206 12 Integral Formulae and Distribution Functions
Likewise, up to second order inβ 1 , the distribution function is equal to
feq=( 4 π)−^1
(
1 +β 1 Fμφμ+
√
3
10
β^21 FμFνφμν+...
)
. (12.22)
In this high temperature approximation, the first two moments are given by
aμ=〈φμ〉=β 1 Fμ, aμν=〈φμν〉=
√
3
10
β 12 FμFν =
√
3
10
aμaν. (12.23)
These relations imply
〈uμuν〉=c(^2 |^11 )〈uμ〉〈uν〉. (12.24)
The numerical factorc( 2 | 11 )is equal to 3/5 in the high temperature approximation.
For a very strong orienting field which induces a practically perfect orientation,
c(^2 |^11 )=c(^2 |^11 )(β 1 F)approaches 1.
In the case of a dipolar orientation, the distribution and consequently all its
moments have uniaxial symmetry. Letebe a unit vector parallel to the fieldF,
viz.Fμ=Feμ, whereFis the strength of the field. Then the energy can be writ-
ten asHdip=−dFeμuμ =−dFx, wherex =eμuμ|is the cosine of the angle
betweenuand the direction of the field. Furthermore, one haseμφμ=
√
3 xand
−Hdip/kBT=
√
3 β 1 Fx. The average〈xn〉is evaluated according to
〈xn〉=
1
2
Z−^1
∫ 1
− 1
xnexp
[√
3 β 1 Fx
]
dx,
with the state functionZgiven by
Z=L
(√
3 β 1 F
)
, L(z)=
1
2 z
(exp[z]−exp[−z])= 1 +z^2 / 6 +z^4 / 120 +....
(12.25)
LetLkbe thek-derivative ofL(z)with respect toz,viz.Lk=dkL/dzk. The average
〈xk〉is then determined byLk(z)/L(z), withz=
√
3 β 1 F=βdF. The results for the
lowest moments are
〈x〉=
1 +z+exp[ 2 z](z− 1 )
zexp[ 2 z]−z
=z/ 3 −z^3 / 45 ±... , (12.26)
〈x^2 〉−
1
3
=
1 +z+exp[ 2 z](z− 1 )
zexp[ 2 z]−z
−
1
3
= 2 z^2 / 45 − 4 z^4 / 945 ±...(12.27)
Forz→∞, both〈x〉and〈x^2 〉approach 1.