Are all lattices Bravais lattices?

Question

Can all lattices be described as one of the fourteen Bravais lattices? Is the hexagonal close packed structure also one of the fourteen Bravais lattices?

Answer 1

From Ashcroft and Mermin's Solid State Physics:

A fundamental concept in the description of any crystalline solid is that of the Bravais lattice, which specifies the periodic array in which the repeated units of the crystal are arranged. The units themselves may be single atoms, groups of atoms, molecules, ions, etc., but the Bravais lattice summarizes only the geometry of the underlying periodic structure, regardless of what the actual units may be.

So, a real crystal structure is defined by the Bravais lattice, and the unit placed at each point on the Bravais lattice. Now, it turns out that the hexagonal close packed structure (hcp) is not, in fact, a Bravais lattice. Hcp is based on the simple hexagonal Bravais lattice, but with two atoms as the basis (one could also call it two interpenetrating simple hexagonal Bravais lattices). The bottom line is the hcp is not itself a Bravais lattice (see further details in Ashcroft and Mermin). I'll skip any discussion of point groups and space groups, since those are just further refinements of what it means to combine the symmetry of a Bravais lattice with the symmetry of the repeating unit.

There is a hierarchy of symmetry - 7 crystal systems, 14 Bravais lattices, 32 crystallographic point groups, and 230 space groups. For hcp, the point is that it can be represented as a simple hexagonal Bravais lattice with a two-atom unit, so having hcp as another "Bravais lattice" would totally change the definition of what a Bravais lattice is - the fundamental symmetry is simple hexagonal.

The general idea is to start with something you can repeat in a specified way to fill space. For example, starting with a cube, it is pretty easy to see how to stack them up to fill space. If, instead of being cubes they are stretched along one axis to get a rectangular solid, you can still fill space, but it will look different depending on what direction you are looking.

So, we get the seven crystal systems: Cubic (with simple cubic, body centered, and face centered cubic Bravais lattices); Tetragonal, which is a cube stretched along one side (with simple tetragonal and centered tetragonal Bravais lattices); Orthorhombic, a unit with perpendicular faces but unequal lengths on each side (with 4 Bravais lattices); Monoclinic, which tilts an orthorhombic unit to make one angle non-90 degrees (2 Bravais lattices); Triclinic, which has no two faces perpendicular to each other (1 Bravais lattice); Trigonal, which is obtained by taking a cube and stretching it along a body diagonal (1 Bravais lattice), and Hexagonal, which has one Bravais lattice.

You get those 7 crystal systems and 14 Bravais lattices by being able to stack the appropriate units to fill space. Now, while this concept is pretty fundamental, there are further symmetries imposed on the crystal by what you stack up. Lets consider something fairly simple, simple cubic. If you start with a big pile of white cubes, the resulting object you put together will look the same regardless of which direction you look at it, or how you rotate it around. BUT - if you start with a pile of white cubes with one face painted red, you just changed things. To conform with the Bravais lattice, you would stack them so that all the red faces pointed in the same direction. You would still get a big cube made of the smaller cubes. But now it would look different depending on what direction you looked at it - you have reduced the symmetry of the crystal.

For simple hexagonal, one starts with a hexagonal right prism (hexagon base, some height perpendicular to the base). Stacking those one on top of each other results in a simple hexagonal Bravais lattice. Lets make the hexagonal prisms out of clear acrylic, but place a marble at the center of the prism. Now when you have stacked them up you would see a simple hexagonal lattice of marbles, or heck, lets call them atoms. Now, exactly where you put the marble in the clear prism doesn't matter - you would still see the exact same arrangement of the marbles in space. OK, lets place the first model in the center of the base of the prism, and place a second marble halfway up one of the edges. When you stack them up, you will now get an arrangement of marbles in space that looks like the hexagonal close packed structure. But you stacked the prisms following the rules of the simple hexagonal Bravais lattice. Placing that second marble in the Bravais prism generated a two atom basis. It clearly is different from simple hexagonal, but the reason is not how you stacked the prisms, it is what the prism contains. The underlying Bravais lattice does not change, but the symmetry of the resulting crystal is changed by the basis unit used.

Now, use different colored marbles in the two positions, and it changes once again. Throw three marbles in in different places in the prism and it changes yet again.

So, the Bravais lattice occupies a fundamental level in understanding the crystal structure - it is the rules for stacking things up (and only the rules for stacking things). The objects you stack up may have their own properties, meaning that two things stacked using the same Bravais lattice may in the end look very different. This combination of the lattice and the unit are what create the 230 space groups.