STUDENT: Hello.
Today we are going to talk about real and reciprocal space
in two dimensions and three dimensions.
Crystallography is a major topic within material science.
A crystal is a highly ordered solid material
made up of a lattice and a periodic arrangement of atoms.
Atoms are located on every lattice site.
Due to the periodic nature of the structure,
all crystals display what's called translational symmetry.
Crystals are defined by their lattice vectors,
which are different in two dimensions and three
dimensions.
In two dimensions, lattice vectors
are the shortest and next to shortest translations
from a given point.
These lattice vectors are called a and b.
Four types of general 2D lattices exist--
oblique, rectangular, square, and hexagonal.
In an oblique lattice, a and b are different.
And the angle between them is greater than 90 degrees.
In a rectangular lattice, a and b are different.
And the angle between them is 90 degrees.
In a square lattice, a and b are the same.
And the angle between them is 90 degrees.
And in a hexagonal lattice, the a and b are the same.
And the angle between them is 120 degrees.
On the other hand, lattices in three dimensions
are defined by the three lattice vectors a, b,
and c, which have angles of alpha, theta,
and gamma between them.
There are seven types of simple crystal systems,
which you can see here.
These structures range from simple cubic,
where all the lattice vectors are equal
and all the angles are 90 degrees
to triclinic, where all the lattice vectors
and angles are different.
One thing to note about this hexagonal crystal system
is it's actually made up of three unit
cells, which you can see here.
In 3D, there are also non-simple crystal systems.
With the non-simple crystal systems added,
there are 14 total possible crystal structures.
For example, body centered cubic and face centered cubic
are non-simple cubic crystals.
Crystals exist in real space.
This is the physical world around us.
Unfortunately, not all concepts in materials science
can be properly represented in real space.
Reciprocal space is a non-physical space.
In material science, it is mainly
used to portray diffraction phenomena.
For those of you who know, reciprocal space
is the four-year transform of real space.
However, this information is not important to understand
the rest of the video.
Reciprocal lattice vectors relate to sets
of planes in real space.
Reciprocal space has some key properties
that related to real space.
These properties include the units
of reciprocal space or an inverse length.
The volume of a reciprocal unit cell
is inverse of the real space volume.
And the plane spacing is inverted.
See the attached notebook for more information
about reciprocal space and how to calculate reciprocal lattice
vectors.
Consequently, a reciprocal space is a very important concept.
For example, it is a key property
in understanding X-ray diffraction.
In X-ray diffraction, a beam of incident X-rays
are diffracted in a material.
And the diffraction angle is measured.
This produces an electron diffraction pattern seen here.
Here the electron diffraction pattern is of silicon carbide.
Points in the pattern originate from a set
of planes in the crystal.
And each point represents a reciprocal lattice vector.
Therefore, electron diffraction patterns
exist in reciprocal space.
Now we are going to compare unit cells
in real and reciprocal space to better visualize the change
from real to reciprocal space.
As I said earlier, there are four types of 2D lattices.
In this image, you can see the real space lattice.
And in this image, you can see the reciprocal space lattice.
In real space, there are lattice vectors a and b.
And in reciprocal space, there are lattice vectors a star
and b star, which are perpendicular
to their real counterpart.
As you can see here, a change in real space
produces an inverse result in reciprocal space.
As the real unit cell shrinks or expands,
the reciprocal unit cell does the opposite.
The same holds true as the angles change.
In three dimensions, it is a little more complicated.
As I said before, there are seven types
of simple crystal systems.
This image is the real space unit cell.
And this image is the reciprocal space unit cell.
Similar to two dimensions, the real lattice
vectors are a, b, and c.
And the reciprocal lattice vectors
are a star, b star, and c star.
The points at the vertices represents
atoms in the simple unit cell.
As you can see, as I change the size of the real unit cell,
the volume between the two cells move in opposite directions.
As the lattice vectors change, the change
in shape and volume of the unit cells are clearly different.
For the simplest example in cubic,
if we shrink the unit cell in real space,
it expands in reciprocal space.
You can see this kind of behavior
in any of the unit cells.
Hopefully this information helps you
to visualize reciprocal space.
Thank you.