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STUDENT: Hello.

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Today we are going to talk
about real and reciprocal space

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in two dimensions
and three dimensions.

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Crystallography is a major
topic within material science.

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A crystal is a highly
ordered solid material

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made up of a lattice and a
periodic arrangement of atoms.

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Atoms are located on
every lattice site.

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Due to the periodic
nature of the structure,

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all crystals display what's
called translational symmetry.

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Crystals are defined by
their lattice vectors,

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which are different in
two dimensions and three

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dimensions.

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In two dimensions,
lattice vectors

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are the shortest and next
to shortest translations

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from a given point.

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These lattice vectors
are called a and b.

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Four types of general
2D lattices exist--

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oblique, rectangular,
square, and hexagonal.

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In an oblique lattice,
a and b are different.

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And the angle between them
is greater than 90 degrees.

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In a rectangular lattice,
a and b are different.

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And the angle between
them is 90 degrees.

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In a square lattice,
a and b are the same.

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And the angle between
them is 90 degrees.

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And in a hexagonal lattice,
the a and b are the same.

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And the angle between
them is 120 degrees.

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On the other hand, lattices
in three dimensions

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are defined by the three
lattice vectors a, b,

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and c, which have
angles of alpha, theta,

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and gamma between them.

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There are seven types of
simple crystal systems,

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which you can see here.

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These structures range
from simple cubic,

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where all the lattice
vectors are equal

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and all the angles
are 90 degrees

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to triclinic, where
all the lattice vectors

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and angles are different.

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One thing to note about this
hexagonal crystal system

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is it's actually
made up of three unit

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cells, which you can see here.

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In 3D, there are also
non-simple crystal systems.

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With the non-simple
crystal systems added,

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there are 14 total possible
crystal structures.

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For example, body centered
cubic and face centered cubic

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are non-simple cubic crystals.

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Crystals exist in real space.

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This is the physical
world around us.

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Unfortunately, not all
concepts in materials science

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can be properly
represented in real space.

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Reciprocal space is
a non-physical space.

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In material science,
it is mainly

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used to portray
diffraction phenomena.

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For those of you who
know, reciprocal space

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is the four-year
transform of real space.

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However, this information is
not important to understand

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the rest of the video.

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Reciprocal lattice
vectors relate to sets

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of planes in real space.

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Reciprocal space has
some key properties

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that related to real space.

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These properties
include the units

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of reciprocal space
or an inverse length.

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The volume of a
reciprocal unit cell

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is inverse of the
real space volume.

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And the plane
spacing is inverted.

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See the attached notebook
for more information

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about reciprocal space and how
to calculate reciprocal lattice

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vectors.

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Consequently, a reciprocal space
is a very important concept.

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For example, it
is a key property

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in understanding
X-ray diffraction.

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In X-ray diffraction, a
beam of incident X-rays

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are diffracted in a material.

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And the diffraction
angle is measured.

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This produces an electron
diffraction pattern seen here.

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Here the electron diffraction
pattern is of silicon carbide.

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Points in the pattern
originate from a set

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of planes in the crystal.

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And each point represents a
reciprocal lattice vector.

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Therefore, electron
diffraction patterns

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exist in reciprocal space.

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Now we are going to
compare unit cells

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in real and reciprocal space
to better visualize the change

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from real to reciprocal space.

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As I said earlier, there are
four types of 2D lattices.

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In this image, you can see
the real space lattice.

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And in this image, you can see
the reciprocal space lattice.

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In real space, there are
lattice vectors a and b.

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And in reciprocal space, there
are lattice vectors a star

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and b star, which
are perpendicular

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to their real counterpart.

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As you can see here,
a change in real space

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produces an inverse result
in reciprocal space.

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As the real unit cell
shrinks or expands,

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the reciprocal unit
cell does the opposite.

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The same holds true
as the angles change.

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In three dimensions, it is
a little more complicated.

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As I said before,
there are seven types

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of simple crystal systems.

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This image is the
real space unit cell.

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And this image is the
reciprocal space unit cell.

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Similar to two dimensions,
the real lattice

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vectors are a, b, and c.

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And the reciprocal
lattice vectors

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are a star, b star, and c star.

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The points at the
vertices represents

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atoms in the simple unit cell.

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As you can see, as I change
the size of the real unit cell,

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the volume between the two cells
move in opposite directions.

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As the lattice vectors
change, the change

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in shape and volume of the unit
cells are clearly different.

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For the simplest
example in cubic,

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if we shrink the unit
cell in real space,

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it expands in reciprocal space.

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You can see this
kind of behavior

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in any of the unit cells.

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Hopefully this
information helps you

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to visualize reciprocal space.

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Thank you.