Operator tensor, compound irreducible

As an example, consider the product of two arbitrary first-rank tensor operators 0 and It is nine-dimensional and can be reduced to a sum of compound irreducible tensor operators of ranks 2, 1, and 0, respectively. Operators of this type play a role in spin-spin coupling Hamiltonians. In terms of spherical and Cartesian components of 0 and J2, the resulting irreducible tensors are given in Tables 8 and 9, respectively.70... 

The compound irreducible tensor operators of the second rank are constructed from the first-rank tensors (spherical vectors) as follows... 

Table 8 Irreducible Spherical Compound Tensor Operators Resulting from a Product of Two First-Rank Tensor Operators J T and. 9.T...

---