Cubic crystal system 248 INDEX

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... 

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. 

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. 

Measurements on crystalline powders lead to a solution of only relatively small and simple structures Most of the reflections must be resolved, so that intensities can be associated with them unequivocally. Even theoretically, this is often impossible, e.g., when several reflections occur at precisely the same glancing angle. In the cubic crystal system this is so when the sum of the squares of the indices does not permit a single indexing e.g., (221) and (300), where the sum of the squares of the indices is 9 for each case. Measurements on crystalline powders that give unambiguous values for all reflections should be inherently preferable to single-crystal measurements, because many experimental difficulties (e.g.. extinction and absorption) either do not occur at all. or are easier to handle. 

With the exception of those solids in the cubic crystal system, most ceramic crystals possess some degree of optical anisotropy. The index of refraction is usually defined in terms of two indices using plane-polarized light Mx, where the E field is perpendicular to the optic axis, and n, where the E field is parallel to the optic axis. The term optic axis refers to an axis of optical and crystallographic symmetry. A consequence of optical anioso-tropy is that when a ray of unpolarized light enters a birefringent crystal, the beam will be split into two rays, the ordinary ray (o ray) and the extraordinary ray (e ray). 

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. 

We note that the benchmarks listed above must be applied altogether. For example, it is nearly always possible to choose the highest symmetry crystal system (i.e. cubic) and a large unit cell to dubiously assign index triplets to all observed Bragg peaks and obtain acceptable e. This happens because the density of points in the reciprocal lattice is proportional to the volume of the... 

The reciprocal space indexing can be implemented in several different ways. Two of them are trial-and-error and zone search methods. The first one is more efficient in high symmetry crystal systems (from cubic to orthorhombic) but becomes slow for low symmetry crystal systems (especially triclinic), while the second method works quite effectively and is fast with low symmetries (from triclinic to orthorhombic). 

Crystalline index of refraction As different polymorphs have different internal structures, they belong to different crystal systems therefore, polymorphs can be distinguished using polarized light and a microscope. The crystals can be either isotropic or anisotropic. In isotropic crystals, the velocity of light is the same in all directions, whereas anisotropic crystals have two or three different light velocities or refractive indices. In terms of crystal systems, only the cubic system is isotropic and the other six are anisotropic. 

Index of refraction Values are given for the three coordinate axes in the order of least, intermediate, and greatest index. For cubic crystals there is only a single value. See Reference 1 for details on the axis systems. Variations of several percent, depending on the origin and exact composition... 

Because the material properties are direction-dependent in a cubic crystal, they have to be stated together with the corresponding direction. According to the definition, the load direction has to be stated for Young s modulus Ei. Because the shear stress Tij and shear strain -y j have two indices, two indices are needed for the shear modulus Gij. Poisson s ratio relates strains in two directions. Here the second index j denotes the direction of the strain that causes the transversal contraction in the direction marked by the first index i eu = If the coordinate system is aligned with the axes... 

The simple method just described is applicable as it stands only to isotropic solids, that is, to glasses and amorphous solids in general, and to crystals belonging to the cubic system. In all other crystals the refractive index varies with the direction of vibration of the light in the crystal the optical phenomena are more complex, and it is necessary to disentangle them. 

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