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PROFESSOR: I know I'm not Dr. Sadoway.
I'm one of Dr. Sadoway's younger colleagues so--
but class is beginning, so let's get started.
Dr. Sadoway, of course, would like you to know that he would
very much love to be with you here today and he would
probably like that very much more than what he's doing now,
which is sitting in a secured area secured by the Secret
Service about to meet the President.
So you can ask him all about it when he comes back.
Associated with that, I have one request for you.
Today, when you're leaving, some of you may be used to
leaving that way.
Don't leave that way.
Leave any other way.
This way's fine.
In the back is fine.
Just don't leave that way.
OK.
So last time, we talked about x-rays
defraction and Bragg's Law.
And x-ray defraction and Bragg's Law has a lot to do
with perfect crystals.
So perfection, right?
We've been dealing with perfect crystal so far and
today, we're going to be dealing with imperfections.
So defects.
And in the field of material science, the field that I work
in along with Dr. Sadoway.
there's a saying due to one of the most famous material
scientists that says that crystals are like people.
It's the defects that make them interesting.
So now you've heard that saying and you can roll your
eyes from now on every time you hear it because it's
repeated it very often.
So let's see.
What do we mean by defects?
Well, there's two types of defects, broadly speaking,
that we're going to be talking about, and we can classify
those into two categories.
One of them is chemical.
So far, we've been dealing with
chemically perfect materials.
So ones that are made up of one element or ones that are
made up in stoichiometric quantities of two elements,
but you can also have chemical imperfections.
You can have impurities.
You can have alloying elements and we're also going to be
talking about atomic arrangement.
And in the case of atomic arrangement, we're dealing
with structure, right?
Here we're dealing with chemistry.
Here we're dealing with structure.
You know about crystal structure so you know what
perfect crystals look like, but real materials are not
perfect, neither in the chemical sense nor in the
structural sense.
So perfect crystals don't really exist. You have always
some forms of imperfections and we'll go over those today.
So in the case of chemical imperfections, we have,
broadly speaking, two types.
Good ones--
Dr. Sadoway labels them with a smiley face so I
will do that, too.
And bad ones are a sad face.
So both are defects and whether they're good or bad
depends on their utility.
Are they useful or are they detrimental?
If they are detrimental, we call them impurities.
And if they are good, we call them other things.
For example, dopants.
We studied dopants, right?
Dopants are a type of impurity, they're a kind of
imperfection, but they're good.
We use dopants.
We use them in semiconductors.
Also, alloying elements, you can put in
the elements yourself.
Those are examples of good chemical imperfections.
When it comes to atomic arrangements, we're going to
spend a little bit of time talking about that in more
detail in just a minute.
But broadly speaking, they are situations where you don't
have perfect crystal in order.
You have disruptions and that perfect crystal in order--
either in the form of missing atoms or extra atoms or
differently oriented unit cells of the
crystal and so forth.
So we'll talk about that a little bit
more in just a minute.
So one thing that I forgot to mention maybe is that we have
a test coming up--
That is, I'm sorry.
Celebration of learning, second
celebration of learning.
And those are your room assignments.
You don't have to necessarily write them down now, but
you'll see them again.
So I just wanted you to see them.
A through Ha in 10-250, He through Sm, 26-100,
So through Z, 4-270.
OK, so let's go into the taxonomy of defects.
So I mentioned that there are a number of different kinds of
defects, and we can do better than to say simply, defects
exist. We can actually start to classify them.
So there are, broadly speaking, four types of
defects, which we classify based on dimensionality.
So there's 0-dimensional defects and 0-dimensional
defects are point defects or point defect clusters.
It's just like in math.
A point has 0 dimension, right?
1-dimensional defects or line defects, things that thread
through a material like a shoestring or something, and
we'll have an example of that in the form of dislocations.
Then we have 2-dimensional defects.
Those are interfacial defects.
So if you have interfaces between two different kinds of
materials or if you have two crystals that are misoriented
with respect to each other, those are examples of
interfacial defects.
