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 ?
BZ is the first Brillouin zone. K is the electron wave vector.
Both the optical and the acoustical phonons due to this profile were quantized in Brillouin zones. The temperatures used in our experimental study for iron- exchanged samples of zeolite remain below 510K where the iron ions remain itinerant (Ising model is applicable).
We compute the solutions for each [[bar.k].sub.i] in the Brillouin zone. For each [[bar.k].sub.i], we compute the low wavenumber part of [g.sub.R]([k.sub.L], [[bar.k].sub.i]; [[bar.[rho]].sub.m], [[bar.[rho]].sub.n]) and [Z.sup.(L).sub.mn].
In the theory of the lattice fields [[phi].sub.L](n), the momentum integration with respect to the wave-vector components [k.sub.[mu]] is restricted by the Brillouin zones k [member of] [-[k.sub.0[mu]]/2, [k.sub.0[mu]]/2], where [k.sub.0[mu]] = 2[pi]/[a.sub.[mu]].
An example of this phenomenon occurs in Solid State Physics where the translational symmetry of atoms in a solid resulting from the regular lattice spacing, is equivalent to an effective sampling of the atoms of the solid and gives rise to the Brillouin zone for which the valid values of k are governed by (35).
To ensure that this EBG material has a band gap around 30 GHz along all the directions of the square lattice, the dispersion diagram should be calculated along the three sides of the irreducible Brillouin zone triangle.
It is evident from Figure 1(b) that for increase in frequencies, the hyperbolic curves are shrinking towards the r point of the Brillouin zone. Moreover, the band slope is negative at this frequency regime.
During the numerical simulations the symmetry of the proposed structure have been considered and the dispersion diagrams (DDs) on the whole GXMG border of the first irreducible Brillouin zone (Brillouin 1953) have been calculated.
Thus they are well ordered but aperiodic, meaning that standard techniques for the determination of their structures and physical properties, very often based on the presence of a Brillouin zone, are not applicable.
In this paper, we use the four-band model that has three conduction sub-bands centered at the [GAMMA], L, and X symmetry points in the Brillouin zone and one equivalent valence band centered at the [GAMMA] symmetry point.
For each section, the propagation vector [Beta] in the wave vector space is varied along the edges of the irreducible Brillouin zone (BZ) (shaded triangle in the inset).
In the case of a square lattice the vectors k = ([k.sub.x], [k.sub.y]) (in units [a.sup.-1]) are restricted to be within the Brillouin zone, i.e., a square defined by the following four points ([+ or -][pi]/a, [+ or -][pi]/a).