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There are 17 two-dimensional space groups and 230 three-dimensional space groups.

There are 17 two-dimensional space groups and 230 three-dimensional space groups.

If, on the other hand, this stacking is carried out by placing the third layer in a third position and the fourth layer above the first, and so forth, the result is an A-B-C-A-

B-C-A----sequence, and the structure is FCC, as explained in Chapter 2. The latter arrangement is more commonly found in nanocrystals.

Some properties of nanostructures depend on their crystal structure, while other properties such as catalytic reactivity and adsorption energies depend on the type of exposed surface. Epitaxial films prepared from FCC or HCP crystals generally grow with the planar close-packed atomic arrangement just discussed. Face-centered cubic crystals tend to expose surfaces with this same hexagonal two-dimensional atomic array.

### 3.2.2. Crystallography

To determine the structure of a crystal, and thereby ascertain the positions of its atoms in the lattice, a collimated beam of X rays, electrons, or neutrons is directed at the crystal, and the angles at which the beam is diffracted are measured. We will explain the method in terms of X rays, but much of what we say carries over to the other two radiation sources. The wavelength X of the X rays expressed in nanometers is related to the X-ray energy E expressed in the units kiloelectronyolts (keV) through the expression

Ordinarily the beam is fixed in direction and the crystal is rotated through a broad range of angles to record the X-ray spectrum, which is also called a diffractometer recording or X-ray-diffraction scan. Each detected X-ray signal corresponds to a coherent reflection, called a Bragg reflection, from successive planes of the crystal for which Bragg's law is satisfied as shown in Fig. 3.1, where d is the spacing between the planes, 0 is the angle that the X-ray beam makes with respect to the plane, A is the wavelength of the X rays, and n = 1,2,3,... is an integer that usually has the value n = 1.

Each crystallographic plane has three indices h,k,l, and for a cubic crystal they are ratios of the points at which the planes intercept the Cartesian coordinate axes x, y, z. The distance d between parallel crystallographic planes with indices hkl for a simple cubic lattice of lattice constant a has the particularly simple form

d sin 9

Figure 3.1. Reflection of X-ray beam incident at the angle 6 off two parallel planes separated by the distance d. The difference in pathlength 2d sin 6 for the two planes is indicated. (From C. P. Poole Jr., The Physics Handbook Wiley, New York, 1998, p. 333.)

d sin 9

Figure 3.1. Reflection of X-ray beam incident at the angle 6 off two parallel planes separated by the distance d. The difference in pathlength 2d sin 6 for the two planes is indicated. (From C. P. Poole Jr., The Physics Handbook Wiley, New York, 1998, p. 333.)

so higher index planes have larger Bragg diffraction angles 9. Figure 3.2 shows the spacing d for 110 and 120 planes, where the index 1 = 0 corresponds to planes that are parallel to the z direction. It is clear from this figure that planes with higher indices are closer together, in accordance with Eq. (3.3), so they have larger Bragg angles 0 from Eq. (3.2). The amplitudes of the X-ray lines from different ciystal-lographic planes also depend on the indices hkl, with some planes having zero amplitude, and these relative amplitudes help in identifying the structure type. For example, for a body-centered mcmatomic lattice the only planes that produce observed diffraction peaks are those for which h + k + I = n, an even integer, and for a face-centered cubic lattice the only observed diffraction lines either have all odd integers or all even integers.

To obtain a complete crystal structure, X-ray spectra are recorded for rotations around three mutually perpendicular planes of the crystal. This provides comprehensive information on the various crystallographic planes of the lattice. The next step in the analysis is to convert these data on the planes to a knowledge of the positions of the atoms in the unit cell. This can be done by a mathematical procedure called Fourier transformation. Carrying out this procedure permits us to identify which one of the 230 crystallographic space groups corresponds to the structure, together with providing the lengths of the lattice constants a,b,c of the unit cell, and the values of the angles a,fi,y between them. In addition, the coordinates of the positions of each atom in the unit cell can be deduced.

As an example of an X-ray diffraction structure determination, consider die case of nanocrystalline titanium nitride prepared by chemical vapor deposition with the grain size distribution shown in Fig. 3.3. The X-ray diffraction scan, with die various lines labeled according to their crystallographic planes, is shown in Fig. 3.4. The fact that all the planes have either all odd or all even indices identifies the structure as face-centered cubic. The data show that TiN has the FCC NaCl structure sketched in Fig. 2.3c, with the lattice constant a = 0.42417 nm.

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