The reciprocal lattice representation

The total binding energy of a NFE metal can be evaluated within second-order perturbation theory. In the presence of a perturbation fi to the Hamiltonian operator of a system, the energy of state is given by 

Substituting eqn (6.44) into (6.43) and interchanging the ordering of integration and summation, we have 

This can be expressed in a form which is familiar in X-ray diffraction, namely 

the screened pseudopotential form factor of aluminium normalized by the Fermi energy will approach q = 0 at —2/3 as observed in Fig. 5.12. 

The zeroth- and first-order contributions to the perturbed energy level, E[2 are independent of the particular arrangement of the atoms. The structure dependence resides in the second-order contribution. Summing over all the occupied energy levels the structure-dependent contribution takes the form 

---

Second-order perturbation theory has allowed us to write the band-structure energy of a perfect crystal as a sum over reciprocal lattice vectors, since the structure factor vanishes unless q = G (cf eqn (6.59)). Thus, within the reciprocal lattice representation we have... 

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. 

Figure 16.11. (a) bm is a vector from the origin O to a lattice point P in the reciprocal lattice representation, and plane 1 is normal to bm. The lattice translation a is a vector from O to another lattice point P2 on plane 1. Plane 0 is parallel to plane 1 through O. (b) a intersects plane 1 at one of the other lattice points in plane 1. If a lies along ai, n2 and n3 are zero and a = ttiHi. Similarly for a2, a3. 

In the Bragg formulation of diffraction we thus refer to reflections from lattice planes and can ignore the positions of the atoms. The Laue formulation of diffraction, on the other hand, considers only diffraction from atoms but can be shown to be equivalent to the Bragg formulation. The two formulations are compared in Fig. 2B for planes with Miller indices (110). What is important in diffraction is the difference in path length between x-rays scattered from two atoms. The distance si + s2 in the Laue formulation is the same as the distance 2s shown for the Bragg formulation. The Laue approach is by far the more useful one for complicated problems and leads to the concept of the reciprocal lattice (Blaurock, 1982 Warren, 1969) and the reciprocal lattice vector S = Q 14n that makes it possible to create a representation of the crystal lattice in reciprocal space. 

Figure 5.6. Schematic representations of the fractions of the volume of the sphere (r = 1/X) in the reciprocal space in which the list of hkl triplets should be generated in six powder Laue classes to ensure that all symmetrically independent points in the reciprocal lattice have been included in the calculation of Bragg angles using a proper form of Eq. 5.2. The monoclinic crystal system is shown in the alternative setting, i.e. with the unique two-fold axis parallel to c instead of the standard setting, where the two-fold axis is parallel to b. ...

Figure 18 A schematic representation of data collection with an electronic area detector. The reciprocal lattice plane / = 0 is shown as black dots. The direct beam and four scattered beams with their respective indices are shown three of them produce diffraction spots on the detector, while the fourth (-1,4,0) falls outside the detector area.

The reciprocal lattice was invented by crystallographers as a simple and convenient representation of the physics of diffraction by a crystal. It is an extremely useful tool for describing all kinds of diffraction phenomena occurring in powder diffraction (Figure 1.5). 

Ewald [EWA 17] suggested a geometric representation in the reciprocal lattice of the results we have just shown. The Ewald sphere is defined as a sphere centered in O, the origin of the direct and diffracted wave vectors, and with radius fX. Vectors kg and k are co-linear to vectors Sg and S, respectively. 

The reciprocal lattice described in Chapter 2 consists of an array of points. Because a diffraction pattern is a direct representation of the reciprocal lattice, it is often useful to draw it as a weighted reciprocal lattice, in which the area allocated to each node is proportional to the structure factor FQikt) of each reflection, where... 

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. 

Fig. 46. (a) Schematic representation of the LEED pattern from the Si(l 11)"5x5"-Cu surface. The open circles correspond to the first-order Si(l 11) reflections, the closed circles correspond to the "5x5" reflections. The size of the circles approximately indicates their intensities. The ratio of the reciprocal lattice vectors a. b is 0.816 0.03 [87K2]. (b) STM image of the Si(l 1 l)"5.5x5.5"Cu. The surface shows up as an array of round shaped clusters with 5.4-5.7 periodicity [97S9]. 

Figure 6.9a-f illustrates a variety of the accepted band structure representations for nearly-free electron model. The Figure introduces the repeated-zone, extended-zone and reduced-zone images. The original free-electron parabola E = fi k Klme) is shown in Figure 6.9a. To leading order in the weak one-dimension periodic potential this curve remains correct except the value of k near the reciprocal lattice vector g. One can imagine that in this point the Bragg plane reflects the electron wave since the Bragg condition holds. Another free-electron parabola is centered at fe = g, and two parabolas are crossed each other at the...

Figure 2.23. Graphical representation of the five independent Fourier components of the dielectric tensor for a given wavevector q(hkl) from the reciprocal lattice.

---