Lecture 19: Crystal Defects: Point Defects, Line Defects

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The weekend is over. We are now in 3.091. So, announcement: this Wednesday there will be no lecture, instead, celebration of learning: test number two. These are the room assignments. We moved them just a little bit, moved the dividing lines, rather.

It's the same rooms, but we wanted to make a little more room in here and in the Walker gym. So, there are more people being asked to go to room 1-190. And then, this is the group to go to the Walker gym, third floor of the Walker Memorial building.

And then, the front half of the alphabet please report here. And, if you take the test that one o'clock, don't go to 6-120 because we won't be there. We are putting everybody in 26-100. That way there are plenty of vacancies.

So, I'm just going to go quickly over the comments that I give before every test just to put everybody in the right frame of mind. The coverage is lecture 7-17. So, that will start with ionics, and go through the generation of x-rays, but not Bragg's law or anything like that.

So, remember, you've got to bring five things with you: your periodic table, table of constants, something to write with, a calculator, and your aid sheet. And, you know, I'm going to make a bold suggestion.

For some of you, why don't you write your recitation instructor's name on your aid sheet? Because I'm going to tell you what I'm going to do. If I don't have a recitation instructor's name on the exam, I'm not going to grade it.

You know why? Because I've got 630 of these. If you don't put the recitation instructor's name on the exam, it means you are in no hurry to get it back. That will save me time. We'll just take all those, and just put them aside, and we'll just grade the ones for the people who really want to get them back the next day.

And, we'll get around to the other ones, I mean, someday. So, I think after two months, it's not unreasonable for us to expect you to put the recitation instructor's name on the exam. The purpose of the test is to provide you with feedback.

So, we want to just tell you, are you mastering the material? Are your study habits effective or not? That's all we're trying to do. So, this is not an aptitude test. It's not an admissions test.

I just want to see if you're mastering the material. And if you are, I'm going to say congratulations, keep up the good work. And if you're not, I'm going to send you a message saying, if you keep up at this level, you're going to fail this class.

You need to get that information on a timely basis. What are the test taking strategies? Read the paper first, identify the easy questions, work in your schedule, don't do number one because I made it number one.

Do number one only because it's the easiest one for you. It will be a confidence builder, soothe the nerves, and so on. Show your work. Show your work. Don't write cryptic answers because if you get them wrong, we can't give you part score.

But, if you give us lots of information, I don't mean volumes of information, but if you give us a few ideas as to how you are approaching the problem, then we can give you partial credit. Work parametrically.

Try not to immediately start going for your calculator. And, please bring something to compute with. I was astonished in the first test the number of people that had no calculator, or the calculator was not working and so on.

We have a few, but this is a large class. I don't have enough calculators for everybody. Don't grade the test. If you feel you are not doing as well as you'd like, don't give up. You just leave the grading to us.

And, you are not competing with your classmates. 50 is a pass. Everybody can pass. There's no success coming at the expense of the classmate. And, as always, do your own work. You've got an aid sheet, you've got your tables of constants, you've got your periodic tables, there's no reason to resort to any kind of intellectual dishonesty.

And, our goal is to have them back to you on Thursday in recitation. So, good luck. I want you just to show me what you know. And, I think it's appropriate that we don't have a weekly test tomorrow.

So, I know there's a lot of disappointment when I say this, but there will be no weekly test. We'll catch up afterwards. OK, so last day we looked at the selection rules for reflection in cubic crystals and saw that we could identify, distinguish different crystals from one another on the basis of the patterns in the line spectrum.

Towards the end we talked about symmetry. And I think I gave you the page here on the lab book of Danny Schechtman when he first started to see that he's got fivefold symmetry. And, so the music at the beginning I was playing, Take Five by Dave Brubeck, I was looking for songs in 5/4 time.

So, I had this one as well. Remember this one? [MUSIC PLAYS] Count it: one, two, three, four, five, one, two, three, four, five. But then they wimp out and they go to 4/4 here. Now it's a disco song.

