Tensors for Physics

204 12 Integral Formulae and Distribution Functions

The expansion tenors pertaining to= 1 , 2 ,3 and=4are

φμ=

√

3 uμ,φμν=

√

15

2

uμuν,φμνλ=

1

2

√

70 uμuνuλ,

φμνλκ=

3

4

√

70 uμuνuλuκ.

The first few terms off(u)= 41 π( 1 +aμφμ+aμνφμν+...)of the expansion can
also be written as

f(u)=

1

4 π

(

1 + 3 〈uμ〉uμ+

15

2

〈uμuν〉uμuν+...

)

. (12.18)

The second rank alignment tensoraμν∼〈uμuν〉has the same symmetry as the

quadrupole moment tensor, cf. (10.29). For this reason the tensor〈uμuν〉is also
denoted byQμνand referred to asQ-tensor, in the liquid crystal literature, [67].

12.2.3 Anisotropic Dielectric Tensor

The anisotropy of the dielectric tensor gives rise to birefringence, cf. Sect.5.3.4.In
fluids of optically anisotropic, uniaxial particles, the symmetric traceless part of the
dielectric tensor is proportional to the alignment tensor:

εμν =εaaμν. (12.19)

The proportionality coefficientεadepends on the densitynof the system and on the
differenceα‖−α⊥of the molecular polarizabilities parallel and perpendicular tou,
cf. Sect.5.3.3.
The electric polarizationPis the average of the induced dipole moment, which is
proportional to the electric field ‘felt’ by the particle. In dilute systems, this is equal
to the applied fieldE. Then one has

Pμ=ε 0 n〈αμν〉Eν

and consequently

εμν =n(α‖−α⊥)〈uμuν〉.

In dense systems, the particles ‘feel’ a local fieldEloc, modified by the surrounding
particles. The Lorentz field approximation

Eloc=ELor≡E+

1

3 ε 0

P,