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PROFESSOR: The final quiz is scheduled for a week from today, on December 8. Following that weekend, we'll have one class on the 13th. And the last days of classes are Monday, Tuesday, and Wednesday, the 12th, 13th, and 14th.
So I know a number of you spoke up last time and said that you have a conflict on Thursday, the eighth, with a quiz in another class. So could I see a show of hands again of how many of you have this conflict? So three, six people, seven people, you're going to enroll in the other course just so you can put off taking the quiz. That's unfortunate, I don't think anybody should have to take two quizzes in one day.
We can't move it up. We'll have to move it back. So I don't know if I am violating any Institute rule, but I know that it is strictly illegal to give assignments that are due after the last day of classes, let alone have a quiz after the last day of classes, which has not been previously scheduled as a final examination. So we can't give it after Wednesday, the 14th, the last day of classes. So, do any of the six impacted people have a strong preference for when we should schedule the quiz?
PROFESSOR: Due it Monday? It doesn't have to be that soon. Monday?
AUDIENCE: Why not Tuesday?
PROFESSOR: Because Tuesday we have a class.
AUDIENCE: Or Wednesday? Well we would have a Thursday class instead. Correct?
PROFESSOR: No. If we're doing it after the quiz next week, there's a class. We have a last meeting here on the 13th. And I'm not going to shift it for other people, just for the impacted. So of the six who are entitled to vote, how many would prefer to have it Monday, the 12th? Three and I know how it's going to turn out. And for Wednesday, the 14th? Three.
AUDIENCE: I can't take the test because-- not because of another test-- I'm out of town. And I might come back Saturday. And Monday is just a little quick to take it after we get back.
PROFESSOR: I think I'm going to have to make an executive-- yes
AUDIENCE: Professor, when you mean by conflict, do you mean having an exam at the exact time?
PROFESSOR: No, it's on the exact same day. Considering this is a two hour examination, to have another examination exactly the same day, if you're not brain dead after an hour or two, you will be pretty close to it. So I think that is an unfair penalty to pay.
I think, to keep it as close as possible to the quiz that the rest of the people will be taking, why don't we make it-- since it's a tie vote-- on Monday, the 12th. Do it sooner, rather than later. And I'll let you know next time of where we will hold it. I'll have to arrange a room for it. So that will not be a make-up quiz. It would be a made-up-- invented-- quiz.
And, as I say on Monday, I should have all of the homework and the previous quiz to return to you. Reason you don't have it now is all the little crabbed handwriting that you see before you in the form of these notes, which takes forever.
All right, so since we've satisfied the unpleasant aspects of the end of the term, let's get back to discussing some of the other piezoelectric effects that we have defined. And then also ask the question rhetorically, is there anything like a representation surface for a third-ranked tensor property? And it doesn't look promising. But there are some things that we can do to discuss variation of properties with direction, and we'll see directly what those are.
But first let's look at the converse piezoelectric effect. And this again is a third-ranked tensor property, but what we do is to have the elements of strain, epsilon ij, a second-ranked tensor. And to take advantage of this curious relation between the directed converse effects, we define the converse piezoelectric effect as giving you nine elements of strain in terms of a third-ranked tensor dijk times e sub i.
So what is not standard is our convention for the order of the subscripts on the moduli. And we make up the rules, we can do it any way we like. And the advantage of defining it this way, in nonstandard tensor notation, is that we can use the same coefficients for both the direct and the converse effects.
So let me-- to illustrate what these equations look like-- write out a few examples. Epsilon 11 would be d1jk times e sub 1. I'm writing it in simple fashion, and not expanding fully, just to save space and time. Epsilon 22 would be d2jk times e2. Epsilon 33 will be d3jk times epsilon 3.
And let me stop after these six terms, and write, at least for these, an expansion. Because this gives us some interesting information. I'm not doing the equal signs in here. So this would say that the element of strain epsilon 11 is d111 times e1 plus d112 times e2 plus d113 times e3. And I want to say this is 11k. And I want to say that this is 22k. The next term would be epsilon 22, and that would be d122 times epsilon 2 plus d212.