I guess they feel that people can't dance to 5/4 time, and it had to be danceable or else it wouldn't be marketable. But that's 5/4, very strange time. OK, today we're going to move on to our discussion of solids, ordered solids.

Up until now, I've been treating crystals as though they're perfect. I've been treating crystals as though they're perfect in two ways, perfect crystals in terms of chemical composition, perfection in terms of chemical composition.

And that's to say that we're looking at pure materials, and also perfection in terms of atomic arrangement. And so, that means we are treating them as though they are defect free. Two types of perfection: see, we are talking about chemical composition and we are talking about atomic arrangement.

Both are critical in determining the performance of material. But, real engineering materials are not perfect. They contain impurities. And, we've seen that there are desirable in impurities such as dopants, and there are undesirable impurities such as contaminants.

And, part of what were doing in the last unit was studying how to manage impurities in such a way as to get desirable properties. Well, the same thing happens with defects. In some cases, we have atoms not always found at their lattice points.

Atoms sometimes misbehave. They don't go where they're told. And that can lead to consequences, mechanical failure. But, if we are clever about it, if we learn how to manage defects, we can engineer performance that is minimally damaging.

And, in some cases, we can enhance the performance of a material. So, we are going to get tailored properties through defect engineering. So, that's what I want to introduce today. So, we're going to go to the handout because this is a lot of taxonomy here, and sort of definitions.

So I thought we could just read along together. So, the defects are classified on the basis of dimensionality. So we're going to look at zero dimensional point defects. Then we will go to one-dimensional line, two dimensional interfacial, and three dimensional bulk defects.

So, the first thing is point defects, start at zero dimension. And, these are localized disruptions in the regularity of the lattice. How local at the level of one lattice site, at the level of one lattice site.

And, so there's three types we're going to look at. The first one is the substitutional impurity. And, if you flip over on the backside, I've copied this cartoon. And so, here's a substitutional impurity atom.

We've seen this already when we put phosphorus into silicon. One example: it occupies a normal lattice site. So, there is the dopant that we've seen before. And, I'll show you towards the end of the lecture, boron in diamond.

Substitutional, it actually sits at the normal carbon lattice site. They are also alloying elements such as magnesium in aluminum. There are very few metals that are used in commerce in pure form. Most metals are used in the form of alloys, that is to say solid solutions.

So, for example, this aluminum can is dominantly aluminum and contains a certain amount of magnesium and a certain amount of silicon. And, this is designed to give it the properties it needs. One of the main properties that we need in this can stock is the ability to have very, very rapid glide of the atoms.

If you look very carefully, there's no seam on the bottom of this can. This is made out of a single sheet of aluminum. And so, the sheet moves over a dye, press punches down, up, down, and forms the U-shaped right circular cylinder, and then a second piece is crimped onto the top.

You can imagine the high speed at which the atoms are gliding over one another, and part of what gives us the ability to deform is the alloying. So, magnesium is purposely added to aluminum. We have seen colored golds.

And, one example is nickel going into gold, which will turn it from its very buttery, almost buttery ductility into something that's got a little bit of strength to it. I mean, you don't want a ring that if you put it down on the dresser it starts to deform.

So, we have to give it some strength. And, in this case, we can alter its color. So, this is nickel atom sitting on the FCC site normally occupied by gold, again, a substitutional impurity. And, here's the last one.

We learned about 7-Up, and the lithium citrate in 7-Up, suppose you're on call and you're administering saline in the hospital, and it's supposed to be 5% sodium chloride in water. But if there's a lithium impurity substituting for sodium in the sodium chloride lattice, it's going to dissolve.

And if you've got some sodium deficient patient and you give them some lithium chloride, it's going to be a bad day at the office. So, substitutional impurities can have deleterious effects. So, those are examples of substitutional impurities.

The second type is the interstitial impurity. And, it occupies the position between the lattice sites. And, if you go to the cartoon, you'll see this highlighted. And, the interstitial impurity, obviously this atom is small.

