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PROFESSOR: All right.

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The quiz on Thursday will
cover up through

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piezoelectricity.

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You've had a set of notes in
your hands that cover just

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about everything that I wanted
to say about piezoelectricity.

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There are other modulae that
one could talk about.

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But these follow quite directly
from the one or two

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that we will do.

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So we will not have anything to
say about elasticity until

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the last lecture of the term,
which is a nice outcome

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because you certainly don't
want to take a quiz on

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forthright tensors.

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You'll spend the entire hour
just writing out all these

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cumbersome equations.

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All right.

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So the quiz will cover up
through piezoelectricity,

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including a few of the
representation surfaces that

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we'll examine today.

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The thing that we'll
be looking at is--

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I'm sorry.

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You have a question?

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AUDIENCE: Yes.

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A question.

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Well, you said [INAUDIBLE]

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example.

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PROFESSOR: I'm sorry.

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Third-rank tensors.

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AUDIENCE: OK.

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PROFESSOR: Also, we had just a
little bit of that going into

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the second quiz.

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And we didn't really ask
anything about that on the

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second quiz.

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So it'd be third-rank tensors.

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But you have to use second-rank
tensors to define

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third-rank tensors.

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So it really will be not the
emphasis, but certainly you

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should be familiar with the
early part of what we did with

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second-rank tensors.

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All right.

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So today we'll look at some
representation surfaces to

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your wonder and delight at how
incredibly anisotropic the

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variation of these third-rank
properties are with direction.

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One of the things that I love to
do for problems when we get

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to these different piezoelectric
effects is make

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up hypothetical devices
just for fun.

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So among those that I've
invented, the first one is a

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earthquake sensing device
because it's well known that

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California is going to split
in half, and half will fall

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into the sea any day now.

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So if this is the San Andreas
fault, I have developed large,

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prismatic monoclinic crystals
that I embed into the San

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Andreas fault at regular
intervals.

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The bottom of these crystals
is grounded.

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There's an electrode
at the top.

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And if the San Andreas fault
starts to move, and there is

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shear on these crystals,
there will be a charge

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developed on the top.

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And I ask you to relate the
charge on the top in terms of

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the piezoelectric modulae to
the shear along the San

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Andreas fault.

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So there's a very clever little
device that clearly is

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going to be lucrative because
these crystals will have to be

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huge in size.

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And they'll cost more
even than silicon.

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Another device that I've
invented is used down at the

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Boston fish pier.

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This is a crystal on which
I hang a pan, and

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you put fish in it.

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That creates a tensile stress
in this direction.

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We measure the charge
on this face.

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And we put a little meter that
measures the charge that's

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accumulated.

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So this is how the fishermen
can weigh their fish after

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they haul them in
off the boat.

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So you see this is really
practical material we're

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dealing with.

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This has applications in
all realms of life.

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So today I'd like to show
you, also, another

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problem that sets up.

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And I pass this out just so you
can, again, have a look at

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some of the questions that you
should be equipped to answer.

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So here--

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not fully expecting anybody to
do it, but you can see the

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sorts of problems
one might ask.

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Here is problem set number 16,
which asks you, should you be

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so inclined, to think
about manipulations

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of third-rank tensors.

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But then the second question is
an idea for a device which

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the Center for Material Science
and Engineering is

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considering employing
here in building 13.

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And you can read all
about that..

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This is the famous soup cell.

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I think probably you will see
some sort of for fun modulus

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for a particular device
on the quiz.

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I haven't made one up yet.

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But I think you'll have a look
at something like that.

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And when we do the longitudinal
piezoelectric

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modulus for quartz, which we'll
do momentarily, this

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will give you an idea of how to
set up these expressions.

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The problem, basically, is that
the direct piezoelectric

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effect measures a polarization
in terms of all of the

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elements of applied stress.

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So there is a modulus dijk times
all of the elements of

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stress, sigma jk.

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So this, until you use the
condensation of subscripts,

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contains nine terms going
this way and three

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equations going this way.

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As we discussed earlier, since
only six of the nine elements

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of stress are independent, you
can condense this down into a

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3 by 6 array of terms.

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The problem in doing that,
though, even though one has

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only 18 different modulae to
work with, instead of 27--

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that's a considerable economy
in notation, and an

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elimination of a great
deal of redundancy--

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the matrix form of this
relation-- and I emphasize

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that it is a matrix, and it's
no longer a tensor--

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the matrix form cannot
be transformed.

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Again, there are only
six terms in the

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matrix elements of stress.

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And there are three equations,
again, one for each component

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of polarization.

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So instead of having 27,
one has only 18.

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But if you are considering
changing the reference axes--

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and that is one of the things
that it's interesting to do

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for these various modulae
that one can define--

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change the orientation of a
particular rod-shaped specimen

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that you cut out of a crystal to
different crystallographic

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orientations and then ask how
the scalar modulus changes as

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you change the direction in
which you've sliced out the

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wafer or the rod of material.

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In order to do that, you want to
transform the piezoelectric

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tensor to a new set
of reference axes.

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And you cannot transform the
dij's because they are a

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matrix and not a tensor.

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And no law of transformation
is defined.

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So what you have to do in any
generic problem of this sort

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is, for a particular single
crystal, look up the form of

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the dij matrix that will have
the equalities between tensor

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elements written in and the 0's,
those modulae which are

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identically 0, entered
into the array.

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And then you have to work your
way backwards to get to the

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full tensor notation.

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So we'll see this when we
look at the modulae for

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symmetry three two.

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You'll have to write in the
exact matrix subscripts

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without absorbing the equalities
in the notation.

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And then you'll have to expand
the matrix terms into full

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three-subscript tensor terms.

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Then if you want to transform
the axes, which is to say you

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want to cut out your specimen,
be it a plate or a rod, in a

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different orientation, you have
to transform the full

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three subscript tensor elements
to a new setting.

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And then if you want to continue
to work in that

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setting in the compact matrix
form, collapse it back down to

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matrix form, insert the
equalities, and then you're

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back to where you
started from.

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So this problem of seeing how
modulae that describe

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different phenomena vary with
crystal symmetry, you have to

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go through this problem of
expanding the compact form and

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then collapsing back down when
you've got it as a function of

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some orientational angle or in
terms of a coordinate system

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that you want to work in.

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The problem is exactly the
same for elasticity.

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And we'll look at some of
these modulae next term.

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You've heard these names
before, I'm sure--

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Young's modulus, shear
modulus, and so on.

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We'll take a look on next
Tuesday, a week from today, at

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Young's modulus, which is one
of the more important ones.

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And probably are used to seeing
this in the form of

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information for a
polycrystalline material,

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which is essentially
isotropic.

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The modulus is much more
interesting for

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single crystal materials.

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And then the surfaces that are
defined particularly for the

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lower symmetries are absolutely
wild things with

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lumps and wiggles and lobes
and things of that sort,

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nothing like the dumb old,
uninteresting ellipsoids that

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we encountered for second-rank
properties.

