Brillouin zone construction

For many metals, the "nearly free" electron description corresponds quite closely 10 the physical situation. The Fermi surface remains nearly spherical in shape. However, it may now he intersected by several Brillouin zone boundaries which break the surface into a number of separate sheets. It becomes useful to describe the Fermi surface in terms not only of zones or sheets filled with electrons, but also of zones or sheets of holes, that is. momentum space volumes which are empty of electrons. A conceptually simple method of constructing these successive sheets, often also referred lo as "first zone. "second zone." and so on was demonstrated by Harrison. An example of such construction is shown in Fig. 2. 

Figure 16.13. Brillouin zone of the reciprocal lattice of the strictly 2-D square lattice, demonstrating the construction of 1 k = ki, C2Zki = k2, C zki = k3, Q"zki = k4.

The boundaries of the Brillouin zone can be constructed with the aid of the Bragg reflection conditions... 

If we now detach ourselves from this viewpoint and go back and construct the orbitals of the one CH per unit cell linear chain 71a, we get 74. The Brillouin zone in 71b (73) is half as long as it is here because the unit cell is twice as long. 

Figure 3.9. (a) The diamond structure viewed as two interlocking FCC sublattices displaced by I a along 1 1 1). In this projection along the [0 01] direction, only the top face of each cube is shown, (b) The unit cell, (c) Some possible sign combinations of the basis atomic orbitals used to construct LCAO COs from two Bloch sums, (d) A qualitative CO energy-level diagram for the center of the Brillouin zone, F = k(0, 0, 0). 

The magnitude of the wave function and therefore also the probability density at any point do not change with time. To discuss electron dynamics we must consider linear combinations of energy eigenstates of different energy. The convenient choice is a trave packet. In particular, we construct a packet, using states with wave numbers near ko and parallel to it in the Brillouin Zone ... 

The sec )iid-band Fermi surface from Fig. 16-6 displayed in the periodic zone sclicme. A Brillouin Zone can be constructed at any point in order to specify the Fermi surface two possibilities are shown as dashed-line and solid-line squares. 

We saw in detail in Section 16-D, and in Fig. 16-6 in particular, how nearly-free-electron bands are constructed. The diamond structure has the translational symmetry of the face-centered cubic structure, so the wave number lattice, the Brillouin Zone, and therefore the nearly-free-electron bands for the diamond structure are identical to the face-centered cubic nearly-free-electron bands and are those shown in Fig. 3-8,c. The bands that are of concern now are redrawn in Fig. 18-l,b. The lowest energy at X, relative to the lowest energy at F, is (fi /2m) x InjaY as shown clearly it is to be identified with the lower level... 

In contrast, neutron spectroscopy is a more powerful probe, its results are directly proportional to the phonon density of states (DOS) (see Fig. 2) which can be vigorously calculated by lattice dynamics (LD) and molecular dynamics (MD). Applying these simulation techniques provide an excellent opportunity for constructing and testing potential functions. Because optical selection rules are not involved, INS measures all modes (IR/Raman measure the modes at the Brillouin Zone (BZ) q = 0, see Fig. 2) and is particularly suitable for studying disordered systems (or liquids). It hence provides direct information on the hydrogen bond interactions in water and ice. 

Fig. 9.19. Calculations used in the construction of an effective Hamiltonian for the W(OOl) surface (adapted from Roelofs et al. (1989)). Panel (a) shows the surface Brillouin zone while panels (b)-(e) show the energy associated with various displacement patterns, with the solid lines corresponding to the energetics of the effective Hamiltonian while the discrete points are the energies associated with first-principles total energy calculations.

However, in our convention, see Eq. (2.55), g(fiv) integrates to unity across its local spectral range, we thus avoid the extra step of Eq. (4.58) and work directly with Li Xaxv). First we simphfy Eq. (4.56), replacing the integral by a sura limited to a selection of discrete k points. For example in the case of Na[FHF], as we shall see below, we construct a three-dimensional grid of 16x16x16 points evenly distributed across the k space of the first Brillouin zone. Individual values (at each... 

