Brillouin zone simple cubic

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 NFE behaviour has been observed experimentally in studies of the Fermi surface, the surface of constant energy, F, in space which separates filled states from empty states at the absolute zero of temperature. It is found that the Fermi surface of aluminium is indeed very close to that of a spherical free-electron Fermi surface that has been folded back into the Brillouin zone in a manner not too dissimilar to that discussed earlier for the simple cubic lattice. Moreover, just as illustrated in Fig. 5.7 for the latter case, aluminium is found to have a large second-zone pocket of holes but smaller third- and fourth-zone pockets of electrons. This accounts very beautifully for the fact that aluminium has a positive Hall coefficient rather than the negative value expected for a gas of negatively charged free carriers (see, for example, Kittel (1986)). 

Table 16.6. Symmetry points in the Brillouin zones (see Figure 16.12) of the reciprocal lattices of (a) the primitive cubic space lattice (simple cubic, sc), for which the reciprocal lattice is also sc, and (b) the fee space lattice, which has a bcc reciprocal lattice.

Because of the translational symmetry of the reciprocal lattice (Section 16.3) and the definition of the Brillouin zone (BZ), the BZ faces occur in pairs separated by a reciprocal lattice vector. For example, the cubic faces of the first BZ of the simple cubic (sc) lattice occur in pairs separated by the reciprocal lattice vectors b (2rc/a)[[l 0 0]] (see eq. (16.3.27)). In general, for every k vector that terminates on a BZ face there exists an equivalent vector k (Figure 17.1) such that... 

Fig. 4 a, b Energy bands of a orientationally ordered simple-cubic solid C60 and b of the fee solid C60 shown in the same simple-cubic Brillouin zone. The generalized tight-binding method is used [20]... 

Fig. 6. Energy of an electron in a crystal plotted against wave number kx for a simple-cubic lattice with Brillouin-zone boundaries at kx = (7r/a), zt(2ir/a). Solid line is for reduced wave number — (w/a) < kx < (w/a). Heavy dashed line is for kx defined in range — o < kx < . Light dashed line is solution for a constant potential. Quantum number m defines the band. Effective mass of interest is...

For a simple cubic metal the first zone is given by hkl = 100 (and equivalent values 010, 001, 100, 010 and 001). In the reciprocal lattice 100 is the point on the (reciprocal) x axis at a distance 1 a from the origin. The Brillouin zone is now produced as the locus of the general condition (Fig. 26a, drawn there for 110) ... 

As a simple (indeed over-simplified) example, we take (as did Clapp) the case of the alloy CuAu we employ an essentially geometrical treatment of strain and ignore electronic effects (the effect of apbs on the Brillouin zones of the alloy, which controls the scale of the final periodicity). CuAu I has a simple superstructure of the cubic-close-packed (c.c.p.) arrangement of metal atoms in pure Cu or Au metals. In the parent, f.c.c. unit cell alternate (001) A layers of atoms contain exclusively Au and exclusively Cu (Fig. 23). Au is larger than Cu and hence, pmely in terms of size effects, the Cu layers must be under tension and the Au layers under compression if the layers are to be perfectly commensurate (as they are). The size effect is in fact seen, for the structure is metrically as well as symmetrically tetragonal the (now distorted) f.c.c. unit cell is face-centred tetragonal, with da = 0.93s instead of 1.000. In the CuAu II structure this strain is relieved (in one direction only ) by the introduction of apbs at every fifth cube plane normal to the layers (Fig. 24). 

Fig. 2 Dispersion relation of the orbital interaction in (7) represented in the Brillouin zone for a simple cubic lattice...

It now remains to construct a mesh in the irreducible wedge of the Brillouin zone and define the corresponding tetrahedra. This is done in the subroutine TGEN, and we shall briefly illustrate the procedure with the example of a simple cubic lattice. 

The irreducible wedge of the Brillouin zone of the simple cubic structure... 

Magnetoacoustic quantum oscillations have been observed in this compound for the Cji mode (Liithi et al. 1984) and for the C44, and the Cjj - Cjj modes (Suzuki et al. 1985a, Ewert et al. 1987). The observed frequencies are very small, however, typically F 5-6 T. They arise from tiny flat electron ellipsoids within the necks along the [110] directions in the simple cubic Brillouin zone (Suzuki et al. 1985a, Onuki et al. 1988). Until now, higher frequencies have not been observed in MAQO techniques. 

Fig. 3.1. Primitive unit cell and Brillouin zone for simple cubic lattice...

Table 4.1. Special points of the Brillouin zone for the simple cubic lattice generated by the symmetrical transformation Bi = - (1,0,0), B2 = (0,1,0), B3 = (0,0,1) Kg = ctiBi -f OC2B2 + OC3B3 = (oi, 02,03)...

Fermi suifaces in the simple cubic Brillouin zone... 

The rare earth hexaborides RBg crystallize in the cubic (CaBe type) structure which possesses a CsCl type arrangement of R atoms and Be octahedra. Figure 7 shows the crystal structure of RBe and its simple cubic Brillouin zone. LaBe is a reference non-f compound. CeBg is a typical Kondo-lattice compound undergoing two magnetic ordering... 

Fig. 7. (a) RBj-cubic crystal structure. Large spheres without pattern and small spheres with pattern show the R atoms and the B atoms, respectively, (b) Brillouin zone of the simple cubic crystal lattice. 

The Brillouin zone for the simple (primitive) cubic lattice is a (simple ) cube. The Brillouin zone for the face-centred cubic lattice is pictured in Fig. 17.11 that for the body-centred cubic is shown in Fig. 17.16. Show that all three have Of, symmetry. 

Note in Figure 16.3, that increasing k beyond tt/a simply repeats the curve between —ti/a and 0, so that aU relevant information is contained within the region from—it/a and tt/a. Recall that the reciprocal lattice vector for a simple cubic direct lattice has magnitude 2tt/a and that the first Brillouin zone is formed by planes that are perpendicular bisectors... 

The dispersion relation simply repeats itself between — tt/a > fc > tt/a, which is the first Brillouin zone for a simple cubic lattice, so that all of the relevant information is contained in this zone. When a finite length L of the chain of N atoms is specified, the dispersion relation is quantized into N modes that are spaced at iTrfL apart throughout the interval -TT/fl>fc>Tr/fl. Each state has energy ho) where the ) is related to k by the dispersion relation. Traveling waves with these discrete values of k are called phonons. Since a wave can have three polarization states, two transverse and one longitudinal, there are 3N normal modes. 

A contour map of the Fermi surface of the simple cubic system in the first Brillouin zone that was described in Figure 19.12. The contours go in steps of 0.1 C with the outer contour corresponding to kz = 0 and the inner to k, = l. 

The effect of the energy gap in the vicinity of the Brillouin zone boimdary is to distort the Fermi surface such that a face will be formed at the zone interface as we saw for the case of the simple cubic system. This schematic in Figure 19.15 illustrates a better approximation of the actual Fermi surface of the noble metals as calculated from Equation 19.39 than that shown in Figure 19.5. 

The presence of energy gaps near the Brillouin zone boundaries distort the Fermi surface causing it to penetrate into the zone interface. The tight-binding approximation allows a better approximation of the actual shape of the Fermi for simple cubic, bcc, and fee structures for simple metals with only s-electrons. 

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