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So let's, then, take a look at
how we had set up and defined

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the direct piezoelectric
effect.

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We set this up as a proper
tensor relation, saying that

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P1 is equal to d1.

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V1 And then, you'll recall, we
have nine elements of strain--

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sigma 11, sigma 12, sigma 13,
sigma 21, sigma 22, sigma 23,

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sigm a 31, sigma 32,
and sigma 33.

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But the tensor is symmetric,
so we really only need to

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enter into our relation six of
these nine terms explicitly.

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And what we did was to replace
the two subscripts on the

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elements of strain, which,
again, you need if you want to

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refer to those elements of
stress through a different

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coordinate system.

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You have to know the elements
of stress in tensor form.

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But we convert it to six terms
by going and replacing the

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pairs of subscripts with a
single one, two, and three,

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00:12:02,580 --> 00:12:05,820
marching down the diagonal of
the tensor this way and then

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00:12:05,820 --> 00:12:09,080
marching up the right-hand
side, calling two, three,

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00:12:09,080 --> 00:12:14,300
four, and calling one, three,
five, and finally ending up in

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this slot here.

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And we call that six.

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00:12:18,860 --> 00:12:22,570
So that was the notation we
used to get to a matrix

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representation.

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00:12:23,520 --> 00:12:26,560
But the place where all
this started is--

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00:12:26,560 --> 00:12:30,190
and I'll write just one line
of this to be merciful--

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00:12:30,190 --> 00:12:37,340
d 111 times sigma 11, so this
pair of subscripts goes with

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00:12:37,340 --> 00:12:41,270
this pair, plus d 122.

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00:12:41,270 --> 00:12:43,870
I'm putting that one in next
because we're going to number

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00:12:43,870 --> 00:12:48,630
the terms for stress in this
order, one through six.

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00:12:48,630 --> 00:13:02,230
Times sigma 22 plus d 133 times
sigma 33 plus d 123

213
00:13:02,230 --> 00:13:23,130
times sigma 23 plus d 132 times
sigma 32 plus d113 times

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00:13:23,130 --> 00:13:31,340
sigma 23 plus d 132
times sigma 32.

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00:13:31,340 --> 00:13:35,040
And finally we end up
in slot number six.

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00:13:35,040 --> 00:13:42,470
And we have a d 112 times
sigma 12 plus a d

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00:13:42,470 --> 00:13:46,780
121 times sigma 21.

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00:13:46,780 --> 00:13:47,110
Look at that.

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00:13:47,110 --> 00:13:50,280
There's an equation that covers
two whole blackboards.

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So if we now condense this down
to matrix form we would

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00:13:57,670 --> 00:14:05,840
say that P1 is d 11 times sigma
1 plus d 12 times sigma

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00:14:05,840 --> 00:14:11,320
2 times d 13 times
sigma 3 plus--

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00:14:11,320 --> 00:14:13,910
and now we have this messy
problem with the 2's--

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00:14:13,910 --> 00:14:24,020
we have a d 14 times sigma
4 plus, again, a d 14

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00:14:24,020 --> 00:14:25,560
times a sigma --

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00:14:25,560 --> 00:14:29,890
2 3 is equal to 32, so
I can call this 4--

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00:14:29,890 --> 00:14:36,580
and then these terms become 15
sigma 5 and, again, a 15 times

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00:14:36,580 --> 00:14:45,595
sigma 5 plus a 16 times sigma
6 plus d 16 times sigma 6.

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00:14:49,850 --> 00:14:52,690
Now we have to make a choice.

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00:14:52,690 --> 00:14:57,190
Either we are going to have, in
a general matrix relation,

231
00:14:57,190 --> 00:15:06,220
that P sub i is equal to dij
times sigma j, if j is equal

232
00:15:06,220 --> 00:15:08,740
to 1, 2, or 3.

233
00:15:08,740 --> 00:15:14,490
But it's equal to 2 dij
times sigma j if j is

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00:15:14,490 --> 00:15:17,500
equal to 4, 5, or 6.

235
00:15:17,500 --> 00:15:20,940
And that is something we like
to avoid, if possible.

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00:15:20,940 --> 00:15:23,470
That's ugly.

237
00:15:23,470 --> 00:15:25,070
That's ugly.

238
00:15:25,070 --> 00:15:28,120
And it's going to be a hell of a
matrix if we have factors of

239
00:15:28,120 --> 00:15:31,040
two in front of some of the
matrix elements but not in

240
00:15:31,040 --> 00:15:32,090
terms of others.

241
00:15:32,090 --> 00:15:34,355
So this is something
we could do.

242
00:15:34,355 --> 00:15:35,700
Hey, it's our ballgame.

243
00:15:35,700 --> 00:15:36,750
We make up the rules.

244
00:15:36,750 --> 00:15:39,790
But that's going to be an ugly
thing to have to deal with.

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00:15:39,790 --> 00:15:44,740
So instead, as we mentioned last
time, what we will do is

246
00:15:44,740 --> 00:15:53,410
to lump together these terms and
define d 14 not as these

247
00:15:53,410 --> 00:15:57,360
individual tensor elements,
but define those matrix

248
00:15:57,360 --> 00:16:04,700
elements as the sum of
these two elements.

249
00:16:04,700 --> 00:16:07,450
And then we saw before that--

250
00:16:07,450 --> 00:16:10,470
we saw last time in our
earlier meeting--

251
00:16:10,470 --> 00:16:14,140
that from the converse
piezoelectric effect, which

252
00:16:14,140 --> 00:16:18,270
expresses strain in terms of
an applied field, where the

253
00:16:18,270 --> 00:16:22,280
elements of strain
1, 23, and 32

254
00:16:22,280 --> 00:16:24,300
appear in separate equations.

255
00:16:24,300 --> 00:16:29,380
And knowing that the same
array of piezoelectric

256
00:16:29,380 --> 00:16:32,700
coefficients amazingly describes
the converse

257
00:16:32,700 --> 00:16:36,020
piezoelectric effect as well
as the direct piezoelectric

258
00:16:36,020 --> 00:16:42,920
effect, we know that
d 123 equals d 132.

259
00:16:42,920 --> 00:16:44,933
And that is from the
converse effect.

260
00:16:51,700 --> 00:16:59,335
So equivalent to saying this is
to define d 14 as twice d

261
00:16:59,335 --> 00:17:03,080
123 because these two
elements are equal.

262
00:17:03,080 --> 00:17:06,869
So we're eating the factor of
two here so that we can write

263
00:17:06,869 --> 00:17:10,589
a nice matrix relation
that doesn't

264
00:17:10,589 --> 00:17:12,500
involve a factor of two.

265
00:17:12,500 --> 00:17:18,890
So making this combination of
terms for the shear stresses,

266
00:17:18,890 --> 00:17:25,355
we would have simply d 14 sigma
4 plus d 15 times sigma

267
00:17:25,355 --> 00:17:30,580
5 plus d 16 times sigma 6.

