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GLORIA CHYR: Hi there.

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My name's Gloria Chyr.

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And for the next
few minutes, I'll

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be talking about how to
find the perfect diamond

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and why that might
actually be impossible.

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Let's start with what a
perfect diamond looks like.

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A quick Google image search
for the perfect diamond

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yields this as the
first result. And it

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does look pretty perfect.

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But just because
it looks perfect

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doesn't mean that
it actually is.

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To determine if a diamond
is really perfect,

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we have to look
at its structure.

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A diamond consists of carbon
atoms arranged in what

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we call a crystal structure.

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More specifically, diamond form
a cubic crystal structure made

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up of tetrahedraly
arranged atoms

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as shown in this 2D depiction.

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An important thing to note
is that diamonds are not

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a stable form of carbon.

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They're what we call
metastable, which

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means that they're
energetically unfavorable,

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but there is a high
activation energy for it

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to decompose so it stays how
it is for a very long time.

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Besides the fact that
diamonds aren't forever

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what really makes finding
the perfect one impossible

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is that it probably
doesn't even exist.

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What real diamonds look
like is something like this.

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It's the same structure
as the last slide,

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but there's a
carbon atom missing.

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This is called a vacancy, which
is a type of defect that can

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exist in crystals like diamond.

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Defects are imperfections
and crystals,

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and vacancies are
not the only kinds.

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All crystals in the
real world have a number

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of defects ranging from
vacancies, to grain boundaries,

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to edge dislocations.

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There's quite a number of them.

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But for the purpose
of this video,

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I'll be referencing
vacancies in diamonds.

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Keep in mind that
anything I say about them

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also applies to other
kinds of defects

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in other kinds of crystals.

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OK, so now that we know why
crystals aren't perfect,

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we want to know why this
happens to begin with.

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The reason that
these defects form

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is because a crystal will always
be tending to equilibrium,

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also known as the state of
lowest Gibbs Free Energy.

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The formation of
defects in a crystal

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may seem counter-intuitive
because it costs

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energy, which is not favorable.

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But the formation
of defects also

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increases the amount of
entropy in the system which

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is favorable.

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The entropy gain can sometimes
outweigh the energy cost

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of forming a defect.

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So a perfect crystal can
actually decrease its Gibbs

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Free Energy by forming defects.

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This can be seen in the
following equation where

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delta G equals delta
H minus T delta S.

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So formation of defects
becomes a battle

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of enthalpy versus entropy.

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If the enthalpy cost
of forming a defect

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is greater than the entropy
increase of the system,

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then the defect does not form.

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If the entropy
increase of the system

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is greater than the cost
of forming a defect,

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then the defect does form.

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The change in Gibbs Energy
of forming vacancies

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depends on several
factors, including

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a number of vacancies
already present

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in the size of the crystal.

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So the energy cost and
entropy contribution

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of vacancy formation
actually changes

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with each defect formed.

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What we see is that
there's actually

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happy spot for the equilibrium
amount of vacancies.

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This here is a visualization
of how the number of vacancies

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affect a Gibbs Free
Energy of a crystal

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at various temperatures.

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On the left is a plot of
Gibbs Free Energy where

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the x-axis represents a
number of vacancies documented

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in a crystal, and
the y-axis represents

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the Gibbs Energy of the system.

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On the right is a visualization
of what the lattice

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looks like at equilibrium.

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In this demo, the crystal
is a square lattice

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made up of 400 atom sites.

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I am using a 2D square
lattice instead of a three

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dimensional diamond
lattice for ease of viewing

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and understanding this topic.

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These concepts
apply the same way

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to diamond crystals, which
are much more complicated.

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As you can see already,
there is a vacancy present

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in this lattice.

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It may be a bit difficult
to see on the left,

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but this is because at
equilibrium the Gibbs Free

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Energy plot is actually
not linear, but curved.

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This becomes more apparent as
we increase the temperature.

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Higher temperatures provide
greater entropy increase

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of defect formation.

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So we see vacancies are
more likely to form.

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Here we can see that
the lowest Gibbs Free

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Energy doesn't occur at 0, but
actually in around 50 vacancies

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per 400 sites.

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This seems pretty high.

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But if you think about it,
this is at 5,000 Kelvin.

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So the atoms in
the diamond crystal

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are actually moving
around quite a bit.

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So it makes sense
that you won't find

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all of the atoms in
the right places.

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If you would like to see how
this visualization was made,

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the code is accompanied
with this video.

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In summary, what we found is
that the perfect crystals don't

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exist because the equilibrium
state of lowest Gibbs

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Free Energy occurs with
some number of defects.

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Since diamonds are
crystals, it turns out

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that finding a perfect
one is impossible.

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But that doesn't
mean we can't find

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a diamond that looks perfect.

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Thanks for listening.

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And I hope this quick lesson
has made understanding why

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defects form a little clearer.

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The notebook that
accompanies this video

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is a written version
of this video.

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And with that, I
would like to thank

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Professor Carter
and Professor Keenan

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and the help of 31.6 TA's
Emma, Burhan, and Philippe.