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POOJA REDDY: Hi.

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I'm Pooja.

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And I will be talking to you
about how you can visualize

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the energy of
defects, specifically

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focusing on screw dislocations.

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So crystals have an
underlying lattice structure.

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But in reality,
they're rarely perfect.

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They usually have mistakes in
them which we call defects.

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A point defect can be a
vacancy where an atom is

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missing from the lattice.

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And it can also be an
interstitial, where there's

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an extra atom in the lattice.

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Another type of defect
are dislocations,

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which are linear defects.

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Here is an example of
an edge dislocation.

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The red arrow here
shows a line direction

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in the defect, which is at the
edge of this extra plane here

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that's in the lattice.

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Another type of dislocation
is a screw dislocation.

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The blue arrow here shows a
line direction of this defect.

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Here, items on one side
are higher than the other.

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Here, it's lower.

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And you see this almost
spiral shape that

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forms kind of like a screw.

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From these pictures
you can really

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see how defects
distort the 3D lattice.

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So why do we care about defects?

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Turns out that they really do
affect material properties.

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Some properties depend
heavily on defects.

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And those include
conductivity, especially

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in semiconductors, hardness and
ductility in metals like brass,

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and diffusion
coefficients of materials.

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So lattices exists because they
are low-energy configurations.

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It would make sense then
that errors in the lattice,

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like defects, should increase
the energy of the lattice.

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So how can you think of the
energy of atoms in a lattice,

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or just in general?

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One way we can calculate
the energy of atoms

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is using Lennard-Jones
potential.

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Lennard-Jones potential
approximates the interaction

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between a pair of neutral atoms.

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In this graph here, the x-axis
is the distance between atoms,

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and the y-axis is a measure
of potential energy.

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And you can really
see in this function

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how there is a energy
well or minima.

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So what happens is
when atoms are too far

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away from each other,
there's an attractive force,

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and they come together
to reduce energy.

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And when they're
too close, there's

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a repulsion force, which
is used to reduce energy.

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And at the energy
minimum the atoms

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are actually this specific
distance away from each other.

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What you actually find is that
the bond length corresponds

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to this distance found at the
minimum of potential energy.

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But how about in the lattice
where there are a lot of atoms?

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Energy still does
want to be minimized.

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So these atoms try
to minimize energies

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by sitting in multiple energy
wells with their neighbors.

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Also, it's good to remember
that an atom will be interacting

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with this one and
this one and this one,

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and just all the
atoms around it.

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Although the effects
will probably

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become smaller as the distance
between atoms increases.

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So an interesting
question to then ask

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is how would defects affect
the energy of particles

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in a lattice when a lattice
is a low-energy configuration?

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In this notebook I
will aim to visualize

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the energy of defects.

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I will mainly focus on first,
vacancies in a 2D lattice,

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and then move on to screw
dislocations in a 3D lattice.

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So here is a 2D
lattice with a vacancy.

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What I've done is I've used
Lennard-Jones potential

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to calculate the
energy of each atom.

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I then use this energy
to color the atoms.

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And blue is low energy,
and red is high energy.

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What you find is
interior is mostly blue,

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while the surface has these
yellow and red energies which

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are higher.

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One way to think about
this is that these atoms

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have fewer neighbors, so they
sit in fewer energy wells.

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This higher energy we
find on the surface

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is actually known
as surface energy.

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Also, notice how the energies
immediately around the vacancy

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are also higher.

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For fun, here's
a cool simulation

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of multiple vacancies
moving around a lattice.

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You can see here how the
energies around vacancies

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change when vacancies
are next to each other

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or near the surface.

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Now that we've
had a taste of how

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you can color atoms
to indicate energies,

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let's move on to the good
stuff-- screw dislocations.

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First, here's a cubic lattice.

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Like with the 2D lattice,
there's a surface energy.

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This higher surface
energy can make

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it difficult to see how the
screw dislocation changes

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energies of atoms
in the lattice.

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So what I'm going to do is
only visualize the inner atoms

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excluding the surface ones.

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To really see how screw
dislocation works,

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here is an inner
layer of the lattice.

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The Burgers vector represents
a magnitude and direction

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of the lattice distortion
resulting from a dislocation.

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As I increase the magnitude
of the Burgers vector,

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you can really see this
kind of spiral staircase

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shape happening around
the dislocation.