It has to fit into the void space between the atoms. So, we did the calculations. We know that a close packed FCC lattice is 74% matter, and 26% void. Well, that 26% void is not just randomly distributed.

It is the nothing in between the atoms. So, I could take all the atoms away, and I could give you a lattice comprising the interstitial sites. You understand the concept. A lattice is simply points in space.

So, instead of putting gold atoms there, let's put the space between the gold atoms. And, imagine what we can do. So, here's an example of an interstitial atom. It's small, and it's obviously an impurity because it's different from the composition of the parent material.

And, so, there are desirable and undesirable. So, an example of a desirable element is carbon in iron. Carbon in iron is the basis for steel. Without carbon and iron, we wouldn't have steel. And, the interesting thing about carbon in iron is that the radius of the carbon atom is actually greater than the radius of the interstitial site in iron, greater than the radius of the interstitial site in iron.

So, therefore, when the carbon goes into that interstitial site, it actually distorts the lattice. It presses back on the parent iron lattice. And, through this distortion, strengthens the material.

Another example is hydrogen in lanthanum nickel five. It goes into the interstices in this compound. Lanthanum nickel five has voids that are large enough to accommodate hydrogen. And, hydrogen can go in this, acts as a sponge, and hydrogen at peak loading can go in at this level.

We have a huge amount of hydrogen going in here. The density of hydrogen in lanthanum nickel five at saturation is greater than liquid hydrogen itself. And, this material is under investigation along with others like it in the context of hydrogen powered vehicles.

If we're going to put hydrogen in cars, if at some point fuel cells become practical, what is going to be the mode of storing hydrogen? Do we want compressed hydrogen? We're going to have to have all sorts of protection in case of a collision.

Well, this might be a safer way of putting hydrogen in the vehicle. So, interstitial packing is advantageous in this case. But, if we get our wires crossed, and instead of putting hydrogen in lanthanum nickel five, we put hydrogen in iron, that leads to deleterious effects.

Hydrogen in iron embrittles iron. And it loses its ductility, becomes very brittle, and can fail prematurely. And this is also germane to the hydrogen economy. In this country, we can deliver energy through the electric grid, and we can deliver gas through gas pipelines.

You cannot send hydrogen through the gas pipelines because the gas pipelines are stable with hydrocarbon contents, but not stable with hydrogen. Hydrogen under pressure will embrittle those pipelines.

That means that we would have to build a new infrastructure to pipe hydrogen over long distances. And, that infrastructure cannot be made out of steel because steel will be embrittled. It's not going to be cheap.

These are cost issues. You need to know this because people are going to ask you. You're from MIT. They're going to ask you help make policy. You can't just get on board and say, well, we'll fix it somehow.

Hydrogen is very small. It goes places where it's not supposed to be. OK, then the third type of defect is called the vacancy. And that's simply an unoccupied lattice site different from the interstitial.

It's a site that's not occupied. And so, here's the cartoon. You see this is a regular atom that hasn't been placed on its proper site. And, we have a vacancy in that location. And so, there's two types of vacancy formations.

The first is when the material is being solidified, whether it's solidifying from the liquid, whether it's vapor deposition, or whether it's a solid-solid transformation. So, in some instances, all of the sites don't get filled.

And that there's an extreme case, such as the wall of a nuclear reactor vessel where under high bombardment of neutron flux, over time, atoms are literally dislodged from their lattice positions. And, you put this on a regular maintenance schedule because you can imagine over a certain period of time the material starts to look, at the atomic level, more nearly like Swiss cheese.

And so, that's not good. So, vacancies are present. And, they're always going to be present. Tom, do you think we could switch the video over to the document camera, please? OK, what I've got here is, well, actually it's an object of art.

This was made by an artiste in MontrÈal back in the 1960s. And, all it is, is two pieces of poly methyl methacrylate that have been milled and joined. And, in between here are 2,000 ball bearings.

So, we just have 2000 ball bearings, and I'm going to try illuminating from the bottom, see that make things better. Much better. And now, if I learned how to focus, OK, good, and now we're going to zoom.

