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34 3j, 6j, 9j SymbolsProperties

§34.4 Definition: 6⁒j Symbol

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

34.4.1 {j1j2j3l1l2l3}=βˆ‘mr⁒msβ€²(βˆ’1)l1+m1β€²+l2+m2β€²+l3+m3′⁒(j1j2j3m1m2m3)⁒(j1l2l3m1m2β€²βˆ’m3β€²)⁒(l1j2l3βˆ’m1β€²m2m3β€²)⁒(l1l2j3m1β€²βˆ’m2β€²m3),

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

Except in degenerate cases the combination of the triangle inequalities for the four 3⁒j 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 6⁒j symbol. Magnify

The 6⁒j symbol can be expressed as the finite sum

34.4.2 {j1j2j3l1l2l3}=Δ⁑(j1⁒j2⁒j3)⁒Δ⁑(j1⁒l2⁒l3)⁒Δ⁑(l1⁒j2⁒l3)⁒Δ⁑(l1⁒l2⁒j3)Γ—βˆ‘s(βˆ’1)s⁒(s+1)!(sβˆ’j1βˆ’j2βˆ’j3)!⁒(sβˆ’j1βˆ’l2βˆ’l3)!⁒(sβˆ’l1βˆ’j2βˆ’l3)!⁒(sβˆ’l1βˆ’l2βˆ’j3)!Γ—1(j1+j2+l1+l2βˆ’s)!⁒(j2+j3+l2+l3βˆ’s)!⁒(j3+j1+l3+l1βˆ’s)!,

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


34.4.3 {j1j2j3l1l2l3}=(βˆ’1)j1+j3+l1+l3⁒Δ⁑(j1⁒j2⁒j3)⁒Δ⁑(j2⁒l1⁒l3)⁒(j1βˆ’j2+l1+l2)!⁒(βˆ’j2+j3+l2+l3)!⁒(j1+j3+l1+l3+1)!Δ⁑(j1⁒l2⁒l3)⁒Δ⁑(j3⁒l1⁒l2)⁒(j1βˆ’j2+j3)!⁒(βˆ’j2+l1+l3)!⁒(j1+l2+l3+1)!⁒(j3+l1+l2+1)!Γ—F34⁑(βˆ’j1+j2βˆ’j3,j2βˆ’l1βˆ’l3,βˆ’j1βˆ’l2βˆ’l3βˆ’1,βˆ’j3βˆ’l1βˆ’l2βˆ’1βˆ’j1+j2βˆ’l1βˆ’l2,j2βˆ’j3βˆ’l2βˆ’l3,βˆ’j1βˆ’j3βˆ’l1βˆ’l3βˆ’1;1),

where F34 is defined as in Β§16.2.

For alternative expressions for the 6⁒j 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).