12.2 Orientational Distribution Function 209
Phenomenologically, theelectro-optic Kerr effectis described byεμν = 2 KEμEν, (12.35)where theKerr coefficient Kquantifies this effect. Due to (12.35), the difference
δν=ν‖−ν⊥of the indices of refraction of the optical electric field parallel and
perpendicular to the applied electric fieldEis determined byν‖^2 −ν^2 ⊥= 2 KE^2 =
(ν‖+ν⊥)δν. Thus, with average index of refractionν ̄=(ν‖+ν⊥)/2, one has
νδν ̄ =KE^2.In the corresponding relations for the Cotton-Mouton effect, the applied electric field
EandKare replaced by the magnetic field and another characteristic coefficient.
Sometimes,K/ν ̄^2 is referred to as Kerr coefficient. The term ‘electro-optic Kerr
effect’ is used in order to distinguish this effect from another one associated with
Kerr,viz.themagneto-optic Kerr effect.
Several mechanisms contribute to these effects. Firstly, strong fields influence
the electronic structure and change the ‘shape’ of atoms and molecules. Secondly,
the field-induced orientation of optically anisotropic particles, in fluids, gives rise
to birefringence. While the first contribution is independent of the temperatureT,
the second one, in general, contains contribution proportional toT−^1 andT−^2 ,in
the low-field and high-temperature limit. The key is the relation (12.19) between the
symmetric traceless part of the dielectric tensor and the alignment tensor, viz.εμν=
εaaμν,for=εasee (12.20). In the high temperature approximation, permanent
dipole moments yield a contribution proportional toT−^2 ,cf.(12.23). Induced dipole
moments linked with an anisotropic polarizability lead to a contribution toKwhich
is proportional toT−^1 , as given by the first term on the right hand side of (12.33).
In (12.35) it is understood, thatEis an applied electric field which is to be
distinguished from the electric fieldElightof the light. When the optical electric field
Elightis strong enough and denoted byE, the Kerr effect gives rise to a nonlinear
susceptibility, cf. (2.59), where the third order contribution to the electric polarization
isP(^3 )=ε 0 χμνλκ(^3 ) EνEλEκwith the susceptibility tensorχμνλκ(^3 ) = 2 KΔμνλκ.
12.2.6 Orientational Entropy
The entropy of a system in an ordered state is lower than that in an isotropic state.
The entropy, per particle, associated with the orientation, characterized by the orien-
tational distribution functionf(u), is determined by the Boltzmann like expression
−kB
∫
flnfd^2 u. This should be compared with the corresponding expression for the
isotropic distributionf 0 ,viz.s 0 =−kB
∫
f 0 lnf 0 d^2 u=−kB∫
flnf 0 d^2 u. The sec-
ond equality follows from the fact that lnf 0 is a constant and that the normalization
imposes
∫
fd^2 u=∫
f 0 d^2 u=1. Thus the difference between the entropy, per particle,