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PROFESSOR: OK, ready for more?
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I defined a problem for you.
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Now let's address it.
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Said that if we have one
coordinate system, and if we
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have some vector, q, that's
defined as a second-rank
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tensor, aij times some other
vector p sub j--
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and let me digress in passing.
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I am very careful to say "a
second-rank tensor" and not a
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"second-order tensor," because
higher-order order terms means
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negligible and non-important.
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And when I say "second-order
tensor," I don't mean to say
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it's not important
and negligible.
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It's very important, so I say
"rank," which has some sort of
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dignity to it.
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So I don't like the term
"order," because it has
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another meaning.
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OK, so here is a tensor that
relates a vector pj to give us
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the components of a vector qi.
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If we change coordinate system,
the components of p,
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representing exactly one in the
same vector, wink on and
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off and take different values.
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The values for q take on
different values, and
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therefore, of necessity, the
three by three array of
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coefficients, which relates
these different numbers, must
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also change its numerical
values.
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And I hopefully convinced you
at the end of last hour that
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there's times when you might
actually want to do this when
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you're cutting out a particular
sample from a
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single-crystal specimen.
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How do we get the new tensor
in terms of the direction
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cosine scheme that specifies
the change of axes and the
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original tensor?
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So I am going to refer to my
notes quite closely here,
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because I want it to come out
pretty and not have to
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redefine variables
when I'm done.
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So let's start with the original
tensor relation, q
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sub i equals aij
times p sub j.
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Now, what we want is something
of the form q sub i prime
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equals aij prime times
p sub j prime.
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And we know the relations
forward and reverse in terms
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of the direction cosine scheme
cij So let's begin by writing
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qi prime in terms of
qi And we know how
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that is going to transform.
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It's going to be elements of
the direction cosine scheme
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cim times q sub m.
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So that will give me the i-th
component of q in the new
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coordinate system.
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I know how q arises from
the applied vector p.
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So let me write qm in terms
of the applied pj.
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And this is going to
be aij times--
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be careful of my variables
here--
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this is going to be a
ml times p sub l.
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And that is going to, from my
definition of a second-rank
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tensor, give me the m-th
component of q.
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So far, so good.
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I've got two different repeated
subscripts here, so
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this is a double summation.
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Now, I'll have what I want to
have, namely a q sub i prime
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on the left-hand side and a p
sub j prime on the right-hand
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side if I can express the
original components of p in
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terms of the new components
of p.
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And I do that by the reverse
transformation.
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So let me now write a ml.
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And then in place of p sub
l, I will write cjl
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times p sub j prime.
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Notice the inverted order
of the subscripts.
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That is the reverse
transformation that's going to
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give me p sub l.
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So you really have now
what I'm after.
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This is a triple summation
in m, l, and j.
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I can write the terms that are
in what is going to be a
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triple summation over
a product of terms.
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I could write these terms
in any order.
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So to simplify it, let me write
q sub i prime is equal
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to cim cjl times a ml,
times p sub j prime.
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And now, hotcha, I've got an
expression that has q prime on
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the left and p prime on the
right, and paying close
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attention to my notes so that
the subscripts all came out
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the way I would like them to.
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So what this says is that the
transform tensor aij prime is
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going to be equal to,
by definition--
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or what we've shown here,
it's going to be equal--
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just picking off terms--
it's going to be
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cim cjl times a ml.
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Just picking off these terms.
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m has no physical meaning,
because m simply is an index
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of summation.
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l has no specific meaning.
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That is just an index
of summation.
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But the i and the j
do have meaning.
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They go with the i and j on the
particular tensor element
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that we were attempting
to evaluate.
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So in other words, to be
specific, if we want the new
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value of the tensor element a1
2 prime, it's going to be c1
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something, c2 something, and
those somethings m and l would
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vary from 1 to 3.
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So if I write this out not in
the reduced subscript notation
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but put a summation sign in
there, so a12 prime is going
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to be the sum over m and the
sum over l of terms c1m--
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the first index is always 1--
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c2 something--
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first index is always 2--
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and then m and l take on
all possible values.
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OK, we've got two results.
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We learned how to transform
a vector.
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And a vector, if you will, is
simply a tensor of first rank.
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And transforming the vector,
we summed over all three
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components of the
original vector.
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And the coefficients in
that summation were
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one direction cosine.
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Now we're transforming
a second-rank tensor.
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Again, each new element is a
linear combination of all nine
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of the elements in the original
tensor, and the
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coefficients are a product
of two direction cosines.
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So we've got two points.
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Let's draw a line through them,
and we can say that, in
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general, any new tensor
element of any rank--
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and you could prove it through
exactly this method by going
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up, now, to the third rank,
fourth rank, and so on, and
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writing substitutions
of this form--
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it turns out that a new tensor
element aijkl however far you
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want to go, is going to be given
by a linear combination
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of direction cosine element ciI,
cj capital J, ck capital
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K, cl capital L, times however
far you have to go times
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aijkl, and so on.