And we also have bulk defects--
3-dimensional defects like inclusions.
We'll go over some of those.
So let's start out with point defects.
Point defects are those 0-dimensional defects.
They're points, just like in math.
And there are a number of different point defects that
we can look at.
We have our substitutional impurities, interstitial
impurities.
The general overall recurring theme in these point defects,
regardless what type they are, is that they are localized
disruptions.
So a lattice, a crystalline lattice, is a regular
arrangement of atoms, and a point defect is a very local
disruption in that regular crystalline arrangement.
And those disruptions can occur on lattice sites.
So basically, on the positions where you would see atoms
normally, they can also occur between lattice sites.
So they're localized rifts, you can say, in the
periodicity of a crystalline material.
And here's just an illustration of the way that
these different point defects play out.
So let's start out by looking at this one.
This is a substitutional impurity atom.
So it's an impurity.
So it's a type of chemical imperfection.
You can imagine that this is, for example, some kind of FCC
material like copper, for instance, and suppose that you
have an impurity of some sort like iron, and it sits there
and it replace one of the copper atoms. So this is
called a substitutional impurity.
It substitutes for the regular atom that would have sat at
that location.
Substitutional impurities.
There's another picture.
There's your substitutional impurity up there.
We'll get to the other ones in just a moment.
We already talked about some substitutional impurities.
So dopants.
If you have boron or phosphorus or silicon, that
boron or that phosphorus replaces the silicon that
would have sat at a certain lattice site.
So the dopants we've studied so far are types of
substitutional impurities and we call them good impurities
because they give us desirable properties.
They are dopants.
They're good impurities.
Then we have alloying elements.
I am not a very big fan of sodas, but if you were
somebody else who is a big fan of sodas, then you might, for
example, drink a lot of sodas out of aluminum cans.
Those aluminum cans are really not pure aluminum.
They're alloys.
They're alloys with other metals to give them the good
properties that they need in order for the
forming process to occur.
So if you ask yourself, how do you actually
make aluminum cans?
They're stamped out from sheets of aluminum, but it's
not just aluminum by itself.
Aluminum by itself would just tear if you do that.
So if you alloy it, you give it better
properties so it's ductile.
You can deform it to very large strains and that's a
good thing in the case of alloying elements.
Another one that he showed you was nickel and gold.
That's if you want to change the color of gold, if you want
to be very creative when you're proposing.
So that's a good kind of substitutional impurity.
There are also contaminants.
Contaminants are bad kinds of impurities, but before we get
to the contaminants, let me just show you what good
impurities can do.
So this is the Hope Diamond.
It's in the American gem collection.
You can actually find out more about it.
I'm not a big gem expert, but anybody who looks at this Hope
Diamond can immediately see that it's a
really pretty diamond.
But you guys are ahead of everybody, because in addition
to knowing that it's a pretty diamond, the fact that it has
boron impurities in it already tells you what the majority
charge carrier is.
So when you go to this gem collection, you can educate
everybody else.
So those are all good impurities.
There are also bad impurities--
contaminants.
Lithium in sodium chloride is an example.
So if you were using saline solution, for instance, in an
IV and you had lithium in there, that is very
detrimental to the person who's receiving that.
That's definitely a contaminant.
You don't want those.
That could make you die.
So let's move on to some other types of point defects.
Here's another type of point defect.
So just now we talked about the
substitutional impurity atom.
Now we're going to talk about the interstitial.
The interstitial's very different from the
substitutional.
In the case of the substitutional, we have a
lattice site, which instead of being occupied by the regular
atom that would have occupied it in a perfect crystal, is
occupied by a chemically distinct atom.
In the case of an interstitial impurity, that impurity can
sit in the space in between lattice sites, so-called
interstitial sites.
That's why we call them interstitial impurities.
And interstitial impurities or interstitial atoms can be both
chemical impurities, and obviously, they are rifts in
the atomic arrangement so they are structural
impurities as well.
Why do I say chemical?
The reason is because if this is, for example, iron, if this
is an iron matrix and you have a carbon atom sitting in the
interstitial site which makes steel, then that's an example
of a structural defect, but it's also a chemical defect.
