
from the axis by some
distance delta.

406
00:26:50,130 --> 00:26:54,590
And that theorem is written out
for you in the notes on

407
00:26:54,590 --> 00:26:56,230
the very first page.

408
00:26:56,230 --> 00:27:00,430
You take a glide plane, and this
is a specific example, a,

409
00:27:00,430 --> 00:27:05,490
pi, if you have a glide plane
that misses a twofold axis by

410
00:27:05,490 --> 00:27:10,480
some distance delta, then the
effect of those two successive

411
00:27:10,480 --> 00:27:15,060
operations is a glide plane at
90 degrees to the first, and

412
00:27:15,060 --> 00:27:19,340
it's removed from the twofold
axis by two delta.

413
00:27:19,340 --> 00:27:20,980
It has a glide component,
excuse me, has a glide

414
00:27:20,980 --> 00:27:22,620
component two delta.

415
00:27:22,620 --> 00:27:26,380
So there's a general theorem
for the twofold rotation.

416
00:27:26,380 --> 00:27:29,860
The general theorem for a
fourfold or a threefold

417
00:27:29,860 --> 00:27:33,210
rotation is considerably more
complex to derive, but that's

418
00:27:33,210 --> 00:27:34,920
contained for you in
the notes as well.

419
00:27:38,180 --> 00:27:41,390
All right, we've come to the end
of one other major part of

420
00:27:41,390 --> 00:27:43,490
our story of symmetry.

421
00:27:43,490 --> 00:27:49,280
These are the periodic
patterns in a

422
00:27:49,280 --> 00:27:51,480
two dimensional space.

423
00:27:51,480 --> 00:28:01,935
And these are then the 17
crystallographic plane groups.

424
00:28:06,010 --> 00:28:09,610
Crystallographic is rather
redundant because they are

425
00:28:09,610 --> 00:28:11,740
groups that have [? in ?]
translation and therefore

426
00:28:11,740 --> 00:28:14,820
these are the symmetries, the
two dimensional symmetries

427
00:28:14,820 --> 00:28:16,570
that are suitable
for the atomic

428
00:28:16,570 --> 00:28:17,820
arrangements in crystals.

429
00:28:22,680 --> 00:28:27,290
As a little bonus, we also have,
as a result, 17 of the

430
00:28:27,290 --> 00:28:30,100
three dimensional
space groups.

431
00:28:30,100 --> 00:28:33,190
Just imagine each of these
plane groups with a

432
00:28:33,190 --> 00:28:39,960
translation perpendicular to the
plane of the plane group,

433
00:28:39,960 --> 00:28:42,760
and that would leave all the
symmetry elements coincidence.

434
00:28:42,760 --> 00:28:45,210
So you could pick a third
translation at right angles to

435
00:28:45,210 --> 00:28:47,610
the blackboard of any arbitrary
length, and you

436
00:28:47,610 --> 00:28:51,400
would have a three dimensional
lattice that contain the

437
00:28:51,400 --> 00:28:53,700
symmetries of a plane group.

438
00:28:53,700 --> 00:28:57,140
I think I mentioned earlier that
the way you distinguish a

439
00:28:57,140 --> 00:29:00,400
plane group with a twofold axis
in it from a space group

440
00:29:00,400 --> 00:29:05,770
with a twofold axis is that
a lowercase letter for the

441
00:29:05,770 --> 00:29:17,300
lattice type implies plane
group, and an uppercase

442
00:29:17,300 --> 00:29:23,110
symbol, capital P stands
for a primitive

443
00:29:23,110 --> 00:29:24,450
lattice in a space group.

444
00:29:34,290 --> 00:29:43,165
So something like lowercase p4mm
is a plane group, capital

445
00:29:43,165 --> 00:29:50,180
P4mm is a space group with a
third translation of arbitrary

446
00:29:50,180 --> 00:29:52,630
length at right angles
to the plane group.

447
00:29:58,130 --> 00:30:01,770
We talked about plane groups and
space groups, let's take a

448
00:30:01,770 --> 00:30:04,080
giant leap backwards.

449
00:30:04,080 --> 00:30:08,650
How about the one dimensional
space groups?

450
00:30:08,650 --> 00:30:10,910
Bet you were wondering about
them all along and

451
00:30:10,910 --> 00:30:13,870
were afraid to ask.

452
00:30:13,870 --> 00:30:18,090
So what would be the story
for one dimension?

453
00:30:21,500 --> 00:30:24,470
It's nontrivial.

