Brillouin-Zone

The Brillouin zones (after Léon Brillouin ) is a term used in solid state physics. They describe symmetric polyhedra in the reciprocal lattice. The first Brillouin zone is the primitive Wigner- Seitz cell of the reciprocal lattice of a crystal, ie, a (generally irregular ) polyhedra in reciprocal space. After the first Brillouin zone, the entire structure is repeated periodically, ie it reaches all processes in the first Brillouin zone to describe.

Construction

For the construction similar to the Wigner-Seitz cell choosing a grid point of the reciprocal lattice and halved all links to all other points by normal levels, i.e., levels at which the connection lines are perpendicular. By inscribing the central vertical line ( or plane in 3D) to all the points, you get around the grid point an area ( or volume in 3D). The polyhedron, which is limited by the normal levels, the Brillouin zone.

Within the first Brillouin zone ( BZ 1 ) are named some important high-symmetry points of the fcc lattice. With the marked coordinate system (x, y, z):

  • Grid points of the first BZ of the fcc lattice: (0, 0, 0); (1, 1, 1); ( -1, 1, 1); ( -1, -1, 1); (1, -1, 1); (1, 1, -1); (-1, 1, -1); (-1, -1, -1); (1, -1, -1)
  • Γ point ( 0, 0, 0): The center of the first BZ
  • X point ( 0, 1, 0): The point of intersection of the axis with the edge of the first BZ
  • L- point (0.5, 0.5, 0.5 ): The point of intersection of the body diagonal to the edge of the first BZ
  • K point ( 0.75, 0.75, 0): The point of intersection of the diagonals in a plane with the edge of the first BZ
  • U- point (0.25, 1, 0.25)
  • Wireless point (0.5, 1, 0)

Application

In solid-state physics of crystal momentum of a particle or quasi-particle (eg electron and hole, and others) is specified as a vector in the reciprocal lattice. A quasi-particles with a given wave vector behaves exactly like one whose wave vector is different from a reciprocal lattice vector. Therefore you need for sizes, which depend on the crystal momentum to determine the values ​​for crystal pulses within the first Brillouin zone. The background is that waves ( particle waves ) are scattered back to the so-called Bragg planes (see also Laue condition ).

Wigner–Seitz cell Quasiparticle Electron hole Laue equations
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