Properties of the Reciprocal Lattice

The important characteristics of the reciprocal lattice that should be noted are (1) that the vector r kl is normal to the crystallographic plane whose Miller indices are (hkl), and (2) that the length r%kl of the vector is equal to the reciprocal of the interplanar spacing dh]d. 

We prove that r k( is perpendicular to the selected (hkl) plane by showing that two nonparallel vectors lying in it are perpendicular to r%kl. For this purpose we choose the vectors HK (= OK — OH) and HL (= OL — OH). Thus we have 

Using (C.3), we find the above two products to be both zero, thus indicating that r%kl is perpendicular to HK and HL and therefore to the plane HKL, which is the (hicl) plane. 

to prove that r kl is equal to Udhki, we note that dhki is given by the dot product of a/h with the unit vector normal to (hkl) and hence parallel to rkkl. We therefore have 

It is left to the reader, as an exercise, to figure out that the unit cell volume of the reciprocal lattice is equal to 1/VU. 

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Figure 6.8 summarizes the most important properties of the reciprocal lattice. It is important that the base vectors of the surface lattice form the smallest parallelogram from which the lattice may be constructed through translations. Figure 6.9 shows the five possible surface lattices and their corresponding reciprocal lattices, which are equivalent to the shape of the respective LEED patterns. The unit cells of both the real and the reciprocal lattices are indicated. Note that the actual dimensions of the reciprocal unit cell are irrelevant only the shape is important. 

The experimental data obtained from these patterns allows the determination of the atomic structure of both high- and low-symmetric crystals. Since the crystals of a plate texture are all oriented with a particular plane parallel to the support, the properties of the reciprocal lattice require that its points will be distributed exclusively along straight lines perpendicular to the support (Fig. 10), independent of the symmetry of the crystals forming the texture. As a result the rings of the reciprocal lattice lie on coaxial cylinders whose axis is texture axis. This distribution of the rings is the most important characteristic of the reciprocal lattice of plate textures. 

The last condition is identical to Eq. (A.10). Therefore, g must be a reciprocal lattice vector. This shows us another important property of the reciprocal lattice it is the Fourier transform of the real space lattice. 

Two properties of the reciprocal lattice make it of value in diffraction theory. We state these without proof. 

The two-dimensional example illustrating the relationships between the direct and reciprocal lattices (or spaces), which are used to represent crystal structures and diffraction patterns, respectively, is shown in Figure 1.40. Pin important property of the reciprocal lattice is that its symmetry is the same as the symmetry of the direct lattice. However, in the direct space atoms can be located anywhere in the unit cell, whereas diffraction peaks are represented only by the points of the reciprocal lattice, and the unit cells themselves are "empty" in the reciprocal space. Furthermore, the contents of every unit cell in the direct space is the same, but the intensity of diffraction peaks, which are conveniently represented using points in the reciprocal space, varies. 

Figure 6.8 Definition and properties of the two-dimensional reciprocal lattice a, and a2 are base vectors of the surface lattice, and a and a2 are the base vectors of the reciprocal lattice. The latter is equivalent to the LEED pattern.

Since the variation of any physical property in a three dimensional crystal is a periodic function of the three space coordinates, it can be expanded into a Fourier series and the determination of the structure is equivalent to the determination of the complex Fourier coefficients. The coefficients are indexed with the vectors of the reciprocal lattice (one-to-one relationship). In principle the expansion contains an infinite number of coefficients. However, the series is convergent and determination of more and more coefficients (corresponding to all reciprocal lattice points within a sphere, whose radius is given by the length of a reciprocal lattice vector) results in a determination of the stmcture with better and better spatial resolution. Both the amplitude and the phase of the complex number must be determined for any Fourier coefficient. The amplitudes are determined from diffraction... 

Thus, the scattering of a periodic lattice occurs in discrete directions. The larger the translation vectors defining the lattice, the smaller a i=1 3, and the more closely spaced the diffracted beams. This inverse relationship is a characteristic property of the Fourier transform operation. The scattering vectors terminate at the points of the reciprocal lattice with basis vectors a i=1>3, defined by Eq. (1.21). 

Figure 4.8. Equivalence of the photodynamic properties of the naphthalene lattice with a lattice with one molecule per cell. A The naphthalene lattice with two molecules per cell (a, b), and its reciprocal lattice A. B The equivalent lattice with one molecule per cell, and its reciprocal lattice B. ...

Use of the reciprocal lattice unites and simplifies crystallographic calcnlations. The motivation for the reciprocal lattice is that the x-ray pattern can be interpreted as the reciprocal lattice with the x-ray diffraction intensities superimposed on it. See Section 14.2 for the definition of the reciprocal lattice vectors a b and c in terms of the direct basis vectors a, b, and c. Table 14.2 shows the parallel between the properties of the direct lattice and the reciprocal lattice, and Table 14.3 relates the direct and reciprocal lattices. 

How do we know the relationship between the crystal and the diffraction pattern that we will obtain from it Often it is easier to think of the diffraction experiment with respect to a reciprocal lattice rather than the crystal lattice planes. The reciprocal lattice is a real physical property of a crystal, and rotation of the crystal will cause a rotation of the reciprocal lattice. 

