What is the first Brillouin zone and how do I construct it?

[Sunnyocean]

HI,

I read the other threads in this forum but I still didn't understand (and I have read Aschroft and Mermin as well as Kittel).

So, could you pleae explain to me (preferably in a VERY detailed manner) what is the first Brillouin zone and how do I construct it?

Thank you :)

 

[ZetaOfThree]

The first Brillouin zone is a primitive cell of the Fourier transform of the lattice. It is found in the same way that you find a primitive cell of a lattice, using the Wigner-Seitz procedure.

 

[M Quack]

The first Brillouin zone is the region of reciprocal space where point are closer to the origin than to any other point of the reciprocal lattice.

To construct it, you find the nearest neighbor reciprocal lattice points around the origin.

Then for each of these reciprocal lattice points you find all the points that have equal distance to the origin and this RL point. These points form a plane perpendicular to the RL vector of your RL point, intersecting it at - you guessed it - half distance.

After you have done this the RL origin is surrounded by a set of intersecting planes. In each direction, first Brillouin zone boundary is given by the first (innermost) plane you hit going outward from the origin. Which plane you hit first depends on the direction. The first BZ is what is within the BZ boundary.

Try and draw this out in 2 dimensions on a piece of paper. It is a very geometrical concept, but a bit messy to visualize in 3D.