Tensors for Physics

208 12 Integral Formulae and Distribution Functions

and similarly, from (12.10) follows

FμνFλκFστFαβ〈φμνφλκφστφαβ〉 0 =

15

7

(FμνFμν)^2.

Thus up to terms of fourth order in the field tensor, the partition functionZis

Z=( 4 π)−^1

(

1 +

1

2

β 22 FμνFμν+

1

27

√

30 β^32 FμνFνκFκμ+

5

56

β 24 (FμνFμν)^2 +...

)

.

(12.31)

ForHgivenby(12.30), the equilibrium averageaμν=〈φμν〉can also be evaluated
according to

aμν=β− 21 Z−^1

∂Z

∂Fμν

=β− 21

∂lnZ

∂Fμν

. (12.32)

With the help of (11.59) andφμνλκ=^34

√

70 uμuνuλuκ, the second order term in
the expression forfeqis rewritten as

FμνFλκφμνφλκ=

15

2

FμνFλκ

(

3

4

√

70

)− 1

φμνλκ+

4

7

√

15

2

FμλFλνφμν+FμνFμν.

Thus in high temperature approximation, up to second order inβ 2 , the second and
fourth rank alignment tensors are given by

aμν=β 2 Fμν+

1

7

√

30 β 22 FμλFλν, aμνλκ=

1

2

√

10

7

β 22 FμνFλκ. (12.33)

A relation similar to (12.24) links the fourth rank alignment tensor with the product

of two second rank alignment tensors, viz.aμνλκ∼aμνaλκ and

〈uμuνuλuκ〉=c(^4 |^22 )〈uμuν〉〈uλuκ〉. (12.34)

The coefficientc(^4 |^22 )depends on the field strength. The high temperature approxi-
mation isc(^4 |^22 )= 5 /7.

12.2.5 Kerr Effect, Cotton-Mouton Effect, Non-linear

Susceptibility

The birefringence induced by an applied electric field or by a magnetic field are
calledKerr effect, named after J. Kerr, andCotton-Mouton effect, named after A.
Cotton and H. Mouton, who discovered these effects in 1875 and 1907, respectively.