If, for example, you have, on the other hand, iron sitting
in a nuclear reactor and it's getting bombarded all the time
by energetic neutrons, then interstitials are being
created by atoms getting knocked out of their lattice
sites and they're getting put into interstitial sites.
So that's creating defects that are chemically the same
as the surrounding matrix material, but which are
structurally distinct.
So those are interstitial atoms.
So here's another picture of interstitial
atoms. There it is.
It's not sitting on a regular lattice site.
It's sitting in between lattice sites.
It's sitting in the space, in the interstitial space between
atoms. And I already mentioned to you the fact that if you
put carbon into iron, those atoms go into interstitial
sites and there would give iron some of its beneficial
properties, which we look for in steel.
So some of the good mechanical properties.
Here's another example of a situation where an atom goes
into a lattice and create an interstitial impurity.
So lanthanum-nickel 5 is a prototype
hydrogen storage material.
It takes up a huge amount of hydrogen.
It takes up a greater density of hydrogen than liquid
hydrogen, actually, so if you expose this to hydrogen, the
hydrogen just goes right in, and it sits in the
lanthanum-nickel 5 lattice as an interstitial atom.
So this is considered to be as a prototype
hydrogen storage material.
Unfortunately, it's extremely expensive so it's not being
used very much these days, but on the other hand, it goes
well with our gem theme in this particular lecture in
terms of expensive things.
Here's another example of an interstitial impurity.
This one is not an alloying element.
It's a contaminant.
So hydrogen, but this time in iron.
So hydrogen in lanthanum-nickel 5 is good.
We want to store it.
Hydrogen in iron is bad because it actually degrades
the mechanical properties of the iron, unlike carbon, which
gives us steel, which is good.
You put hydrogen in, it embrittles it.
So hydrogen embrittlement in steels is a big problem.
And it's actually one of the challenges
to a hydrogen economy.
If you have steel pipelines or valves or various pieces of
machinery, structural components that are made out
of iron that are constantly exposed to hydrogen, over
time, they're going to brittle.
They're going to become very difficult to use.
It's one of the challenges in that whole undertaking.
OK.
So we're going fairly at a clip here through these
taxonomy of point defects.
So in the taxonomy of point defects, perhaps the easiest
defect to understand is the vacancy.
So when you think of the vacancy, think of the hole
that we talked about in the case of semiconductors.
Vacancy is nothing.
It's a missing atom.
So if you have a crystalline lattice and it's FCC or it's
BCC or it's simple cubic, whatever you like, and you
know that there's supposed to be an atom at a certain
lattice site and it's not there, that's a vacancy.
That's a situation where you have an unoccupied lattice
site and there are different ways to form these vacancies.
They can be formed during crystallization.
If you heat up a material, the number of vacancies decreases.
So if you quench it really quickly, you can actually trap
the vacancies before they can leave in service under extreme
conditions.
I mentioned just a moment ago that interstitials can be
created if you irradiate a material, if you bombard it
with energetic particles, like neutrons for instance, you'll
create interstitials, sure, by knocking atoms out of their
atomic sites, but what's left behind after you
knock that atom out?
Vacancies.
So actually, you create two defects at the same time:
vacancies and interstitials.
So I think we probably have a picture here of a vacancy.
A vacancy is nothing.
It's just empty space.
That's a vacancy.
There you go again.
Nothing.
So we've gone through a bunch of taxonomy, right?
So we know now that there are a number of different kinds of
point defects in crystals.
We've talked about interstitials.
We talked about vacancies.
We've talked about substitutionals.
What can we say about these defects quantitatively?
So let's take the vacancy and derive or write down what is
the number of vacancies that you can expect to find in a
given material at a certain temperatures?
So to do that, let's be a little bit more quantitative.
Suppose that you have a crystal.
I'm showing you a plane, for example, a 1 1 1 plane in an
FCC material and how do you create a vacancy?
You simply remove an atom.
So you had an atom.
