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ALEXIS GERVAIX:
Hello everyone, and

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welcome to this little
video presenting

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you my project on Mathematica.

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My name is Alex.

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And I'm currently
studying material science

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in EPFL in Switzerland.

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And we had this
personal challenge

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to do something you
wanted with Mathematica.

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So I took an interest
into crystallography,

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because this is a quite
essential field of material

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science.

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Well, actually, it
was on the request

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of my girlfriend, who has quite
trouble to see things in 3D

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that I did this.

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And my idea was to
create a tool that

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could be used by anyone
that doesn't especially

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know about Mathematica, and
that, without coding anything,

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he could create a
couple of structures,

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have a look at the--

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have an interactive
part with it,

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and choose the atom, et cetera,
and visualize more easily

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these structures.

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So let's have a look
first at a bit of theory,

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and first with the
Bravais lattice.

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So Bravais lattice is
one of the fundamentals

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of crystallography.

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It defines the
different structure

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in which the atom can
organize themselves.

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So there is 14 of
them, where we will

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vary the axes, the primitive
cells, which are A, B,

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and C, the different angle.

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And these are the
different axes.

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And these axes can change
their length, all the angle,

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as you can see here.

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And there is also
a place where you

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will have more
atoms in the middle,

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or in the same-- in the faces.

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So in my program, I will
focus on the simple cubic,

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the face-centered cubic, and
the body-centered cubic, which

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are some of the most, I wouldn't
say basic, but some of the ones

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you will encounter
more often in solids.

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There is also
[INAUDIBLE] compact

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that is quite often represented,
but it was quite difficult

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implement.

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So it could be nice to
do it in the future,

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but for now, it's not there.

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And I also had a look
at interstitial sites.

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So interstitial
sites are atoms that

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would come to put themselves in
some void area of your lattice.

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So here is the FCC lattice
with an atom on each summit

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and one in the
middle of the faces.

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And we have in pink
an octahedral sets.

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So it's an octahedron
of void place.

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And in the center, you have then
the possibility to put an atom.

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So this is interesting
when you are

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looking for alloys
in metallurgy,

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for example, because it
gives a lot of possibilities

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to put new atoms in your metal.

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And so there are two
categories, as I mentioned,

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the octahedron and
the tetrahedron.

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And for each of the
three structures

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that I have been picking,
there is a different number

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of octahedron.

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For example, for cubic
simple, there is only one,

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so in the middle.

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But for BCC, as we
can see here, there

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is a lot of tetrahedraces,
which are here on the faces.

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So yeah, well,
this is a 2D graph.

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And it's not always easy to
really see how things are done.

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And I remember having
a really hard time

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to find where the tetrahedron
sites were for the BBC.

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So this is why I
really wanted also

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to put in my 3D graph
the interstitials.

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And I hope it will be
useful for you as well

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if you use my program.

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So yeah, so about
crystallography in general,

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this is a field that gives
you a lot of information

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about the properties of matter,
because when you understand how

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the atoms organize
themselves, you

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can start to understand
things like metal transition.

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And a more advanced
thing, you can

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start to look at the
growth of monocrystals.

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For example, if you're
working in semiconductors

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and you're trying
to do nanowires,

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then you need to
understand how the--

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what are the different
plane, how the atoms organize

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in plane, themselves.

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And when you're working
on aircraft engineering,

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you have the turbine
that you need to do

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in a monocrystal of nickel.

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This is very important.

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So yeah, there is a
lot of application

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that you can have
from this subject.

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And I thought it was an
interesting one to observe.

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So let's have a
look at the program.

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So this is some reminder
about how it works.

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And this is the
interactive part.

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So my program
consists of two cells

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that you simply have to click
on them and press Shift, Enter.

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The first parts are
the function that

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will define the
position of the atoms.

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And the second part is
the interactive tool,

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which we call
Manipulate Mathematica,

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and gives a lot
of possibilities.

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So you will see, this is a
huge cell with a lot of code.

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I know that Mr. Carter
doesn't like it at all.

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But I had to keep it like
that to have my Manipulate.

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And let's simply run it.

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So it takes a bit of time.