454
00:30:24,470 --> 00:30:32,300
A one dimensional space group
would be a lattice row.

455
00:30:32,300 --> 00:30:34,600
You have just one dimension to
play with, and you can make

456
00:30:34,600 --> 00:30:38,750
that space periodic
by translation.

457
00:30:38,750 --> 00:30:40,210
So there's one lattice type.

458
00:30:43,160 --> 00:30:46,580
Just the lattice row.

459
00:30:46,580 --> 00:30:48,780
And what sort of symmetries
could you

460
00:30:48,780 --> 00:30:50,150
place in that lattice?

461
00:30:53,070 --> 00:30:57,220
Now you have to define
your ground rules.

462
00:30:57,220 --> 00:31:01,010
In our two dimensional
symmetries, we did not permit

463
00:31:01,010 --> 00:31:04,890
any operation which would pick
the two dimensional space up

464
00:31:04,890 --> 00:31:11,220
and rotate it and flip it over
and put it down again, because

465
00:31:11,220 --> 00:31:14,310
that is a transformation that
takes you out of that space

466
00:31:14,310 --> 00:31:16,790
and pops you back into it.

467
00:31:16,790 --> 00:31:19,230
And we will not in our
discussion of three

468
00:31:19,230 --> 00:31:23,030
dimensional crystallography
consider symmetries operations

469
00:31:23,030 --> 00:31:27,740
that take something, suck it
up into a fourth dimension

470
00:31:27,740 --> 00:31:30,410
that we can't comprehend, and
then all of a sudden pop it

471
00:31:30,410 --> 00:31:33,640
down in again, and suddenly
it appears.

472
00:31:33,640 --> 00:31:35,000
I mean, that sounds bizarre.

473
00:31:35,000 --> 00:31:35,960
We wouldn't want to do that.

474
00:31:35,960 --> 00:31:39,910
So for plane groups, we did not
allow for any operation,

475
00:31:39,910 --> 00:31:44,380
say a twofold axis that would
flip the object over and turn

476
00:31:44,380 --> 00:31:45,910
it upside down.

477
00:31:45,910 --> 00:31:47,430
But why not?

478
00:31:47,430 --> 00:31:49,340
I mean this, is mathematics.

479
00:31:49,340 --> 00:31:52,190
If it's your ballgame, you
can make up the rules.

480
00:31:52,190 --> 00:31:56,330
So why couldn't we have a
twofold axis in the plane or

481
00:31:56,330 --> 00:31:57,000
the plane group?

482
00:31:57,000 --> 00:31:59,330
It would have to leave
the net invariant.

483
00:31:59,330 --> 00:32:01,970
Well actually such entities
do exist, and

484
00:32:01,970 --> 00:32:03,770
they have been derived.

485
00:32:03,770 --> 00:32:06,990
They're called the two
sided plane groups.

486
00:32:06,990 --> 00:32:10,710
And if you want to allow that
when you make up the game, you

487
00:32:10,710 --> 00:32:14,150
can actually do that if
you wish to allow it.

488
00:32:14,150 --> 00:32:16,600
And also there could be a mirror
plane in the plane of

489
00:32:16,600 --> 00:32:20,460
the space that would take the
top side and relate it to the

490
00:32:20,460 --> 00:32:21,140
bottom side.

491
00:32:21,140 --> 00:32:22,940
So you can do that, you
can permit that.

492
00:32:22,940 --> 00:32:27,010
That's a different beast
entirely, but you can allow

493
00:32:27,010 --> 00:32:30,310
that to be one of the
transformations.

494
00:32:30,310 --> 00:32:35,450
So for our one dimensional
space, I would submit to be

495
00:32:35,450 --> 00:32:38,550
consistent with what we've done
and just finished in two

496
00:32:38,550 --> 00:32:39,110
dimensions.

497
00:32:39,110 --> 00:32:42,130
And what we will do in three,
we will not allow for any

498
00:32:42,130 --> 00:32:46,020
operation that will take the
space and transform it into a

499
00:32:46,020 --> 00:32:49,100
second or third dimension, and
then put it down again.

500
00:32:49,100 --> 00:32:52,810
So that being the case, the
only operation that is

501
00:32:52,810 --> 00:32:56,310
possible is a mirror point
that would reflect

502
00:32:56,310 --> 00:32:59,860
things left to right.

503
00:32:59,860 --> 00:33:02,380
And let me illustrate now
with some patterns.

504
00:33:06,710 --> 00:33:07,970
There's the lattice point.

505
00:33:07,970 --> 00:33:10,710
So let me put in some motifs.

