Indexing hexagonal crystal system

Cinnabar is a red crystalline or powdery substance hexagonal crystal system refractive index 2.854 density 8.10 g/cm sublimes at 583.5°C color changes to brown at 250°C and converts to black sulfide at 386°C reverts to red color on cooling insoluble in water, alcohol and nitric acid soluble in aqua regia and solutions of alkali metal sulfides decomposed by hot concentrated sulfuric acid. 

We will begin with the cubic crystal system, where the assignment of indices is nearly transparent and then consider the theory behind the ab initio indexing in crystal systems with tetragonal and hexagonal symmetry. Indeed, as with any kind of experimental work, experience is paramount, and we hope that the contents of this section may help the reader to achieve accurate solutions of real life indexing tasks successfully. 

At this point (or after the whole pattern has been indexed) the analysis of the observed values of A enables one to establish whether we deal with the tetragonal or hexagonal crystal systems. As seen in Table 5.19, the whole multipliers of a are 1 and 3, and 3 is only possible in the hexagonal crystal system for A =, k= 1. After a simple calculation using the average values of a and c listed in Table 5.19 we find approximate values of a and c as 5.046 and 4.015 A, respectively. A least squares refinement of the lattice parameters using the entire array of indexed Bragg peaks obviously yields the same lattice parameters as were established before (see Table 5.6). 

Fortunately, but also evidently, many simplifications arise when considering the uniaxial indicatrix of crystals belonging to the trigonal, tetragonal, and hexagonal crystal systems. First, there is only one optic axis which, by convention, always lies along the z crystallographic axis hence, X, Y, and Z disappear, as z suffices to define this direction. The refractive index associated with this direction is called or n. The plane perpendicular to the optic axis is necessarily a circular section whose diameters all have the same refractive index denoted or n. Thus, and Hy disappear. Optically positive means... 

The orientations of planes for a crystal structure are represented in a similar manner. Again, the unit cell is the basis, with the three-axis coordinate system as represented in Figure 3.5. In all but the hexagonal crystal system, crystallographic planes are specified by three Miller indices as hkl). Any two planes parallel to each other are equivalent and have identical indices. The procedure used to determine the h, k, and I index numbers is as follows ... 

For powder photographs, the use of the charts described on p. 143 and in Appendix 3 will show whether the substance is cubic, tetragonal, or hexagonal if it is not, the numerical methods of indexing the patterns of crystals of low symmetry may be tried or, if it is. possible to pick out single crystals, or if the specimen can be recrystallized to give suitable crystals, the unit cell dimensions may be determined by the methods described earlier. A search may then be made in the tables of Donnay and Nowacki (1954), in which, for each crystal system, the species are arranged in order of the axial ratios. 

The Miller-Bravais index system for identifying planes and directions in hexagonal crystals is similar to the Miller index system except that it uses four axes rather than three. The advantage of the four-index system is that the symmetry is more apparent. Three of the axes, ai, a2, and a3, he in the hexagonal (basal) plane at 120° to one another and the fourth or c-axis is perpendicular to then, as shown in Figure 3.1. 

For hexagonal crystal planes, a slightly different indexing nomenclature is used relative to cubic crystals. To index a plane in the hexagonal system, four axes are used, called Miller-Bravais indices. In addition to both a and b axes, another axis... 

Porter, M. W. and Spiller, R. C. (1951). Crystals of the tetragonal, hexagonal, trigonal and orthorhombic systems. In The Barker index of crystals. A method for the identification of crystalline substances, Vol. 1, W. Heffer Sons, Cambridge. [14, 95] Porter, M. W. and Spiller, R. C. (1956). Crystals of monoclinic system. In The Barker index of crystals. A method for the identification of crystalline substances, Vol. 2, W. Heffer Sons, Cambridge. [14]... 

In hexagonal and trigonal crystal systems, the fourth index is usually introduced to address the possibility of three similar choices in selecting the crystallographic basis as illustrated in Figure 7.5 7. In addition to the unit cell based on the vectors a, b and c, two other unit cells, based on the vectors a, -(a + b) and c, and -(a + b), b and c are possible due to the six-fold or the... 

Figure 5.14. The relationship between the lattice parameters of the hexagonal unit cell (solid and dotted lines) and the related orthorhombic unit cell (dashed lines). The unit cell parameter perpendicular to the plane of the projection is identical in both crystal systems. The smaller orthorhombic unit cell found using the DICVOL91 indexing program is indicated by the thick solid vectors (a o and c o). Open circles show lattice points and the dash-dotted vector illustrates the C-translation in the conforming orthorhombic lattice.

Fig. Al-5 The reciprocal lattice of a hexagonal crystal which has = 4 A. (Here the three-symbol system of plane indexing is used and 2l is the axis usually designated c.) The axes as and bs are normal to the drawing.

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

Compounds belonging to the cubic crystal system have only a single refractive index value, but other systems are anisotropic, so that the crystal is characterized by two or three unique indexes. Hexagonal, rhombohedral, and tetragonal crystals have two unique indexes which are traditionally labeled and for ordinary ray and extraordinary ray . Orthorhombic, monoclinic, and triclinic crystals are characterized by three indexes which are here called n, and n. The table indicates the crystal system for each entry in order to identify the material uniquely. 

As with Miller indices, to specify directions in hexagonal crystals a four-index system, u v t w ] is sometimes used. The conversion of a three-index set to a four-index set is given by the following rules ... 

For crystals having hexagonal symmetry, it is desirable that equivalent planes have the same indices as with directions, this is accomplished by the Miller-Bravais system shown in Figure 3.8. This eonvention leads to the four-index (hkil) scheme, which is favored in most instances because it more clearly identifies the orientation of a plane in a hexagonal crystal. There is some redimdaney in that i is determined by the sum of h and k through... 

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