Brillouin zone unit cell

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

The region within which k is considered (—n/a first Brillouin zone. In the coordinate system of k space it is a polyhedron. The faces of the first Brillouin zone are oriented perpendicular to the directions from one atom to the equivalent atoms in the adjacent unit cells. The distance of a face from the origin of the k coordinate system is n/s, s being the distance between the atoms. The first Brillouin zone for a cubic-primitive crystal lattice is shown in Fig. 10.11 the symbols commonly given to certain points of the Brillouin zone are labeled. The Brillouin zone consists of a very large number of small cells, one for each electronic state. 

In the left panel of Figure 8 we show the band structure calculation of graphite in the repeated zone scheme, together with a drawing of the top half of the first Brillouin zone. The band structure is for the 1 -M direction. As the dispersion is very small along the c-axis we would find a similar result if we add a constant pc component to the line along which we calculate the dispersion [17]. The main difference is that the splitting of the a 1 and % band, caused by the fact that the unit cell comprises two layers, disappears at the Brillouin zone boundary (i.e. if the plot would correspond to the A-L direction). 

Polydiacetylene crystals. The enhancement of x because of one-dimensional electron delocalization is strikingly corroborated in the polydiacetylene crystals. Their structure is that of a super alternated chain with four atoms per unit cell and the Huckel approximation yields four bands for the ir-electrons, two valence and two conduction bands. When depicted in the extended Jones zone, each pair can be viewed as arising by a discontinuity at the middle of the Brillouin zone of the polyene chain. The dominant contribution to X(2n 1) comes from the critical point at the edge of the extended Jones zone (initially at the center of the reduced B.Z.). The complete expressions are derived in (4,22) and calculated for different polydiacetylenes. We reproduce the values of x 2 for TCDU and PTS in table IV. The calculated values are in good agreement... 

The first Brillouin zone for vectors k, being determined by the crystal unit cell, it can be larger than that for the adsorbate lattice, and hence the sum over R entering into the Eq. (4.1.6) is found as... 

A is the volume of the unit cell in the direct lattice of the crystal The range of integration is restricted to the first Brillouin zone of the crystal, and the volume of the zone is (27t)3/A. 

Thus, the reciprocal lattice of a simple cubic lattice is also simple cubic. It is shown in Fig. 5.7 in the xy plane, where it is clear that the bisectors of the first nearest-neighbour (100) reciprocal lattice vectors from a closed volume about the origin which is not cut by the second or any further near-neighbour bisectors. Hence, the Brillouin zone is a cube of volume (2n/a)2 that from eqn (2.38) contains as many allowed points as there are primitive unit cells in the crystal. The second, third, and fourth zones can... 

In particular, let us consider the band structure along where kr = (0,0,0) and kx = (2n/a)(l, 0,0) with a the edge length of the face-central cubic unit celL (Note that the X point for fee is In/a not nfa like for simple cubic.) In this direction the two lowest free-electron bands correspond to Ek = (H2/2m)k2 and k+I = (H2/2m)(k + g)2 respectively. The term g is the reciprocal lattice vector (2n/a)(2,0,0) that folds-back5 the free-electron states into the Brillouin zone along so that Ek and k+l... 

The N q vectors allowed by the boundary conditions just fill the first Brillouin zone (BZ) of volume equal to vb, the volume of the primitive unit cell of the reciprocal lattice. Because of this dense, uniform distribution of q vectors it is possible to treat q as a continuous variable and thus replace... 

Figure 6.3a presents a unit cell of graphene sheet comprising two kinds of carbon atoms a and (1, where a = 2.46 A and x2 are the lattice constant and the vector connecting two carbon atoms, respectively. The corresponding Brillouin zone (BZ) in the reciprocal lattice is shown in Figure 6.3b. Assuming that only the nearest neighbors overlap and resonance integrals work in the system, the Jt-band energies are calculated from the secular equation as expressed by...

Outside of a small region around the center of the Brillouin zone, (the optical region), the retarded interactions are very small. Thus the concept of coulombic exciton may be used, as well the important notions of mixure of molecular states by the crystal field and of Davydov splitting when the unit cell contains many dipoles. On the basis of coulombic excitons, we studied retarded effects in the optical region K 0, introducing the polariton, the mixed exciton-photon quasi-particle, and the transverse dielectric tensor. This allows a quantitative study of the polariton from the properties of the coulombic exciton. 

Therefore k is generally restricted (hence referred to as the reduced wave vector) to a region of k space such that no two points in this region are separated by any vector K. This is a unit cell in reciprocal space, and is referred to as the first Brillouin zone. 

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