A construction in reciprocal space identical to that used to delineate the Wigner-Seitz cell in direct space gives a cell known as the first Brillouin zone, (Figure 2.7). The first Brillouin zone of a lattice is thus a primitive cell. 

Figure 2.7 The first Brillouin zone of a reciprocal lattice (a) the real lattice and Wigner-Seitz cell (b) the reciprocal lattice and first Brillouin zone. The zone is constructed by drawing the perpendicular bisectors of the lines connecting the origin, 00, to the nearest neighbouring lattice points, in an identical fashion to that used to obtain the Wigner-Seitz cell in real space...

The COR is constructed to calculate those extra structure constants which may be used to correct the ASA for approximate treatment of the region between the sphere and the atomic polyhedron, and for the neglect of higher components as described in Sect.6.9. The programme produces and stores on disk or tape a set of correction-term matrices distributed on a suitable grid in the irreducible wedge of the Brillouin zone. Whenever requested the correction matrices may be retrieved by LMTO and used together with the canonical structure constants to set up the corrected LMTO matrices. 

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. 

We ought to discuss the mathematics connected with construction of the reciprocal lattice and those of the Brillouin Zone. We will do so in the following Section. 

Using all coinbinations of the values allows us to construct the 1st Brillouin Zone of the bcc lattice. 

The 1st Brillouin zone follows from the reciprocal lattice by construction of the planes which are perpendicular to the lines connecting neighbouring points in the reciprocal lattice at their midpoints. The smallest closed volume which is bounded by these planes is the 1st BZ. For the naphthalene crystal, we find from the lattice parameters at T = 300 K (Table 2.3) the following magnitudes for the reciprocal lattice vectors a = lit 0.145 A ) = 2jt 0.167 A c = 2jr 0.138 A and for the volume V of the primitive... 

This is usually shortened to 0 < k < j, and this part of the reciprocal space is called the Brillouin zone the (first) Brillouin zone may be looked upon as a unit cell in reciprocal space. Its construction goes as follows. One first chooses one reciprocal lattice point, then connects it with all nearest points and bisects all vectors by planes the enclosed volume is called the (first) Brillouin zone. The theorem by Kramers [53] assures that the energies of wave functions at either negative or positive k values are the same. 

The calculation of the diffraction pattern for a periodic system revolves around the construction of the reciprocal lattice and subsequent placement of the first Brillouin zone however, in this case the aperiodicity of the pentagonal array requires a different approach due to the lack of translational symmetry. The reciprocal lattice of such an array is densely filled with reciprocal lattice vectors, with the consequence that the wave vector of a transmitted/reflected light beam encounters many diffraction paths. The resultant replay fields can be accurately calculated by taking the FT of the holograms. To perform the 2D fast Fourier transform (FFT) of the quasi-crystalline nanotube array, a normal scanning electron micrograph was taken, as shown in Fig. 1.13. 

When the unit cell Is doubled, the Brillouin zone will be half as long, because k exists in reciprocal space. Hence, the first Brillouin zone will range from —kHo to +7rlla. Let us again consider the example of a linear chain of H atoms. If the unit cell is doubled, as shown in Figure 11.38, then we must first take linear combinations of the two I s AOs in the unit cell to make MOs and then take linear combinations of the MOs at the extreme edges of the first Brillouin zone to construct the Bloch orbitals. 

The band calculation of Dimmock and Freeman was done for paramagnetic Gd, i.e. by putting equal numbers of 4f electrons in spin-up and spin-down states in constructing the potential. Two different starting potentials were used in the non-relativistic APW calculation, but the results were remarkably similar. The energy bands obtained from one of these calculations are plotted along high symmetry axes of the Brillouin zone as shown in fig. 3.1. The two bottom bands... 

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