AUDIENCE: Wouldn't the 2's be balanced?
AUDIENCE: Wouldn't that be d2's?
PROFESSOR: Yeah, you're right. This should be epsilon jk equals dijk times e sub i. So this is all e1. This is e1. And this one goes with this one. This 2 goes with this one. You're right. d2 and this is the d311 times e11. This would be 122 times e1 plus 222 times e2 plus d322 times e3. And the fourth one would be e33 equals d133 times e1 plus d233 times e2 plus d333 times e3.
OK, these elements here are the three that appear in the box that I had indicated for the terms djk. When we write the direct piezoelectric effect, this would be the box of coefficients. When we write the converse effect, this is the box of coefficients.
And now what I wanted to point out is that delta v over v is equal to episilon 11 plus epsilon 22 plus epsilon 33, which is the trace of the strain tensor. And we're going to get one set of terms which depends on the x1 component of the field, another set of three terms that depend on the x2 component of the field, and another one on the x3 component. If these expressions sum to 0, then there would be no volume change.
So this first set of 3-- 3 of the first 6 equations-- give you an indication of when the application of a field will result in no volume change in the sample. And that again, I remind you, can be for two reasons. It could be because all of these six of the nine piezoelectric moduli are 0.
Or alternatively, if you examined the form of the piezoelectric moduli matrices that is required by symmetry constraints, you'll find that there are a substantial number of point groups for which some terms are 0. But then there is, in addition, an equality between some of the other terms, which make the volume change zero. Even though, not all of the elements within this 1/2 box of moduli are zero. So that is an interesting effect.
And it turns out that the majority of the non-centrosymmetric point groups do not have a volume change, when you apply in a field in any way you choose. A few do, but it's a minority. Alright, but this is not the main point of writing this. I want to-- at this point-- point out that this tells you about the volume change.
And then we would have additional terms, we would have a term of the form e1 something like e123 or e132. And this would be e14. This would also be e14 since we replace both of those subscripts by a single subscript.
And the tensor elements that would go in here would be d123. We just want one of the them. d123 times e1, and then we want d223 times e2, and then d323 times e3. We're writing one of the specific equations for the shear strings. If we write the expression for d132, this is going to be d132 times e1 plus d232 times e2 plus d332 times e3.
There are two interesting consequences of this. The strain tensor is symmetric. And this element of strain-- why have I got three subscripts in here? Don't want that one in there.
These two strains are equal. And therefore, if we would apply just an e1 for example, let e be equal to just a component of field along x1. The strains have to be equal. But we have two different tensor elements here. And the only way that strain can be symmetric, and it's defined as such, is that d123 be identical to d132. And that resolves the issue that came up in connection with the direct case electric effect.
We said the direct piezoelectric effect depends just on the sum of the elements dijk and dikj. And since they're lumped together, all we can measure is the sum. And, so, we'll just have to call that a single matrix element.
The converse piezoelectric effect tells us these tensor elements have to be equal, if the strain tensor is to be symmetric. So that says that, since we defined d16 as the sum of d123 plus d132, this says that d132 is equal to d123 is equal to 1/2 of d16, just making the equality in the reverse direction. The converse effect let's us say that 123 has to be 132, that any ijk has to be equal to a dijk.
So if I tried now to write this first expression in the reduced subscript notation, e23 is what we let e5 be. And now we have this equal to and in our reduced subscripts, we have this as 1/2 of d16. And the field that's multiplying this is piezoelectric modulus is e1.
And that messy factor of 2 has come back to haunt us again. It's like trying to stuff a jack-in-the box back in the box. It keeps popping up. We ate the factor of 2 in defining the matrix representation of the piezoelectric electric modulus. And now when we try to go to a reduced subscript notation for the converse piezoelectric effect, we've got a 1/2 in there. And similarly, the second equation would be e5-- same result epsilon 5-- and it's 132, but 132 is 1/2 of d16 times e1.
