12.2 Orientational Distribution Function 205
leads to
εμν=n(
εiso+ 2
3) 2
〈αμν〉+... ,whereεisois the orientationally averaged dielectric coefficient. This result is equiv-
alent to (12.19) with
εa=n(
εiso+ 2
3) 2
(α‖−α⊥)ζ 2 −^1 , (12.20)whereζ 2 =
√
15
2 is the normalization factor occurring in the definition of the align-
ment tensor.
12.2.4 Field-Induced Orientation.
In the presence of electric or magnetic fields, the energyHof a particle with per-
manent or induced electric or magnetic dipole moments depends on its orientation
relative to the direction of the applied field. For a uniaxial particle, one hasH=H(u).
In thermal equilibrium, the orientational distribution functionf=feqis proportional
to exp[−H/kBT]=exp[−βH]:
feq=Z−^1 exp[−βH(u)], Z=∫
exp[−βH]d^2 u,β=1
kBT. (12.21)
It is assumed that〈H〉 0 =0 where〈...〉 0 =( 4 π)−^1
∫
...d^2 uindicates the unbiased
orientational average in an isotropic state. Then thehigh temperature expansionfor
the state functionZreads
Z=( 4 π)(
1 +
1
2
β^2 〈H^2 〉 0 −1
6
β^3 〈H^3 〉 0 +1
24
β^4 〈H^4 〉 0 ∓...)
.
First, the interaction of an electric or magnetic dipole moment is considered, which
is parallel touand subjected to an electric or magnetic field, which is denoted byF.
ThenHis equal to
H=H(^1 )≡Hdip=−dFμuμ,wheredstands for the magnitude of the relevant dipole moment. In this case,−βH
is written asβ 1 Fμφμwithφμ=
√
3 uμandβ 1 =βd/√
3, andZ, in lowest order in
β 1 , reduces to
Z=( 4 π)( 1 +1
2
β^21 FμFμ+...).