Dispersion relations for liquid crystals using the anisotropic Lorentz model with geometrical effects I. ABDULHALIM Department of Electrooptics Engineering, Ben Gurion University, P.O. Box 653, Beer Sheva 84105, Israel; e-mail: abdulhlm@bgu.ac.il (Received 24 February 2006; accepted 5 April 2006 ) The dispersion relations for the refraction indices and extinction coefficients of an ordered system of anisotropic molecules are derived, taking into account absorption near the resonance frequencies and the molecular geometrical form factor. The derivation is based on a combination of the anisotropic Lorentz oscillator dispersion model and the generalized Lorentz–Lorentz (LL) relationship. This relationship is shown to be consistent with the isotropic limit. The geometrical form tensor (GFT) distinguishes this relationship from the LL and the Calusius–Mossotti equations valid for isotropic media. Far from the absorption bands the dispersion relationships are shown to converge to those of Sellmeier-type dispersion relationships for transparent dielectrics that can be expanded into generalized Cauchy-type series. The principal values of the GFT are shown to cause a red shift to the resonance wavelengths that can be found from measurements in the disordered phase. Experimental data are presented based on published works as well as measurements using a spectroscopic retardation technique, and excellent agreement is obtained with the theoretical calculations on several LC materials. 1. Introduction Linear and nonlinear optical properties of liquid crystals (LCs) are a subject of growing interest, because of their strong electro-optic effects useful for a variety of applications as well as for the study of their physical phenomena [1]. Therefore it is of great importance to have accurate dispersion dependences of their refractive indices. Such relationships have been derived in the past by several investigators [2–9] and used for fitting to the experimental data [2, 9–12]. The derivation of the dispersion relationship relies strongly on the relation between the polarizability tensor and the dielectric tensor. Such relationships have to take into account the anisotropy of the constituent particles and their geometrical form as well as the damping factor of the absorption resonance peaks, in particular in the visible range of the spectrum, which is not far enough from the UV electronic absorption. Attempts to generalize the Lorentz–Lorentz (LL) or the Calusius–Mossotti (CM) equations to crystals taking into account the geometrical form tensor (GFT) in the depolarization fields existed since the first half of the 19th century [13–15], and some were considered for application to liquid crystals [16–22] in particular to explain the temperature dependence of their polarizability [22–25]. Neugebauer [14] obtained an equation similar to the LL equation: e q {1 e q z2 ~ 4pN 3 b q ð1Þ where the parameter b q was called the apparent polarizability and e q with q51, 2, 3 being the principal values of the dielectric tensor. Although Neugebauer took the crystal shape into account, he did not consider the anisotropic geometrical shape of the molecules or constituent particles. Saupe and Maier [16] have modified the Neugebauer relationship and obtained the following formulae: e q {1~4pNH q a q ð2Þ where H q 5(12NL q a q ) 21 are called the local field factors and L q are the internal field constants; N is the number density of molecules and a q the corresponding principal values of the polarizability tensor. Equation (2) may then be designated as the Neugebauer–Saupe–Maier or simply NSM relationship. Although equation (2) is in fact the most accurate within the mean field theory, it did not become popular because it did not show a good fit with experimental data. In an attempt to improve the fit between measured and calculated data for several molecular crystals, Vuks [15] derived a slightly modified form of equation (1). Liquid Crystals, Vol. 33, No. 9, September 2006, 1027–1041 Liquid Crystals ISSN 0267-8292 print/ISSN 1366-5855 online # 2006 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/02678290600804896