And we have a similar mess for the other coefficients here. So what do we do? Do we say that the relation between strain epsilon j equals dij e sub j has 1/2 in front of several of the coefficients and not in others? Well, we can't really absorb the factor of 2 in the definition of the piezoelectric moduli, because we've already done that. So the only thing we can do is to say that we will have to take the off diagonal strains, and define them as having a 1/2 in front, and we add these up.
So we will have to write matrix strain in this reduced subscript notation. We'll have to take e11, epsilon 12, epsilon 13, epsilon 21, epsilon 22, epsilon 23, epsilon 31, epsilon 32, and epsilon 33. And in converting this to matrix form, we'll call this epsilon 1, analogous to what we did for the tensile stresses. We'll call this epsilon 2 and this epsilon 3.
And then for all of the off-diagonal elements of strain, in order to avoid the factor of 2 popping up in front of the matrix representation of the piezoelectric moduli, we're going to have to put in here 1/2 of epsilon 4, 1/2 of epsilon 5, and 1/2 of epsilon 6, and same for the off diagonal terms 1/2 of epsilon 5, 1/2 of epsilon 4, and 1/2 of epsilon 6. So the moral of this story is that you can't win, but if you play it right, you can come out even. So only if we define the reduced subscript strains in this fashion, can we write an expression of this form.
So this algebra is carried through for you for the other elements in the notes. But this is the way we are forced, unless we want to have a factor of 2 in some terms, and not in others, is the way we have to define matrix strain. All this is formalism and definition, but I'd like to now do two things. First of all, give you some examples of real numbers for piezoelectric moduli, and then ask the question about representation surfaces. Once again, these numbers are in the handout for you, so you don't have to make note of them.
But one of the very important piezoelectric materials is the quartz form of SiO2. SiO2 has many polymorphic forms. Quartz is the form that's stable at room temperature, and it has point group 32 asymmetric. And there are higher temperature polymorphs of SiO2. There's a phase transition in quartz to a more symmetric form, and then there are cubic forms at the highest temperatures.
Now, quartz is not the material that displays the largest piezoelectric moduli. But it has the following advantages. One is it is a naturally occurring material that is very inexpensive. So it's not an exotic expensive material. Very stable. It's not water soluble. Extremely tough. You can take a thin wafer of quartz, for example, if you want to make a monochromator for a diffraction experiment.
You can take a thin wafer of quartz, and bend it like this, and it does not break, very elastic. And if you want a material that's going to earn its living by being squished, you want something that doesn't plastically deform and something that is very hard and resistant to stress. So quartz, even though the moduli are not the largest, is a very attractive material, and is used in a variety of devices.
There was a time when CB radios were very, very popular. Everybody had to have one in their car. I guess so they could pretend that they were truck drivers. But anyway, you don't have them anymore now that cellphones have come in in existence.
But for your CB radio, you needed something called a crystal. And they were fairly expensive. And the number of channels on which you could communicate dependent on the number of crystals that you could plug into your CB radio.
The so-called crystal was exactly that. It was a little black box that looked almost like a transistor. And there were two leads coming out of it. If you ever got curious and broke this thing open, what you found was a nice wafer of quartz. And on the wafer of quartz-- brazed onto it-- were two wires. And that's all there was in the box. The crystal really was a crystal. And the crystals had been very precisely ground to thicknesses such that when a field caused these wafers to hit a resonance, that resonance would be at exactly a particular frequency. And that was the frequency of that channel.
One of the crises, during the Second World War, is that the highest quality natural crystals of quartz come from Brazil, and during the conflict the sea channels were essentially blocked. And so, people-- in order to make all these communication devices-- had to learn how to synthesize crystals of quartz synthetically. And there are a number of companies, such as Sylvania up on the North Shore, that developed entire buildings devoted to growing single crystals of quartz.
