DERIVATION OF THE INERTIAL AC RESPONSE FOR THE KERR-EFFECT RELAXATIONFROM THE LANGEVIN EQUATION IN 3-DIMENSIONAL SPACE

The Kerr-effect response of an assembly of noninteracting dipolar and anisotropically polarizable spherical molecules is calculated from the Euler-Langevin stochastic differential equations. The solution is obt ained using combinations of the Legendre polynomials and the associate d Legendre functions as angular variables, together with the Hermite p olynomials as angular velocity variables. The molecules are compelled to rotate in three-dimensional space and are acted on by a de bias fie ld, superimposed on which is an ac electric field in the same directio n. When the response is restricted to second order in the applied fiel d, the coupling of the fields gives rise to two distinct nonlinear har monic components of the complex birefringence, varying at the fundamen tal (omega) and the second harmonic (2 omega) of the ac field. Small i nertial effects are considered, and their importance is illustrated in numerous dispersion and Cole-Cole plots for various values of the par ameter P measuring the balance between induced and permanent dipole mo ments and a fixed value of the inertial parameter gamma. Special empha sis is placed on the phase angles between in-phase and out-of-phase ha rmonic terms whose values may be multiplied by a factor of 2 at high f requencies, compared to those obtained in the rotational diffusion lim it (Debye's model), thus allowing possible practical applications. The transition matrix that is appropriate to dielectric relaxation is als o given, since a knowledge of it in the nonlinear case, unlike the aft ereffect solution, is needed for the description of the dynamic Kerr e ffect.