OK, so now, what do you see? You see atoms packing. These are hard spheres. And, they're packing in ways that indicate the way atoms would pack. You see, here's regular crystal arrays. Here's some disorder.

Oh, this is even the gas phase. Look at this, see, we have everything. OK, so now, what I'm going to do is I'm going to anneal. So, I'm going to lift this up. And, annealing would be the equivalent of tapping this.

OK, I'm giving thermal energy to this because thermal energy causes atom motion. Look at the impact. I have annealed out so much of the defects. And now, look at large, regular regions. There's a surface.

I see a few gas atoms up there. And, look, vacancies. There will always be vacancies above zero Kelvin. You can come to my office anytime you want. I defy, I challenge anyone to anneal all of the vacancies out.

There will always be a few vacancies in here. And, sometimes vacancies coalesce. There is a divacancy. There's another divacancy. This is a good day for divacancies. The Red Sox win the World Series.

A lot of unusual things are happening here. OK, so let's take a look. There is an expression for the vacancy population in any crystal. So, what I'm showing you, this is what's going on in steel.

This is what's going on in that aluminum beverage container. This is what's going on in the airplane wings that you look out over when you fly: vacancies. OK, so what's the population of vacancies? And, the calculation of vacancy population involves what I'm going to give as equilibrium, meaning it's in equilibrium with the surroundings in terms of temperature.

It's at thermal equilibrium, equilibrium population of vacancies. And, this is in a crystal. And so, what we're going to do is it's a comparison of the binding energy, comparison of the binding energy to the energy of disruption because you can imagine the binding energy of an atom is the negative of the energy to form a vacancy.

How you form a vacancy? You can imagine, you go in with atomic tweezers and you pull an atom out. And, what bonds do you have to break? The bonds that bonded that atom to others. So, this is minus the energy of vacancy formation.

And, the disruptive energy is, in our case, a thermal energy. And, so, there's a way of expressing this. And, we're going to say, it's given by the following. We're going to compare these two in an exponential not unlike the one you saw earlier in the course.

So, I'm going to define the fraction of sites that are unoccupied, which will be the ratio of the number of vacancies per unit volume to the number of sites per unit volume. So, the nv is number of vacancies per unit volume, and big N is the number of atomic sites, number of atomic sites per unit volume.

OK, and that's going to be given by the ratio mediated through this exponential of the energy. And, I loosely move energy and enthalpy back-and-forth. This is the enthalpy of vacancy formation energy over the thermal energy.

And, how do we get a rough measure of the thermal energy? Product of the Boltzmann constant and absolute temperature. So, what are we doing? We are taking this onto the temperature distribution, and we are saying, where is this required energy in relation to the average energy of the system, and in particular, the high end tail.

So, and then, it has a pre-factor, A, and A is just an entropic pre-factor. And, it's not equal to one. A will take values between a tenth up to about ten. It's typically on the order of, oh, four, five, something like this, but in some cases it can even be less than one.

So, this is a relationship that tells us what the fraction of unoccupied sites is as a function of temperature. So, we could go in and calculate what the effective bond energy is because we know what the thermal energy is.

So, we could work backwards and calculate that. So, I did a sample calculation. I was wondering, what should be the vacancy population in, say, a piece of copper? So, I said, all right, copper, it happens that the enthalpy of formation of a vacancy is 1.29 electron volts per atom.

And, the pre-factor is equal to 4.5. I had to look this up in tables. This is not from first principles calculation. So, I said, OK, let's look at room temperature. At room temperature, I plugged these numbers in and I got the fraction of vacant sites as ten to the minus 21.

Well, if you multiply by the density of copper, that turns out to be 85 vacancies per cubic centimeter. I mean, this is vanishingly small. This is almost the inverse. This is almost none, right, because we have roughly Avogadro's number of atoms per cubic centimeter for condensed matter.

