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MARCELO GONZÁLEZ: This video is
about crystals and structures

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presented by me, Marcelo
Alejandro González.

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Well, first of all,
I would like to say,

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don't focus on the code.

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And second of all, one
of the basic things--

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most basic concepts--
that we need

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to see for
crystalline structures

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is first, what is a crystal
structure and second,

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what is a unit cell?

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Well, a crystal
structure is just

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the way in which atoms, ions,
and molecules are spatially

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arranged in 3D.

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And a unit cell is its
smallest repetitive volume,

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which means that it has to
contain the complete lattice

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pattern.

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As you can see here, we have
a crystalline structure,

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of course, but what do
you think that is the unit

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cell of this crystal structure?

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This is the unit cell.

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The unit cell is just
a simple cubic crystal,

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where the lattice constants
and the interfacial angles

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are the same, which means
that the unit cell is actually

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represented by a cube.

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This is just one of the
seven different crystal

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systems that actually exist.

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However since we've
already started

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talking about the simple
crystal structure,

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let's not concentrate on
different crystal systems

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and let's go deeper into
this types of structure.

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A simple lattice is indeed just
a cube, as you can see here.

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It's a cube that has atoms
in all of its corners,

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so it has eight atoms.

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These eight atoms
have 1/8 of themselves

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inside the unit cell.

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and at the end, it
makes the unit cell

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have one complete atom.

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Well, another important concept
in crystalline structures

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is the coordination number.

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The coordination number is just
the number of nearest neighbors

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each atom has which are
the closest to them.

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In this simple cubic structure,
the coordination number is six.

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It's hard to see it in just one
unit, so what we're going to do

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is create another visualization
with eight unit cells put

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together with different
colors, so it is a lot easier

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to see which ones are
the nearest neighbors.

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Here, you will be
able to see it.

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Now here, you see how the red
atom has six nearest neighbors,

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which are the yellow atoms.

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Actually even though this is
the simplest crystal structure,

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there is only one
type of atom that

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can arrange in this structure,
and that is alpha polonium.

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I am sure none of you guys
has ever used polonium before.

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One of the reasons why is
because it's radioactive.

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But the main reason
why only alpha polonium

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goes into this
crystal structure is

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because of the packing factor.

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This type of crystal structure
has a very low packing factor,

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and that's why
not a lot of atoms

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would crystallize into
this type of structure.

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But actually, what is the
atomic packing factor?

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Well, the atomic packing
factor, as we can see here

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in this formula, is just the
volume of atoms in a unit cell

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if we assume they are hard
spheres and the volume

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of the entire unit cell.

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So to calculate the
atomic packing factor,

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I just created a
function to use it,

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which means the number of atoms
and the radius of this atoms

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if we actually think
about the mass spheres.

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Thinking of the atoms
as spheres is actually

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the easiest way to
calculate the atomic packing

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factor because the
formula to calculate

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the volume of the
sphere is pretty easy.

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It's just 4/3 of pi
times radius cubed.

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Now, what we need
to find is what

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is the radius of these spheres?

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In this case, the
easy way to do it

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is having two atoms
in two corners.

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If you have two
atoms in two corners,

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you can see that from the
center of each of the atoms,

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you have one length
of the lattice.

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Here, the lattice
is just a cube,

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so its volume would be
the length of the lattice

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to the power of three.

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And since we have two
atoms in one length,

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the radius of each
of these atoms

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is just going to be half of
the length of the lattice.

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Therefore using the function
to calculate the atomic packing

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factor, knowing that
there is only one atom

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and that the radius is half
the length of the lattice,

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you get an APF of just 0.52.

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Well, we've already
gone through the basics

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of a simple cubic structure,
but let's go a little bit more

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complicated now.

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We're now going to the
body-centered cubic lattice.

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Well in a body-centered
cubic lattice,

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imagine you're an atom.

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You want to be like
alpha polonium,

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so you want to have your space.

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You know, you want to not have
a very large packing factor.

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You're trying to get together
with seven other atoms

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in a unit cell, and
then just an atom

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comes in between all of you.

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This is what happens.

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This is as you can see
here in the visualization.

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And just an atom is in
between the unit cell.

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We can see it a little better
with a lot more separation.

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So there is an atom
in between all of you.

