Monday, October 17, 2011

A lot of Brillouin Zones

In condensed matter physics we deal a lot with crystals, which is a repeating structure of atoms.  Every crystal is associated with a lattice,* which is an infinitely repeating set of points.

For example, if the points shown above were to repeat infinitely over the plane, they would make a lattice.

A useful concept is the first Brillouin Zone, which is the set of all points which are closer to the center point of the lattice than to any other point of the lattice.  In the above example, the first Brillouin Zone would simply be a square around the center.

Interestingly (but less usefully) there is also a definition for the Nth Brillouin Zone.  The Nth Brillouin Zone is the set of all points such that the center point of the lattice is the Nth closest point in the lattice.  I thought it might be pretty to draw the first few Brillouin Zones, and then I went totally overboard.

The first 27 Brillouin Zones.  The first Brillouin Zone is the red square in the center, and successive Brillouin Zones are each in a different color, moving outwards.  Click to enlarge.

The Brillouin Zones have the interesting property that each one has equal area.  And for any given Brillouin Zone, you can translate the different pieces by integer distances and form a square out of them.

I think one thing that attracted me to condensed matter physics is that there's all this geometry in it... and it's actually useful for describing reality.  Not this Nth Brillouin Zone stuff though, that's useless.  Hardly anyone thinks about the second Brillouin Zone, much less the third or twenty-seventh.

You can get Brillouin Zones of different shapes if you have different kinds of lattices!  And you can even have 3D lattices, producing 3D Brillouin Zones.  Note that even though real crystals exist in 3-dimensional space, they can be layered in 2D structures and thus be associated with 2D lattices.  For example, the 2D lattice I showed is the correct lattice to use for the cuprate superconductors that I study.

*The locations of the atomic nuclei in a crystal often make a lattice, but for complicated reasons, this is not the one I'm talking about.  Every crystal is also associated with a completely different "reciprocal lattice", which is the one relevant to Brillouin Zones.

1 comment:

Eduard said...

Recently the Nobel price for chemestry was given for quasicristals. Here too I thougt that quasi cristals are essentially 2-dimensional Penrose tilings. But now I think that you have to add al lot of thinking to pass from 2 dimensions to 3 dimensions in this case. I was a little bit prepared because I know the very interesting dissection of the triacontahedron in Amman polyhedra.