268
00:17:30,580 --> 00:17:33,790
And we can say, in general, for
the other two equations,

269
00:17:33,790 --> 00:17:41,370
by analogy to this one, that P
sub i is dij times sigma j--

270
00:17:41,370 --> 00:17:45,510
a nice, neat matrix relation
but one for which,

271
00:17:45,510 --> 00:17:49,550
unfortunately, there's no law
of transformation for the

272
00:17:49,550 --> 00:17:52,900
matrix modulae dij.

273
00:17:52,900 --> 00:17:55,270
If you want to change to another
coordinate system, we

274
00:17:55,270 --> 00:17:58,990
have to be prepared to resurrect
this full three

275
00:17:58,990 --> 00:18:02,030
subscript notation on the
piezoelectric modulae.

276
00:18:09,320 --> 00:18:09,620
OK.

277
00:18:09,620 --> 00:18:11,573
Comments or questions
at this point?

278
00:18:18,475 --> 00:18:18,970
All right.

279
00:18:18,970 --> 00:18:23,910
If not, let me remind you that
in the notes which I

280
00:18:23,910 --> 00:18:30,570
distributed last time, there
are summarized all of the

281
00:18:30,570 --> 00:18:36,020
constraints imposed on the
piezoelectric modulae for

282
00:18:36,020 --> 00:18:37,270
single crystals.

283
00:18:40,120 --> 00:18:47,100
And, again, these constraints,
these requirements that the

284
00:18:47,100 --> 00:18:54,500
tensors remain invariant for the
change of axes produced by

285
00:18:54,500 --> 00:19:00,120
a symmetry element that the
crystal possesses, these

286
00:19:00,120 --> 00:19:05,880
transformations show that no
third-rank tensor property can

287
00:19:05,880 --> 00:19:08,510
exist in a crystal that
has inversion.

288
00:19:08,510 --> 00:19:12,980
So the 11 [INAUDIBLE] group,
so-called, that possess

289
00:19:12,980 --> 00:19:17,360
inversion have absolutely no
property and can be not

290
00:19:17,360 --> 00:19:20,150
considered further for
third-rank properties.

291
00:19:20,150 --> 00:19:26,730
And then one must consider all
of the 32 minus 11 21 point

292
00:19:26,730 --> 00:19:29,400
groups that lack conversion
separately.

293
00:19:29,400 --> 00:19:32,890
There's no reason why they
should behave the same way.

294
00:19:32,890 --> 00:19:36,320
Remember that for second-rank
tensor properties we pulled

295
00:19:36,320 --> 00:19:41,480
the argument that inversion
imposes no restrictions or

296
00:19:41,480 --> 00:19:43,960
constraints whatsoever on
second-rank properties so,

297
00:19:43,960 --> 00:19:48,250
therefore, two different
symmetries that differ only by

298
00:19:48,250 --> 00:19:51,010
the presence or absence of
an inversion center.

299
00:19:51,010 --> 00:19:54,460
That is to say, you
change 2 to 2 over

300
00:19:54,460 --> 00:19:56,150
m if you add inversion.

301
00:19:56,150 --> 00:19:59,240
But the argument was inversion
requires nothing, so the

302
00:19:59,240 --> 00:20:01,590
constraints or symmetry,
too, look exactly

303
00:20:01,590 --> 00:20:02,750
like those for symmetry.

304
00:20:02,750 --> 00:20:06,450
And here you've got to plod
through every single one of

305
00:20:06,450 --> 00:20:09,860
the non-centrosymmetric point
groups separately.

306
00:20:09,860 --> 00:20:14,080
And they all have tensors that
have different forms.

307
00:20:14,080 --> 00:20:17,840
So what I'd like to do is look
at one specific one.

308
00:20:17,840 --> 00:20:24,750
And that is the matrix
for symmetry 32.

309
00:20:24,750 --> 00:20:28,510
And that is a point group that
you'll recall has a threefold

310
00:20:28,510 --> 00:20:33,665
axis and twofold axes at
intervals of 60 degrees.

311
00:20:37,060 --> 00:20:45,760
And in your list of the
qualities and absences, 32,

312
00:20:45,760 --> 00:20:51,570
where the 3 is parallel to the
axis x3, and the twofold axis

313
00:20:51,570 --> 00:20:53,700
is parallel to x1.

314
00:20:53,700 --> 00:21:01,560
So we're defining this as x1,
this as x3, and x2 comes out

315
00:21:01,560 --> 00:21:03,720
halfway between the two.

316
00:21:03,720 --> 00:21:10,130
So let me draw this looking down
along the threefold axis.

317
00:21:10,130 --> 00:21:14,000
These are all twofold axes.

318
00:21:14,000 --> 00:21:16,960
And we'll take x1 in
this direction.

319
00:21:16,960 --> 00:21:21,600
x2 pokes out in between
two twofold axes.

320
00:21:21,600 --> 00:21:26,510
And x3 comes straight up along
the threefold axis.

321
00:21:26,510 --> 00:21:32,050
With that coordinate system, the
form of the piezoelectric

322
00:21:32,050 --> 00:21:44,884
modulus matrix has this form-- d
11 minus d 11 0 d 14 0 0 0 0

323
00:21:44,884 --> 00:21:56,680
0 d 15 minus d 14 0 and
in the bottom row d 31

324
00:21:56,680 --> 00:22:04,780
d 31 d 33 0 0 0.

325
00:22:04,780 --> 00:22:11,080
So this matrix with the
equalities put in is obtained

326
00:22:11,080 --> 00:22:15,720
by looking at the full
three-subscript tensor and

327
00:22:15,720 --> 00:22:19,810
requiring that it look the same
before and after any of

328
00:22:19,810 --> 00:22:26,790
the rotations involved by these
three distinct axes--

329
00:22:26,790 --> 00:22:28,500
the threefold and the
pair of twofolds.

330
00:22:32,870 --> 00:22:33,090
OK.

331
00:22:33,090 --> 00:22:37,530
Now, there are many different
scalar modulae that one could

332
00:22:37,530 --> 00:22:42,220
define, some of them serious
and worth the consideration

333
00:22:42,220 --> 00:22:47,940
because of the application in
devices or other practical

334
00:22:47,940 --> 00:22:49,740
situations.

335
00:22:49,740 --> 00:22:58,100
The modulus that I'd like to
examine is something called

336
00:22:58,100 --> 00:23:01,280
the longitudinal piezoelectric
effect.

337
00:23:12,960 --> 00:23:17,390
And let's emphasize, again, that
it is impossible to come

338
00:23:17,390 --> 00:23:22,250
up with one representation
surface that fits every need

339
00:23:22,250 --> 00:23:28,410
because, again, the direct
piezoelectric effect relates

340
00:23:28,410 --> 00:23:36,170
the components of a vector
to a tensor, sigma ij.

341
00:23:36,170 --> 00:23:40,470
There are six independent
tensor elements.

342
00:23:40,470 --> 00:23:46,110
So how can you describe how this
vector is going to change

343
00:23:46,110 --> 00:23:51,320
as you change orientation of
a crystal whose behavior is

344
00:23:51,320 --> 00:23:53,840
described by all of
these modulae?

