Brillouin zone

First Brillouin zone of a face-centered cubic lattice (the reciprocal lattice is body-centered cubic )

The Brillouin zones (after Léon Brillouin ) describe symmetrical polyhedra in the reciprocal lattice in solid-state physics . The first Brillouin zone is the primitive Wigner-Seitz cell of the reciprocal lattice of a crystal , i.e. a (generally irregular) polyhedron in reciprocal space . After the first Brillouin zone, the entire structure repeats itself periodically, i.e. That is, it is sufficient to describe all processes in the first Brillouin zone.

construction

1. Brillouin zone in the two-dimensional square and hexagonal lattice

Animated construction of the 1st Brillouin zone of a body-centered cubic lattice (krz lattice)

For the construction analogous to the Wigner-Seitz cell, one selects a lattice point of the reciprocal lattice and bisects all connecting lines to all other points by normal planes , i.e. H. by levels on which the connecting lines are each vertical. By drawing in the mid- perpendicular (or plane in 3D ) to all points, you get an area (or volume in 3D) around the grid point. The polyhedron bounded by the normal planes is the Brillouin zone.

Within the first Brillouin zone (1st BZ) some important highly symmetrical points of the fcc lattice are named. With the drawn coordinate system (x, y, z) the following applies:

• Grid points of the 1st BZ of the fcc grid: (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 1st BZ
• X point (0, 1, 0): The point of intersection of the axis [010] with the edge of the 1st BZ
• L point (0.5, 0.5, 0.5): The point of intersection of the space diagonal [111] with the edge of the 1st BZ
• K point (0.75, 0.75, 0): The intersection of the diagonals in a plane [110] with the edge of the 1st BZ
• U point (0.25, 1, 0.25)
• W point (0.5, 1, 0)

application

In solid-state physics , the crystal momentum of a particle or quasiparticle (e.g. electron and hole and others) is given as a vector in the reciprocal lattice. A quasiparticle with a certain wave vector behaves exactly like one whose wave vector differs from a reciprocal lattice vector . Therefore, for quantities that depend on the crystal momentum, only the values ​​for crystal momentum within the first Brillouin zone need to be determined. The background is that waves ( particle waves ) are backscattered at so-called Bragg planes (see also Laue condition ). ${\displaystyle \hbar {\vec {k}}}$ ${\displaystyle {\vec {k}}_{1}}$ ${\displaystyle {\vec {k}}_{2}}$ ${\displaystyle {\vec {k}}_{1}}$ ${\displaystyle {\vec {K}}}$ ${\displaystyle {\vec {k}}_{1}={\vec {k}}_{2}+{\vec {K}}}$

literature

• Charles Kittel: Introduction to Solid State Physics . 10th edition. Oldenbourg, Munich 1993, ISBN 3-486-22716-5 .
• Neil W. Ashcroft, N. David Mermin: Solid State Physics . 2nd Edition. Oldenbourg, Munich 2005, ISBN 3-486-57720-4 .
• Konrad Kopitzki: Introduction to Solid State Physics . 6th edition. Teubner, Wiesbaden 2007, ISBN 3-8351-0144-7 .