Reciprocal unit-cell vectors

Figure Bl.21.4. Direct lattices (at left) and reciprocal lattices (middle) for the five two-dimensional Bravais lattices. The reciprocal lattice corresponds directly to the diffraction pattern observed on a standard LEED display. Note that other choices of unit cells are possible e.g., for hexagonal lattices, one often chooses vectors a and b that are subtended by an angle y of 120° rather than 60°. Then the reciprocal unit cell vectors also change in the hexagonal case, the angle between a and b becomes 60° rather than 120°.

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... 

Figure 3.2.1.6 Relation between real and reciprocal unit-cell vectors for (a) the general case and (b) the special case 5] 02- In panel (c) the three-dimensional reciprocal lattice is displayed.

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. 

Figure 2. Ewald construction for X-ray (soUd sphere) and electron (dotted sphere). ( kO, k wave-vectors, X - wave-length, a, b - parameters of reciprocal unit cell).

Figure 7.7a shows the extended-zone electronic band structure for a one-dimensional crystal - an atom chain with a real-space unit cell parameter a and reciprocal lattice vector Tr/n - containing a half-filled (metallic) band. In this diagram, both values of the wave vector, +k, are shown. The wave vector is the reciprocal unit cell dimension. The Fermi surface is a pair of points in the first BZ (Fig. 7.7c). When areas on the Fermi surface can be made to coincide by mere translation of a wave vector, q, the Fermi surface is said to be nested. The instability of the material towards the Peierls distortion is due to this nesting. In one dimension, nesting is complete and a one-dimensional metal is converted to an insulator because of a Peierls distortion. This is shown in Figure 7.7b, where the real-space unit cell parameter of the distorted lattice is 2a and a band gap opens at values of the wave vector equal to half the original values, 7r/2a.

The reciprocal lattice, like the lattice of the crystal, may also be divided into unit cells with the reciprocal unit vectors a, b, andc as edges. Since reciprocal space of a crystal is zero everywhere except at lattice points, however, the interiors of the reciprocal unit cells will be vacant. The relation between orthorhombic and monoclinic unit cells, and the corresponding reciprocal unit cells derived from them are shown in Figures 3.20 and 3.21. The type of reciprocal unit cell will be the same as the real cell from which it arises, and the reciprocal unit cell, hence the reciprocal lattice, will manifest the symmetry and centering properties of the real crystal lattice. 

The standard way to describe the orientation of a plane is by a vector d perpendicular (normal) to it. An equivalent description for Bragg planes is in terms of how many times they intersect each of the three unit cell axes in one lattice repeat (Figure 11). These Miller indices h (for axis a), k (for axis b), and / (for axis c) uniquely define the plane and its X-ray reflection for instance, 1 (1,0,0) plane intersects the x-axis once, a (2,1,0) plane, the x-axis once, and the j-axis twice, and so on. In principle, it is possible to calculate the vector Akki knowing the Miller indices h,k,l and the unit cell vectors a, b, and c. In practice, this may not be easy. As we want to have a simple description of the normal vectors Akki (which determine when Bragg s law will hold) we adopt a different set of basis vectors (a, b, c ), called the reciprocal lattice and the space they define is called reciprocal space. Each plane can be described by a vector ... 

As was shown in Equation (34) of Chapter 1, Bragg s law dictates that the scattering vectors for a Bragg peak, h, correspond to these reciprocal lattice vectors. The three-dimensionality of the diffraction pattern makes the identification of the three vectors a, b, c, straightforward, from which the direct space unit cell vectors ... 

The concept of the reciprocal lattice is very useful in discussing the diffraction of x-rays and neutrons from crystalline materials, especially in conjunction with the Ewald sphere construction discussed in Section 1.5.3. The regular arrangement of atoms and atomic groupings in a crystal can be described in terms of the crystal lattice, which is uniquely specified by giving the three unit cell vectors a, b9 and c. It turns out that the diffraction from a crystal is similarly associated with a lattice in reciprocal space. The reciprocal lattice is specified by means of the three unit cell vectors a, b, and c in the same way as the crystal lattice is based on a9 b, and c. In fact, the crystal lattice and the reciprocal lattice are related to each other by the Fourier transform relationship. 

Figure C.l illustrates how the reciprocal vector c is related to the unit cell vectors a, b9 and c. The relationships a c =0 and b c = 0 show that the vector c is perpendicular to both a and b and hence to the base OACB of the unit cell. The relationship c c = 1 means that its length c is equal to the reciprocal of OP, the projection of c to c. In the special case in which the directions of c and c coincide, i.e., when c is perpendicular to both a and b9 c is equal to 1/ c. Figure C.2 illustrates the relationship between the crystal and reciprocal lattices, drawn for a monoclinic crystal where b and b are normal to the plane of the drawing. Note that...

Figure C.l Reciprocal lattice vector c in relation to the unit cell vectors a,b, and c in real space.

In this system, we use a, b, c (not unit-cell vectors) to represent the lengths of intercepts which define the planes within the unit cell. Miller Indices are the reciprocals of the intercepts, a, b c, of the chosen plane on the x, y, z - directions in the lattice. 

The reciprocal lattice is just what the name says it is. If we have unit cell vectors in the real lattice, the imaginary reciprocal lattice vectors are 90° to the real ones. These imaginary vectors define the Brillouin Zone for that structure. 

Note that some nonequivalent points of BZ-3 become equivalent in BZ-2 (for example, the points X for the surface (001) and the points L for the (110) surface). Some points of BZ-2 have higher qrmmetry than in BZ-3 (for example, the vertices of BZ-2 for the (111) surface). These properties of BZ-2 arise because the unit-cell vectors in the two-dimensional reciprocal space B are not the lattice vectors of the three-dimensional reciprocal lattice. 

Figure BT2T4 illustrates the direct-space and reciprocal-space lattices for the five two-dimensional Bravais lattices allowed at surfaces. It is usefiil to realize that the vector a is always perpendicular to the vector b and that is always perpendicular to a. It is also usefiil to notice that the length of a is inversely proportional to the length of a, and likewise for b and b. Thus, a large unit cell in direct space gives a small unit cell in reciprocal space, and a wide rectangular unit cell in direct space produces a tall rectangular unit cell in reciprocal space. Also, the hexagonal direct-space lattice gives rise to another hexagonal lattice in reciprocal space, but rotated by 90° with respect to the direct-space lattice.

Note that the denominator in each case is equal to the volume of the unit cell. The fact that a, b and c have the units of 1/length gives rise to the terms reciprocal space and reciprocal latlice. It turns out to be convenient for our computations to work with an expanded reciprocal space that is defined by three closely related vectors a , b and c, which are multiples by 2tt. of the X-ray crystallographic reciprocal lattice vectors ... 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

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