345
00:23:53,840 --> 00:23:57,250
But I would point out, however,
that there are only

346
00:23:57,250 --> 00:23:59,160
two distinct--

347
00:23:59,160 --> 00:24:00,073
oops, I'm sorry.

348
00:24:00,073 --> 00:24:01,890
This is d 14.

349
00:24:01,890 --> 00:24:04,260
I don't know how I
made that d 15.

350
00:24:04,260 --> 00:24:09,470
There are only, in
this array--

351
00:24:09,470 --> 00:24:10,730
and I slipped a notch.

352
00:24:10,730 --> 00:24:11,980
Excuse me.

353
00:24:15,220 --> 00:24:19,990
Last couple of days are such
that I am not able to even

354
00:24:19,990 --> 00:24:21,420
read from my notes.

355
00:24:21,420 --> 00:24:25,190
So these bottom lines, my
apologies, are all 0.

356
00:24:25,190 --> 00:24:30,600
And there are two modulae,
d 11 and d 14.

357
00:24:30,600 --> 00:24:36,120
So there are two independent
numbers, but they appear as

358
00:24:36,120 --> 00:24:39,540
different matrix elements and,
therefore, different tensor

359
00:24:39,540 --> 00:24:42,040
elements, as well.

360
00:24:42,040 --> 00:24:44,610
And this should be
minus 2 d 14.

361
00:24:47,610 --> 00:24:47,970
I'm sorry.

362
00:24:47,970 --> 00:24:50,910
I slipped down a notch,
and I got some for 32

363
00:24:50,910 --> 00:24:52,630
and some for 6.

364
00:24:52,630 --> 00:24:55,030
And I'm glad I found it at this
point, or I'd really be

365
00:24:55,030 --> 00:24:56,280
in deep trouble.

366
00:24:58,970 --> 00:24:59,270
OK.

367
00:24:59,270 --> 00:25:04,160
So two numbers and that is, in
fact, the form of the matrix

368
00:25:04,160 --> 00:25:06,460
for symmetry 32.

369
00:25:06,460 --> 00:25:10,520
So the longitudinal
piezoelectric effect is one of

370
00:25:10,520 --> 00:25:14,730
the representation surfaces
that gives you the way in

371
00:25:14,730 --> 00:25:19,800
which the polarization will
change for one very, very

372
00:25:19,800 --> 00:25:22,850
specific type of stress.

373
00:25:22,850 --> 00:25:27,080
In particular, what we'll
do is set this up as

374
00:25:27,080 --> 00:25:28,330
a coordinate system.

375
00:25:30,940 --> 00:25:38,830
And we will look at a coordinate
system where this

376
00:25:38,830 --> 00:25:41,790
is the reference axis, x1.

377
00:25:41,790 --> 00:25:45,470
We'll come down with a
compressive stress, sigma 11,

378
00:25:45,470 --> 00:25:48,680
along that axis.

379
00:25:48,680 --> 00:25:53,750
And, therefore, we're applying
a uni-axial stress, which has

380
00:25:53,750 --> 00:25:55,890
just one component of stress.

381
00:25:55,890 --> 00:25:57,470
So that's very specialized.

382
00:25:57,470 --> 00:25:59,160
In general, there would
be six different

383
00:25:59,160 --> 00:26:00,500
components of stress.

384
00:26:00,500 --> 00:26:04,160
But we're looking at one
specific stimulus applied to

385
00:26:04,160 --> 00:26:05,600
this crystal plate.

386
00:26:05,600 --> 00:26:09,960
In response to that sigma 11
there are charges induced on

387
00:26:09,960 --> 00:26:11,210
all of these surfaces.

388
00:26:16,170 --> 00:26:20,210
And these charges, this charge
per unit area, is proportional

389
00:26:20,210 --> 00:26:26,840
to the component of polarization
P1, the component

390
00:26:26,840 --> 00:26:31,590
of polarization P2 along the
surface out of which x 2

391
00:26:31,590 --> 00:26:35,610
comes, and the charge
per unit area or the

392
00:26:35,610 --> 00:26:37,420
polarization along x3.

393
00:26:37,420 --> 00:26:38,880
So this would be P3.

394
00:26:38,880 --> 00:26:42,560
And this would be the
direction of x3.

395
00:26:42,560 --> 00:26:47,600
Now I deliberately tried to
show this sample as a thin

396
00:26:47,600 --> 00:26:53,900
wafer, which has a much larger
surface area here than it does

397
00:26:53,900 --> 00:26:57,950
on the other two surfaces--
a much smaller area there.

398
00:26:57,950 --> 00:27:01,350
Therefore, since polarization
is charge per unit area, if

399
00:27:01,350 --> 00:27:06,090
this area normal to x1 is a very
large area, there's a lot

400
00:27:06,090 --> 00:27:07,810
of charge accumulated there.

401
00:27:07,810 --> 00:27:09,330
It's going to be easy
to measure.

402
00:27:09,330 --> 00:27:14,620
If we make the wafer vanishingly
thin, then the

403
00:27:14,620 --> 00:27:18,340
charge per unit area is high,
but the total area is small.

404
00:27:18,340 --> 00:27:22,500
So there's going to be a
negligible accumulation of

405
00:27:22,500 --> 00:27:25,240
charge on these two
side surfaces.

406
00:27:25,240 --> 00:27:28,150
So the longitudinal
piezoelectric effect and the

407
00:27:28,150 --> 00:27:33,710
longitudinal piezoelectric
electric modulus is an effect,

408
00:27:33,710 --> 00:27:38,400
where we look at the component
of polarization, P1, in

409
00:27:38,400 --> 00:27:42,050
response to an applied
stress, sigma 11.

410
00:27:44,900 --> 00:27:45,910
So it's that simple.

411
00:27:45,910 --> 00:27:48,070
Look at all the terms
we've thrown out.

412
00:27:48,070 --> 00:27:51,920
We've thrown out a whole bunch
of elements of stress, which

413
00:27:51,920 --> 00:27:55,800
we could impose if
we wanted to.

414
00:27:55,800 --> 00:27:58,690
And we've thrown away two of
the three components of

415
00:27:58,690 --> 00:28:02,510
polarization by designing
a specialized sample.

416
00:28:02,510 --> 00:28:06,580
So all that's left then is
that P1 equals sigma 11.

417
00:28:06,580 --> 00:28:13,430
And the relation between those
two parameters is the 111.

418
00:28:13,430 --> 00:28:17,920
So all this is going to hinge
on one single piezoelectric

419
00:28:17,920 --> 00:28:21,940
electric modulus, d
111, and how that

420
00:28:21,940 --> 00:28:23,335
changes with direction.

421
00:28:26,410 --> 00:28:32,830
So this is for one orientation
of a plate.

422
00:28:32,830 --> 00:28:37,110
And I had not specified how the
orientation of this plate

423
00:28:37,110 --> 00:28:39,600
is related to the
symmetry axes.