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So these are the true indices.
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These have physical meaning.
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These have relevance
to how a particular
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property will behave.
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The capital I, capital J,
capital L, and so on, are what
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we referred to last time
as dummy indices.
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These are indices
of summation.
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But the first index on
the direction cosines
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has specific meaning.
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They are tied to the indices on
the subscript of the term
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that you would like
to evaluate.
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So we've got now a very profound
relation for a tensor
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of any rank.
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And really, it just involves
substitutions using the
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reverse or the forward
transformation until you get
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one element on the one side
related to another element on
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the right-hand side.
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And this is a specific element
in the new tensor, in our
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case, of the second-rank
tensor, aml.
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This is all very abstract.
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It is something that we'll have
to do a couple of times
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for a real problem before you
see how it works out.
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The number of elements that
figure into these
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transformations is really
astronomical.
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Suppose, for example, the tensor
involved were something
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like the elastic stiffness
tensor, which is
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represented by c.
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And we need four subscripts.
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This is a tensor
or fourth rank.
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If we wanted to transform a
particular stiffness to a new
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coordinate system, we would need
a summation ci capital I,
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cj capital J, ck capital K, cl
capital L, times all of the
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elements in the original
tensor, aijkl.
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A fourth-rank tensor consists
of an array of 9 by 9 terms.
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So there are 81 of these.
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We'd have four direction cosines
out in front, and
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there would be a total of 81
times 5 characters that we
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would have to write.
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So to do the complete tensor
transformation, we would have
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to write on the order of 400
quantities to get just one of
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the 81 elements in the
new transform tensor.
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So the total number of elements
we'd have to write to
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do this would be 81
squared times 5.
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That's a lot of elements.
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We'll do a few of these
transformations directly, but
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let me assure you that if we
do a transformation that is
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going to involve symmetry, a lot
of the direction cosines,
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if we're lucky, will be 0.
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So it's not quite as onerous
as it seems.
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So we would make use of this
sort of formalism if we wanted
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to go from one set of reference
axes to a new set
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that might represent a special
specimen that we
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cut out of a crystal.
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But there's another formal way
in which we could make very
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profound and non-intuitive
use of these relations.
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Crystals, except for the
abominable triclinic crystals,
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have symmetry.
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If a crystal has symmetry, you
can transform the solid
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physically by that symmetry
operation.
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And you have to measure
the same
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property before and after.
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So suppose we have a crystal
that has a twofold axis.
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And this crystal is something
that looks like this.
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So this is side A, and this is
side B. We could move the
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crystal by a 180-degree
rotation.
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Put it down.
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00:14:03,800 --> 00:14:05,960
I won't draw it, because it's
going to look exactly the
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00:14:05,960 --> 00:14:09,860
same, except now this thing--
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I won't draw, and
I do draw it--
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00:14:11,730 --> 00:14:15,230
this is face A, and this is face
B. If we had electrodes
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00:14:15,230 --> 00:14:18,860
on the crystal before and after
that transformation, we
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00:14:18,860 --> 00:14:21,390
have to measure, let's say,
the same electrical
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00:14:21,390 --> 00:14:24,350
conductivity for both
orientations of the crystal.
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00:14:24,350 --> 00:14:29,980
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00:14:29,980 --> 00:14:34,430
Now, moving a crystal relative
to some coordinate system,
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00:14:34,430 --> 00:14:37,430
relative to a pair of electrodes
that we're
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00:14:37,430 --> 00:14:40,710
fastening onto the crystal, is
exactly the same thing as
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00:14:40,710 --> 00:14:43,390
doing the reverse transformation
of the
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00:14:43,390 --> 00:14:46,080
coordinate system.
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00:14:46,080 --> 00:14:48,540
That's a vague, strange-sounding
term.
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00:14:48,540 --> 00:14:52,410
So suppose we have a crystal
with a fourfold axis with four
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00:14:52,410 --> 00:14:59,980
faces, A, B, C, D. And here
are our electrodes.
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00:14:59,980 --> 00:15:03,015
To move the crystal relative
to the electrodes by a
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90-degree rotation would involve
rotating face D up to
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this location.
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A would move to this location.
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C would move to this location.
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B would move to this location,
and we'd fasten electrodes on
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the crystal again.
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If the crystal originally had a
coordinate system such that
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this were X1 and this is X2,
moving the electrodes onto a
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different direction on the
crystal is the same as moving
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the crystal in the
opposite sense.
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00:15:37,980 --> 00:15:43,010
So we could either envision
moving the crystal relative to
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the electrodes like this, or we
could move the electrodes
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relative to the crystal by the
reverse transformation.
