Reciprocal lattice and Brillouin zones

The reciprocal lattice is generated from the fundamental translations bi b2 b3 defined by 

The MR of the scalar product (SP) bm a (which conforms with the laws of matrix multiplication) is 

In the hexagonal net a lattice vector a — nlal I n2a2, where ai = a2 = a, and 

The crystal lattice and the reciprocal lattice representations have different purposes. The crystal lattice describes, and enables us to visualize, the crystal structure. The reciprocal lattice will provide a means of describing electron states and phonon states in crystals. 

Applying periodic boundary conditions, eq. (16.2.18), to the Bloch functions of eq. (16.2.26) yields 

---

Therefore, since the crystal is composed by bonded periodically atoms it can be, in turn, characterized by two basic orthogonal states, namely that of core and that of valence, being also characterized by the wave vector associated to the wave packets of those states, according to the relation (3.53), and intimate bonds (described in previous section) with crystallographic planes by reciprocal lattice and Brillouin zone(s). 

Concepts of the reciprocal lattice and Brillouin zones are introduced in the chapter on X-rays along with a more formal freatment of the scattering of X-rays or wave-like particles. This formal treatmenf for obfaining the Laue conditions for constructive interference leads to a deeper understanding of the scattering process and gives students the ability to calculate the structure factors for various materials. The applications of X-ray diffraction to material identification, structure determination, and crystal characterization are briefly discussed. 

Periodic Boundary Conditions, Reciprocal Lattices and Brillouin Zones... 

Fig. 26b. Reciprocal lattice and one plane of the 111 Brillouin zone (octahedron)--------------.

Fig. 14. Reciprocal lattice and 1st Brillouin zone for the square lattice (upper part) and triangular lattice (lower part). The c(2x2) structure is described by the single wavcvcctor q0 id reciprocal space, while the (2x1) structure on the square lattice is described by a star (q, q2), as well as the V3xV R30o structure on the triangular lattice. The star of the (2x2) structure on the triangular lattice contains three members q, q 2 and q 3.

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...

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. 

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. 

If we now move from real space into reciprocal space, the Brillouin zone associated with the crystal lattice is also hexagonal and it shows characteristic high-symmetry points the centre is called F point, while two consecutive corners are denoted as K and K points. Fig. 2b. 

Before we can get as far as discussing the adaptations we must make to the calculations, we must introduce two concepts commonly used in crystallography and materials science the reciprocal lattice and the Brillouin zone. If you are not already familiar with the basic concept of the crystallographic unit ceU, you should read Section 10.5 and its sub-sections before continuing with this section. 

As the GNRs C-C bond contracts by up to 1 = 0.3, the corresponding reciprocal lattice and the Brillouin zone edge will expand by 0.7 ( L4), the... 

To that we would add the reciprocal lattices, fc-space, and Brillouin zones of crystallography. Kittel (1966) says The crystal lattice is a lattice in real or ordinary space the reciprocal lattice is a lattice in Fourier space , and It is as important to be able to visualize the reciprocal lattice as it is to visualize the real crystal lattice. ... 

Vgiec and Vxc represent the electron-nuclei, electron-electron and exchange-correlation dionals, respectively. The delta function is zero unless G = G, in which case it has lue of 1. There are two potential problems with the practical use of this equation for a croscopic lattice. First, the summation over G (a Fourier series) is in theory over an rite number of reciprocal lattice vectors. In addition, for a macroscropic lattice there effectively an infinite number of k points within the first Brillouin zone. Fortunately, e are practical solutions to both of these problems. 

The reciprocal lattice is useful in defining some of the electronic properties of solids. That is, when we have a semi-conductor (or even a conductor like a metal), we find that the electrons are confined in a band, defined by the reciprocal lattice. This has important effects upon the conductivity of any solid and is known as the "band theory" of solids. It turns out that the reciprocal lattice is also the site of the Brillouin zones, i.e.- the "allowed" electron energy bands in the solid. How this originates is explciined as follows. 

A Bloch function for a crystal has two characteristics. It is labeled by a wave vector k in the first Brillouin zone, and it can be written as a product of a plane wave with that particular wave vector and a function with the "little" period of the direct lattice. Its counterpart in momentum space vanishes except when the argument p equals k plus a reciprocal lattice vector. For quasicrystals and incommensurately modulated crystals the reciprocal lattice is in a certain sense replaced by the D-dimensional lattice L spanned by the vectors It is conceivable that what corresponds to Bloch functions in momentum space will be non vanishing only when the momentum p equals k plus a vector of the lattice L. 

Fig. 3.4. The ground-state configuration of dipoles on a complex honeycomb lattice and the corresponding triangular basic and dense lattices (a) the first Brillouin zones for the reciprocal basic and dense lattices (b).

Here it is our intention to show that for a system constituted by substrate phonons and laterally interacting low-frequency adsorbate vibrations which are harmonically coupled with the substrate, the states can be subclassified into independent groups by die wave vector K referring to the first Brillouin zone of the adsorbate lattice.138 As the phonon state density of a substrate many-fold exceeds the vibrational mode density of an adsorbate, for each adsorption mode there is a quasicontinuous phonon spectrum in every group of states determined by K (see Fig. 4.1). Consequently, we can regard the low-frequency collectivized mode of the adsorbate, t /(K), as a resonance vibration with the renormalized frequency and the reciprocal lifetime 7k-... 

The first Brillouin zone, formed by the bisector lines between the center and the nearest lattice points in reciprocal space, is a square bounded with lines h = ula and ky - nia. The lowest Fourier components of the sum of the local density of states (LDOS) over a range of energy A should have the form ... 

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... 

---