424
00:28:39,600 --> 00:28:40,850
So let's do that now.

425
00:28:45,290 --> 00:28:50,270
What I'm going to assume is
that this is a crystal.

426
00:28:50,270 --> 00:28:52,280
It doesn't look like
it's hexagonal.

427
00:28:52,280 --> 00:28:55,130
But imagine that this is
a crystal of quartz.

428
00:28:55,130 --> 00:29:00,270
And we could look at an x1
that's in this direction.

429
00:29:00,270 --> 00:29:07,690
And imagine that we have cut
out of this crystal a wafer

430
00:29:07,690 --> 00:29:10,070
that has a normal along x1.

431
00:29:10,070 --> 00:29:13,450
And then relative to this
coordinate system, if this is

432
00:29:13,450 --> 00:29:20,120
x1, the modulus d 111 would
tell us what charges

433
00:29:20,120 --> 00:29:21,725
accumulated on these
two surfaces.

434
00:29:26,300 --> 00:29:29,960
But now, what we could do if
we wanted to know how this

435
00:29:29,960 --> 00:29:35,030
modulus changed with direction
would be to cut out a plate, a

436
00:29:35,030 --> 00:29:37,880
thin plate, in another
orientation,

437
00:29:37,880 --> 00:29:43,300
where this is x1 prime.

438
00:29:43,300 --> 00:29:51,160
And this has changed relative to
the orientation of the cell

439
00:29:51,160 --> 00:29:52,240
edges in the crystal.

440
00:29:52,240 --> 00:29:53,220
The crystal is fixed.

441
00:29:53,220 --> 00:29:56,060
We're just cutting a wafer out
in a different orientation.

442
00:29:56,060 --> 00:29:58,280
So this is x1 prime.

443
00:29:58,280 --> 00:30:03,360
We're going to, again, squeeze
it with a tensile

444
00:30:03,360 --> 00:30:06,490
stress sigma 11 prime.

445
00:30:06,490 --> 00:30:12,090
And we'll ask how the
polarization P1 prime is

446
00:30:12,090 --> 00:30:13,820
related to sigma 11 prime.

447
00:30:13,820 --> 00:30:19,460
And the answer is that P1 prime
will be a tensor element

448
00:30:19,460 --> 00:30:22,440
d 11 prime times
sigma 11 prime.

449
00:30:26,220 --> 00:30:33,460
So what we are asking,
essentially, is how does d 11

450
00:30:33,460 --> 00:30:38,280
transform when we take the
direction of x1 in a different

451
00:30:38,280 --> 00:30:43,225
orientation and, thus, change
the value of d 11 prime?

452
00:30:43,225 --> 00:30:44,560
It's going to change all of the

453
00:30:44,560 --> 00:30:46,590
piezoelectric matrix elements.

454
00:30:46,590 --> 00:30:50,800
But we're looking at an effect
in a sample that is

455
00:30:50,800 --> 00:30:56,540
deliberately prepared such that
we will measure only the

456
00:30:56,540 --> 00:30:58,780
surface charge given
by P1 prime.

457
00:30:58,780 --> 00:31:01,780
And, therefore, the way in which
the properties of this

458
00:31:01,780 --> 00:31:06,040
plate change as we vary the way
in which we've sliced it

459
00:31:06,040 --> 00:31:09,570
out of the single crystal is
going to be simply the

460
00:31:09,570 --> 00:31:13,970
variation of d 11 prime
with direction.

461
00:31:13,970 --> 00:31:18,170
So this is the general nature
of what we will do when we

462
00:31:18,170 --> 00:31:26,100
define any of the scalar
modulae related to the

463
00:31:26,100 --> 00:31:27,350
piezoelectric modulus tensor.

464
00:31:30,600 --> 00:31:37,500
We can change our notation a
little bit in that we have a

465
00:31:37,500 --> 00:31:41,100
modulus which I'll define
as a scalar modules d.

466
00:31:41,100 --> 00:31:45,330
And that d is going
to be d 111 prime.

467
00:31:49,800 --> 00:31:53,620
And I know how to
evaluate that.

468
00:31:53,620 --> 00:32:00,990
d 111 priime will be C 1l, C1m,
C 1n, where these are

469
00:32:00,990 --> 00:32:06,220
direction cosines, times all
of the elements in the

470
00:32:06,220 --> 00:32:12,330
original tensors, dlmn, in the
tensor referred to the

471
00:32:12,330 --> 00:32:13,810
original coordinate system.

472
00:32:13,810 --> 00:32:15,550
So even though this
looks simple--

473
00:32:15,550 --> 00:32:17,500
it's just one modulus--

474
00:32:17,500 --> 00:32:20,210
when we transform it, we've
got a product of three

475
00:32:20,210 --> 00:32:25,460
direction cosines out in front
at every single one of the 27

476
00:32:25,460 --> 00:32:29,760
tensor elements in the
original tensor.

477
00:32:29,760 --> 00:32:32,350
So it's not as trivial
as it seems.

478
00:32:32,350 --> 00:32:35,750
So this is how this modulus
that relates compressive

479
00:32:35,750 --> 00:32:38,630
stress to induced surface
charge will change with

480
00:32:38,630 --> 00:32:39,620
orientation.

481
00:32:39,620 --> 00:32:42,930
But what are these direction
cosines?

482
00:32:42,930 --> 00:32:44,850
These are the direction
cosines--

483
00:32:44,850 --> 00:32:47,650
not the full direction
cosine matrix.

484
00:32:47,650 --> 00:32:52,590
These are the direction
cosines for x1 prime.

485
00:32:52,590 --> 00:32:54,530
OK?

486
00:32:54,530 --> 00:33:00,970
So we can get rid of this
two-subscript notation if it's

487
00:33:00,970 --> 00:33:13,190
understood that these are the
direction cosines of x1 and

488
00:33:13,190 --> 00:33:22,270
simply call these l l, l m, and
l n, just as we did for

489
00:33:22,270 --> 00:33:24,910
the direction cosines of a
vector because we're only

490
00:33:24,910 --> 00:33:27,990
concerned about the orientation
of one of the

491
00:33:27,990 --> 00:33:30,790
axes, namely x1 prime.

492
00:33:30,790 --> 00:33:35,260
We don't care diddly-bop about
x2 prime or x3 prime because

493
00:33:35,260 --> 00:33:38,400
these don't enter into the
modulus that we have defined.

494
00:33:38,400 --> 00:33:39,940
And this will be times dlmn.

495
00:33:44,216 --> 00:33:44,960
OK.

496
00:33:44,960 --> 00:33:46,210
Is what we're doing clear?

497
00:33:49,660 --> 00:33:57,810
So we have defined this
particular effect in terms of

498
00:33:57,810 --> 00:34:03,900
those of the 27 piezoelectric
modulae which are necessary to

499
00:34:03,900 --> 00:34:05,350
describe it.

500
00:34:05,350 --> 00:34:09,370
And then we've established how
they will change with a change

501
00:34:09,370 --> 00:34:12,929
of the direction of one
particular direction.