506
00:33:10,710 --> 00:33:14,810
and I have to make a one
dimensional motif, but to

507
00:33:14,810 --> 00:33:18,230
distinguish the ends I'll take
a little artistic license and

508
00:33:18,230 --> 00:33:21,690
make one end of the motif a
little fatter than the other.

509
00:33:21,690 --> 00:33:24,630
Or alternatively, I could put
a little one dimensional

510
00:33:24,630 --> 00:33:26,250
headlight on this
thing that would

511
00:33:26,250 --> 00:33:27,830
shine just in one direction.

512
00:33:27,830 --> 00:33:32,980
So here's a motif hung on every
lattice point, and so

513
00:33:32,980 --> 00:33:34,940
this is no symmetry at all.

514
00:33:34,940 --> 00:33:39,220
So the symmetry part of the
symbol would be one, and the

515
00:33:39,220 --> 00:33:42,430
lattice is primitive, and what
do you think people use?

516
00:33:42,430 --> 00:33:46,050
We've used lowercase symbols,
we used uppercase symbols,

517
00:33:46,050 --> 00:33:47,870
what's left?

518
00:33:47,870 --> 00:33:48,320
Yes?

519
00:33:48,320 --> 00:33:49,840
AUDIENCE: Greek?

520
00:33:49,840 --> 00:33:51,000
PROFESSOR: Good try.

521
00:33:51,000 --> 00:33:52,250
Gothic.

522
00:33:54,700 --> 00:33:58,510
Actually, what you do is you use
a Gothic symbol with all

523
00:33:58,510 --> 00:34:03,360
these nice little shaded
angular shapes.

524
00:34:03,360 --> 00:34:04,560
That's a Gothich P, isn't it?

525
00:34:04,560 --> 00:34:07,800
Yeah, that's a Gothic
P. It is.

526
00:34:07,800 --> 00:34:10,179
It actually has thick and
thin parts, curved parts

527
00:34:10,179 --> 00:34:10,650
[INAUDIBLE].

528
00:34:10,650 --> 00:34:11,900
So that's P1.

529
00:34:14,580 --> 00:34:17,500
What about another symmetry?

530
00:34:17,500 --> 00:34:24,760
Well if we have a mirror point
in there, another possibility

531
00:34:24,760 --> 00:34:27,620
would be to have these
motifs point

532
00:34:27,620 --> 00:34:30,040
alternately left and right.

533
00:34:30,040 --> 00:34:34,380
So I'll take some non one
dimensional license and say

534
00:34:34,380 --> 00:34:37,970
that there would be mirror
points at these locations.

535
00:34:42,830 --> 00:34:46,610
Really nice symmetry, but
actually it would be totally

536
00:34:46,610 --> 00:34:51,489
wasted on these poor denizens
in a one dimensional world.

537
00:34:51,489 --> 00:34:53,320
Life would be dull and
uninteresting because they

538
00:34:53,320 --> 00:34:57,300
couldn't see anything except
the motif on either side of

539
00:34:57,300 --> 00:35:01,250
them, because everything is
constrained to one dimension.

540
00:35:01,250 --> 00:35:07,090
There's an old saying about sled
dogs that says if you are

541
00:35:07,090 --> 00:35:12,240
not the lead dog, the scenery
is not very interesting.

542
00:35:12,240 --> 00:35:14,830
And the same is true for these
poor people in a one

543
00:35:14,830 --> 00:35:15,560
dimensional world.

544
00:35:15,560 --> 00:35:18,030
They could never see the
elegance and beauty of that.

545
00:35:18,030 --> 00:35:23,070
But in any case, that would be
Gothic P, no snickers, please,

546
00:35:23,070 --> 00:35:25,960
and that would be m.

547
00:35:25,960 --> 00:35:27,930
And those are the only two
possibilities and a one

548
00:35:27,930 --> 00:35:29,180
dimensional space.

549
00:35:44,530 --> 00:35:48,180
We have a couple minutes to go,
but I think this is a good

550
00:35:48,180 --> 00:35:50,530
place to pause.

551
00:35:50,530 --> 00:35:54,650
And I'll give just a brief
indication of coming

552
00:35:54,650 --> 00:35:57,190
attractions.

553
00:35:57,190 --> 00:36:00,940
We're now going to look at
point groups in three

554
00:36:00,940 --> 00:36:05,080
dimensions, and a good way of
entering this much more

555
00:36:05,080 --> 00:36:10,640
complex situation is to go back
to something analogous to

556
00:36:10,640 --> 00:36:14,580
what we did for two dimensional
symmetries.