So, this is really, really low, ten to the 21, ten to the 23. There's 100 difference, and sure enough, there it is, 85. There's so few you could give each of them a name. And then, so then I said, well, what happens if we go to the melting point, at the melting point of copper? OK, I'm not going to melt copper.

It's like ice at 0° C. It's copper at the threshold of melting but still solid. If you do the calculations at the melting point of copper, you get 7.35 times ten to the minus five, which now turns out to be about 70 parts per million on an atom basis.

So, for every million copper sites, you can expect about 70 of them to be unoccupied. So, now we are starting to get up to real numbers because that would give us a vacancy population of six times ten to the 18th per cubic centimeter.

That's substantial. This is the kind of stuff that would be perceptible if this were a dopant. If this were a dopant concentration, this would alter the electrical properties of silicon. And, here's the way to look at this.

What's going on here? What's going on is this critical ratio, this is so important. The ratio of the binding energy to the thermal energy, at room temperature, this ratio is about 50 to one. It's about 50 to one.

At the melting point, it's about 10 to one. And, that's what's going on. You are shifting the distribution way over, and giving yourself the ability to rip the copper atoms out of their sites. So, that's interesting to know.

OK, Tom, may we go back to the computer please for graphics? Thank you. OK, so now, I want to talk about point defects in ionic crystals because there's a special twist here. With an ionic crystal, if you've got sodium chloride, you can't just remove a sodium ion because you violate charge neutrality.

So, when you make vacancies in an ionic crystal, you have to take them out. You have to make those vacancies by removing atoms in neutral units. So, let's look at the various types. The first one is called Schottky imperfection.

And, it's the formation of equivalent, not equal, but "equi-valent" numbers of cationic and anionic vacancies. And, there's the other cartoon on the backside of the handout. So, here's the Schottky imperfection where you see - the video graphic is not so good.

If you squint just a little bit, you'll see, this is not just an array of dark spheres, but there are light spheres in between the dark spheres. And, the light spheres represent the anions, and the dark spheres represent the cations.

And, what you are seeing here is the pair. We are missing both the anion here in the midst of four cations, and the cation over here, in order to preserve charge neutrality. Now, they don't necessarily have to be side by each.

I mean, the cation vacancy can be way over here, and anion vacancy could be way over here. But globally, it's the same number. And, so let's look at how that works. One way to describe that is to think about, instead of removing the cation and anion, the other way to think about this mathematically is to say, suppose I take, here's nothing.

I'm going to take some nothing. But, it's not any old nothing. It's FCC nothing. And, I'm going to dissolve the FCC nothing into sodium chloride. So, let's do that. You'll get the idea. Conceptually, it's so cool.

So, how do I make, I want to make a sodium chloride vacancy pair. So, what I do is, instead of ripping out a sodium, and ripping out a chloride, what I do is I take nothing, which I represent as null, and I dissolve it.

So, I'm going to take nothing, but I'm going to have some nothing that goes to the cationic site, and nothing that goes to the anionic site. So, this is V, is a vacancy on a sodium site, and a vacancy on a chloride site.

So, this is charge neutral now. OK, we said, well, it's only nothing anyways. Well, let's think about it. If you look at sodium vacancy, sodium vacancy is charge zero, but it's on a cationic sub lattice.

It's zero, but everywhere around it, it's got neighbors of comparable charge, except this is in a land of plus one for V sodium. So, if this means it has an apparent, I'm going to put quotation marks here, it has an apparent charge of minus one.

Zero in a land of plus one is relatively negative. And, what about the chloride? The chloride vacancy is zero in a land of minus one. So, a zero in a land of minus one is effectively a plus one. So, let's put that down.

So, this is, I want to put here, this indicates that this is locally minus, and this indicates it's locally plus. The way I remember these is, this slash reminds me of a minus sign. And, the plus is sort of like the crosshairs of the plus sign.

That's how I remember it. OK, and we can do the same as we did for the vacancy fraction, and compute a Schottky fraction. What's the fraction of Schottky defects? It's going to be the number of Schottky defects divided by the total number of sites.

