Clebsch-Gordan coefficient, triangular

The expansion parameters that appear in Eqs. 4.7 and 4.8 as arguments of the Clebsch-Gordan coefficients satisfy the triangular relationship,... 

In angular momentum theory a very important role is played by the invariants obtained while summing the products of the Wigner (or Clebsch-Gordan) coefficients over all projection parameters. Such quantities are called 7-coefficients or 3ny-coefficients. They are invariant under rotations of the coordinate system. A j-coefficient has 3n parameters (n = 1,2,3,...), that is why the notation 3nj-coefficient is widely used. The value n = 1 leads to the trivial case of the triangular condition abc, defined in Chapter 5 after formula (5.25). For n = 2,3,4,... we have 67 -, 9j-, 12j-,. .. coefficients, respectively. 3nj-coefficients (n > 2) may be also defined as sums of 67-coefficients. There are also algebraic expressions for 3nj-coefficients. Thus, 6j-coefficient may be defined by the formula... 

Performing the summation of the products of Clebsch-Gordan coefficients over all projection parameters, we obtain the invariants, called 3n/-coefficients. Their trivial cases were presented in Figs 8.6b-e (Kro-necker and triangular deltas). The first non-trivial case is represented by formula (6.16) or the transformation matrix of three angular momenta, connected with 6/-coefficient (see formula (6.42) and Fig. 8.7)). 

Selection rules for Ek-radiation in the case of one-electron configurations consist of the triangle conditions, which the parameters j j2 and k in the Clebsch-Gordan coefficients must satisfy. The triangular condition il l2k) with the extra requirement that h + h + k be even takes care of the selection rule with respect to the parities of the configurations. 

The numerical value of the Clebsch-Gordan coefficient is zero unless the triangular condition A(J k J) is satisfied. Thus we have... 

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