The Helmholtz dispersion equations have been rewritten in a form that enables the optical constants of both transparent and opaque media to be calculated from their spectra. Both Helmholtz equations are used to describe the optical properties of opaque media, and to obtain values of reflectance, refractive index, and absorption coefficient. The Sellmeier dispersion equation is a special case of the dispersive Helmholtz equation applicable to weakly absorbing media (including the great majority of minerals studied in thin section): it is used to derive the wavelength-, composition-, direction-, and volume-dependence of the principal indices in mixed crystals of monoclinic or higher symmetry. The treatment can be extended to triclinic crystals.
The birefringence of transparent phases is the expression in the visible region of the pleochroism of absorption bands in the ultra-violet. The bireflectance of opaque phases depends also upon the pleochroism of bands in the visible and near infra-red, resulting in extreme sensitivity of the optics of opaque materials to changes in wavelength, composition, and structure. The optical anisotropy of both transparent and opaque phases may be calculated if the dependence of the spectra on structure can be established by measurement or by calculation from structure data. The quantitative application of Bragg's method is restricted to phases of very simple chemistry and structure (e.g. calcite and rutile), but it may be applied qualitatively to rationalize the optic orientations of phases containing only closed-shell ions of neon or argon configuration, including many pyroxenes, amphiboles, micas, and chain aluminosilicates. When open-shell (transition metal) ions enter the structure, the general rule is that the anisotropy becomes more closely related to the distortion of the coordination polyhedron about the metal ion, and for ions of formal charge ≥ 3, this source of anisotropy is usually dominant.