And they're big tanks like something out of the aquarium. And in the center of the tank is a rod, and seeds of quartz are placed on the rod. And the thing very slowly rotates around in this solution. And on the rod, eventually, are single crystals of quartz that are this size. And its a very spectacular thing to see.
In any case, symmetry 3 2, and the moduli that are 0 and non-zero. If we refer the reference axes to a set of coordinates with x1 in this direction, and x2-- since it has to be orthogonal to x1-- in between the two-fold axes, and x3. And for all materials of commerce that are anisotropic, there has to be some standard for defining the choice of axes. For example, crystallographers would say the unique axis should be along the z direction, the x3 direction. But why isn't x1 and x2 in between the two-fold axis?
There's some professional society that is responsible for giving standards for representing property measurements in some mutually agreed upon form. And this is the standard set of axes for the quartz and symmetry 3, 2. The moduli, dij, not dijk, but dij, these two are constrained to be equal. This one is 0. This is minus 0.67, 0, 0, 0, 0, 0, 0, 0.67, 4.6, 0, 0, 0, 0, 0, 0, 0. One of the strange materials for which no field can create a strain-- field along x3 cannot create a strain. And these are all in units of 10 to the minus 12 coulombs per Newton.
An example they give you here, if you apply a field of 100 volts per centimeter, which is not terribly large, but would be comparable to what you have in some electronic device, perhaps. So this, since our units are MKS, this would correspond to 10 to the 4 volts per meter. The strain epsilon 1, which is d11 times e1. It turns out to be minus 2.3, which means it contracts. That's the significance of the negative times 10 to the 4. And that turns out to be 10 to the minus 12 times 10 to the fourth. That turns out to be a strain of minus 2.3 times 10 to the minus eighth.
10 to the minus eighth is not exactly a large point strain. You're not going to see the crystal wafer twitch and jump, if you apply a field of 100 volts on it. But yet, even a strain of this sort is more than enough to be useful. But this is just to illustrate that quartz is not the most sensitive of piezoelectric materials.
Another one that I give you data for is so-called ADP. And this is a widely used material. This is ammonium dihydrogen phosphate. And this is one of the family of salts that have very large piezoelectric responses. The nice part about it is that it's water soluble, so you can grow very, very large crystals easily from solution. The nasty part about it is that it is water soluble, so you have to be careful to protect this material from moisture if you're going to use it in any sort of device.
But if we look at the moduli, relative to the standard axes, and this has point group 4-bar 2m. So x3 is taken along the 4-bar access. Then there are two-fold axes and orientations like this. And since they are orthogonal, you can take both x1 and x2 along the two-fold axes. And the numbers here for dij, are 0, 0, 0, 1.7, 0, 0, 0, 0, 0, 0, 1.7, 0. This is one of the interesting tensors where there's a diagonal row of non-zero terms off on the right hand side. And finally, the big surprise is the third modulus this is 51.7.
So you can see this has a very strong effect. This is over 10 times the maximum piezoelectric modulus in quartz. So this is a material that's very commonly used in transducers. This is again times ten to the minus 12 coulombs per Newton.
One of the very, very exciting developments in recent years is a class of materials that are perovskites. And they are very, very new. I gave you the reference to the first one that was reported, and that was just in the spring of 2000. And these are perovskites.
Perovskites are materials that in the type form are cubic. But depending on composition and temperature, they can transform to a distorted version of this very simple cubic structure that is tetragonal. And this material can exhibit piezoelectric effects. And whether it distorts or not depends on the relative sizes of what goes into the perovskite.
Perovskite has a composition ABO3 like barium titanate is one example. And the material has two different cations. And they have different valences so they have different sizes. And I won't bother to describe the structure, but it is only a very restricted locus in the field RA versus RB, where both the A and the B can remain in contact with the oxygen without distortion. And it turns out to be a line that does something like that.
