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JURGIS RUZA: In
this demonstration,

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I'm going to show you how to
construct 2D Brillouin zones

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and hopefully achieve a
better understanding of them

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and what they are.

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So a Brillouin zone is
an important concept

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in material science
and solid state physics

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alike because it
is used to describe

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the behavior of an electron
in a perfect crystal system.

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So what is a Brillouin zone?

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A Brillouin zone is a
particular choice of the unit

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cell of the reciprocal lattice.

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It is defined as the
Wigner-Seitz cell

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of the reciprocal lattice.

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It is constructed
as the set of points

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enclosed by the Bragg planes--

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the planes perpendicular
to a connection line

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from the origin to each
lattice point passing

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through the midpoint.

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Alternatively, it is
defined as the set

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of points closer to the origin
than to any other reciprocal

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lattice point.

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The whole reciprocal space
may be covered without overlap

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with copies of such
Brillouin zone.

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OK.

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So that was a rather
convoluted definition.

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Let's do it again.

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Let's brush up and be sure that
we understand this definition.

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We're going to define the
reciprocal space and the Bragg

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planes as well.

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But to define both
of these, we'll

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also do a quick revision of
what is the erect lattice so

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that we can go from there
to the reciprocal lattice.

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The microscopic
perfect crystal is

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formed by adding
identical building blocks.

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So to say unit cells consisting
of atoms and groups of atoms.

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A unit cell is the
smallest component

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of a crystal that,
once tacked together

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with pure translational
repetition,

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reproduces the whole
crystal, which essentially

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means that you can
take the same thing

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over and over and over again
and get the whole system done.

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So the groups of atoms,
these unit cells that

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form the microscopic crystal
by infinite repetition,

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is called the basis.

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OK.

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That seems quite clear.

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And the basis is
formed in such a way

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that it forms the lattice,
more commonly known

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as the Bravais lattice.

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Every point of a
Bravais lattice is

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equivalent to every
other point, which

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means that the arrangement
atoms in a crystal

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is the same when viewed from
different lattice points.

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OK.

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That also seems
quite understandable,

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and you should probably
know that by now.

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So any fundamental
lattice must be

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definable by three primitive
translational vectors--

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a1, a2, and a3.

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The combination of
these vectors is usually

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to find the crystal
translational vector r,

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such that r is equal to a1
n1 plus a2 n2 plus a3 n3,

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where n are just
arbitrary integers to show

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the size of our lattice.

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The crystal lattice is repeated
an infinite amount of times

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to create the perfect
crystal structure,

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and each of those lattice are
translationally symmetric.

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Another way to look at it
is that one cannot tell

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their position in the crystal
structure because every lattice

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looks the same.

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OK.

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So that seems to make sense.

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So now, let's go through
reciprocal space.

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So every lattice has
a reciprocal lattice

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associated to it.

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In crystallography terms,
the reciprocal lattice

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is the fraction
prior of a crystal,

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or in quantum mechanics
it's describe as k space,

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with k being for k wave vectors.

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In 3D lattice, the vectors
would be b1, b2, and b3.

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And they can be denoted
as-- we'll look at just b1.

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b1 is equal to two parts of the
cross-products of the vectors

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a2 and a3 from our
direct lattice divided

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by the triple cross
scalar product of a1, a2,

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and a3, in which case this
cross-product of a2 and a3

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is the area of our vector
of our two vectors,

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and the triple scalar product
is the volume of our system.

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By simplifying it,
we can just get

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2 pi over the height
of our unit cell,

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or we can put it this way.

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The larger our direct lattice,
the smaller in comparison

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our reciprocal lattice becomes.

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Another observation
that could actually

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be made by the
reciprocal lattice

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is that the reciprocal lattice
of the reciprocal lattice

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is the direct lattice.

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But OK, for simplicity's
sake, let's look

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at a transformation
from 2D lattice

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to a reciprocal lattice.

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So we have a
visualization here where

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we can change the length of
our x vector in direct space

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and the length of our y
vector in direct space.

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And we can change
whether or not we're

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seeing this as a direct
lattice or whether we're

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seeing this pattern as
a reciprocal lattice.

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So as we can see, by
increasing the length

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of our direct vector,
we change the sizes

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of our reciprocal lattice
vectors, and the other way

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around.

