# Brillouin zone

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## Brillouin zone

In the propagation of any type of wave motion through a crystal lattice, the frequency is a periodic function of wave vector k . This function may be complicated by being multivalued; that is, it may have more than one branch. Discontinuities may also occur. In order to simplify the treatment of wave motion in a crystal, a zone in k -space is defined which forms the fundamental periodic region, such that the frequency or energy for a k outside this region may be determined from one of those in it. This region is known as the Brillouin zone (sometimes called the first or the central Brillouin zone). It is usually possible to restrict attention to k values inside the zone. Discontinuities occur only on the boundaries. If the zone is repeated indefinitely, all k -space will be filled. Sometimes it is also convenient to define larger figures with similar properties which are combinations of the first zone and portions of those formed by replication. These are referred to as higher Brillouin zones.

The central Brillouin zone for a particular solid type is a solid which has the same volume as the primitive unit cell in reciprocal space, that is, the space of the reciprocal lattice vectors, and is of such a shape as to be invariant under as many as possible of the symmetry operations of the crystal. See Crystallography

## Brillouin zone

[brēy·wan ¦zōn]
(solid-state physics)
A fundamental region of wave vectors in the theory of the propagation of waves through a crystal lattice; any wave vector outside this region is equivalent to some vector inside it.
References in periodicals archive ?
Caption: FIGURE 1: Calculated phonon dispersion curves of bulk Mg along high-symmetry directions in the Brillouin zone.
As we know, the high-symmetry points in the Brillouin zone have the coordinate as [GAMMA] = (0, 0, 0), X = (2[pi]/a, 0, 0), W = (2[pi]/a, [pi]/a, 0), K = (1.
This is called the reduced zone scheme and [pi]/a is called the Brillouin zone boundary [21, see p.
This was also done to ensure that this EBG structure has a band gap at all directions of the irreducible Brillouin zone.
The energies of the extrema of the conduction and valence sub-bands are referenced to the bottom of the conduction sub-band at the [GAMMA] symmetry point in the Brillouin zone of the reciprocal lattice space.
1 [it is the same Figure 2 but showing the dispersion relation inside the first Brillouin zone for the selected [beta] value].
3) in the Y-M section of the Brillouin zone contains both x and y-directed components.
The periodic structure enables folding of the waveguide mode in the homogeneous MIM structure into the first Brillouin zone.
1(b) with the frequency normalized quantity ofa/[lambda] or [omega]a/2[pi]c (a is the lattice constant), the inset represents the first Brillouin zone of triangular PhC with three high-symmetry lattice points, and the point T is the Brillouin zone center.
The first one represents the band structure in the first Brillouin zone.

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