Indices of planes

The formula shown in Eq. 1.11 becomes more complicated for non-orthogonal crystal systems, i.e. hexagonal, monoclinic and especially triclinic, because inter-axial angles here are different from 90°, and should be included in the calculations. The complete description of the corresponding mathematical relationships will be given later in the book (Eqs. 2.29 to 2.37 in section 2.8.1). 

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The Miller indices of planes in crystals with a hexagonal unit cell can be ambiguous. In order to eliminate this ambiguity, four indices, (hkil), are often used. These are called Miller-Bravais indices and are only used in the hexagonal system. The index i is given by... 

Fig. 11. A crystal of ammonium sulphate Fig. 12. Indices of planes of hexagonal (class mmni). (After Tut ton.) crystals. ABCDEFA BC D E Fhexa-...

It is not coincidental that I use the variable names h, k, and I for both the indices of planes in the crystal and the indices of reflections in the diffraction pattern (Chapter 2, Section V). I will show later that in fact the set of planes (hkl) produces the reflection hkl of the diffraction pattern. In the terms used in Chapter 2, each set of parallel planes in the crystal produces one reflection, or one term in the Fourier series that describes the electron density within the unit cell. The intensity of that reflection depends upon the electron distribution and density along the planes that produce the reflection. 

Consider a crystal having a n-fold rotation axis, often simply referred to as an n-axis. In Fig. 9.1.5, OE, OE, OE" represent the horizontal projections of three vertical planes generated by successive rotations of 2n/ about the symmetry axis, which passes through point O. As reference axes, we adopt the n-axis, OE, and OE", and the latter two are assigned unit length. Next, we consider a plane E F which is drawn parallel to OE. The intercepts of E F on the three axes are oo, 1, and (sec 2n/ri)/2. The Miller indices of plane E F are therefore (0,1,2 cos 2n/n) which must be integers. Since cos 2n/n < 1, the only possible values of 2 cos 2n/n are 2, 1, and 0. Hence... 

Fig. 2-11 (a) The hexagonal unit cell (heavy lines) and (b) indices of planes and directions. 

The Miller indices of planes in hexagonal lattices can be ambiguous. For example, three sets of planes lying parallel to the c-axis, which is imagined to be perpendicular to the plane of the diagram, are shown in (Figure 2.15). These planes have Miller indices A, (110), B, (120) and C, (210). Although these Miller indices seem to refer... 

Figure 2.5 Miller indices of planes in the cubic system. The distance d is the interplanar spacing. (From B.D. Cullity, Elements of X-ray Diffraction, 1978 by Addison-Wesley Publishing Company. Reprinted with permission of the publisher.)...

FIGURE 5.6 Indices of planes (using Miller-Bravais notation) and directions (using three-index Miller notation) in the hexagonal unit cell. 

In terms of indices of planes and directions, the condition that the plane (hkl) is parallel with the (zone axis) direction [uvw] is analytically translated by the equation ... 

If the detection system is an electronic, area detector, the crystal may be mounted with a convenient crystal direction parallel to an axis about which it may be rotated under tlie control of a computer that also records the diffracted intensities. Because tlie orientation of the crystal is known at the time an x-ray photon or neutron is detected at a particular point on the detector, the indices of the crystal planes causing the diffraction are uniquely detemiined. If... 

Figure B2.3.5. Typical time-of-flight spectra of DF products from the F + D2 reaction [28]- The collision energies and in-plane and out-of-plane laboratory scattered angles are given in each panel. The DF product vibrational quantum number associated with each peak is indicated. Reprinted with pennission from Faiibel etal [28]. Copyright 1997 American Chemical Society.

Because of Bragg s explanation of diffraction of x-rays from a crystal as being like reflections from famihes of planes, the diffraction spots ate usually called "reflections." Each reflection is identified with three integer indices, h, k, and / For the set of planes shown in Figure 7, the indices of the corresponding reflection are /i = 1, = 0, and I = 2. 

Vitrinite Reflectance. The amount of light reflected from a poHshed plane surface of a coal particle under specified illumination conditions increases with the aromaticity of the sample and the rank of the coal or maceral. Precise measurements of reflectance, usually expressed as a percentage, ate used as an indication of coal rank. 

We could make scale drawings of the many types of planes that we see in all unit cells but the concept of a unit cell also allows us to describe any plane by a set of numbers called Miller Indices. The two examples given in Fig. 5.5 should enable you to find the... 

These are typical of ionic liquids and are familiar in simulations and theories of molten salts. The indications of structure in the first peak show that the local packing is complex. There are 5 to 6 nearest neighbors contributing to this peak. More details can be seen in Figure 4.3-3, which shows a contour surface of the three-dimensional probability distribution of chloride ions seen from above the plane of the molecular ion. The shaded regions are places at which there is a high probability of finding the chloride ions relative to any imidazolium ion. 

In the perfect lattice the dominant feature of the electron distribution is the formation of the covalent, directional bond between Ti atoms produced by the electrons associated with d-orbitals. The concentration of charge between adjacent A1 atoms corresponds to p and py electrons, but these electrons are spatially more dispersed than the d-electrons between titanium atoms. Significantly, there is no indication of a localized charge build-up between adjacent Ti and A1 atoms (Fu and Yoo 1990 Woodward, et al. 1991 Song, et al. 1994). The charge densities in (110) planes are shown in Fig. 7a and b for the structures relaxed using the Finnis-Sinclair type potentials and the full-potential LMTO method, respectively. 

The third (3x) harmonic of running speed is a critical monitoring parameter. Most, if not all, generators tend to move in the axial plane as part of their normal dynamics. Increases in axial movement, which appear in the third harmonic, are early indicators of problems. 

Draw the meso form of each of the following molecules, and indicate the plane of symmetry in each ... 

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