Reciprocal unit cell

The amplitude and therefore the intensity, of the scattered radiation is detennined by extending the Fourier transfomi of equation (B 1.8.11 over the entire crystal and Bragg s law expresses die fact that this transfomi has values significantly different from zero only at the nodes of the reciprocal lattice. The amplitude varies, however, from node to node, depending on the transfomi of the contents of the unit cell. This leads to an expression for the structure amplitude, denoted by F(hld), of the fomi... 

Figure Bl.21.3. Direct lattices (at left) and corresponding reciprocal lattices (at right) of a series of connnonly occurring two-dimensional superlattices. Black circles correspond to the ideal (1 x 1) surface structure, while grey circles represent adatoms in the direct lattice (arbitrarily placed in hollow positions) and open diamonds represent fractional-order beams m the reciprocal space. Unit cells in direct space and in reciprocal space are outlined.

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°.

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... 

The sums in Eqs. (1) and (2) run, respectively, over the reciprocal space lattice vectors, g, and the real space lattice vectors, r and Vc= a is the unit cell volume. The value of the parameter 11 affects the convergence of both the series (1) and (2). Roughly speaking, increasing ii makes slower the convergence of Eq. (1) and faster that of Eq. (2), and vice versa. Our purpose, here, is to find out, for an arbitrary lattice and a given accuracy, the optimal choice, iiopt > tbal minimises the CPU time needed for the evaluation of the KKR structure constants. This choice turns out to depend on the Bravais lattice and the lattice parameters expressed in dimensionless units, on the... 

The prindple of a LEED experiment is shown schematically in Fig. 4.26. The primary electron beam impinges on a crystal with a unit cell described by vectors ai and Uj. The (00) beam is reflected direcdy back into the electron gun and can not be observed unless the crystal is tilted. The LEED image is congruent with the reciprocal lattice described by two vectors, and 02". The kinematic theory of scattering relates the redprocal lattice vectors to the real-space lattice through the following relations... 

Figure 4.26. Schematic drawing of the LEED experiment on a single costal with a unit cell given by vectors a-i and O2. The LEED pattern corresponds to the reciprocal lattice described by vectors and 02. ...

In this diagram, a series of hexagon-shaped planes are shown which are orthogonal, or 90 degrees, to each of the corners of the cubic cell. Each plane connects to cuiother plane (here shown as a rectangle) on each fiace of the unit-cell. Thus, the faces of the lattice unit-cell and those of the reciprocal unit-cell can be seen to lie on the same pltme while those at the corners lie at right angles to the corners. 

The magnitude of k corresponds to a wave number 2n/X and therefore is measured with a unit of reciprocal length. For this reason k is said to be a vector in a reciprocal space or k space . This is a space in a mathematical sense, i.e. it is concerned with vectors in a coordinate system, the axes of which serve to plot kx, ky and kz. The directions of the axes run perpendicular to the delimiting faces of the unit cell of the crystal. 

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 6.10 illustrates LEED patterns of the clean Rh(lll) surface, and the surface after adsorption of 0.25 monolayers (ML) of NH3 [22]. The latter forms the primitive (2x2) overlayer structure (see Appendix I for the Wood notation). In the (2x2) overlayer, a new unit cell exists on the surface with twice the dimensions of the substrate unit cell. Hence the reciprocal unit cell of the adsorbate has half the size of that of the substrate and the LEED pattern shows four times as many spots. 

In crystallography, the difiiraction of the individual atoms within the crystal interacts with the diffracted waves from the crystal, or reciprocal lattice. This lattice represents all the points in the crystal (x,y,z) as points in the reciprocal lattice (h,k,l). The result is that a crystal gives a diffraction pattern only at the lattice points of the crystal (actually the reciprocal lattice points) (O Figure 22-2). The positions of the spots or reflections on the image are determined hy the dimensions of the crystal lattice. The intensity of each spot is determined hy the nature and arrangement of the atoms with the smallest unit, the unit cell. Every diffracted beam that results in a reflection is made up of beams diffracted from all the atoms within the unit cell, and the intensity of each spot can be calculated from the sum of all the waves diffracted from all the atoms. Therefore, the intensity of each reflection contains information about the entire atomic structure within the unit cell. 

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).

From a comparison of various spot electron diffraction patterns of a given crystal, a three-dimensional system of axis in the reeiproeal lattice may be established. The reeiproeal unit cell may be eompletely determined, if all the photographs indexed. For this it is sufficient to have two electron diffraction patterns and to know the angle between the seetions of the reeiproeal lattice represented by them, or to have three patterns which do not all have a particular row of points in common (Fig.5). Crystals of any compound usually grow with a particular face parallel to the surface of the specimen support. Various sections of the reciprocal lattice may, in this case, be obtained by the rotation method (Fig.5). 

If unit cell is orthogonal there are layers lines on oblique texture electron diffraction pattern. These lines occur when certain reciprocal lattice planes lie perpendicular to the texture axis. In this case period c may be more accurately determined by measuring the minor semi-axis R of any ellipse (in the presence of layer lines it is measured directly, since there is a zero line with /=0) and H of any reflection on that ellipse (preferably with a large l) ... 

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