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So now we had a definition
of the reciprocal space

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and direct space.

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Let's go back to our definition.

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So the first
Brillouin zone can be

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defined as a set of
points in reciprocal space

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that can be reached from
a specific point of origin

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without crossing
any Bragg planes.

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So what are Bragg planes?

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A Bragg plane, or in
this case, a Bragg line,

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is a Bragg line which
perpendicularly bisects

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a reciprocal lattice vector--

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a vector which connects
two lattice points.

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And the closest Bragg
planes are essentially

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crossing the Brillouin zone.

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Now, we can show
the Bragg planes

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with the closest neighbors, with
this being our original lattice

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point.

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And these are the four
closest neighbors in a simple,

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I'd say cubic, but it's
actually just the square lattice

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because it's in 2D.

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And if we add the
second largest vectors,

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for the second closest neighbors
you can see these ones.

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And essentially, it conveys
the same information.

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When we go to a higher
order of closest neighbors,

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we can see that the system
gets a lot more complex.

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So now we've seen
what are Bragg planes,

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we can go towards
Brillouin zones.

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So this is the first
Brillouin zone.

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It is what it seems it is.

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As you can imagine, the
Bragg planes just go here.

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And the Brillouin
zone shows us the area

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in reciprocal space that is
closer to our lattice point

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than any other lattice point,
which are essentially the Bragg

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planes as well.

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As we can see, our reciprocal
lattice origin point

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is in here.

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The square is closer to this
point than to any other point.

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And after these lines, it
gets the other way around.

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So we can move on
forward to a higher

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order of Brillouin zones.

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And this is the Brillouin
zone for the second closest

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neighbor.

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As you can see, it
is rather similar.

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It's just takes the
second closest neighbors

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and essentially
draws another square.

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But it's a bit
tilted to the edge.

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So now let's look
at the third one.

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For the third Brillouin zone,
it gets a bit more complex

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because if we scroll
a bit backwards,

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we can see that the Bragg planes
for the third closest neighbors

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are these ones.

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So we might think that
this whole thing would

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be the third Brillouin
zone, but it's actually not,

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because with every next system
it gets a bit more complex.

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And it's actually taking
account both the third Bragg

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planes and the first ones.

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So now let's do another thing.

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Let's turn on that we can
see all the Bragg planes

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and turn up another notch.

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So here we can see that
the fourth Brillouin

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zone gets a lot more complex.

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And we can see all
of the Bragg planes

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for the closest neighbors.

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And these lines get quite
difficult to understand

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by drawing themselves, but
we can help ourselves out

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with this visualization.

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So yeah, we can
see that they start

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to interact with each
other, and thus make

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a more difficult structure.

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And if we go to the
fifth one and show

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that we can see all
the Brillouin zones--

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so here we can see that it
becomes quite a nice drawing.

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If we were to keep adding them--

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I mean, let's just do it.

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Let's add until the ninth one.

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So here we can see a high
order Brillouin zone.

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It actually looks kind of
nice, which is also interesting

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because it's also an art form
drawing high order Brillouin

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zones.

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The higher you go, the more
complex and more discrete

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the system gets.

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And it essentially looks nicer.

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So after looking
at this, we can get

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a short definition of how to
construct 2D Brillouin zones.

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So n-th Brillouin zone can be
defined as the area, or volume

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if we look in 3D, in
reciprocal space that

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can be reached from
the origin by crossing

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exactly n minus 1 Bragg planes.

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So we can look also at a
Brillouin zone for a system

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where the atoms
are not perfectly

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in a perfect square lattice but
are offset a bit, making, so

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to say, a triangular lattice.

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And as we can see here,
the first Brillouin zone

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is a hexagon.

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And the second one already
gets a bit more difficult

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by forming a star shape.

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And the third one is,
again, so to say, a hexagon.

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And it goes on like that.

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And then, so what information
does the Brillouin zone hold

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and what does it give us?

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In short, vectors in
the Brillouin zone

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or on its boundary characterize
states in the system

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with lattice periodicity.

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For example, phonon
or electron states.

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But for that, a
whole other video.

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This code for this demonstration
was taken and edited

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from mathematical demonstrations
made by Jaroslaw Klos.