Tensors for Physics

12.2 Orientational Distribution Function 207

The coefficientc(^2 |^11 )defined in (12.24) is given by

c(^2 |^11 )=〈P 2 (x)〉/〈P 1 (x)〉^2 =

3

2

(

〈x^2 〉−

1

3

)

/〈x〉^2. (12.28)

The high-temperature approximation is

c(^2 |^11 )= 3 / 5 + 4 z^2 / 175 + 4 z^4 / 2625 +..., (12.29)

forz→∞one hasc(^2 |^11 )→1.
Next the case is considered where the orientational energy involves the scalar
product of the symmetric traceless tensor fieldFμνand the tensorφμν=

√

15 / 2

uμuμ,viz.

H=H(^2 )∼−Fμνuμuμ, −H(^2 )/kBT=β 2 Fμνφμν. (12.30)

Interactions of this type occur for linear molecules with anisotropic polarizability in
the presence of an electric fieldE, cf. Sect.5.3.3, and for particles with an electric
quadrupole moment in the presence of an electric field gradient∇E. In the first case,

the field tensorFμν=EμEν is uniaxial, in the second case, whereFμν= ∇μEν
applies, the field tensor is biaxial, unless the gradient is parallel to theE-field.
The distribution function pertaining to (12.30)is

feq=Z−^1 exp[β 2 Fμνφμν]

=Z−^1

(

1 +β 2 Fμνφμν+

1

2

β 22 FμνFλκφμνφλκ+...

)

,

with

Z=( 4 π)−^1

(

1 +

1

2

β^22 FμνFμν+z(^3 )+z(^4 )+...

)

.

The third and fourth order terms are

z(^3 )=

1

6

β 23 FμνFλκFστ〈φμνφλκφστ〉 0 ,

z(^4 )=

1

24

β 24 FμνFλκFστFαβ〈φμνφλκφστφαβ〉 0.

The relation (12.7) implies

FμνFλκFστ〈φμνφλκφστ〉 0 =

√

30

2

7

FμνFνκFκμ