Body-centered cubic lattice Brillouin zone

A theoretical interpretation relating the valence electron concentration and the structure was put forward by H. Jones. If we start from copper and add more and more zinc, the valence electron concentration increases. The added electrons have to occupy higher energy levels, i.e. the energy of the Fermi limit is raised and comes closer to the limits of the first Brillouin zone. This is approached at about VEC = 1.36. Higher values of the VEC require the occupation of antibonding states now the body-centered cubic lattice becomes more favorable as it allows a higher VEC within the first Brillouin zone, up to approximately VEC = 1.48. 

First Brillouin Zone of the Body-Centered Cubic Lattice. The Figure is a Regular Rhombic Dodecahedron... 

The term first Brillouin zone is applied only to the k-space cell. Because the reciprocal of the body-centered cubic lattice is a face-centered cubic lattice, the first Brillouin zone of the bcc is just the fee Wigner-Seitz cell (Figure 4.8). Conversely, the first Brillouin zone of the fee lattice is just the bcc Wigner-Seitz cell. 

Fig. 3.2. Primitive unit cells (a, c) and Brillouin zones (b, d) for face-centered and body-centered cubic lattices...

Table 4.3. Special points of the Brillouin zone for the body-centered cubic lattice generated by the symmetrical transformation...

Figure 4.8 The Wigner-Seitz cells (a) and the first Brillouin zones (b) for the ce-centered cubic (fee) and body-centered cubic (bcc) crystal lattices.

Figure 4.13 First Brillouin zone of the body-centered cubic crystal lattice, k, ky,kz are the axes of a Cartesian coordinate system in fe-space. The symmetry points and symmetry lines are indicated. See Table 4.1 for details.

Table 4.1 Points and directions of high symmetry in the first Brillouin zones, fee is the face-centered cubic crystal lattice bcc is the body-centered cubic crystal lattice hep is the hexagonal close-packed crystal lattice.

In Figure 3.5, the body-centered-cubic (bcc) lattice and the corresponding Brillouin zone are illustrated. The primitive translation vectors are... 

Fig.3.5. a) Unit cell and primitive translation vectors of the body centered cubic (bcc) lattice, b) Brillouin zone of the bcc lattice [3.5]... 

Figure 2.5 (a) The method for construeting the Brillouin zone for a square planar lattice and the first Brillouin zone for (b) a faee-eentered eubic crystal and (c) a body-centered cubic crystal. For the fee erystal the diamond-shaped faees of the Brillouin zone are along cube axes, [100]-type direetions, while the hexagonal faees are along [111] cube diagonals. For discussion of the [100], [111], and other erystal indiees, see Chapter 4. 

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