Tensors for Physics

12.1 Integrals Over Unit Sphere 201

The orientational average of the corresponding multipole potential tensorsX···rather
than theY···-tensors is given by the right hand side of (12.5), multiplied byr−^2 (+^1 ).

12.1.2 Multiple Products of Irreducible Tensors

Integrals involving products of more than two irreducible tensors can also be evalu-
ated with the help of symmetry considerations and the contraction of tensors.
For instance, the orientational average of the fourfold product̂rμ̂rν̂rλ̂rκof com-
ponents of the unit vector must be proportional to the isotropic fourth rank tensor
with the appropriate symmetry, viz.δμνδλκ+δμλδνκ+δμκδλν. By analogy to the
consideration given above, one finds

1

4 π

∫

̂rμ̂rν̂rλ̂rκd^2 ̂r=

1

15

(δμνδλκ+δμλδνκ+δμκδλν). (12.6)

To check the numerical factor on the right hand side, putλequal toκand compare
with (12.3).
The integral of three irreducible second tensors

̂rμ̂rν ̂rλ̂rκ ̂rσ̂rτ ,

over the unit sphere must be proportional to the sixth rank isotropic tensorΔμν,λκ,σ τ,
defined by (11.36). The resulting equation is

1

4 π

∫

̂rμ̂rν ̂rλ̂rκ̂rσ̂rτd^2 ̂r=

8

105

Δμν,λκ,σ τ. (12.7)

The numerical factor 8/105 can be verified in the following exercise.
The integral of four irreducible second tensors

̂rμ 1 ̂rν 1 ̂rμ 2 ̂rν 2 ̂rμ 3 ̂rν 3 ̂rμ 4 ̂rν 4

can be found in the same spirit. Here, the result must be proportional to an isotropic
tensor of rank 8 with the required symmetry properties. There are two expressions
of this type, viz.
Δμ 1 ν 1 ,λκΔμ 2 ν 2 ,κσΔμ 3 ν 3 ,σ τΔμ 4 ν 4 ,τ λ,

which is a generalization of theΔ(^2 ,^2 ,^2 )-tensor, cf. Sect.11.4, and

Δμ 1 ν 1 ,μ 2 ν 2 Δμ 3 ν 3 ,μ 4 ν 4 +Δμ 1 ν 1 ,μ 3 ν 3 Δμ 2 ν 2 ,μ 4 ν 4 +Δμ 1 ν 1 ,μ 4 ν 4 Δμ 2 ν 2 ,μ 3 ν 3 ,

which is constructed by analogy to the fourth rank tensor occurring on the right hand
side of (12.6). Multiplication of these isotropic tensors withSμ 1 ν 1 Sμ 2 ν 2 Sμ 3 ν 3 Sμ 4 ν 4 ,