Many of the physical properties of crystals, as well as the geometry of the three-dimensional patterns of radiation diffracted by crystals, (see Chapter 6) are most easily described by using the reciprocal lattice. The two-dimensional (plane) lattices, sometimes called the direct lattices, are said to occupy real space, and the reciprocal lattice occupies reciprocal space. The concept of the reciprocal lattice is straightforward. (Remember, the reciprocal lattice is simply another lattice.) It is defined in terms of two basis vectors labelled a and b. ... 

In solving three-dimensional triperiodic diffraction problems the concept of the reciprocal lattice (128) helps greatly. Reciprocal space constructions are useful for diperiodic structures also. In the simplest case of a strictly two-dimensional single layer grating, the reciprocal space construction is an array of parallel rods normal to the plane of the grating. These rods cut the plane at the points of a reciprocal net generated by translations of unit reciprocal vectors a and b having properties defined below in terms of the real-space imit vectors d and b (Section IIB). 

It has been shown also (Boulesteix et al., 1972) that only one third of the nodes of the reciprocal lattice have homologues in twinned crystals both for a type I twin and for a type II twin, which makes these twins indeed reticular pseudo-merohedral twins (or nearly superlattice conserving twins NSLCT) with multiplicity 3 (appendix). This is shown in fig. 10 for the (132) reciprocal plane and for a type I twin. This cannot be explained by the description of B as a mere deformation of A but we shall explain this property later on it is related to the existence of antiphase domains. 

The sets of integers hkl) can be imagined to label the points of intersection of three sets of equally spaced parallel planes. It is straightforward to show that these planes can be chosen so that this new lattice, which is then called the reciprocal lattice, has the following property for all values of h, k and / the line joining the origin of the reciprocal lattice to the point (hkl) is of length l/dh i and is normal to the hkt) planes of the real lattice. 

Instead of the Lagrangian weight functions W2p(x), the Cardinal B-splines M (x) [117,118] were utilized in later versions of the PME method [110,112,113]. These weight functions are continuously differentiable and allow the forces to be obtained from analytical differentiation of the approximation of the reciprocal lattice sum. The Cardinal B-spline of the second order M2(x) gives the linear hat function M2(x)= 1 — x — 1 in the interval 02. The nth order B-spline satisfies the following properties ... 

Physical properties of blue phases are due to such a structure. From the point of view of the light diffraction, the pertinent quantity is the dielectric tensor j,(r) which is a periodic function of r which, therefore, can be decomposed into a Fourier series. This decomposition was done for the first time by Dick Hornreich et al. [10] who used the traceless part of the dielectric tensor as the order parameter for a Landau-type theory of blue phases. The traceless part of the symmetric dielectric tensor has five independent coefficients. For this reason, for each q(Ak/) vector of the reciprocal lattice, five independent Fourier components have to be considered. We already know one of them from the section on cholesteric liquid crystals (2.25). Let... 

To understand and describe the electrical and optical properties of a semiconductor, it is essential to have knowledge of its electronic band structure, which exhibits the relation between energy and momentum E k) of electrons and holes in the different possible states of the conduction and valence bands at the various symmetry points of the first Brillouin zone of the reciprocal lattice. In particular, the band gap between the valence and conduction bands is important, because it determines, e.g., the optical transition energy and the temperature dependence of the intrinsic conductivity. In the case of the complex boron-rich solids with large numbers of atoms per unit cell, the agreement between theoretical calculations of the band gaps and the experimental results has not yet been satisfactory. 

The reciprocal unit-cell vectors gi,2 define the lateral periodicity of the reciprocal lattice. As the real-space vectors 5i,2, they He within the surface. The reciprocity is mirrored by the property... 

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. 

The symmetry properties of the density show up experimentally as properties of its Fourier components p. If those components vanish except when the wave vector k equals one of the lattice vectors K of a certain reciprocal lattice, the general plane wave expansion of the density,... 

We want to introduce the properties of the crystal and of the X-rays and solve f or the electric displacement or flux density, D. Hart gives a careful discussion of the polarisability of a crystal, showing that a sufficient model of the crystal for X-ray scattering is a Fourier sum of either the electron density or the electric susceptibility over all the reciprocal lattice vectors h. Thus the crystal is represented as... 

Shmueli, U. and Wilson, A. J.C. (1996). Statistical properties of the weighted reciprocal lattice. In International Tables for Crystallography, Shmueli, U., ed., Vol. B, pp. 184-200. Kluwer Academic Pubhshers, Dordrecht. 

Dynamic smdies of the alloy system in CO and H2 demonstrate that the morphology and chemical surfaces differ in the different gases and they influence chemisorption properties. Subnanometre layers of Pd observed in CO and in the synthesis gas have been confirmed by EDX analyses. The surfaces are primarily Pd-rich (100) surfaces generated during the syngas reaction and may be active structures in the methanol synthesis. Diffuse scattering is observed in perfect B2 catalyst particles. This is attributed to directional lattice vibrations, with the diffuse streaks resulting primarily from the intersections of 111 reciprocal lattice (rel) walls and (110) rel rods with the Ewald sphere. 

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