Now you have no atom.
You've created a vacancy.
When you created that vacancy, you broke all the bonds
between the atom that used to be there and the neighboring
atoms. Breaking bonds costs energy.
So it costs energy to create a vacancy.
It costs energy to remove an atom from the place where it
would have been because you're breaking bonds.
So in this case, I have six bonds that I broke.
If I were looking at some material, for example, placing
our cubic material in 3D, I would find that I would be
breaking 12 bonds to the 12 nearest neighbors, and so on
for all the different crystal structures.
So we actually then take that information, the fact that
we're breaking bonds, and encapsulate it in a single
descriptions of how much energy it
costs to create a vacancy.
And we call that the vacancy formation energy.
So this is an energy and so it's expressed in eV.
Its units are electron volts.
You can convert them to joules, anything you like.
And if you wanted to then compute how many vacancies
there are in a given crystal, well, first of all, it costs
energy to make them, so why would you ever even have a
vacancy in the material?
Well, no material is perfect.
We know that from studying materials, but what causes it
is the fact that at finite temperature because of the
Boltzmann distribution that Dr. Sadoway told you about a
few lectures ago, just like in the case of intrinsic charge
carrier promotion in semiconductors, you can get
thermal formation of vacancies.
So that Maxwell Boltzmann distribution can actually
cause there to be vacancies despite the
fact that there are--
that it costs energy to do that.
So how can we actually use that to express how many
vacancies we have in a given material?
So I'm going to write down an expression for how many
vacancies you can expect to find at a given temperature on
the basis of their formation energy, OK?
So let's do that.
This is going to be the fraction of vacant sites.
So if you have a given material--
FCC, BCC, whatever you like-- you know that there's a
certain number of lattice sites per unit volume, and you
learn how to calculate those things.
And to quantify the number of vacancies, you have to
basically say, what fraction of those sites does not
contain an atom?
Is that 1/100 of a percent or is it 1% or how many?
So this is actually going to be expressed as a ratio, which
I'm going to call this.
This is the number of vacancies per unit volume and
this is the number of atomic sites, also per
unit volume, OK.
So this is the definition of the fraction of vacant sites.
And how do we express it?
Well, we express it using a very simple formula.
This formula contains a factor here which is experimentally
determined.
This is an empirical factor and then an exponential.
So the exponent we take here, the vacancy formation energy,
and we divide it by the thermal energy at the given
temperature of interest.
So this is actually telling you that to form vacancies,
you actually have two competing factors.
On the one hand, you have the bonding energy that makes the
difficult-to-form vacancies because you're breaking bonds,
you're taking it an atom out.
On the other hand, you have this thermal energy, Boltzmann
constant times the temperature, and the thermal
energy is competing with that bonding energy.
And when that thermal energy is high enough, you can
actually start knocking atoms out despite the fact that it
costs you some energy, and obviously when this ratio is
very large, that means that the bonding predominates, and
when it gets smaller, that means the thermal energy is
more and more sufficient to actually knock atoms out of
their positions and cause there to be vacancies.
So when you do these calculations, make sure to use
the absolute temperature in Kelvin, and dimensional
analysis will tell you the units of the Boltzmann
constant have to be energy units per degrees.
So eV's per Kelvin, for instance.
So this is the absolute temperature.
This is the vacancy formation energy and this is the
Boltzmann constant.
So what that means is that at any given temperature, you'll
actually expect to see some fraction of vacancies.
So let's actually try to do a calculation with an actual
vacancy formation energy and see how many vacancies we get
at a given temperature.
So to do that, we've been provided with what is the
common currency in science, which is published
experimental data, which we find from published journals
which we find online through, for example, Web of Science.
And from this journal, we find that for copper, the vacancy
formation energy is 1.03 electron volts.
Furthermore, in the same journal in the abstract there,
you'll find the magnitude of this constant A, which as I
told you is experimentally determined.
This quantity A we call the entropy factor.
And even though there's no way to very easily derive it--
I can't tell you what's the entropy factor for iron and
you can't really tell me just by thinking about it--
nevertheless, it turns out that this factor usually fall
within a certain range.
