# §34.2 Definition: $3j$ Symbol

The quantities $j_{1},j_{2},j_{3}$ in the $3j$ symbol are called angular momenta. Either all of them are nonnegative integers, or one is a nonnegative integer and the other two are half-odd positive integers. They must form the sides of a triangle (possibly degenerate). They therefore satisfy the triangle conditions

 34.2.1 $|j_{r}-j_{s}|\leq j_{t}\leq j_{r}+j_{s},$ Symbols: $j,j_{r}$: nonnegative integer, $r\in 1,2,3$, $t\in 1,2,3$ and $s$: nonnegative integer Referenced by: §34.10, §34.2, §34.3(iv) Permalink: http://dlmf.nist.gov/34.2.E1 Encodings: TeX, pMML, png

where $r,s,t$ is any permutation of $1,2,3$. The corresponding projective quantum numbers $m_{1},m_{2},m_{3}$ are given by

 34.2.2 $m_{r}=-j_{r},-j_{r}+1,\dots,j_{r}-1,j_{r},$ $r=1,2,3$, Symbols: $j,j_{r}$: nonnegative integer and $r\in 1,2,3$ Referenced by: §34.4 Permalink: http://dlmf.nist.gov/34.2.E2 Encodings: TeX, pMML, png

and satisfy

 34.2.3 $m_{1}+m_{2}+m_{3}=0.$ Referenced by: §34.10, §34.2, §34.3(iv), §34.4 Permalink: http://dlmf.nist.gov/34.2.E3 Encodings: TeX, pMML, png

See Figure 34.2.1 for a schematic representation.

If either of the conditions (34.2.1) or (34.2.3) is not satisfied, then the $3j$ symbol is zero. When both conditions are satisfied the $3j$ symbol can be expressed as the finite sum

 34.2.4 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}={(-1)^{j_{1}-j_{2}-m_{3}}}\Delta(j_{1}j_{2}j_{3% })\left((j_{1}+m_{1})!(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!(j_{3}+m_{3})!% (j_{3}-m_{3})!\right)^{\frac{1}{2}}\*\sum_{s}\frac{(-1)^{s}}{s!(j_{1}+j_{2}-j_% {3}-s)!(j_{1}-m_{1}-s)!(j_{2}+m_{2}-s)!(j_{3}-j_{2}+m_{1}+s)!(j_{3}-j_{1}-m_{2% }+s)!},$ Defines: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$: $3j$ symbol Symbols: $!$: $n!$: factorial, $j,j_{r}$: nonnegative integer, $\Delta(j_{1}j_{2}j_{3})$: product and $s$: nonnegative integer Permalink: http://dlmf.nist.gov/34.2.E4 Encodings: TeX, pMML, png

where

 34.2.5 $\Delta(j_{1}j_{2}j_{3})=\left(\frac{(j_{1}+j_{2}-j_{3})!(j_{1}-j_{2}+j_{3})!(-% j_{1}+j_{2}+j_{3})!}{(j_{1}+j_{2}+j_{3}+1)!}\right)^{\frac{1}{2}},$ Symbols: $!$: $n!$: factorial, $j,j_{r}$: nonnegative integer and $\Delta(j_{1}j_{2}j_{3})$: product Permalink: http://dlmf.nist.gov/34.2.E5 Encodings: TeX, pMML, png

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

Equivalently,

 34.2.6 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}={(-1)^{j_{2}-m_{1}+m_{3}}}\frac{(j_{1}+j_{2}+m_% {3})!(j_{2}+j_{3}-m_{1})!}{\Delta(j_{1}j_{2}j_{3})(j_{1}+j_{2}+j_{3}+1)!}\left% (\frac{(j_{1}+m_{1})!(j_{3}-m_{3})!}{(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})% !(j_{3}+m_{3})!}\right)^{\frac{1}{2}}\*{\mathop{{{}_{3}F_{2}}\/}\nolimits\!% \left(-j_{1}-j_{2}-j_{3}-1,-j_{1}+m_{1},-j_{3}-m_{3};-j_{1}-j_{2}-m_{3},-j_{2}% -j_{3}+m_{1};1\right)},$

where $\mathop{{{}_{3}F_{2}}\/}\nolimits$ is defined as in §16.2.

For alternative expressions for the $3j$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series $\mathop{{{}_{3}F_{2}}\/}\nolimits$ of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).