Liquid Crystals. Iam-Choon Khoo

(3.33)

More generally, one can write Eq. (3.33) as

(3.34)

where , the local field “factor,” is a second‐rank tensor, which states that the local field is linearly related to the macroscopic applied field . In general, experimental measurements show that the treatments by Vuks and Dunmar are qualitatively in agreement.

      3.4.2. Equilibrium Temperature and Order Parameter Dependences of Refractive Indices

The two principal refractive indices n_⊥ and n_|| of a uniaxial liquid crystal and the anisotropy n_|| − n_⊥ have been the subject of intensive studies for their fundamental importance in the understanding of liquid crystal physics and for their vital roles in applied electro‐optic devices. Since the dielectric constants (ε_⊥ and ε_||) enter directly and linearly into the constitutive equations (Eqs. (3.30a)–(3.30c)), it is theoretically more convenient to discuss the fundamentals of these temperature dependences in terms of the dielectric constants.

From Eq. (3.34) for the local field and Eq. (3.31) for the induced dipole moments, we can express the polarization by

      (3.35)

      where is the polarizability tensor of the molecule, N is the number of molecules per unit volume, and the parentheses denote averaging over the orientations of all molecules.

      The dielectric constant (in units of ε₀) is therefore given by

      (3.36)

      and

      (3.37)

      From these considerations and from observations by deJeu and Bordewijk [13] that

      (3.38)

and

      (3.39)

      we can write ε_|| and ε_⊥ as

(3.40)

      and

(3.41)

      respectively, where K_l and K_t are the values of