Brillouin zone (idea)

The Brillouin zones of crystals are immensely important to solid state physics and crystallography. See the nodes crystal and reciprocal lattice for requisite background information.

The Brillouin zones are volumes in a crystal's reciprocal lattice space. The Brillouin zones are defined by their boundaries, which are called Bragg planes in the field of x-ray diffraction. The first Brillouin zone is the most important one.

The rule for constructing Brillouin zones is straightforward, though their shapes can be very complex in three dimensions.

The first Brillouin zone is the first volume enclosed by the planes that perpendicularlly bisect all reciprocal lattice vectors, with the origin taken as a point in the reciprocal lattice. It is easy to show that the first Brillouin zone is simply the Wigner-Seitz cell of a crystal's reciprocal lattice.

It is easier to draw 2-d Brillouin zones than 3-d zones. A square lattice in real space has a corresponding square lattice in reciprocal space. The first Brillouin zone for the square lattice is drawn below. The point O is taken as the origin of the crystal's reciprocal space. The points Y are the points whose perpendicular bisectors enclose the first Brillouin zone. The dashed lines are the Bragg planes that enclose the first zone.

              


First Brillouin zone of a square lattice
  
  
                    
     X     X     X     Y     X     X     X
                        
                     ----- 
                    |     |     
     X     X     Y  |  O  |  Y     X     X
                    |     |     
                     ----- 

     X     X     X     Y     X     X     X

The nth Brillouin zone is the volume of the reciprocal lattice reached by crossing exactly (n-1) Bragg planes. The second Brillouin zone of the square lattice is drawn below. I had to stretch the lattice vertically to make the ASCII art fit correctly. The number 1 refers to the first Brillouin zone. The second Brillouin zone is the combination of the areas labeled 2--it does not contain the first Brillouin zone. The points Z (and also Y again) are the points whose perpendicular bisectors enclose the second Brillouin zone.

First and second Brillouin zones of a square lattice


         X      Z      Y      Z      X            
                     /   \
                    /  2  \     
                   / _____ \
                  / |     | \ 
                 /  |     |  \    
         X      Y 2 |  O  | 2 Y      X            
                 \  |1    |  /    
                  \ |_____| / 
                   \       /
                    \  2  /
                     \   /
                      \ /
         X      Z      Y      Z      X                 
     

The following are important properties of Brillouin zones:

• The volume of each Brillouin zone is equal to (2π)ⁿ / V, where V is the volume of the Wigner-Seitz cell in the crystal's real space, and n is the dimensionality of the crystal (usually 3).
• An incident x-ray with wavevector k will strongly diffract from the crystal only if k lies on a Brillouin zone boundary (Bragg plane). See x-ray diffraction for more on this topic.
• Electron waves in the crystal with Bloch wavevectors k (see Bloch's theorem) can experience strong diffraction by the crystalline potential if k lies on a Brillouin zone edge. This diffraction leads to energy bandgaps in solids.
• All of the Bloch wavevectors k can be confined to the first (or any, but the first is most convenient) Brillouin zone. This follows from the fact that e^iKr, where K is a reciprocal lattice vector, has the same periodicity as the lattice and from the fact that e^i(k+K)r = e^ikre^iKr.

Some of the best solid-state physicists in the world have no idea how to pronounce Brillouin.