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34 3j,6j,9j SymbolsProperties

§34.4 Definition: 6j Symbol

The 6j symbol is defined by the following double sum of products of 3j symbols:

34.4.1 {j1j2j3l1l2l3}=mrms(-1)l1+m1+l2+m2+l3+m3(j1j2j3m1m2m3)(j1l2l3m1m2-m3)(l1j2l3-m1m2m3)(l1l2j3m1-m2m3),

where the summation is taken over all admissible values of the m’s and m’s for each of the four 3j symbols; compare (34.2.2) and (34.2.3).

Except in degenerate cases the combination of the triangle inequalities for the four 3j symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths j1,j2,j3,l1,l2,l3; see Figure 34.4.1.

See accompanying text
Figure 34.4.1: Tetrahedron corresponding to 6j symbol. Magnify

The 6j symbol can be expressed as the finite sum

34.4.2 {j1j2j3l1l2l3}=Δ(j1j2j3)Δ(j1l2l3)Δ(l1j2l3)Δ(l1l2j3)×s(-1)s(s+1)!(s-j1-j2-j3)!(s-j1-l2-l3)!(s-l1-j2-l3)!(s-l1-l2-j3)!×1(j1+j2+l1+l2-s)!(j2+j3+l2+l3-s)!(j3+j1+l3+l1-s)!,

where the summation is over all nonnegative integers s such that the arguments in the factorials are nonnegative.

Equivalently,

34.4.3 {j1j2j3l1l2l3}=(-1)j1+j3+l1+l3Δ(j1j2j3)Δ(j2l1l3)(j1-j2+l1+l2)!(-j2+j3+l2+l3)!(j1+j3+l1+l3+1)!Δ(j1l2l3)Δ(j3l1l2)(j1-j2+j3)!(-j2+l1+l3)!(j1+l2+l3+1)!(j3+l1+l2+1)!×F34(-j1+j2-j3,j2-l1-l3,-j1-l2-l3-1,-j3-l1-l2-1-j1+j2-l1-l2,j2-j3-l2-l3,-j1-j3-l1-l3-1;1),

where F34 is defined as in §16.2.

For alternative expressions for the 6j symbol, written either as a finite sum or as other terminating generalized hypergeometric series F34 of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).