Any other perovskite-- and there are lots of them in this field of radii-- has to have one of the ion sort of flopping around. And if that gets too serious, the structure distorts so that all these ions can remain in contact with the oxygen. OK in order to do that, many of them distort to tetragonal forms. Others distort to super structures, which have very, very large unit cells.
But in any case, when you're right at the phase boundary between the distorted structure and the true perovskite structure, these materials sometimes have very, very complicated x solutions of the two phases, on a very sort of fine scale. And it's not known exactly why they have this property. But they have absolutely enormous piezoelectric moduli, very close to this phase boundary.
And the references that I give you-- here-- is a compound that is a lead titanium zinc niobate. And the complicated composition is to get you close to this phase boundary. This has a d333 that is greater than 2,000 picocuries per Newton. And pico is 10 to the minus 12.
So this is a piezoelectric electric modulus that is 10 to the 3 times d11 for quartz. So 3 orders of magnitude stronger than this very commonly used piezoelectric material. Some of these materials have strains getting close to 1%.
So this is something that will actually twitch on the lab bench, when you apply a field to it. So these are entirely new. People still don't know the origin of this behavior. And it's still under considerable study. So this is a new family of materials. It's very exciting, and undergoing a lot of investigation and development work at the moment.
Alright, we are almost out of time. Time goes fast when you're having fun. Let me raise the question that we'll consider next time, which will be one of our last lectures. And that is, is it possible to create representation surfaces that tell you how the piezoelectric properties of a particular material will vary with direction? Well, vary with what?
Well, we talk about the direct piezoelectric effect. This gives us components of-- well let's look at the simpler one, in terms of what we apply. We have the converse piezoelectric effect that says that epsilon ijk is going to be dijk times e sub i. So we've got a piece of material, and we apply a field, e sub i. So we can vary this in space relative to a coordinate system x1, x2, x3. But how in the world are we going to show what happens? So I always do an extra thing in here.
There are 9 components to the strain tensor, which is symmetric. So they're really six responses that are unique. So yes, we can define the direction of the applied field. But they're going to be 6 different strains. So we're going to need six representation surfaces. One for each of the three tensile strains, and one for each of the three shear strains. So you can't do it with a single surface. So you can't do much other than say, there are certain responses which are intended to emphasize one particular sort of strain or one particular sort of polarization.
So one of the things we might do is to cut a very thin plate of something like quartz, and subject it to a uniaxial stress. So let's say sigma along the x3 axis. So we're looking at a very restricted strain tensor. That's 0, 0, 0, 0, 0, 0, 0, 0, sigma 3. And in response to that strain, there are going to be three different components of the polarization.
Polarization is manifested as a charge per unit area. So if we make a very thin plate-- to be sure there will be charges induced on these thin edges-- but if it's got a surface area that's a large compared to the area of these thin edges, we are going to be measuring primarily p3-- the component of polarization that's normal to this surface-- and that might have a charge per unit area that is comparable to these other two charged surfaces. But because the area by design of our specimen is so large, the response that would be most easy to detect, and which would be the largest response, by design, would be p3, which is the charge per unit area on this surface. So this is an effect we can define for a particular sample, and for a particular special form of the generalized force.
And we then can ask, what is the value of the single modulus that relates p3 to sigma 3? And that's a question we can ask. And we can plot that response as a function of direction of a plate that we consider as being cut out of a single crystal, and different orientations, and then ask how this modulus-- which connects the two-- changes with the orientation of x3. So that is a question we can ask.
And these surfaces are absolutely wild, highly anisotropic, can be identically zero in certain special directions, and they are very interesting, and a lot of fun to look at. So we'll take a quick look at a couple of those, which won't come as a surprise because they're already worked out for you in the notes. So we'll take a look at one of those. And in the problem set, I invite you to amuse yourself by looking at such representation surfaces for two other point groups. And with that, having kept you til five after the hour I will quit.