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So as compared
with before, there

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is not only one atom per unit
cell, but now there's two--

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the one that's right in the
middle, which is complete,

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and still the 1/8 of
the whole eight atoms

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there are in the corners.

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Now, we knew that
the coordination

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number for the simple
cubic cell was six.

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And now, for the
body-centered cubic,

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it's eight, which means that
it has a lot more atoms closer

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to him.

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That would, in principle,
say that the atomic packing

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factor would be higher,
but that we will see later.

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We talked about the
coordination number being eight.

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And here, we can see by putting
eight unit cells together,

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we can see the eight nearest
neighbors for the red atom

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here in the middle.

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It's actually pretty hard
to see it with bonding,

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so now we can see
it a lot better.

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So it has eight
layers neighbors--

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this red atom in the middle.

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And these eight
nearest neighbors

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are the yellow ones
that can be seen here.

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Well now, we can start talking
about the atomic packing

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factor of a BCC structure.

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It is actually not as trivial
as the simple cubic structure

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because since you have an atom
in the middle of the unit cell,

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the corners of the same unit
select don't actually touch.

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So it's not as simple
as just saying that it's

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half the length of a lattice.

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Now, we actually
need three atoms,

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as you can see here in
the visualization, that

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are going through the
inner diagonal of the cube.

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If you remember from
your old math classes,

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the length of the inner
diagonal of a cube

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is actually just square
root of three times

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longer than the
length of a lattice.

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And as you see here,
we have four radii

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going through this inner
diagonal of the cube,

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so actually the radius
of all of these spheres

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is square root of
3 times the length

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of a lattice divided by 4.

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Why?

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Because we have four radii
through this whole diagonal.

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So now using the
function we used

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before to calculate the
atomic packing factor,

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we know that there is two atoms.

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I've already explained how
to calculate the radius.

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So then using this function,
we get an APF of just 0.68,

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which is considerably
higher than 0.52

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for the simple cubic lattice.

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Some of the examples for these
body-centered cubic structure

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are chromium, tungsten, iron,
tantalum, and also molybdenum.

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Lastly, we can talk about a
face-centered cubic lattice.

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In a face-centered
cubic lattice,

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you can see that the unit cell
is a lot more densely packed

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than the other ones.

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Why?

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Because here, we have eight
atoms which are in the corners.

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That means one complete
atom in the unit cell.

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But now, we have
six atoms in all

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of the faces of a
cube, which are six.

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And half of these atoms
are inside the unit cell,

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which means that we have
six atoms per six faces--

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half of them as an inside 6/2.

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It means three more
atoms, so the total number

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of atoms in this
unit cell is four.

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That will mean that the atomic
packing factor will actually

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be higher than all the
other two we've seen,

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but that we'll actually
talk about later.

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Now if we want to see the
coordination number here,

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it is actually pretty hard to
do it with eight unit cells

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together, as you can see here.

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So what I'm going to do is put
two of these unit cells one

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on top of the other, so you can
see the coordination number,

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which is actually 12.

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That is also goes together
with the atomic packing factor.

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As you can see it here,
even with the bonding--

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I changed the colors now--

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the atom in yellow has
12 nearest neighbors,

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which are the atoms in red.

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That's the coordination number.

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Now finally, for the
atomic packing factor

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of an FCC structure, it's
actually also not so trivial.

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Now instead of using
the diagonal of a cube,

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we're going to use the
diagonal of a square.

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Not so hard to remember
from your math classes,

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since you know that the
diagonal of the square

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is just square root of 2
times larger than the length

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of this same square.

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So using the same principle,
the radius of these spheres--

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since we know that we
have four radii here--

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would be square root
of 2 times the length

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of the lattice over 4.

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And of course, we
know that now there

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are four atoms per unit cell.

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And this APF gives
us 0.74, which

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is the highest one you
can get from atoms,

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as I said before, that
have the same radii.

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Then you can try to concentrate
on mixtures of atoms,

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not of pure atoms and elements,
as I have mentioned here.

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That's a little bit
more complicated.

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Some of the examples
for this FCC structure

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is aluminum, copper,
nickel, silver, or gold--

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fairly well-known metals.

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That's all for
this lecture, guys.

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Thank you very much
for watching the video.

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I would like to acknowledge
Bianca Eifert and Christian

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Heiliger because they're
the authors of the package

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Crystallica, which made this
Mathematica notebook possible.

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Thank you.

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Goodbye.