And, it'll be an equation of this form, some pre-exponential times the exponential of the ratio of the energy to form the Schottky defect divided by the product of the Boltzmann constant and temperature.

And, for sodium chloride, the Schottky energy is on the order of about 2.5 electron volts. And, that's about double what it is for copper. So, and that makes sense. It should take more energy to make a vacancy pair than it does to make a vacancy.

And, Coulombic forces of attraction in ionic crystals are more intense than metallic bonding forces in metal crystals. I want to show you one other one. This is what's going on in the oxygen sensor of your car.

It's zirconia. And, I'm going to show you later how you manage the defects in zirconia to advantage. So, how do I make a vacancy in zirconia? The dominant vacancy is Schottky. And, I'm going to write again null.

I'm going to dissolve nothing into zirconia. So, going to make a zirconium vacancy and two oxygen vacancies. I have to preserve the stoichiometry of nothing. Nothing counts here, OK, because what's the local charge? This is zero in a land of plus four.

So, it's apparently minus four. And, every oxygen is a zero in a land of minus two. All of the oxide ions are minus two. This oxide ion vacancy is zero. Zero in a land of minus two is plus two, apparently.

So, I have minus four apparent, two times minus two apparent, charge neutrality, and this is now equivalent numbers of cation and anion vacancies. Very cool. OK, so there's another way, though. There's another way to make vacancies, and that's called the Frenkel imperfection.

And, it's the formation of an ion vacancy and ion interstitial. So, that's shown over here where a little cation, you see the cation? See this cation's gone over to another cation. There's an interstitial in here.

You see the little cation? Cation left and went over here? But see, it didn't leave. It didn't leave the crystal. It moved to another place in the crystal. So, I don't have to worry about compensating with the ion of opposite charge because the total cation population hasn't changed.

But, I have created a cation vacancy. So, this is what the issue is here. And so, with Frenkel, clearly with Frenkel, Frenkel occurs in crystals where you have wide variations in the radii. So, for Frenkel, the condition is that the radius of the cation must be either substantially larger than the radius of anion or it must be substantially smaller than the radius of the anion.

If you have ions that are roughly nearly equal in dimension, the interstitial site isn't big enough to take the ion. So, let's look at that one. A good example of this is silver bromide. Silver bromide forms Frenkels because, as you know, bromine is big.

Bromide is huge, and silver is small. And when it loses an electron, it's smaller. So, I've got a small cation, and a honking big anion. And so, there's lots of room in there. And, here's the way we'd write that: silver sitting on a silver site now wants to move.

And see, it's neutral because silver is plus one in a land of plus one. So, I can even put a little x there to show that it's neutral. So, silver on a silver site now becomes silver on an interstitial site.

And, an interstitial site is zero. And, silver is plus one. So, now, it's apparently plus one. And, what's going to do to charge compensate? It's going to leave behind a vacancy. Obviously, it jumped off this site.

So, there is a vacancy. And that vacancy is what? The vacancy is zero in a land of plus one. So, zero in a land of plus one is apparently minus. This thing makes sense. It's consistent. And, likewise, we can write a formula for the fraction of Frenkel sites in a crystal.

It would be the number of Frenkels divided by the total number of atomic sites for that, which then will equal A times some exponential expression of the ratio of the formation energy for the Frenkel defect divided by the product of the Boltzmann constant and temperature.

And, the Frenkel energy, what do you think? It's going to be closer to that of the Schottky energy, or closer to that of the single atom vacancy? It's about one electron volt. It's closer to that of the single atom.

And, it's typically the cations. It's typically the cations because normally the anions are the big ones. Fluoride is the one exception, OK? But generally, it's the cations that are going to go interstitially.

The last one is the F center. The F center, they don't talk about in your books, but it's a really cool one. I like this one. The F center occurs in non-stoichiometry. It may surprise you, but it's possibly to have sodium chloride that isn't 1:1 sodium and chlorine.