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And the result is the same.
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So what I'm saying is that if
a crystal has symmetry--
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and let me be specific and
suppose that our crystal has a
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00:16:08,260 --> 00:16:12,435
twofold rotation
axis along X3.
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00:16:12,435 --> 00:16:20,130
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00:16:20,130 --> 00:16:24,300
Let's ask how that twofold
access would change the
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coordinate system relative
to the crystal.
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That's the same as moving the
crystal relative to the
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coordinate system.
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It's going to take X1 and move
it to this location X1 prime.
253
00:16:36,930 --> 00:16:39,610
It's going to take
X2 and move it
254
00:16:39,610 --> 00:16:41,600
through 180-degree rotation.
255
00:16:41,600 --> 00:16:43,810
This is going to be X2 prime.
256
00:16:43,810 --> 00:16:47,620
And if the twofold axis
is along X3, X3 prime
257
00:16:47,620 --> 00:16:51,480
is the same as X3.
258
00:16:51,480 --> 00:16:54,060
So what is the direction
cosine scheme for
259
00:16:54,060 --> 00:16:56,070
this change of axes?
260
00:16:56,070 --> 00:17:00,200
You might immediately start
working and saying, well, C11
261
00:17:00,200 --> 00:17:03,940
is the cosine of the angle
between X1 prime and X1.
262
00:17:03,940 --> 00:17:05,430
That's 180 degrees.
263
00:17:05,430 --> 00:17:08,230
Cosine of 180 degrees
is minus 1.
264
00:17:08,230 --> 00:17:12,359
But let me remind you that the
direction cosine scheme, c ij,
265
00:17:12,359 --> 00:17:16,660
simply gives us the relation
between the new axes x sub i
266
00:17:16,660 --> 00:17:19,200
prime and the old
axes, x sub j.
267
00:17:19,200 --> 00:17:22,900
So let me, just by inspection,
write down the relation
268
00:17:22,900 --> 00:17:26,599
between these two sets of
coordinate systems.
269
00:17:26,599 --> 00:17:29,755
So X1 prime is equal
to minus X1.
270
00:17:29,755 --> 00:17:33,610
X2 prime is equal to minus X2.
271
00:17:33,610 --> 00:17:37,200
X3 prime is equal to X3.
272
00:17:37,200 --> 00:17:41,350
So the direction cosine scheme
for this particular
273
00:17:41,350 --> 00:17:49,170
transformation is simply minus
1 0 0, 0 minus 1 0, 0 0 1.
274
00:17:49,170 --> 00:17:52,070
So I just evaluated a
nine-element direction cosine
275
00:17:52,070 --> 00:17:55,540
scheme by inspection, if I can
write the relation between the
276
00:17:55,540 --> 00:17:57,790
coordinate system before and
after the transformation.
277
00:17:57,790 --> 00:18:02,750
278
00:18:02,750 --> 00:18:04,000
OK?
279
00:18:04,000 --> 00:18:06,400
280
00:18:06,400 --> 00:18:09,630
And as we examine higher
symmetry, the same is going to
281
00:18:09,630 --> 00:18:13,330
be true for the threefold axis,
let's say, along the 1,
282
00:18:13,330 --> 00:18:16,880
1, 1 direction of a cubic
crystal, for a sixfold
283
00:18:16,880 --> 00:18:20,130
axis, and so on.
284
00:18:20,130 --> 00:18:22,810
So we'll be able to write the
direction cosine schemes for a
285
00:18:22,810 --> 00:18:25,455
symmetry transformation
simply by inspection.
286
00:18:25,455 --> 00:18:29,500
287
00:18:29,500 --> 00:18:34,950
So for a twofold axis along X3,
this is the form of the
288
00:18:34,950 --> 00:18:36,490
direction cosine scheme.
289
00:18:36,490 --> 00:18:47,670
290
00:18:47,670 --> 00:18:52,640
So now let me transform the
elements of a second-rank
291
00:18:52,640 --> 00:18:58,590
tensor term by term and
see what we get.
292
00:18:58,590 --> 00:18:59,840
Suppose I want--
293
00:18:59,840 --> 00:19:02,590
294
00:19:02,590 --> 00:19:05,840
let's stick with conductivity
as an example.
295
00:19:05,840 --> 00:19:11,140
Suppose I want the value for
the conductivity element
296
00:19:11,140 --> 00:19:13,640
sigma1 1 prime.
297
00:19:13,640 --> 00:19:18,140
That's going to be c1 something,
c1 something,
298
00:19:18,140 --> 00:19:21,760
because these are the elements
that go in here, times every
299
00:19:21,760 --> 00:19:26,180
element of a conductivity
tensor sigma lm.
