Principles of quantitative absorbance measurements in anisotropic crystals

The accurate measurement of absorbance (A=-log T; T=I/I_0) in anisotropic materials like crystals is highly important for the determination of the concentration and orientation of the oscillator (absorber) under investigation. The absorbance in isotropic material is linearly dependent on the concentration of the absorber and on the thickness of the sample (A=ɛ·c·t). Measurement of absorbance in anisotropic media is more complicated, but it can be obtained from polarized spectra (i) on three random, but orthogonal sections of a crystal, or (ii) preferably on two orthogonal sections oriented parallel to each of two axes of the indicatrix ellipsoid. To compare among different crystal classes (including cubic symmetry) it is useful to convert measured absorbance values to one common basis (the total absorbance A_tot), wherein all absorbers are corrected as if they were aligned parallel to the E-vector of the incident light. The total absorption coefficient (a_tot=A_tot/t) is calculated by (i)a_tot = ∑^(3)_(i=1)(a_(max,i)+a_(min,i))/2, or by (ii) a_tot = a_x + a_y + a_z. Only in special circumstances will unpolarized measurements of absorbance provide data useful for quantitative studies of anisotropic material. The orientation of the absorber with respect to the axes of the indicatrix ellipsoid is calculated according to A_x/A_tot=cos^2 (x < absorber), and analogously for A_y and A_z. In this way, correct angles are obtained for all cases of symmetry. The extinction ratio of the polarizer (Pe=I_crossed/I_parallel) has considerable influence on the measured amplitude of absorption bands, especially in cases of strong anisotropic absorbance. However, if Pe is known, the true absorbance values can be calculated even with polarizers of low extinction ratio, according to A max=−log[(T_(max,obs)−0.5·Pe·T_(min,obs))/(1−0.5·Pe)], and similar for A_min. The theoretical approach is confirmed by measurements on calcite and topaz