502
00:34:12,929 --> 00:34:15,119
And we don't care anything
about x2 or x3.

503
00:34:17,690 --> 00:34:18,040
All right.

504
00:34:18,040 --> 00:34:23,690
Now we go through this process
of inserting for matrix

505
00:34:23,690 --> 00:34:27,100
notation with the equalities
built in.

506
00:34:27,100 --> 00:34:35,679
The proper matrix notation in
the first term is d 11 in

507
00:34:35,679 --> 00:34:37,070
matrix notation.

508
00:34:37,070 --> 00:34:42,030
That's this term up here in the
upper left-hand corner.

509
00:34:42,030 --> 00:34:49,150
The second term, the term that
we've written as minus d 11,

510
00:34:49,150 --> 00:34:51,139
that's not d 11 at all.

511
00:34:51,139 --> 00:34:55,560
This is, by definition, d 12.

512
00:34:55,560 --> 00:34:58,400
And we need the true subscripts,
if we're going to

513
00:34:58,400 --> 00:35:00,140
transform this.

514
00:35:00,140 --> 00:35:04,775
So this really is not even a
matrix because the subscripts

515
00:35:04,775 --> 00:35:08,170
have lost meaning, and we're
just using them to identify

516
00:35:08,170 --> 00:35:09,950
equalities.

517
00:35:09,950 --> 00:35:11,340
Then comes a 0.

518
00:35:11,340 --> 00:35:14,880
And next comes something that
we've labeled d 14.

519
00:35:14,880 --> 00:35:19,880
And the subscripts there
are correct.

520
00:35:19,880 --> 00:35:23,580
That is, indeed, the fourth
term in the first row.

521
00:35:23,580 --> 00:35:26,870
But we're going to want to
convert d 14 into a tensor

522
00:35:26,870 --> 00:35:28,050
element momentarily.

523
00:35:28,050 --> 00:35:32,010
Now let's get the rest of the
terms that are non-zero.

524
00:35:32,010 --> 00:35:33,325
This is really d 25.

525
00:35:37,040 --> 00:35:41,930
So the next term that is
non-zero is d 25, which just

526
00:35:41,930 --> 00:35:45,940
happens, because of symmetry,
to be equal to minus d 14.

527
00:35:49,440 --> 00:35:55,360
But this is the true matrix
subscripts, and this is the

528
00:35:55,360 --> 00:35:57,930
true matrix subscript here.

529
00:35:57,930 --> 00:36:02,430
The next term over to the right
is the fifth and final

530
00:36:02,430 --> 00:36:03,720
non-zero term.

531
00:36:03,720 --> 00:36:06,535
This is minus 2 d 11.

532
00:36:09,970 --> 00:36:15,600
And this is really d 26.

533
00:36:18,920 --> 00:36:22,710
Those are the true
matrix elements.

534
00:36:22,710 --> 00:36:25,960
So we put in the proper
matrix subscripts.

535
00:36:25,960 --> 00:36:30,440
And now the next, final, step in
the expansion is to convert

536
00:36:30,440 --> 00:36:34,710
these terms into actual
tensor elements.

537
00:36:34,710 --> 00:36:38,230
So this is d 111.

538
00:36:38,230 --> 00:36:40,540
And these are tensor subscripts,
so this is

539
00:36:40,540 --> 00:36:42,770
something we can transform.

540
00:36:42,770 --> 00:36:44,460
This is d 12.

541
00:36:44,460 --> 00:36:48,780
In tensor notation
this is d 122.

542
00:36:48,780 --> 00:36:51,764
And that's something that has
a law of transformation.

543
00:36:51,764 --> 00:37:05,060
d 14 is really d
123 plus d 132.

544
00:37:05,060 --> 00:37:07,360
We lumped two tensor
elements together

545
00:37:07,360 --> 00:37:11,020
to define this modulus.

546
00:37:11,020 --> 00:37:14,750
Minus d 14 that appears in
the next to the last

547
00:37:14,750 --> 00:37:17,110
non-zero spot is d 25.

548
00:37:17,110 --> 00:37:27,370
d 25 is really d
231 plus d 213.

549
00:37:27,370 --> 00:37:40,415
Then, finally, d 26 is
d 121 plus d 112.

550
00:37:40,415 --> 00:37:42,690
AUDIENCE: Shouldn't
that be d221?

551
00:37:42,690 --> 00:37:43,070
PROFESSOR: Sorry.

552
00:37:43,070 --> 00:37:44,100
d 26, you're right.

553
00:37:44,100 --> 00:37:51,520
That's down in the second
row. d 221 and d 212.

554
00:37:51,520 --> 00:37:52,950
Now we've got something
we can transform.

555
00:37:56,400 --> 00:37:59,983
The law for transformation
is l sub l, l

556
00:37:59,983 --> 00:38:02,190
sub m, l sub n, dlmn.

557
00:38:02,190 --> 00:38:11,370
So And these are the direction
cosines of x1.

558
00:38:11,370 --> 00:38:19,570
So this term will transform as
l 1, l 1, l1 times d 111 .

559
00:38:19,570 --> 00:38:35,440
1 The next term will transform
as l 1, l 2, l 2 times d 122.

560
00:38:35,440 --> 00:38:39,020
And that will be d 122
prime for different

561
00:38:39,020 --> 00:38:40,560
orientation of x1.

562
00:38:40,560 --> 00:38:42,910
This will be two terms.

563
00:38:42,910 --> 00:38:45,780
This will be l 123.

564
00:38:45,780 --> 00:38:47,250
And I can write them
in any order.

565
00:38:47,250 --> 00:38:54,020
So this is l 123 times
d 123 plus d 132.

566
00:38:54,020 --> 00:38:57,720
And these terms prime, when we
change axes, are going to be

567
00:38:57,720 --> 00:39:19,050
equal to l 2, l 1, l 3 times
d 231 plus d 213.

568
00:39:19,050 --> 00:39:26,350
And this last term will be l
1, l 2 squared times d 221

569
00:39:26,350 --> 00:39:30,690
plus d 212.

570
00:39:30,690 --> 00:39:31,130
All right.

571
00:39:31,130 --> 00:39:34,530
So this now is our new tensor
element, d 11 prime.

572
00:39:38,760 --> 00:39:42,480
And that's given by
this sum of terms.

573
00:39:42,480 --> 00:39:49,860
So we'll have a first term
l 1 cubed times d 111.

574
00:39:49,860 --> 00:39:52,800
And if I look through these
other terms, that's the only

575
00:39:52,800 --> 00:39:56,300
term in l 1 cubed
that I'll have.

576
00:39:56,300 --> 00:40:00,260
The next term will involve the
product of three cosines--

577
00:40:00,260 --> 00:40:05,640
l 1 and l 2 squared
times d 122.

578
00:40:05,640 --> 00:40:10,420
And if I go down here, here's
an l 1, l 2 squared again.

579
00:40:10,420 --> 00:40:20,610
So I have plus d
221 plus d 212.

