Reciprocal space, wave vector, Brillouin zone

1 Reciprocal space, wave vector, Brillouin zone 

Consider a molecular crystal, like, for example, the one whose layers are depicted in Fig. 5.5. Each molecule is a recognizable entity composed of a number of atoms, Aat, each one in turn described by a number, Aorb.i, of atomic orbitals expressed in gaussian expansions as in equations 3.40-3.42. There are Z entire molecules in the unit cell. A reference unit cell (called the Ref-cell) is chosen, which contains a basis set of Nbs atomic orbitals, xj, with Nbs = Z J Aorb.i- The corresponding real and reciprocal space can then be defined (Section 5.4)  

For historical reasons that will become clearer later on, the reciprocal lattice vector r is instead called the wave vector, symbol k, and real to reciprocal-space dot-products are normalized to 2n instead of unity. Besides, the boundaries of the independent volume unit of reciprocal space are defined as follows after choosing a reciprocal lattice point as the origin, vectors are drawn to its nearest-neighbor points, and planes are drawn perpendicular to these vectors and passing through their midpoint. The space enclosed within these planes is called the first Brillouin zone and its content is translationally invariant. 

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The Peierls model explains why a chain of unsaturated carbon atoms with one conduction electron per atom does not exhibit metallic properties. If all the atoms are spaced at eqnal distance, a, the basic cell in reciprocal space is the Brillouin zone in the interval -nlawave vector). With one electron per atom, the band would be half-filled and hence the chain would exhibit metallic behaviour. A periodic distortion of the chains, commensurate with the original structure, generates an -fold super-structure and reduces the Brillouin zone to -nlnaunit cell. The effect of the distortion is to open a gap at the boundaries k = n/na of the new Brilliouin zone (Figure 1.1). Therefore, if only states below the new gap are... 

The wave vector k appearing in (4.18) can always be confined to the first Brillouin zone in reciprocal lattice. The first Brillouin zone has a fe-space volume... 

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. 

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. 

The wave vector forms a reciprocal space as before and has continuous values in the range —rr/a < < tt/u. These limits define the (first) Brillouin zone of a... 

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. 

Thus these points in a small but well-defined region of k space include all possible irreducible representations of the translation group the vectors of the reciprocal lattice transform points in the Brillouin zone into equivalent points. The Brillouin zone therefore contains the whole symmetry of the lattice, each point corresponding to one irreducible representation, and no two points being related by a primitive translation. The smallest value of k ki, k2, kz) belonging to the rep is called the reduced wave-vector. The set oi reduced wavevectors is called the first Brillouin zone. 

The reciprocal cell of a cubic cell with side length L is also a cube, with the side length 2k/L. The equivalent of a unit cell in reciprocal space is called the (first) Brillouin zone. Just as a point in real space may be described by a vector r, a point in reciprocal space may be described by a vector k. Since k has units of inverse length, it is often called a wave vector. It is also closely related to the momentum and energy, e.g. the momentum and kinetic energy of a (free) particle described by a plane wave of the form e is k and respectively. 

In (6.5) the subscript n indicates the band index and fe is a continuous wave vector that is confined to the first Brillouin zone of the reciprocal lattice. The index n appears in the Bloch theorem because for a given k there are many solutions to the Schrodinger equation. Because the eigenvalue problem is set in a fixed finite volume, we generally expect to find an infinite family of solutions with discretely-spaced eigenvalues which we label with the band index n. The wave vector k can always be confined to the first Brillouin zone. The vector k takes on values within the Brillouin zone corresponding to the crystal lattice, and particular directions like r,A,A,Z (see Figures 4.13-4.15). 

Period of the chain is equal to a. Let us suppose the linear relationship between the interaction force between the nearest neighbors and atomic displacement. Every internal motion of the lattice could be represented by the superposition of the mutually orthogonal waves as follows from the lattice dynamic theoiy (see e.g. Bom and Huang 1954 Leibfried 1955). Aiy lattice wave could be described by the wave vector K from the first Brillouin zone in the reciprocal space. Dispersion curve co K) has two separated branches (for 2 atoms in the primitive unit), which could be characterized as acoustic and optic phonons. If we suppose also the transversal waves (along with longimdinal ones), we can get three acoustic and three optical phonon branches. There is always one longitudinal (LA or LO) and two mutually perpendicular transversal (TA or TO) phonons. 

The introduction of reciprocal space allows for both the characterization by unitary projection (planes are represented by dots) and completes projection (both information of orientation and interplanar distance are present) for the crystallographic planes of direct space as well as the possibility of their quantum representation by the characterization of Brillouin Zones of wave vectors associated to the crystal s eigen-states (Pettifor, 1995). 

The conditions described here also define the conditions for diffraction of electron waves at the Brillouin zone boimdaries. Likewise the Brillouin zones described in Chapter 2 are reciprocal-space objects with the symmetry of the reciprocal lattice rather than the real-space lattice. The reciprocal lattice points in Figure 2.5, for example, are located at points hbj, kb2, and lb3. The reciprocal lattice for a simple cubic system with basis vectors ai, a 2, and as has reciprocal lattice vectors parallel to the real space vectors. However, larger distances in real space correspond to shorter distances in reciprocal space. Thus, planes that are widely spaced in real space have closely spaced reciprocal lattice points and vice versa. One may determine by examination of Figure 4.2 that the (100) planes are V3 times farther apart than are the (111) planes. In general, the distance, d, between (hkl) lattice planes in a cubic system may be shown to be ... 

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