It's usually between 0.1 and 10.
That's usually the range and what you find As.
And in the case of copper, it's very much in that range.
It's just 1.1.
So let's use this information to actually figure out how
many vacancies we expect to see in copper at the given
temperature.
OK.
So here's my fraction of vacancies and we know that it
is going to be expressed as 1.1 times exponent minus the
formation energy, which is 1.03 eV, and then we'll put in
Boltzmann's constant, and then let's pick a temperature.
For the moment, let's just take room temperature.
So T, room temperature, and room
temperature's about 300 K.
So when we put in all these numbers, it's just a matter of
calculating.
This is on your sheet of constants.
This is the temperature which we choose at room temperature.
Let's actually write down T, room temperature, is about 300
K, and we find a certain number of vacancies.
And the number of vacancies that we find is 2.19 times 10
to the minus 18 vacancies.
So this is the fraction of vacant sites.
Well, is that a lot?
Is that a little?
How can we determine whether this is a lot or a little?
We compare it to the number of atoms that there actually are,
and in the case of solid materials, we expect there to
be something on the order-- this has to be compared to
something on the order of Avogadro's number, but we can
actually calculate more explicitly that this turns out
to be something like 10 to the 5th vacancies
per centimeter cubed.
And if in a real material, a solid material like silicon,
for instance, or boron or whatever you're interested in,
your number of atoms is something like 10 to the 23rd,
Avogadro's number, right, then this is
actually a very tiny number.
Compare 10 to the 5th to 10 to the 23rd.
The number of atoms that are missing at room temperature is
very low in copper.
We can compute that using this expression.
However, even though it's low, it's not zero.
And if we continue going down in temperature, we'll find
that the number is lower and lower and lower, but it never
really goes to zero because we have this expression which
gives us the total number.
So let's do another one and this time, let's take a
different temperature.
Let's take the melting temperature of copper.
The melting temperature of copper is considerably higher.
It's about 1085 degrees Celsius and we can go through
exactly the same calculation.
1.1 times all these quantities here times melting
temperature, OK?
And when we do this, we get a vacant site fraction which is
1.67 times 10 to the minus 4 and that corresponds to 1.41
times 10 to the 19th vacancies per centimeter cubed.
So what do we know by--
what can we see now by comparing these two
quantities?
Number of vacancies, add room temperature, number of
vacancies at melting temperature.
How many orders of magnitude do they differ by?
Huge difference in the number of vacancies that we find at
two different temperatures.
And why do we see such a huge difference?
Let's actually write down what that difference is.
Let's write down the ratio.
The fraction of vacant sites at melting temperature divided
by fraction of vacant sites at room temperature is something
like 7.6 times 10 to the 13th.
That's the difference in number of vacancies you get
just by increasing the temperature from room
temperature to melting temperature.
Why is that?
Well, one way to look at that is just from the expression
that we have to calculate the number.
Here's where the temperatures go.
So any difference in temperature if it's a factor
of two or if it's a factor of three or if it's a factor of
four, it's going to go into the exponential.
So that exponential is actually making a huge
difference as a function of temperature in terms of number
of defects that you get.
The higher you go up in temperature,
you don't get just--
you go up a factor of four in temperature, you don't just
get a factor of four increase in the number of defects.
You get that in the exponential.
So the factor of four is hugely magnified by the
exponential.
So that means that at any given temperature, even the
lowest temperatures, you expect to see some defects,
but if you increase the temperature, you see a hugely
larger number of defects.
And you can use this sort of expression
for any kind of defect.
So I talked about vacancies right now and vacancies have a
specific formation energy, but interstitials also have
formation energies, substitutionals also have
formation energies.
So you can use this expression to determine the fraction of
defects per lattice site for any kind of defect so long as
you have the formation energy of that defect.
So just to show you how difficult it is actually to
remove defects, if you have a crystalline material, defects
want to stay even at the lowest temperatures.
I have this interesting little piece of art to show you.
