Cell, unit dimensions, reciprocal

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. 

The conductance measured between the cell terminals is multiplied by (he cell constant given in reciprocal units of length to calculate the conductivity. To calculate the resistivity, the measured resistance between the cell terminals is divided by die cell constant. Althuugh the cell constant (in reciprocal units nf length) can he calculated from the dimensions of the conductivity cell by dividing the length of Ihe electrical path through the solution by the cross-sectional area of the path, in practice, these measurements are difficult to make and arc only used to approximate the cell constant, which is determined by use of standard solutions of known conductivity or by comparison with other conductivity cells which have been so standardized. 

Equation (4.7) shows that the number of available reflections depends only upon Vand A. For a modest-size protein unit cell of dimensions 40 x 60 x 80 A, 1.54-A radiation can produce 1.76 X 106 reflections, an overwhelming amount of data. Fortunately, because of cell and reciprocal-lattice symmetry,... 

There are several pitfalls to watch out for. First, there are (maybe obvious) planes that are in one of the planes of the unit cell itself Figure 21.19 shows several. None of these planes intersects the third crystal axis. In this case, the intercepts are considered to occur at infinite dimensions of the unit cell, and the reciprocal of < is zero. Therefore, the Miller indices of the planes in Figure 21.19 are (100), (110), and (200), respectively. (You should satisfy yourself that these designations are correct.)... 

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. 

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. 

Assuming that the equatorial reflections have been shown to fit a rectangular reciprocal lattice net, attention may be turned to the upper and lower layer lines. The values for all the spots are read off on Bernal s chart, and the reciprocal lattice rotation diagram is constructed from these values if the values for the upper and lower layer lines correspond with those of the equator—that is, row7 lines as well as layer lines are exhibited as in Fig. 80—then the unit cell must be orthorhombic. It should be noted that some spots may be missing from the equator, and it may be necessary to halve one or both of the reciprocal axes previously found to satisfy the equatorial reflections. The dimensions of the unit cell, and the indices of all the spots, follow immediately from the reciprocal lattice diagrams. 

If no external evidence is available, it is still possible to determine the unit cell dimensions of crystals of low symmetry from powder diffraction patterns, provided that sharp patterns with high resolution are avail able. Hesse (1948) and Lipson (1949) have used numerical methods successfully for orthorhombic crystals. (Sec also Henry, Lipson, and Wooster, 1951 Bunn 1955.) Ito (1950) has devised a method which in principle will lead to a possible unit cell for a crystal of any symmetry. It may not be the true unit cell appropriate to the crystal symmetry, but when a possible cell satisfying all the diffraction peaks on a powder pattern lias been obtained by Ito s method, the true unit cell can be obtained by a reduction process first devised by Delaunay (1933). Ito applies the reduction process to the reciprocal lattice (see p. 185), but International Tables (1952) recommend that the procedure should be applied to the direct space lattice. 

Because the real lattice spacing is inversely proportional to the spacing of reflections, crystallographers can calculate the dimensions, in angstroms, of the unit cell of the crystalline material from the spacings of the reciprocal lattice on the X-ray film (Chapter 4). The simplicity of this relationship is a dramatic example of how the macroscopic dimensions of the diffraction pattern are connected to the submicroscopic dimensions of the crystal. 

In this chapter, I will discuss the geometric principles of diffraction, revealing, in both the real space of the crystal s interior and in reciprocal space, the conditions that produce reflections. I will show how these conditions allow the crystallographer to determine the dimensions of the unit cell and the symmetry of its contents and how these factors determine the strategy of data collection. Finally, I will look at the devices used to produce and detect X rays and to measure precisely the intensities and positions of reflections. 

The unit-cell dimensions determine the reciprocal-lattice dimensions, which in turn tell us where we must look for the data. Methods like oscillation photography require that we know precisely which reflections will fall completely and partially within a given oscillation angle so that we can collect as many reflections as possible without overlap. So we need the unit-cell dimensions in order to devise a strategy of data collection that will give us as many identifiable (by index), measurable reflections as possible. 

Figure 4.26 Reflection spacings on the film are directly proportional to reciprocal-lattice spacings, and so they are inversely proportional to unit-cell dimensions.

In the diffraction pattern from a crystalline solid, the positions of the diffraction maxima depend on the periodicity of the structure (i.e. the dimensions of the unit cell), whereas the relative intensities of the diffraction maxima depend on the distribution of scattering matter (i.e. the atoms, ions or molecules) within the repeating unit. Each diffraction maximum is characterized by a unique set of integers h, k and l (called the Miller indices) and is defined by a scattering vector h in three-dimensional space, given by h=ha +A b +Zc. The three-dimensional space in which the diffraction pattern is measured is called reciprocal space , whereas the three-dimensional space defining the crystal structure is called direct space . The basis vectors a, b and c are called the reciprocal lattice vectors, and they depend on the crystal structure. A given diffraction maximum h is completely defined by the structure factor F(h), which has amplitude F(h) and phase a(h). In the case of X-ray diffraction, F(h) is related to the electron density p(r) within the unit cell by the equation... 

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.

---