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And here we have it.

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So this is the panel--

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maybe we should put that away--

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where you have, on the left,
the controls, on the right,

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a display with a panel of the
periodic table, the 3D graphic,

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and a little graph that helps
you to visualize more easily

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the parameters of the lattice.

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That helps you when you try
to calculate the densities

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of these particular structures.

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So as you see, we can move it.

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There is different
type of atom here.

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I on purpose put
NaCl, which is salt.

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And now let's have a look
at all the possibilities

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we have with this program.

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Because the thing is that I
want you, when you have a--

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you know already a
structure, and you

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want to know what it looks
like, to be able to create it.

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Well, first, let's look
at the different options.

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So we have these three
structures, the BCC,

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the cubic simple, and the FCC.

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You can change the atom size.

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Here you can display--

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you can choose if there is
the [INAUDIBLE],, and here,

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the choice of element,
because I wanted

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that you can choose what type
of atom you put in your cell.

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This is why there is this panel.

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So I divided my graph
into four parts.

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Basically, atom one and atom
two are the atoms on the edges.

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And in the center are
a different division

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that helps you to create
more complex cells.

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So let's have a try.

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I would like a
body-centered cubic.

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And I would like an alloy
which is iron and aluminum.

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That structures
itself into a BCC.

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So I will put an atom
two there, aluminum.

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And here you will
see it changing.

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So this is normal, because
I asked the software

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to only update this part when I
click on the panel [INAUDIBLE]..

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It can be improved,
this part, but it

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was to have stability
during the initialization

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of the Manipulate.

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And here I have a
way to initialize

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my body-centered cubic
with the iron atom

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on the edges and my
aluminum atom at the center.

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You can, of course,
change the atom size.

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You can click on the
same atom, and this way,

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get a little bit bigger.

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OK, so there is also
FCC that you can do.

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There is one alloy
that works with this,

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which is, if I remember well--

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I wrote it somewhere--

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aluminum and nickel.

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So let's circle the aluminum
and this nickel here.

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And that should be
[INAUDIBLE]---- yeah.

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And here is the FCC.

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And here you can
turn and have a look.

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You see that there is
nothing in the middle.

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And for me, it helps me a lot
to visualize it like that.

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There is also the adapted
density calculation graph,

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where you see that this
diagonal in the middle will--

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no, in the face here, will
help you calculate the density.

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And you know that this line
is a square too of the lattice

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parameter.

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Now let's have a look
about the interstitial

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that I took some time to put--

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add on my graph.

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And I would like to show you
a little example that I had--

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that I forwarded, which is
this slide about the magnesium

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lead, intermetallic Mg2-Pb.

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And they said that this
should look like this.

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And they are giving
us that, so the anion,

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the magnesium are on
the FCC site in blue.

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And the lead is on
the octahedral site.

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So let's try to put
it in my program.

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So first, I will
say, yes, I would

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like to have octahedron
interstitial.

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I will stay in FCC.

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I would put the same atom
here, magnesium, and here.

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And it's a bit long to update.

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My computer is
having a hard time.

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All right, so we see that
it doesn't look like at all

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on what's [INAUDIBLE] on here.

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This is a bit--

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all right.

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OK, and here is where I see
that this guy did a mistake here

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by putting octahedron
in this place,

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because this place
is tetrahedron,

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the other type of interstitial.

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So we will remove
the octahedral,

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put the tetrahedral [INAUDIBLE].

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And here we go.

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[INAUDIBLE]

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OK-- no, no, OK, I should
do things properly.

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Interstitial tetrahedral
and [INAUDIBLE]----

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now it's better.

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And it looks like
what we have here.

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And I find it better
to look like that,

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because now that we can
change a little bit the size,

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it is clear now that
there is nothing

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in the middle, which,
it's not that clear when

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you look at this thing.

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So yeah, I found that it
was one nice application

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of this little program, to
visualize it more easily.

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So here is for now.

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I hope you enjoyed this
little presentation,

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and that it will help
you by using this program

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to see better in 3D your
crystalline structure,

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and that will help
you to have a better

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idea of what you're doing with
your crystallography courses.

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So thank you for watching.