Tensors for Physics

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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+...).
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