11.4 Isotropic Coupling Tensors 191
11.4 Isotropic Coupling Tensors
11.4.1 Definition ofΔð‘;^2 ;‘Þ-Tensors
Isotropic tensorsΔ(···^3 ,···,^2 ,,···^1 )can be constructed fromδ-tensors and zero or oneε-
tensor, such that they generate an irreducible tensor of rank 3 from the product of
irreducible tensors of ranks 1 and 2. These tensors are also referred to asClebsch-
Gordan tensors, [16, 17].
Special cases of coupling tensors, discussed so far, are theΔ()···,···-tensors of
Sect.11.2, and the()···,···-tensors of Sect.11.3, pertaining to 3 = 1 ≡, 2 = 0
and 3 = 1 ≡, 2 =1, respectively. The next and only other case to be considered
here corresponds to 3 = 1 ≡, 2 =2. These tensors are defined by
Δμ(, 12 μ,) 2 ···μ,λκ,μ′
1 μ
′
2 ···μ
′
=Δ()μ 1 μ 2 ···μ,ν 1 ν 2 ···ν− 1 σΔ(στ,λκ^2 ) Δ()τν 1 ν 2 ···ν− 1 ,μ′
1 μ
′
2 ···μ
′
.
(11.34)
LetSandAbe irreducible tensors of rank 2 and. The tensorΔ(,^2 ,)accomplishes
their multiplicative coupling of these tensors to a tensor of rank. This can be
expressed as
(S·A)μ 1 μ 2 ···μ≡ Sμ 1 λAλμ 2 ···μ =Δ(,μ^2 ,)
1 μ 2 ···μ,λκ,μ′ 1 μ′ 2 ···μ′
SλκAμ′ 1 μ′ 2 ···μ′.
(11.35)
For=1, the expression (11.34) reduces to
Δ(μ,λκ,ν^1 ,^2 ,^1 )=Δ(μν,λκ^2 ) ≡Δμν,λκ.Of particular interest is the case=2. Here one has
Δ(μν,λκ,αβ^2 ,^2 ,^2 ) ≡Δμν,λκ,αβ=Δμν,μ′ν′Δμ′λ′,λκΔν′λ′,αβ. (11.36)This tensor is symmetric against the interchange of any pair of subscripts, e.g.
Δμν,λκ,αβ=Δμν,αβ,λκ=Δλκ,μν,αβ. (11.37)Further properties are:
Δμν,λκ,κσ=7
12
Δμν,λσ,Δμν,λκ,λκ= 0 ,Δμν,νκ,κμ=