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PROFESSOR: In this goodie bag, we'll
be exploring atomic packing and solids
through the cubic Bravais lattices.
We'll make three ball-and-stick models of different cubic unit
cells.
For this, you will need 31 balls and 44 sticks--
24 for FCC, 12 for SC, and 8 for BCC.
If you don't have a fancy crystal modeling kit like this,
any ball-and-stick equivalents will work.
A delicious alternative would be using
toothpicks and marshmallows.
The objective of the goodie bag is
to explore the different structures
of crystalline materials.
As you build the models, consider how many
nearest neighbors an atom has in an SC, a BCC, and an FCC
crystal.
So now we're going to construct our Bravais lattices.
We'll start with simple cubing.
Begin by taking 8 balls and connecting them with
sticks such that each ball sits at the corner of a cube.
Then for body-centered cubic, we're
going to start with a single ball in the center of a cube
and connect to 8 other balls at the corners with sticks.
For face-centered cubic, it's going
to be a little more involved.
We'll start with the corner of a cube and form tetrahedra
to other balls, which themselves will form tetrahedra
to another corner of a cube.
If we iterate through this process,
we eventually end up with a full face-centered cubic lattice.
We've now constructed three ball-and-stick models
of cubic Bravais lattices.
The simple cubic lattice has atoms
at each of the 8 corners of the unit cell.
The body-centered cubic lattice has an additional atom
in the body center of the unit cell.
And the face-centered cubic lattice
has an additional atom in each of the 6 face
centers of the unit cell.
Each of these lattices has a different coordination number.
That is, an atom in each lattice has a different number
of nearest neighbors.
If we imagine these atoms as hard spheres
rather than this ball-and-stick model,
we can see that each lattice will also
have a different atomic packing factor.
That is, each lattice will have a different volume of space
occupied in the unit cell.
In general, the coordination number of a lattice
correlates with atomic packing factor, which itself
correlates with stability.
That's why we don't often find simple cubic metallic solids
in nature.