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1: 34.2 Definition: 3 j Symbol
§34.2 Definition: 3 j Symbol
The quantities j 1 , j 2 , j 3 in the 3 j symbol are called angular momenta. …They therefore satisfy the triangle conditions …The corresponding projective quantum numbers m 1 , m 2 , m 3 are given by … When both conditions are satisfied the 3 j symbol can be expressed as the finite sum …
2: 34.10 Zeros
§34.10 Zeros
In a 3 j symbol, if the three angular momenta j 1 , j 2 , j 3 do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the 3 j symbol is zero. Similarly the 6 j symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four 3 j symbols in the summation. …However, the 3 j and 6 j symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. …
3: 34.3 Basic Properties: 3 j Symbol
§34.3 Basic Properties: 3 j Symbol
§34.3(ii) Symmetry
§34.3(iv) Orthogonality
§34.3(vi) Sums
4: 34.14 Tables
§34.14 Tables
Tables of exact values of the squares of the 3 j and 6 j symbols in which all parameters are 8 are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of 3 j , 6 j , and 9 j symbols on pp. … Tables of 3 j and 6 j symbols in which all parameters are 17 / 2 are given in Appel (1968) to 6D. …Other tabulations for 3 j symbols are listed on pp. …
5: 34.13 Methods of Computation
§34.13 Methods of Computation
Methods of computation for 3 j and 6 j symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). …
6: 34.9 Graphical Method
§34.9 Graphical Method
For specific examples of the graphical method of representing sums involving the 3 j , 6 j , and 9 j symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).
7: 34.1 Special Notation
2 j 1 , 2 j 2 , 2 j 3 , 2 l 1 , 2 l 2 , 2 l 3 nonnegative integers.
The main functions treated in this chapter are the Wigner 3 j , 6 j , 9 j symbols, respectively, … An often used alternative to the 3 j symbol is the Clebsch–Gordan coefficient
34.1.1 ( j 1 m 1 j 2 m 2 | j 1 j 2 j 3 m 3 ) = ( 1 ) j 1 j 2 + m 3 ( 2 j 3 + 1 ) 1 2 ( j 1 j 2 j 3 m 1 m 2 m 3 ) ;
For other notations for 3 j , 6 j , 9 j symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).
8: 34.12 Physical Applications
§34.12 Physical Applications
The angular momentum coupling coefficients ( 3 j , 6 j , and 9 j symbols) are essential in the fields of nuclear, atomic, and molecular physics. … 3 j , 6 j , and 9 j symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).
9: 34.6 Definition: 9 j Symbol
The 9 j symbol may be defined either in terms of 3 j symbols or equivalently in terms of 6 j symbols:
34.6.1 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = all  m r s ( j 11 j 12 j 13 m 11 m 12 m 13 ) ( j 21 j 22 j 23 m 21 m 22 m 23 ) ( j 31 j 32 j 33 m 31 m 32 m 33 ) ( j 11 j 21 j 31 m 11 m 21 m 31 ) ( j 12 j 22 j 32 m 12 m 22 m 32 ) ( j 13 j 23 j 33 m 13 m 23 m 33 ) ,
10: 34.8 Approximations for Large Parameters
§34.8 Approximations for Large Parameters
For large values of the parameters in the 3 j , 6 j , and 9 j symbols, different asymptotic forms are obtained depending on which parameters are large. … Uniform approximations in terms of Airy functions for the 3 j and 6 j symbols are given in Schulten and Gordon (1975b). For approximations for the 3 j , 6 j , and 9 j symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.