300
00:19:26,180 --> 00:19:31,230
The only element of the form c1
something that is non-zero
301
00:19:31,230 --> 00:19:37,540
in this row c11, c12, c13,
is the term c11.
302
00:19:37,540 --> 00:19:40,830
In the same way in the same row,
the only term of the form
303
00:19:40,830 --> 00:19:42,990
c1 something is c11.
304
00:19:42,990 --> 00:19:47,850
So this is going to be simply
c11 times c11 times sigma11.
305
00:19:47,850 --> 00:19:50,550
That's the only term
that survives.
306
00:19:50,550 --> 00:19:58,400
c11 has a numerical value of
minus 1, and that says that
307
00:19:58,400 --> 00:20:00,860
sigma11 prime is equal
to sigma11.
308
00:20:00,860 --> 00:20:04,730
309
00:20:04,730 --> 00:20:09,180
So is there any constraint, any
restriction on sigma 11?
310
00:20:09,180 --> 00:20:14,540
No, sigma11 could be
anything it likes.
311
00:20:14,540 --> 00:20:15,790
So there's no constraint.
312
00:20:15,790 --> 00:20:21,290
313
00:20:21,290 --> 00:20:22,750
Let's do another element.
314
00:20:22,750 --> 00:20:25,590
Let's see what sigma12
prime would be.
315
00:20:25,590 --> 00:20:32,430
This will be c1 something times
c2 something times sigma
316
00:20:32,430 --> 00:20:35,380
something something.
317
00:20:35,380 --> 00:20:38,070
The only element of the
form c1 something that
318
00:20:38,070 --> 00:20:42,360
is non-zero is c11.
319
00:20:42,360 --> 00:20:46,790
So I'll put in a 1 for the l.
320
00:20:46,790 --> 00:20:49,430
The only direction cosine
element of the form c2
321
00:20:49,430 --> 00:20:52,810
something which is
non-zero is c22.
322
00:20:52,810 --> 00:20:58,630
So I'll put in c22 and
let n be equal to 2.
323
00:20:58,630 --> 00:21:02,840
c11 is minus 1, c22
is minus 1.
324
00:21:02,840 --> 00:21:04,950
So again, this gives us
something not terribly
325
00:21:04,950 --> 00:21:06,010
interesting.
326
00:21:06,010 --> 00:21:10,690
Sigma 12 prime is equal
to sigma 12.
327
00:21:10,690 --> 00:21:17,900
So there's no constraint, at
which point you're probably
328
00:21:17,900 --> 00:21:20,310
getting very restive,
say, this is
329
00:21:20,310 --> 00:21:22,580
not telling us anything.
330
00:21:22,580 --> 00:21:25,930
So let me shake you up
by doing one further
331
00:21:25,930 --> 00:21:31,940
transformation, and that is to
find the value for c13 prime.
332
00:21:31,940 --> 00:21:34,880
And that would be
c1 something, c3
333
00:21:34,880 --> 00:21:39,050
something times sigma lm.
334
00:21:39,050 --> 00:21:42,330
The only form of this term of
the form c1 something, that's
335
00:21:42,330 --> 00:21:44,720
non-zero is c11,
as we've seen.
336
00:21:44,720 --> 00:21:46,730
So I'll put in just that
single term and
337
00:21:46,730 --> 00:21:49,400
replace l by 1.
338
00:21:49,400 --> 00:21:52,880
The term of the form
c3 something that
339
00:21:52,880 --> 00:21:56,581
is non-zero is c33.
340
00:21:56,581 --> 00:22:02,130
And I'll put in a 3 for m, and
this is then c11 times c33
341
00:22:02,130 --> 00:22:02,790
times sigma13.
342
00:22:02,790 --> 00:22:09,520
1, And c11 is minus 1.
343
00:22:09,520 --> 00:22:12,630
c33 is plus 1.
344
00:22:12,630 --> 00:22:18,950
And that says that sigma13
prime is minus sigma13.
345
00:22:18,950 --> 00:22:22,620
But if this is a symmetry
transformation, the tensor has
346
00:22:22,620 --> 00:22:25,020
to remain invariant.
347
00:22:25,020 --> 00:22:30,630
And if we're to have sigma13
equals minus sigma13, there's
348
00:22:30,630 --> 00:22:35,960
only one number that can make
that claim, and that's 0.
349
00:22:35,960 --> 00:22:41,800
So sigma13 is identically 0.
350
00:22:41,800 --> 00:22:45,850
And that places a rather severe
constraint on the way
351
00:22:45,850 --> 00:22:50,150
in which the crystal is going to
relate an applied electric
352
00:22:50,150 --> 00:22:52,100
field to a current flow.
353
00:22:52,100 --> 00:22:53,470
Sigma11 is anything.