It's possible to have something that is chlorine deficient. So, let's see, what would happen if we were to dissolve excess sodium? I'm going to dissolve excess sodium into sodium chloride. Well, I can't dissolve a neutral.

Sodium chloride will not accept a neutral. So, what's the only way I can make sodium soluble in sodium chloride? I have to convert it into sodium ion. So, I'm going to make a sodium ion. And, I'm going to be left with an electron.

What am I going to do with that electron? Where is that electron going to go? Think. You've got valence bands that are filled. Do you see a parallel between this and putting phosphorus into silicon? You've got this extra electron.

This thing has bonding capability. This is a fifth wheel, didn't know what to do. So, one way to model this is I'm going to now to give you the full notation as to put sodium on a sodium site. So, now this is sodium plus.

I've got an electron, and what? I have to make a vacancy on a chloride site in order to compensate for this excess electron. And now, let's look at the way this thing is going to operate. I've got sodium ions in some FCC lattice.

Here's a chloride vacancy that's been created to compensate here. So, this is VCl. And, where is this electron going to be? Its electron is not attached to any one of these sodiums. Do you know of another example where you ended up with something that was apparently positive and motionless in a crystal with an extra electron that was attracted by Coulombic forces? If you had to model this operation, what model might you choose? You could use a modified Bohr model to model the electron motion around the vacancy in an F center.

Yeah, this is really cool. This is really cool. It's called f center because it comes from the German farbe, which means color. These form color centers because if you go through the Bohr model, the energy levels in here are in the visible.

And so, the different crystals we can identify, if you take sodium chloride, lithium chloride, potassium chloride, they're all white salts. But, if we create vacancies by injecting energy, say in the form of an electrical shock, we will dope the crystal different colors.

And the colors change with the identity of the crystal because the energies are going to be different. OK, good, so that's the end of point defects. Now, let's look at line defects. Line defects are very important because they answer this question.

Data made line defects necessary. People in the early part of the 20th century were trying to figure out what the ultimate yield strength of a metal is. So, let's take a look at, say, two planes of atoms.

And, I'm just going to, for simplicity, connect these with little struts to indicate that there's some bonding going on here. And, I want to ask the question, what is the shear strength, the magnitude of the shear strength necessary because these two planes to slide over one another? This is going to be involved in the yield of a crystal.

Well, one way of doing this calculation is to say, let's look at, say, bond strength between pairs of atoms. So, look at pair bond strength times the bond density, right? If we take pair strength times bond density, that should give us some round number that would be representative of how much energy it takes for us to be able to shear the material.

And, this quantity that we're looking for is the yield strength, denoted sigma sub y. And what we find is that the measured value of the yield strength is on the order of only about 1/10 of the yield strength as determined by theory.

Something is way wrong. Well, this was solved by Orowan. Orowan was a professor in Budapest in the 1930s. After World War II, he ended up here at MIT and he finished his career in the mechanical engineering department.

He was a brilliant theoretician in the area of strength of metals, failure in metals. So, the story goes, this is the apocryphal story, is that Orowan came home one day from the university and went up to his apartment.

And there was a long hallway and a runner, a carpet, a long carpet, along the hallway. And the chambermaid was just finishing cleaning, and she was going to put the carpet back. She had moved it out of the way, and she was going to move it back.

And so, he came up and offered to help her move the carpet. And she told him to get out of the way. And instead, what she did is rather then drag the carpet in order to displace it, what she did is she went to the far end of the carpet and made a little kink.

Made a little kink in the carpet, stamped her foot, and propagated the kink all the way down. And then: made a second kink, stamped her foot, and propagated the carpet all the way down. So, what's the difference? What's the difference? Well, with Orowan, if we stand at this end and pull, we've got the coefficient of friction along the entire length of the carpet.

It takes a huge amount of energy. If, instead, we invest a little bit of energy to make the kink, and then propagate the kink, we can move the carpet in these small amounts. So, what we've done is now all we have to come up with is the amount of energy to lift off that little patch of carpet.