557
00:36:14,580 --> 00:36:21,280
And I'm going to first consider
the arrangement of

558
00:36:21,280 --> 00:36:25,990
rotation axes in three
dimensions.

559
00:36:25,990 --> 00:36:29,540
We do not have the constraint
that all the rotation axes be

560
00:36:29,540 --> 00:36:34,060
perpendicular to the plane of a
two dimensional space, being

561
00:36:34,060 --> 00:36:36,470
therefore more properly
rotation points.

562
00:36:36,470 --> 00:36:39,760
We've got three dimensions we
have to view a rotation axis

563
00:36:39,760 --> 00:36:42,820
as extending infinitely in a
direction, and there's no

564
00:36:42,820 --> 00:36:45,770
reason why we can't have another
rotation axis at an

565
00:36:45,770 --> 00:36:46,890
angle to it.

566
00:36:46,890 --> 00:36:47,930
But you know that already.

567
00:36:47,930 --> 00:36:51,260
Everybody's heard about
cubic crystals.

568
00:36:51,260 --> 00:36:53,920
Even if you think all crystals
are either body centered

569
00:36:53,920 --> 00:36:57,800
cubic, primitive cubic, face
centered cubic, or complex,

570
00:36:57,800 --> 00:37:00,540
four kinds of lattices.

571
00:37:00,540 --> 00:37:02,310
But you've heard of cubic
crystals, in the cubic

572
00:37:02,310 --> 00:37:05,940
crystals clearly the rotation
axes are inclined to one

573
00:37:05,940 --> 00:37:08,310
another and arranged
spacially.

574
00:37:08,310 --> 00:37:11,120
So we're going to ask the
nontrivial question, how can

575
00:37:11,120 --> 00:37:15,820
we combine more than one
rotation axis at a time about

576
00:37:15,820 --> 00:37:18,560
a common point in space?

577
00:37:18,560 --> 00:37:22,880
And the constraint is going to
be that a rotation about the

578
00:37:22,880 --> 00:37:26,960
first axis followed by a
rotation about the second

579
00:37:26,960 --> 00:37:31,410
axis, wherever it is, is going
to have to turn out to be

580
00:37:31,410 --> 00:37:34,610
something that is
crystallographically

581
00:37:34,610 --> 00:37:37,160
compatible with a lattice.

582
00:37:37,160 --> 00:37:38,410
So that's the constraint.

583
00:37:40,690 --> 00:37:42,660
It's not an easy question.

584
00:37:42,660 --> 00:37:47,050
And just to make you feel proud
of yourself when we get

585
00:37:47,050 --> 00:37:52,540
through this, this uses a
geometry that is due to one of

586
00:37:52,540 --> 00:37:57,990
the great mathematicians of
all time, Leonhard Euler.

587
00:37:57,990 --> 00:38:01,630
Probably the most remarkable
thing about Euler was that

588
00:38:01,630 --> 00:38:03,800
he's Swiss.

589
00:38:03,800 --> 00:38:06,860
How many world class scientists
have you heard of

590
00:38:06,860 --> 00:38:08,290
that come from Switzerland?

591
00:38:08,290 --> 00:38:12,720
Not very many, and the reason is
it's such a small country,

592
00:38:12,720 --> 00:38:16,760
and if genius occurs as a
certain fraction of the

593
00:38:16,760 --> 00:38:19,530
population, that's not going
to happen very often in a

594
00:38:19,530 --> 00:38:22,820
country like Lichtenstein
or Switzerland.

595
00:38:22,820 --> 00:38:25,420
Have you ever heard of anybody
prominent in science who came

596
00:38:25,420 --> 00:38:27,660
from Lichtenstein?

597
00:38:27,660 --> 00:38:28,900
No, probably not.

598
00:38:28,900 --> 00:38:32,160
OK, this then is going to
be a nontrivial piece of

599
00:38:32,160 --> 00:38:36,890
mathematics for us, largely
because it involves spherical

600
00:38:36,890 --> 00:38:38,170
trigonometry.

601
00:38:38,170 --> 00:38:41,200
Which I'm sure if you've ever
heard of, you've forgotten,

602
00:38:41,200 --> 00:38:45,130
and probably don't see
any utility in it.

603
00:38:45,130 --> 00:38:49,440
OK, with that exciting prospect
in hand, I look

604
00:38:49,440 --> 00:38:50,690
forward to seeing
you on Thursday.