354
00:22:53,470 --> 00:22:55,890
Sigma12 is anything.
355
00:22:55,890 --> 00:22:59,470
Sigma13 has to be
identically 0.
356
00:22:59,470 --> 00:23:01,330
Now, let's cut to
the bottom line.
357
00:23:01,330 --> 00:23:06,450
The direction cosine scheme is
diagonal, so we can say that
358
00:23:06,450 --> 00:23:14,630
for any element that we pick to
transform sigma ij prime is
359
00:23:14,630 --> 00:23:20,360
going to be cii, the diagonal
term which has the second
360
00:23:20,360 --> 00:23:25,030
subscript equal to the first,
times cjj times sigma ij.
361
00:23:25,030 --> 00:23:29,710
362
00:23:29,710 --> 00:23:39,410
And this says that if we have
i or j is equal to 3, then
363
00:23:39,410 --> 00:23:43,530
sigma ij has to be 0, because
we're going to have a minus 1
364
00:23:43,530 --> 00:23:45,250
times a plus 1.
365
00:23:45,250 --> 00:23:57,280
If neither i or j is equal to 3,
then again, we will have a
366
00:23:57,280 --> 00:23:59,330
minus 1 times a minus 1.
367
00:23:59,330 --> 00:24:00,580
There will be no constraint.
368
00:24:00,580 --> 00:24:04,250
369
00:24:04,250 --> 00:24:10,420
And the only final possibility
is that both i and j
370
00:24:10,420 --> 00:24:11,320
are equal to 3.
371
00:24:11,320 --> 00:24:14,040
That would be the single
element c33.
372
00:24:14,040 --> 00:24:17,810
Then we would have plus 1 times
plus 1 as the product of
373
00:24:17,810 --> 00:24:20,070
direction cosines, and there
will be no constraint.
374
00:24:20,070 --> 00:24:25,460
375
00:24:25,460 --> 00:24:30,980
So for a crystal that has a
twofold axis, and in which
376
00:24:30,980 --> 00:24:35,960
that twofold axis is along the
direction of x3, the form of
377
00:24:35,960 --> 00:24:44,660
the tensor will be sigma11,
sigma12, 0, sigma21, sigma22,
378
00:24:44,660 --> 00:25:00,900
0, sigma31, that's going to be
equal to 0, and sigma33 has no
379
00:25:00,900 --> 00:25:02,150
constraints.
380
00:25:02,150 --> 00:25:04,370
381
00:25:04,370 --> 00:25:13,100
So rather than having nine
elements, there are only five
382
00:25:13,100 --> 00:25:25,485
independent elements
rather than nine.
383
00:25:25,485 --> 00:25:28,100
384
00:25:28,100 --> 00:25:34,400
And there is now another
relation that can occur in a
385
00:25:34,400 --> 00:25:36,390
second-rank tensor.
386
00:25:36,390 --> 00:25:44,370
The off-diagonal terms sigma12
and sigma21 do
387
00:25:44,370 --> 00:25:48,200
not have to be related.
388
00:25:48,200 --> 00:25:49,823
But for most--
389
00:25:49,823 --> 00:25:52,570
390
00:25:52,570 --> 00:25:53,820
but not all--
391
00:25:53,820 --> 00:25:58,200
392
00:25:58,200 --> 00:26:07,035
most second-rank tensor
properties happily have sigma
393
00:26:07,035 --> 00:26:12,440
ij identical to sigma ji.
394
00:26:12,440 --> 00:26:15,710
In other words, the tensor
is symmetric across
395
00:26:15,710 --> 00:26:19,490
its principal diagonal.
396
00:26:19,490 --> 00:26:26,470
That is a condition that does
not arise from symmetry That
397
00:26:26,470 --> 00:26:29,420
depends on the specific
physical property.
398
00:26:29,420 --> 00:26:32,400
399
00:26:32,400 --> 00:26:34,780
So let me emphasize
that this depends
400
00:26:34,780 --> 00:26:36,030
on the tensor property.
401
00:26:36,030 --> 00:26:46,120
402
00:26:46,120 --> 00:26:48,970
And for a great many physical
properties--
403
00:26:48,970 --> 00:26:53,230
conductivity, diffusivity,
permeability, susceptibility--
404
00:26:53,230 --> 00:26:58,270
you can show that the tensor
has to be symmetric.
405
00:26:58,270 --> 00:27:01,310
But there are a lot of tensors,
particularly for the
406
00:27:01,310 --> 00:27:03,800
more obscure physical
properties, where to my
407
00:27:03,800 --> 00:27:08,980
knowledge, this proof has
never been given.
408
00:27:08,980 --> 00:27:12,350
And along the same lines, it
is well known that there is
409
00:27:12,350 --> 00:27:16,020
one physical property for
which this is not true.