This particular piece of art was actually discovered, I
guess, by some scientist at the University of Toronto
where Dr. Sadoway went to school, and this particular--
what do they call it?
They call it The Atomix.
Let me write down the name of it in case you want to look it
up because I think they sold a lot more of these to material
scientists than they sold to anybody else.
And all this is two plastic plates, two PMMA plates,
polymethyl methacrylates, or plexiglass.
And in between them, there's a little hole that's cut, a gap,
and in that gap are a number of ball bearings.
And the ball bearings are kind of like atom, right, so they
move around.
And here we're going to the document camera right now.
So if you shake these things around and the ball bearings
are moving around, that's like introducing temperature.
That's like taking a crystal, melting it, all the atoms are
bouncing around.
You will even see some vapor atoms when these sort of leave
the surface and fly through the air, but then if you stop,
that's like quenching.
That's like suddenly I've taken this crystal, which was
originally molten, and I've dropped the temperature.
Here's what I see.
So can you see that?
How to use this thing--
there we go.
So you can actually see a lot of the defects that we were
talking about just a moment ago.
Here you can see areas of perfect crystalline order.
Here's another area of perfect crystalline order.
So you have many, many crystals that are adjacent to
each other.
They're oriented differently.
So they're forming misorientation defects between
themselves.
So for example, here's a grain boundary.
This is a crystalline grain.
And inside this crystal, you see a vacancy.
It's right there, So here's another
instance of the vacancy.
There's another vacancy.
And what happens if I try to remove some of the disorder?
By the way, interesting thing is that some of the vapor is
also left here.
So there's the vapor.
If I then try to remove some of these defects, I can just
tap on this.
If I just continue to tap on it, I'm removing defects.
The whole thing is crystallizing more and more
and more so now I put it down again.
OK.
You still see the vapor phase.
Now you see a big huge crystal right here with some grain
boundaries around it.
Here's another big huge crystal.
Here's another crystal.
There's a smaller crystal.
Each one of them is bounded by grain boundaries, but what do
you see in each of these crystals?
There's a vacancy.
And you can actually sit here--
well, not here.
Maybe later, but if you want to take a look at one of these
things, you can probably go to Dr. Sadoway's office and pick
it up and he'll let you play with it for a little while.
You can try to get all these vacancies out.
You can try to get all the defects out.
No matter how hard you try, if you spent hours, you'll get
things to be more and more and more perfect by tapping on it,
but there's always going to be a vacancy.
Always.
And even if you are extremely patient, you get things down
to just one vacancy and you think that all you have to do
now is just give it a little bit more of a tap to remove
that one vacancy, well, more often than not, you'll find
that with that tap you'll remove the vacancy, but create
another vacancy somewhere else.
So you'll always find these vacancies in
these crystalline materials.
It's just a consequence of the fact that when you're
agitating a material like you're doing here-- you're
shaking it--
that's the same thing that happens if you have some
finite temperatures, some non-zero temperature in the
material, you're always going to be creating some defect.
So that's exactly what this expression is giving you.
It's telling you that there's going to be defects at any
temperature, but their number goes up dramatically as you
increase the temperature.
So let's go back to the slides real quick now.
OK.
So everything I've told you so far is concerned with defects
in crystals that are of one type, so that are made up of a
single element.
But in the case of, for example, things like ionic
crystals, and ionic crystals are made up of multiple
elements with multiple charge states, you can get new kinds
of point defects that you can't see in a
single element material.
So we'll talk a little bit about those and when it comes
to those defects in ionic materials, we have to expand
the taxonomy a little bit.
The taxonomy is now going to include defects called
Schottky imperfections, Frenkel
imperfections, and F-centers.
So let's go to a visualization of what these are.
Here you see an ionic crystal.
So you have alternating types of atoms. You have these big
ones, which are presumably the negatively charge ones, and
then you have the small positively charged ones, and
here you have an example of a Schottky imperfection.
So a Schottky imperfection, it's kind of like a vacancy.