580
00:40:20,610 --> 00:40:24,100
And then, finally, the other
coefficient that I have is

581
00:40:24,100 --> 00:40:31,880
plus l 1, l 2, l 3--

582
00:40:31,880 --> 00:40:33,730
which is what this should be.

583
00:40:33,730 --> 00:40:41,450
And that will be times the sum
of terms d 123 plus d 132.

584
00:40:41,450 --> 00:40:43,450
Up here, you've got the
same thing again--

585
00:40:43,450 --> 00:40:50,170
plus d 231 plus d 213.

586
00:40:50,170 --> 00:40:56,145
And I have a total of
1, 2, 3, 4, 5--

587
00:40:56,145 --> 00:40:58,820
1, 2, 3, 4, 5 terms.

588
00:41:02,150 --> 00:41:08,180
OK, that is how the longitudinal
piezoelectric

589
00:41:08,180 --> 00:41:12,950
modulus will change as we change
the direction of the

590
00:41:12,950 --> 00:41:16,430
normal to the plate that we have
cut out of the crystal.

591
00:41:16,430 --> 00:41:20,170
So these are direction cosines
relative to the

592
00:41:20,170 --> 00:41:22,960
crystallographic axes.

593
00:41:22,960 --> 00:41:25,860
l 3 is the angle between the
normal to the plate and the

594
00:41:25,860 --> 00:41:27,240
threefold axis.

595
00:41:27,240 --> 00:41:31,200
l 1 is the angle cosine to the
angle between the normal to

596
00:41:31,200 --> 00:41:34,270
the plate and one of
the twofold axes.

597
00:41:34,270 --> 00:41:38,680
And l 2 is the direction cosine
for the normal to the

598
00:41:38,680 --> 00:41:41,930
threefold axis and
the twofold axis.

599
00:41:41,930 --> 00:41:42,240
OK.

600
00:41:42,240 --> 00:41:45,000
So we've got it now in terms
of tensor elements.

601
00:41:45,000 --> 00:41:45,800
And now--

602
00:41:45,800 --> 00:41:46,514
yeah?

603
00:41:46,514 --> 00:41:48,737
AUDIENCE: Is it at all
reasonable to assume instead

604
00:41:48,737 --> 00:41:53,430
of taking those sums in d 123,
d 122, just saying 2 d 123?

605
00:41:53,430 --> 00:41:57,244
Is that OK in assuming?

606
00:41:57,244 --> 00:41:58,048
Or not necessarily?

607
00:41:58,048 --> 00:42:00,500
PROFESSOR: Well, we
could do that.

608
00:42:00,500 --> 00:42:07,430
But I did it the long way to not
obscure what we're doing.

609
00:42:07,430 --> 00:42:08,620
OK?

610
00:42:08,620 --> 00:42:11,830
This is a well-defined summation
over subscripts.

611
00:42:11,830 --> 00:42:16,140
And we're going to collapse
immediately down to the sums.

612
00:42:16,140 --> 00:42:18,420
And we're going to replace
the equalities.

613
00:42:18,420 --> 00:42:22,110
So let's see what comes out
of this, if we now, having

614
00:42:22,110 --> 00:42:25,650
reached the zenith, having
transformed the tensor

615
00:42:25,650 --> 00:42:33,100
elements, go down and replace
this with a consolidation of

616
00:42:33,100 --> 00:42:35,580
terms and an insertion
of the equalities

617
00:42:35,580 --> 00:42:36,830
between the matrix elements.

618
00:42:39,810 --> 00:42:47,820
OK The first term is d 11.

619
00:42:47,820 --> 00:42:53,620
So I will have l 1
cubed times d 11.

620
00:42:53,620 --> 00:42:57,060
Notice I'm getting third powers
of direction cosines,

621
00:42:57,060 --> 00:43:00,530
which is going to be what causes
the exotic nature of

622
00:43:00,530 --> 00:43:02,990
these anisotropies.

623
00:43:02,990 --> 00:43:08,070
And then I have a product
of l 1 and l 2 squared.

624
00:43:10,700 --> 00:43:15,040
And this is d 12.

625
00:43:20,600 --> 00:43:25,800
And this second term is--

626
00:43:25,800 --> 00:43:26,760
where did it go?

627
00:43:26,760 --> 00:43:31,440
This is d 21 plus d 212.

628
00:43:31,440 --> 00:43:35,140
And that is what we call d 26.

629
00:43:42,550 --> 00:43:46,390
And then, finally, this product
of three different

630
00:43:46,390 --> 00:43:48,332
direction cosines--

631
00:43:48,332 --> 00:43:52,720
l 1, l 2, l 3.

632
00:43:52,720 --> 00:43:58,360
And we have d 231 plus d 213.

633
00:43:58,360 --> 00:43:59,680
And this is d 25.

634
00:44:05,740 --> 00:44:08,818
And, again, an l 1, l
2, l 3 times d 14.

635
00:44:13,780 --> 00:44:22,990
And the second term here is d
25, if I've done it correctly.

636
00:44:22,990 --> 00:44:24,240
d 25 --

637
00:44:25,940 --> 00:44:29,080
this is d 24.

638
00:44:29,080 --> 00:44:33,870
And this one is d 24.

639
00:44:33,870 --> 00:44:40,800
2/4 OK.

640
00:44:40,800 --> 00:44:43,830
Let's now insert the
equalities--

641
00:44:43,830 --> 00:44:45,840
back to where we came from.

642
00:44:45,840 --> 00:44:48,202
d 11 is d 11.

643
00:44:48,202 --> 00:44:51,820
d 12, however, for
symmetry 32--

644
00:44:58,551 --> 00:45:02,700
I'm going to my handy-dandy
chart of symmetry

645
00:45:02,700 --> 00:45:03,950
restrictions.

646
00:45:09,000 --> 00:45:09,850
I don't want to do that.

647
00:45:09,850 --> 00:45:12,037
That's fourth rank.

648
00:45:12,037 --> 00:45:13,170
AUDIENCE: It's still
on the board.

649
00:45:13,170 --> 00:45:16,090
PROFESSOR: It's still
on the board?

650
00:45:16,090 --> 00:45:16,880
Yes.

651
00:45:16,880 --> 00:45:17,710
Thank you.

652
00:45:17,710 --> 00:45:21,276
When your nose is in it,
it's hard to see.

653
00:45:21,276 --> 00:45:29,490
d 12 is minus d 11.

654
00:45:29,490 --> 00:45:33,750
1 And d 26 is minus 2 d 11.

655
00:45:38,890 --> 00:45:41,490
So these two terms can
be consolidated.

656
00:45:41,490 --> 00:45:53,380
l 1 cubed plus l 1, l 2 squared
times d 11 minus d 11.

657
00:45:53,380 --> 00:45:55,310
So these two terms die.

658
00:45:55,310 --> 00:46:00,670
And I have a minus 2
d 11 that's left.

659
00:46:00,670 --> 00:46:06,220
If I insert the equalities here,
I'll have l 1, l 2, l 3.