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Now I'm going to
extend this layer

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to show all the inner
atoms in my cubic lattice.

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As I increase the magnitude
of the Burgers vector here,

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you can see how the atoms
right around the dislocation

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become more red
or higher energy.

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So the energy around
the dislocation

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is higher, very similar to
what we saw with the vacancies.

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Now what I'm going to do is
chop this in half and only look

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at half the atoms.

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And as I increase the magnitude
of the Burgers vector,

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it becomes really clear that
the energy becomes higher just

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around the dislocation,
where here is a dislocation,

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here's a very high energy red,
and here's the lower energy.

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The only thing is it's kind of
hard to see how the inner atoms

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energies change.

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So what I've done below is
I've created a simulation

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where the higher the energy,
the more opaque the atom.

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So we start with a pretty
translucent lattice

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because everything's
lower energy.

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So as I increase the magnitude
of the Burgers factor here,

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you start to see that
atoms around the center

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become more opaque.

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And you can really see where the
locations of high energy are.

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Here is kind of this
column of high energy

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right in the center.

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These parts have
higher energy as well

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because they're more exposed.

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They have fewer neighbors.

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So now that we've
looked at cubic lattice,

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let's look at a hexagonal
close packed lattice, or HCP.

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So here I visualize an inner
portion of the HCP lattice.

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Again, I'm not looking
at surface atoms.

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And all atoms here
are pretty dark blue,

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so they're lower energy.

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As I introduce my
Burgers vector,

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the energy does clearly increase
right around the center.

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And what you'll also
notice is that atoms

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are becoming less
blue, which means

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that the energy of all the
atoms is kind of increasing.

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And something cool
happens if I keep

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increasing the magnitude
of the Burgers vector.

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What I find is that I
have this area of yellow,

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kind of higher energy.

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And an area of much higher
energy right outside of that.

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And then the energy
starts to decrease again.

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So let me go down here where I
have colored my particles also

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based on opacity.

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And you can really see
this super high energy

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opaque red lines near the
center where my dislocation is.

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And this is a pretty cool
result of simulating screw

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dislocations.

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I expect energies to be higher
right around the dislocation,

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which I saw pretty clearly
with the cubic lattice.

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But in HCP, in addition
to energies being higher,

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energies become a lot
higher a little ways away.

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So this is a neat example of
how simulations can really

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teach you something
more about a system

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that you didn't know before.

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In the above simulations,
I varied the magnitude

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of Burgers vector.

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But we also know that defects
can propagate through lattices.

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In my simulation
before, I actually

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had the vacancies moving around.

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So now I'm curious to see how my
screw dislocation can propagate

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through a lattice.

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What I've done below is
I've fixed my Burgers vector

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to a magnitude of 1, and
have propagated the terminus

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of the screw dislocation.

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Here is a cubic lattice.

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And as I move the terms of
the Burgers vector through,

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you see this layer of
high and low energy moving

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through the whole system until
it passes all the way through.

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And now you have the
geometry that you can

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tell is of a screw dislocation.

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Now if I only look
at half the atoms,

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and I start moving my Burgers
vector terminus through,

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you can really see how the
blue or low energies are where

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the atoms are being
squished together,

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and the red is where
they're being spread apart.

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And also above where the
terminus has passed through you

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have this geometry of
a screw dislocation,

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and below you still
have the cubic geometry

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of the cubic lattice.

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In hexagonal closed pack I still
do see a really similar thing

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of this layer of lower
and higher energy

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passing all the way
through the lattice.

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At the end, you just
have this geometry

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that we remember before
of a screw dislocation.

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The interesting thing
is you can still

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see that the energy is higher
around the line of the screw

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dislocation, just like
how we saw before.

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So here are some takeaways.

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First, I went over a couple of
basic types of dislocations.

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Then I talked about how lattices
are low-energy configurations.

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I also showed how you can
use Lennard-Jones potential

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to calculate energies of atoms.

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I then visualized
energies of defects

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using Lennard-Jones
potential and focused in

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on screwed dislocations.

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I first varied the magnitude
of the Burgers vector.

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And this was to show you
how energy was higher

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on the position of
the dislocations.

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I then showed you
how a lattice changes

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as you move the terminus of
the Burgers vector through it.

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I hope you've enjoyed learning
about the energies of defects.