It's missing atoms. The main difference between a Schottky
imperfection and a vacancy by itself is the fact that in a
material that involves charged atoms--
ions, right--
in an ionic crystal, when you take atoms out, you have to
make sure to maintain charge neutrality.
So these materials are charge neutral.
They have equal numbers of positive and negative ions,
but you want to make sure that you take these
out in equal numbers.
So in the Schottky defect, you take out one negative and one
positive ion, or basically one unit, one stoichiometric unit.
If you add zirconium oxide, which is ZrO2, you would have
to take out three atoms to maintain charge neutrality.
That would be the Schottky defect.
So here is another example, another visualization of a
Schottky defect.
So when you take these two atoms out, nothing says that
they have to be taken out from right next to each other.
You can take out one here.
You can take out one there.
It's good both ways because in the end, it's just about
charge neutrality.
It's about maintaining a charge neutral material.
So we can actually write down reactions to describe the
formation of these Schottky defects.
So let's do that.
When we write down reactions to describe the formation of
Schottky defects, we first recognize the fact that we're
dealing with empty sites.
We're dealing with void.
We want to see how this void, or in scientific parlance,
null, is decomposed into two vacant sites in the particular
ionic crystal that we're dealing with.
So let's-- to make things specific,
deal with sodium chloride.
So sodium chloride, we have two type of
atoms. Sodium and chloride.
And to maintain charge neutrality, we have to remove
one of each.
So this is actually-- this null space is actually going
to be composed of two vacancies: a vacant site on a
sodium lattice where a sodium atom would have been sitting
and a vacant site where a chlorine
would have been sitting.
And because this entire process takes place in the
presence of materials that are composed of ions--
so ionic solids--
these two vacancies, in fact, have some charge them.
So when we think about the charge of these vacancies,
what we actually have to think of, it's a little bit
counterintuitive.
The sodium in a sodium chloride crystal is charge
positive, but what does that make the vacancy?
So if you have a lattice of atoms that are charge positive
like the sodium atoms and you remove one of them, what is
the effective charge of that vacancy?
You have nothing, as Dr. Sadoway would say,
in the land of plus.
So the charge of this vacancy, the effective charge of this
vacancy, is going to be negative.
And we mark that with this little thing right there.
So void or nothing in land of plus is negative.
We mark that with one of these apostrophes.
So what about this vacancy?
That vacancy is created on a lattice of sites which are
usually charge negative.
So if you have these chlorine atoms that are charged
negative and you remove one of them, now you have void in a
land of plus.
So this is going to be effectively charge positive
and so we have this void in land of minus.
It is positive.
And we do it--
we annotate it with one of these dots.
We can do the same thing for other kinds of crystals.
So I mentioned that in this case, a Schottky defect would
just be composed of these two atoms, because these two atoms
constitute the structural unit for this crystal.
But if we had some other, more complicated crystal, like, for
examples, zirconium oxide, then we would still write down
a similar reaction.
So we could put a Schottky defect into zirconium oxide.
OK.
And so now we have to figure out how this
reaction plays out.
We have to create vacancies.
Here's our vacancy on the zirconium.
And because the structural unit contains two oxygens, we
have to create two oxygen vacancies.
And now we have to think about how do we maintain charge
neutralities?
What is the effective charge on all of these atoms?
So the zirconium sites are usually the positively charged
atoms, so when we remove one of them, that's removing
something, that's putting void basically
into the land of plus.
So we have now a negatively charged vacancy, and this
negatively charged vacancy has four negative charges, an
effective minus 4 negative charge.
And the oxygens are the negatively charged ions, and
so when we create voids there, these two vacancies have an
effective positive charge, and because the charges have to
balance, we know that this has to be effectively two negative
charges and there are two vacancies.
So we have charge neutrality.
So that's how the Schottky defects work.
OK.
So let's move on to Frenkel defects.
In the case of Frenkel defects, it's really not so
different from Schottky defects in some sense, except
now instead of dealing with a situation where we maintain
charge neutrality by removing two atoms, we actually
displace one atom from its initial location to a
different part of the crystal.