660
00:46:06,220 --> 00:46:08,840
d 25 is minus d 14.

661
00:46:12,790 --> 00:46:14,970
And here's a d 14 itself.

662
00:46:14,970 --> 00:46:17,550
So these two terms die.

663
00:46:17,550 --> 00:46:18,800
And then I had d 25.

664
00:46:21,290 --> 00:46:22,980
And that is--

665
00:46:22,980 --> 00:46:24,360
AUDIENCE: You don't have d25.

666
00:46:24,360 --> 00:46:25,120
PROFESSOR: I don't have d 25.

667
00:46:25,120 --> 00:46:26,903
Where did I get the extra one?

668
00:46:26,903 --> 00:46:28,153
AUDIENCE: [INAUDIBLE].

669
00:46:30,607 --> 00:46:31,845
PROFESSOR: OK.

670
00:46:31,845 --> 00:46:34,980
I'll take your word for it.

671
00:46:34,980 --> 00:46:38,320
And I know how it
has to turn out.

672
00:46:40,880 --> 00:46:41,220
OK.

673
00:46:41,220 --> 00:46:43,340
So these two terms
kill each other.

674
00:46:43,340 --> 00:46:48,080
And I'm left with, then,
an l 1, l 2, l 3.

675
00:46:51,330 --> 00:46:54,190
Or have I left something out?

676
00:46:54,190 --> 00:46:55,440
This is d 1--

677
00:47:02,370 --> 00:47:08,870
this is d 15 and d
231 and d 23 --

678
00:47:11,710 --> 00:47:16,470
uh, this is d 2 --

679
00:47:19,380 --> 00:47:28,870
and the combination of
13 and 31 is d 25.

680
00:47:28,870 --> 00:47:30,730
Right?

681
00:47:30,730 --> 00:47:34,090
And if I look at my equalities,
this is l 1, l 2,

682
00:47:34,090 --> 00:47:41,480
l 3 minus d 14 plus
d 14 plus d 25.

683
00:47:41,480 --> 00:47:45,055
And d 25 is minus d 14.

684
00:47:45,055 --> 00:47:47,970
AUDIENCE: Why would
you add this d25?

685
00:47:47,970 --> 00:47:49,210
PROFESSOR: Let me check
my notes and see

686
00:47:49,210 --> 00:47:50,370
what I've got here.

687
00:47:50,370 --> 00:47:51,620
AUDIENCE: [INAUDIBLE].

688
00:48:02,695 --> 00:48:03,681
PROFESSOR: OK.

689
00:48:03,681 --> 00:48:04,931
See what I --

690
00:48:08,140 --> 00:48:11,740
d 25 is minus d 14.

691
00:48:11,740 --> 00:48:13,750
And then I have just a d 14.

692
00:48:13,750 --> 00:48:17,010
I don't know --

693
00:48:17,010 --> 00:48:17,660
I see what I did.

694
00:48:17,660 --> 00:48:18,780
I put it in the wrong slot.

695
00:48:18,780 --> 00:48:20,880
This is d 14.

696
00:48:20,880 --> 00:48:25,150
And d 25 is the one
that's minus d 14.

697
00:48:25,150 --> 00:48:28,870
so this term dies, which
is nice because that

698
00:48:28,870 --> 00:48:30,500
cross term is messy.

699
00:48:30,500 --> 00:48:34,000
So what I'm left with, then, if
I check against my notes,

700
00:48:34,000 --> 00:48:45,400
is d l 1 cubed plus l 1,
l 2 squared times d 11.

701
00:48:45,400 --> 00:48:48,710
And then I have minus
d 1 minus--

702
00:48:48,710 --> 00:48:50,390
this doesn't belong in here.

703
00:48:57,630 --> 00:49:08,840
This wants to end up being l 1
cubed minus 3 l 1, l 2 squared

704
00:49:08,840 --> 00:49:10,290
times d 11.

705
00:49:16,720 --> 00:49:18,310
The first term is l 1 cubed.

706
00:49:18,310 --> 00:49:18,960
That's correct.

707
00:49:18,960 --> 00:49:21,340
The second term is
l 1, l 2 squared.

708
00:49:21,340 --> 00:49:25,260
We've got a minus d 1
plus minus 2 d 11.

709
00:49:33,580 --> 00:49:38,960
So I have minus 3.

710
00:49:38,960 --> 00:49:40,000
It should be a minus.

711
00:49:40,000 --> 00:49:40,855
Yeah, that carries
down to a minus.

712
00:49:40,855 --> 00:49:44,224
So I have minus 3 l 1, l 2
squared all time d 11.

713
00:49:47,170 --> 00:49:47,480
OK.

714
00:49:47,480 --> 00:49:52,630
So what we have ended up with
is an expression for the

715
00:49:52,630 --> 00:49:56,700
longitudinal piezoelectric
modulus as a function of

716
00:49:56,700 --> 00:49:58,110
orientation.

717
00:49:58,110 --> 00:50:02,960
The surprising thing is that l 3
does not appear here at all.

718
00:50:02,960 --> 00:50:07,900
It doesn't depend on the angle
between the normal to the

719
00:50:07,900 --> 00:50:12,210
plate and the threefold axis.

720
00:50:12,210 --> 00:50:15,510
It depends only on one modulus,
and that is a

721
00:50:15,510 --> 00:50:17,970
remarkable thing.

722
00:50:17,970 --> 00:50:23,540
This says that the shape of this
surface is independent,

723
00:50:23,540 --> 00:50:30,810
essentially, of the property,
any property that relates the

724
00:50:30,810 --> 00:50:40,360
one one prime to a uni-axial
stimulus, sigma 11.

725
00:50:40,360 --> 00:50:44,240
And you measure a vectory
component in the same

726
00:50:44,240 --> 00:50:48,310
direction is always going to
have this universal surface.

727
00:50:48,310 --> 00:50:52,770
And it involves just a geometric
term and then one

728
00:50:52,770 --> 00:50:57,600
modulus that changes the
magnitude of the longitudinal

729
00:50:57,600 --> 00:51:01,500
piezoelectric modulus but does
not change the asymmetry.

730
00:51:01,500 --> 00:51:04,550
So let's see what this function
looks like as a

731
00:51:04,550 --> 00:51:05,480
function of direction.

732
00:51:05,480 --> 00:51:08,000
Maybe we better wait for that
until we come back because

733
00:51:08,000 --> 00:51:09,850
that's going to take
a few minutes.

734
00:51:09,850 --> 00:51:12,040
So this is what we found.

735
00:51:12,040 --> 00:51:13,310
That is correct.

736
00:51:13,310 --> 00:51:16,780
And we have to now decide what
this looks like, which will

737
00:51:16,780 --> 00:51:17,750
take a few more minutes.

738
00:51:17,750 --> 00:51:20,330
But let's stop here rather
than run late.

739
00:51:23,160 --> 00:51:23,600
All right.

740
00:51:23,600 --> 00:51:26,120
Let's take our 10-minute
break as usual.