Tables for
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.2, p. 740

Section Refraction

H. Klappera and Th. Hahna Refraction

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The dependence of the refractive index on the vibration direction of a plane-polarized light wave travelling through the crystal can be obtained from the optical indicatrix. This surface is an ellipsoid, which can degenerate into a rotation ellipsoid or even into a sphere. Thus, the lowest symmetry of the property `refraction' is 2/m 2/m 2/m, the point group of the general ellipsoid. According to the three different forms of the indicatrix, three categories of crystal systems have to be distinguished (Table[link]).

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Categories of crystal systems distinguished according to the different forms of the indicatrix

Crystal systemShape of indicatrixOptical character
Cubic Sphere Isotropic (not doubly refracting)
[\!\left.\matrix{\hbox{Tetragonal}\hfill\cr \hbox{Trigonal}\hfill\cr \hbox{Hexagonal}\hfill\cr}\right\}\hfill] Rotation ellipsoid [\!\left.\matrix{\hbox{Uniaxial}\hfill\cr \noalign{\vskip 2pt}\cr \hfill\cr \hbox{Biaxial}\hfill\cr}\right\}\matrix{\hbox{Anisotropic}\hfill\cr\quad\hbox{(doubly}\hfill\cr\quad\hbox{refracting)}\hfill\cr}]
[\!\left.\matrix{\hbox{Orthorhombic}\hfill\cr\hbox{Monoclinic}\hfill\cr\hbox{Triclinic}\hfill\cr}\right\}] General ellipsoid

The orientation of the indicatrix is related to the symmetry directions of the crystal. In tetragonal, trigonal and hexagonal crystals, the rotation axis of the indicatrix (which is the unique optic axis) is parallel to the main symmetry axis. For ortho­rhombic crystals, the three principal axes of the indicatrix are oriented parallel to the three symmetry directions of the crystal. In the monoclinic system, one of the axes of the indicatrix coincides with the monoclinic symmetry direction, whereas in the triclinic case, the indicatrix can, in principle, have any orientation relative to a chosen reference system. Thus, in triclinic and, with restrictions, in monoclinic crystals, the orientation of the indicatrix can change with wavelength λ and temperature T (orientation dispersion). In any system, the size of the indicatrix and, in all but the cubic system, its shape can also vary with λ and T.

When studying the symmetry of a crystal by optical means, note that strain can lower the apparent symmetry owing to the high sensitivity of optical properties to strain.

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