410
00:27:16,020 --> 00:27:19,730
This is the thermal electricity
tensor.
411
00:27:19,730 --> 00:27:24,300
So for at least one property,
you can show for sure that the
412
00:27:24,300 --> 00:27:29,260
tensor does not have to be
symmetric and that for a
413
00:27:29,260 --> 00:27:33,550
crystal of symmetry 2, this
term and this term are
414
00:27:33,550 --> 00:27:34,860
definitely not equal.
415
00:27:34,860 --> 00:27:42,060
416
00:27:42,060 --> 00:27:45,840
All right, let us do another
transformation for another
417
00:27:45,840 --> 00:27:49,390
symmetry, and we can see that
it goes fast when the
418
00:27:49,390 --> 00:27:52,370
direction cosine scheme
is relatively sparse.
419
00:27:52,370 --> 00:27:55,360
Let's ask the restrictions,
if any, that
420
00:27:55,360 --> 00:27:56,730
are imposed by inversion.
421
00:27:56,730 --> 00:28:08,690
422
00:28:08,690 --> 00:28:12,880
So what is the direction
cosine scheme here?
423
00:28:12,880 --> 00:28:18,880
Here's x1, here's
x2, here's x3.
424
00:28:18,880 --> 00:28:22,380
Then operation of inversion at
the intersection of these axes
425
00:28:22,380 --> 00:28:26,035
will invert the direction
of x1 prime to here.
426
00:28:26,035 --> 00:28:29,500
It'll invert the direction
of x2 prime here.
427
00:28:29,500 --> 00:28:33,440
It will invert the direction
of x3 prime to here.
428
00:28:33,440 --> 00:28:36,770
So the relation between the
reference axes is that x1
429
00:28:36,770 --> 00:28:39,870
prime is equal to minus x1.
430
00:28:39,870 --> 00:28:42,990
x2 prime is equal to minus x2.
431
00:28:42,990 --> 00:28:48,090
x3 prime is equal to minus x3,
so that the form of the
432
00:28:48,090 --> 00:28:53,940
direction cosine scheme crj
is minus 1, 00, 0 minus
433
00:28:53,940 --> 00:28:56,300
1 0, 00 minus 1.
434
00:28:56,300 --> 00:29:04,080
435
00:29:04,080 --> 00:29:06,900
Slightly different from that for
a two-fold axis for which
436
00:29:06,900 --> 00:29:09,460
the first two diagonal elements
were minus 1, the
437
00:29:09,460 --> 00:29:11,110
third one was 0.
438
00:29:11,110 --> 00:29:13,350
Well, let's jump right
to it and see if we
439
00:29:13,350 --> 00:29:14,440
can generalize this.
440
00:29:14,440 --> 00:29:19,180
It's a diagonal direction cosine
scheme once again.
441
00:29:19,180 --> 00:29:23,080
And this says that if we
transform a particular element
442
00:29:23,080 --> 00:29:31,840
sigma ij, it's going to be given
by cil, cjm, sigma lm,
443
00:29:31,840 --> 00:29:35,070
where l and m are variables
of summation.
444
00:29:35,070 --> 00:29:39,530
The only ones that survive are
the ones for which i equals l
445
00:29:39,530 --> 00:29:41,590
and for which j equals m.
446
00:29:41,590 --> 00:29:47,330
So it's going to be cii,
cjj times sigma ij.
447
00:29:47,330 --> 00:29:51,530
Regardless of the values of i
and j, the diagonal terms are
448
00:29:51,530 --> 00:29:53,330
always minus 1.
449
00:29:53,330 --> 00:29:57,590
And therefore, sigma ij prime is
always going to turn out to
450
00:29:57,590 --> 00:30:00,460
be equal to sigma ij, so
there's going to be no
451
00:30:00,460 --> 00:30:04,070
constraint on any element.
452
00:30:04,070 --> 00:30:08,170
453
00:30:08,170 --> 00:30:13,470
And this shortens the job that's
facing us immeasurably.
454
00:30:13,470 --> 00:30:18,820
So let me write that down,
because that's important.
455
00:30:18,820 --> 00:30:38,070
Inversion imposes no constraint
on any second-rank
456
00:30:38,070 --> 00:30:39,320
tensor property.
457
00:30:39,320 --> 00:30:46,460
458
00:30:46,460 --> 00:30:51,240
So if we stay with monoclinic
crystals, we looked at
459
00:30:51,240 --> 00:30:53,920
symmetry 2.
460
00:30:53,920 --> 00:30:59,910
Symmetry 2 over m is equal
to 2 with an inversion
461
00:30:59,910 --> 00:31:01,290
center put on it.
