# Brillouin zone

(redirected from First Brillouin zone)

## 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

McGraw-Hill Concise Encyclopedia of Physics. © 2002 by The McGraw-Hill Companies, Inc.

## 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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
BZ is the first Brillouin zone. K is the electron wave vector.
Next applying the Method of Moments gives a linear eigenvalue equation that gives all the multi-band solutions simultaneously for a point in the first Brillouin zone. We label this as "broadband simulations" as the multi-band solutions are calculated simultaneously rather than searching the band solution one at a time.
Let [[bar.k].sub.i] be a wave vector in the first Brillouin zone, where
q stands for lattice wave vector and it is in the first Brillouin zone. [phi](j) indicates the phase difference caused by the damping and it is a constant value when the damping is certain.
The sum and integral are taken over the first Brillouin zone of the 2D PhC waveguide used [1, 5].
In Figure 3 we open artificially the projected bands for TE modes at [beta]a / 2[pi] = 0.1 [it is the same Figure 2 but showing the dispersion relation inside the first Brillouin zone for the selected [beta] value].
where [N.sub.e] is the number of equivalent ellipsoids in the first Brillouin zone, the volume of the unit cell is V= [a.sub.L.sup.3], [a.sub.L] is the lattice constant, m* is one of the effective masses listed in Table 1 for the appropriate band extrema, and [m.sub.0] is the free electron mass.
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.

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