And it will propagate very easily. He looked at this and he said, that could explain what's going on here. So, I've got a little cartoon that shows, actually it's showing the caterpillar moving here.

But, can you imagine if this is the plane of atoms such that there is an extra plane sandwiched in here that ends? And now, if I want to shear the top three rows of atoms versus the bottom three rows, instead of having to break every one of these bonds, I have to break them all.

I have to break them all. Look at here. All I do is I break this one, and then I break this one. And then I break this one, and then I break this one. And so what I do is I break one bond and just keep propagating it.

And that's the dislocation. That's the dislocation. It's a line defect that involves the misregistry of atoms. And through this, it allows us the ability to deform metals at values substantially below those of the entire crystalline, makes sense now.

So, here's another cartoon showing the same thing. And, how do these dislocations move? These dislocations move along close-packed planes in close packed directions. Tom, may we go back to the document camera, please? OK, so here is, I'm going to now go to top view, top lamp.

Look at this. This is face centered cubic. Where are the bonds going to be strongest? This is going to be like rafts of atoms moving. Where are the bonds going to be strongest? They're going to be strongest in the plane with the most bonds.

So, that's going to be the 1-1-1 plane because the 1-1-1, if you look at this atom here, you see the hexagon starting? There's the hexagon. Train your eye on this one. One, two, three, four, and then there will five, six.

So, this is a close-packed plane. It's got six nearest neighbors in the plane, three above, and three below. This is the 1-1-1 plane. It goes up the diagonal, OK? It goes up the diagonal. So, this is the plane that has the most bonds.

If there are the most bonds in the plane, that's the plane that has the greatest strength and will not give. And, doesn't it go axiomatically that if I have the most bonds in the plane, then normal to the plane I'm going to have weakest bonding? So, we have these strong rafts of 1-1-1 planes that want to slide.

Now, within the plane, how does something move? This is like pushing the wet noodle question. So, if I'm in this 1-1-1 plane, which direction am I going to move? I'm going to move along the direction where I have the greatest strength, the greatest rigidity, and that means along the direction where I have the greatest number of bonds.

So, this is along the 1-1-1 direction, see, excuse me, the 0-1-1 direction. I'm in the red plane, and then this is across the diagonal of the face. So, I'm moving in the close packed direction in the close packed plane because that's where the bonds are most dense.

So, Tom, may we go back to the computer video please? So, here's the set of directions. I just showed you FCC. The highest density plane is 1-1-1, and the close packed direction, that's the direction where the atoms touch.

You know this. They touch along 0-1-1. In BCC, it's the complement. The highest density plane is the cube diagonal that contains that central atom. And, the closest packed direction is the body diagonal.

That's where the atoms touch. Simple cubic it's the face and the face. So, this is now the origin of mechanical behavior. This is the origin of mechanical behavior. I can look at this and I can tell you how the material will deform.

So, if I take the set of close packed planes moving in close packed directions. Together, this represents the slip system. And this is how the material deforms. And so, we'll come back to that next day.

I see we're getting close to the end of the hour. So, let me jump ahead here, jump through this, we'll go through that. I'll show you that another time. All right, I'll talk to you about defects and crystals.

I talked to you about boron. You know what happens if you dope silicon with boron? You end up with something that is a p-type semiconductor. It's shy one electron, generates a hole. This is the Hope Diamond.

What makes the Hope Diamond so precious is not only its size, but it's blue. And, how did it get to be blue? It got to be blue by boron doping. It's boron doping at around one part per million. And what this does is it puts an acceptor level about 0.4 eV above the valence band.

And with the acceptor level about 0.4 eV above the valence band, we now have transitions that give us this beautiful blue color. So, what you're looking at is - these are not Hope Diamonds. This is just some 51 little units by Cartier.

And, this is the rest. This is in the Smithsonian. So, if you're in Washington sometime and you want to study boron doping, well, you can go and see this at the national gem collection. And you can stand with your contemporaries and explain to them the boron doping and how it makes the thing blue.

All right, good luck on Wednesday.