462
00:31:01,290 --> 00:31:05,270
But inversion doesn't require
anything, so the symmetry
463
00:31:05,270 --> 00:31:10,620
constraints for 2 over m
have to be the same as
464
00:31:10,620 --> 00:31:14,640
for symmetry 2.
465
00:31:14,640 --> 00:31:18,600
We look at the constraints that
might be imposed by a
466
00:31:18,600 --> 00:31:20,870
mirror plane.
467
00:31:20,870 --> 00:31:26,555
A mirror plane plus inversion
is 2 over m.
468
00:31:26,555 --> 00:31:29,310
2 over m has to be
the same as 2.
469
00:31:29,310 --> 00:31:31,760
So this will be the same as 2.
470
00:31:31,760 --> 00:31:35,240
And now we've shown that for
any monoclinic crystal,
471
00:31:35,240 --> 00:31:37,840
regardless of whether the
symmetry, the point group, is
472
00:31:37,840 --> 00:31:42,050
2m or 2 over m, the form
of the tensor has to
473
00:31:42,050 --> 00:31:43,840
be exactly the same.
474
00:31:43,840 --> 00:31:56,760
So for any monoclinic crystal,
namely 2m or 2 over m, this is
475
00:31:56,760 --> 00:32:00,380
the form of the tensor where
the twofold axis, again, is
476
00:32:00,380 --> 00:32:03,420
along x3, the mirror plane would
have to be perpendicular
477
00:32:03,420 --> 00:32:07,945
to x3, and for 2 over m, both
of the preceding conditions.
478
00:32:07,945 --> 00:32:11,420
479
00:32:11,420 --> 00:32:12,670
So how about that?
480
00:32:12,670 --> 00:32:17,500
481
00:32:17,500 --> 00:32:22,860
Let me issue a caveat, because
we're almost out of time.
482
00:32:22,860 --> 00:32:25,570
There are five independent
elements.
483
00:32:25,570 --> 00:32:27,810
And that's true.
484
00:32:27,810 --> 00:32:45,570
But elements sigma12, sigma21,
sigma13, and sigma31 are 0
485
00:32:45,570 --> 00:32:53,150
only for this arrangement
of axes relative to
486
00:32:53,150 --> 00:32:54,400
the symmetry elements.
487
00:32:54,400 --> 00:33:06,980
488
00:33:06,980 --> 00:33:14,086
If you wanted to take a
different set of axes, you
489
00:33:14,086 --> 00:33:17,400
know how to get the tensor
for that set of axes.
490
00:33:17,400 --> 00:33:20,310
Each tensor element is going
to be given by a linear
491
00:33:20,310 --> 00:33:23,470
combination of these five
non-zero elements.
492
00:33:23,470 --> 00:33:27,520
And if the orientation of the
axes relative to the symmetry
493
00:33:27,520 --> 00:33:35,080
elements is quite general, all
nine elements of the tensor
494
00:33:35,080 --> 00:33:37,790
will be non-zero.
495
00:33:37,790 --> 00:33:40,770
There will be only five
independent numbers, which
496
00:33:40,770 --> 00:33:44,430
composes each of those nine
elements, and they will be
497
00:33:44,430 --> 00:33:47,800
given by a product of two
direction cosines, sometimes
498
00:33:47,800 --> 00:33:50,860
each of these five non-zero
elements.
499
00:33:50,860 --> 00:33:53,750
But there will be no zeros in
this array at all for an
500
00:33:53,750 --> 00:33:55,500
arbitrary set of coordinate
systems.
501
00:33:55,500 --> 00:33:58,620
502
00:33:58,620 --> 00:34:02,460
Something that I think I'll ask
you to do as a problem,
503
00:34:02,460 --> 00:34:06,230
because it's really easy
to do, if the tensor is
504
00:34:06,230 --> 00:34:10,100
symmetric, which most of them
are, one thing that you can
505
00:34:10,100 --> 00:34:20,210
show quite directly is that
a symmetric tensor remains
506
00:34:20,210 --> 00:34:30,375
symmetric for any arbitrary
change of axes.
507
00:34:30,375 --> 00:34:43,780
508
00:34:43,780 --> 00:34:46,900
And that, again, is something
that's fairly easy to prove,
509
00:34:46,900 --> 00:34:49,750
and I'll let you have the fun
and exhilaration of doing that
510
00:34:49,750 --> 00:34:51,000
for yourself.
511
00:34:51,000 --> 00:34:52,980
512
00:34:52,980 --> 00:34:56,980
OK, so this means that for
everything except thermal
513
00:34:56,980 --> 00:35:01,010
electricity, you really have
to transform, at most, only
514
00:35:01,010 --> 00:35:02,800
six elements if you go from one
515
00:35:02,800 --> 00:35:05,040
coordinate system to another.
