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1: 34.4 Definition: 6 j Symbol
§34.4 Definition: 6 j Symbol
The 6 j symbol is defined by the following double sum of products of 3 j symbols: …
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 … where F 3 4 is defined as in §16.2. …
2: 34.14 Tables
§34.14 Tables
Tables of 3 j and 6 j symbols in which all parameters are 17 / 2 are given in Appel (1968) to 6D. … 11-12; for 6 j symbols on pp. … Biedenharn and Louck (1981) give tables of algebraic expressions for Clebsch–Gordan coefficients and 6 j symbols, together with a bibliography of tables produced prior to 1975. … 270–289; similar tables for the 6 j symbols are given on pp. …
3: 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).
4: 34.10 Zeros
§34.10 Zeros
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. …
5: 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, … 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).
6: 34 3j, 6j, 9j Symbols
Chapter 34 3 j , 6 j , 9 j Symbols
7: 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).
8: 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). …
9: 34.5 Basic Properties: 6 j Symbol
§34.5 Basic Properties: 6 j Symbol
§34.5(ii) Symmetry
Additional symmetries are obtained by applying (34.5.8) to (34.5.9) and (34.5.10). …
§34.5(iv) Orthogonality
They constitute addition theorems for the 6 j symbol. …
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. …
34.8.1 { j 1 j 2 j 3 j 2 j 1 l 3 } = ( 1 ) j 1 + j 2 + j 3 + l 3 ( 4 π ( 2 j 1 + 1 ) ( 2 j 2 + 1 ) ( 2 l 3 + 1 ) sin θ ) 1 2 ( cos ( ( l 3 + 1 2 ) θ 1 4 π ) + o ( 1 ) ) , j 1 , j 2 , j 3 l 3 1 ,
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.