# The Bravais Lattices Song

By Walter F. Smith

Sing to the tune of “I Am the Very Model of a Modern Major General”
from “The Pirates of Penzance”.

If you have to fill a volume with a structure that’s repetitive,
Just keep your wits about you, you don’t need to take a sedative!
Don’t freeze with indecision, there’s no need to bust a seam!
Although the options may seem endless, really there are just fourteen!
There’s cubic, orthorhombic, monoclinic and tetragonal,
There’s trigonal, triclinic and then finally hexagonal!
There’s only seven families, but kindly set you mind at ease –
‘Cause four have sub-varieties, so there’s no improprieties!

The chorus  :
‘Cause four have sub-varieties, so there’s no improprieties.
‘Cause four have sub-varieties, so there’s no improprieties.
‘Cause four have sub-varieties, so there’s no impropri-e, prieties!

These seven crystal systems form the fourteen Bravais lattices.
They’ve hardly anything to do with artichokes or radishes –
They’re great for metals, minerals, conductors of the semi kind –
The Bravais lattices describe all objects that are crystalline!

The cubic is the most important one in my “exparience”,
It comes in simple and in face- and body-centred variants.
And next in line’s tetragonal, it’s not all diagonal,
Just squished in one dimension, so it’s really quite rectagonal!
The orthorhombic system has one less degree of symmetry
Because an extra squish ensures that a not equals b or c.
If angle gamma isn’t square, the side lengths give the “sig-o-nal”
For monoclinic if they’re different or, if equal, trigonal!

The chorus (reprovingly) :
Of course for trigonal, recall that alpha, beta, gamma all
Are angles that are equal but don’t equal ninety, tut, tut, tut!
Are angles that are equal but don’t equal ninety, tut, tut, tut!

If you squish the lattice up in every way that is conceivable,
You’ll get the least amount of symmetry that is achievable –
It’s called triclinic, then remains the one that really self explains –
Hexagonal gives us no pains an so we now may rest our brains!

A Bravais lattice is a grouping by symmetries of discrete points in three dimensional space. They describe the geometric arrangement of the atoms or molecules in crystalline structures.

Cubic :

Orthorhombic :

Monoclinic :

Tetragonal :

Trigonal :

Triclinic :

Hexagonal :

Diamonds are face-centred cubic.  Sapphires, rubies and amethysts are trigonal.  Emeralds and aquamarines are hexagonal.  Topaz is orthorhombic.

The Cannonball Problem was first posed by Sir Walter Raleigh to Thomas Harriot. Cannonballs stacked in triangular or rectagonal pyramids produce a face-centred  cubic lattice.  A six-sided pyramid produces a hexagonal close-packing lattice.

Gauss proved that these two systems of close-packing of equal shares occupy the highest density of available space.  It is equal to :

$$\displaystyle \frac {\pi}{3\sqrt2}\simeq 0.74048$$

Close-packing lattice generation :

If a third layer is placed directly over the first layer, then a hexagonal close-packing lattice is built.

If a third layer is placed over holes in the first layer, then a face-centred cubic lattice is built.