Tensors for Physics

12.2 Orientational Distribution Function 203

whered^2 uis the surface element on the unit sphere, the average〈ψ〉of an angle
dependent quantityψ=ψ(u), is given by

〈ψ〉=

∫

ψ(u)f(u)d^2 u. (12.12)

In an isotropic state, where no preferential direction exists, the orientational distrib-
ution functionf 0 does not depend onu. Due to the normalization condition one has
f 0 =( 4 π)−^1. Averages over this isotropic distribution are denoted by〈···〉 0 ,viz.

〈ψ〉 0 =

1

4 π

∫

ψ(u)d^2 u. (12.13)

12.2.2 Expansion with Respect to Irreducible Tensors

In general, the orientational distribution functionf(u)can be expanded in terms of
irreducible tensors constructed from the components ofu. It is convenient to use the
tensors

φμ 1 μ 2 ···μ≡

√

( 2 + 1 )!!

!

uμ 1 uμ 2 ···uμ. (12.14)

The basis functionsφ...are orthogonal and normalized according to

1

4 π

∫

φμ 1 μ 2 ···μφν 1 ν 2 ···ν′d^2 u=〈φμ 1 μ 2 ···μφν 1 ν 2 ···ν′〉 0 =δ′Δ()mu 1 μ 2 ···μ,ν 1 ν 2 ···ν,
(12.15)
cf. (12.1). Then the expansion reads

f(u)=f 0 ( 1 +Φ), f 0 =( 4 π)−^1 ,Φ=

∑∞

= 1

aμ 1 μ 2 ···μφμ 1 μ 2 ···μ. (12.16)

Clearly,Φis the deviation offfrom the isotropic distributionf 0. The expansion
coefficientsa...are the moments of the distribution function, viz.

aμ 1 μ 2 ···μ=

∫

φμ 1 μ 2 ···μf(u)d^2 u≡〈φμ 1 μ 2 ···μ〉. (12.17)

These quantities are referred to asalignment tensorsororder parameter tensors.In
general, they may depend on the timetand the positionrin space.