So we're creating actually a vacancy right there, and this
atom, which was originally sitting on a lattice site, is
now displaced to an interstitial location.
So that's an interstitial.
So a Frenkel defect is a vacancy and an interstitial.
And here's another view of a Frenkel defect.
Here, I've removed an atom from one of the lattice sites
and moved it over to here.
So that's my Frenkel defect.
And in the case of Frenkel defects I can also write down
reactions for Frenkel defects.
Let's do that.
OK.
So Frenkel defects usually occur in crystals with widely
differing atomic radii, and in ionic crystals, the radius of
whatever is positively charged would have to be much, much
less than whatever the radius of whatever's negatively
charged occasionally, or very rarely, but I suppose it's
still possible, you would have to have basically a very big
difference in radius between these two cases.
An example of one of these situations is, for example,
silver bromide, where here the silver's positively charged
and the bromide ion is negatively charged.
And we can write down a reaction to describe the
Frenkel defect formation just like we did in the case of the
Schottky defect formation.
So in the case of a Frenkel defect formation, how do we
actually go about this?
We have, for example, can start out with a silver atom
site and we're going to displace that silver atom away
from its original site so it now sits in an interstitial
position and we're going to leave behind a vacancy.
So let's do that.
OK.
So we're going to have silver interstitial site and let's
just be clear--
say that this is on a regular silver site.
And then we have a vacancy left over on the silver site
and this is the reaction for the formation
of our Frenkel defect.
Frenkel defects, just like Schottky defects, preserve
charge neutrality.
So how do we actually mark down charge
neutrality in this case?
Well, there's no change in charges in the original state
so we'll just mark that with an X.
And the vacancy is sitting in a place that used to be
positive so it's a hole in the land of positive so we know
that this one's going to be negative.
And this silver ion, which was displaced from its original
lattice site, is now going to be sitting in an interstitial
site which was originally a land of zero.
So this one's going to be charged positive.
And now we've balanced, once again, this reaction.
We have charge neutrality.
We've created a defect and this is defect, just like in
the case of vacancies, costs us energy to make.
So we can actually write down a formation
energy for Frenkel defects.
And this formation energy is usually a little bit higher
than the formation energy of vacancies so it's going to be
something more perhaps on the order of 2 eV.
OK.
So we have gone through--
before I go to this one, let me just mention one last
defect that you might run into if you're working in ionically
bonded materials.
An F-center is a situation where you have a vacancy with
a bound electron inside of it.
And F-centers have the property that they give
ceramics a tint, a hue.
They actually scatter light in the visible range
so you can see them.
So with a very short amount of time left, I wanted to tell
you about a couple of the other defects that you will
encounter in crystals and one of the most interesting ones
to me is the dislocation and dislocations are line defects.
They're 1D defects.
So if you look in, for example, a corn ear right
here, you can see dislocations as the termination of a single
atomic plane.
So here's a picture of what a dislocation
looks like in a crystal.
You have these atomic planes and here you have
one that just ends.
That's a dislocation.
Some of you probably have dislocations in your
fingerprints.
So after the class is over, you can look for dislocations
in your fingerprints.
They're line defects.
So here's another example of that.
You see that there's an extraatomic plane which ends.
That's a dislocation.
We mark it with one of these Ts.
Here's a picture of a dislocation.
Dislocations actually help materials to deform easily
without having to shear off.
Here's a little picture of mine which I really like.
This is the only way that people really had to study
dislocations before computers existed.
So now we can study, for example, dislocation by
simulating them in real crystals.
And this is a mid-20th century computer.
It's a raft of bubbles.
It's just a bunch of bubbles that somebody blew on the
surface, soap bubbles, and you can deform them.
You can deform this raft of bubbles by shearing it and
here you see--
what's wrong?
Why isn't it going?
There it is.
There's your dislocation running through and you see
lots of dislocations running through this crystal.
This was, by the way, done by Lawrence Bragg, who you heard
about in connection with x-ray scattering.
You'll hear about Bragg again when we get to DNA.
So keep him in mind.