516
00:35:05,040 --> 00:35:09,270
That's still a lot, but it's
considerably better than
517
00:35:09,270 --> 00:35:11,110
transforming all nine.
518
00:35:11,110 --> 00:35:14,070
So if the tensor originally is
symmetric in one coordinate
519
00:35:14,070 --> 00:35:17,370
system, it stays symmetric in
any other coordinate system.
520
00:35:17,370 --> 00:35:20,720
521
00:35:20,720 --> 00:35:22,380
Now, one thing that
I should mention--
522
00:35:22,380 --> 00:35:24,230
I passed over it rather
quickly--
523
00:35:24,230 --> 00:35:27,350
524
00:35:27,350 --> 00:35:33,780
we said that a property of a
direction cosine scheme is
525
00:35:33,780 --> 00:35:37,860
that it is what's called a
unitary transformation.
526
00:35:37,860 --> 00:35:42,680
And it has the property that
the determinant of the
527
00:35:42,680 --> 00:35:46,030
coefficients is plus
1 if the axis
528
00:35:46,030 --> 00:35:49,580
retains the same chirality.
529
00:35:49,580 --> 00:35:53,140
The determinant is minus 1 if
you change the handedness.
530
00:35:53,140 --> 00:35:59,520
That is only true for what is
called a measure-preserving
531
00:35:59,520 --> 00:36:02,160
transformation.
532
00:36:02,160 --> 00:36:02,920
That's what it's called.
533
00:36:02,920 --> 00:36:05,490
And when it's a
measure-preserving
534
00:36:05,490 --> 00:36:09,690
transformation, then the
direction cosine scheme is a
535
00:36:09,690 --> 00:36:11,440
unitary matrix.
536
00:36:11,440 --> 00:36:17,210
What is measure-preserving
transformation?
537
00:36:17,210 --> 00:36:19,070
If it's a right-handed system
538
00:36:19,070 --> 00:36:21,560
beforehand, there's no squishing.
539
00:36:21,560 --> 00:36:23,740
It doesn't go to an oblique
coordinate system.
540
00:36:23,740 --> 00:36:26,100
Cartesian stays Cartesian.
541
00:36:26,100 --> 00:36:30,470
If the reference axes are of
equal lengths, they don't
542
00:36:30,470 --> 00:36:33,440
stretch upon the
transformation.
543
00:36:33,440 --> 00:36:36,150
You can define transformations
like that if you like, where
544
00:36:36,150 --> 00:36:40,950
the angles between them go from
orthogonal to oblique
545
00:36:40,950 --> 00:36:43,670
after the transformation, and
the units of length along the
546
00:36:43,670 --> 00:36:45,900
three axes change dimension.
547
00:36:45,900 --> 00:36:49,920
But then the determinant of the
coefficients is not unity,
548
00:36:49,920 --> 00:36:52,170
and a lot of the nice,
convenient properties that
549
00:36:52,170 --> 00:36:53,420
we've seen here do not hold.
550
00:36:53,420 --> 00:36:57,205
551
00:36:57,205 --> 00:37:01,620
All right, that is a good
place to stop, I think.
552
00:37:01,620 --> 00:37:02,970
Next week, no quiz.
553
00:37:02,970 --> 00:37:04,650
If you came in late, we're
going to postpone
554
00:37:04,650 --> 00:37:06,070
the quiz for a week.
555
00:37:06,070 --> 00:37:09,710
And let's see, look at all I
can ask you just after one
556
00:37:09,710 --> 00:37:12,930
lecture on tensors.
557
00:37:12,930 --> 00:37:16,700
What we will do next is explore
the form of the
558
00:37:16,700 --> 00:37:18,830
tensors for other crystal
symmetries.
559
00:37:18,830 --> 00:37:21,600
It goes fairly quickly.
560
00:37:21,600 --> 00:37:25,340
And then having done all that,
I'll show you how you can
561
00:37:25,340 --> 00:37:30,970
determine the symmetry
constraints by inspection for
562
00:37:30,970 --> 00:37:31,990
a tensor of any rank.
563
00:37:31,990 --> 00:37:35,240
And you're going to despise me
for that, but this was useful,
564
00:37:35,240 --> 00:37:39,090
because we can get used to
manipulating the notation.
565
00:37:39,090 --> 00:37:40,760
But there is a method called--
566
00:37:40,760 --> 00:37:43,380
appropriately enough-- the
method of direct inspection
567
00:37:43,380 --> 00:37:47,170
where you can very quickly and
very easily do the symmetry
568
00:37:47,170 --> 00:37:48,800
transformations.
569
00:37:48,800 --> 00:37:52,470
So all this and more will be
revealed next time, which is
570
00:37:52,470 --> 00:37:55,120
going to be a lot more
beneficial than taking a quiz.
571
00:37:55,120 --> 00:37:58,167