# §34.4 Definition: $6j$ Symbol

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

 34.4.1 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\sum_{m_{r}m^{\prime}_{s}}(-1)^{l_{1}+m^{\prime% }_{1}+l_{2}+m^{\prime}_{2}+l_{3}+m^{\prime}_{3}}\*\begin{pmatrix}j_{1}&j_{2}&j% _{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&l_{2}&l_{3}\\ m_{1}&m^{\prime}_{2}&-m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&j_{2}&l_% {3}\\ -m^{\prime}_{1}&m_{2}&m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&j_% {3}\\ m^{\prime}_{1}&-m^{\prime}_{2}&m_{3}\end{pmatrix},$ Defines: $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}$: $6j$ symbol Symbols: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$: $3j$ symbol, $j,j_{r}$: nonnegative integer, $l,l_{r}$: nonnegative integer and $r$: nonnegative integer Referenced by: §34.10, §34.4, §34.9 Permalink: http://dlmf.nist.gov/34.4.E1 Encodings: TeX, pMML, png

where the summation is taken over all admissible values of the $m$’s and $m^{\prime}$’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 $j_{1},j_{2},j_{3},l_{1},l_{2},l_{3}$; see Figure 34.4.1.

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

 34.4.2 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})% \Delta(l_{1}j_{2}l_{3})\Delta(l_{1}l_{2}j_{3})\*\sum_{s}\frac{(-1)^{s}(s+1)!}{% (s-j_{1}-j_{2}-j_{3})!(s-j_{1}-l_{2}-l_{3})!(s-l_{1}-j_{2}-l_{3})!(s-l_{1}-l_{% 2}-j_{3})!}\*\frac{1}{(j_{1}+j_{2}+l_{1}+l_{2}-s)!(j_{2}+j_{3}+l_{2}+l_{3}-s)!% (j_{3}+j_{1}+l_{3}+l_{1}-s)!},$ Symbols: $!$: $n!$: factorial, $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}$: $6j$ symbol, $j,j_{r}$: nonnegative integer, $l,l_{r}$: nonnegative integer and $\Delta(j_{1}j_{2}j_{3})$: product Referenced by: Equation (34.4.2) Permalink: http://dlmf.nist.gov/34.4.E2 Encodings: TeX, pMML, png Errata (effective with 1.0.7): Originally the factor $\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})\Delta(l_{1}j_{2}l_{3})\Delta(l_% {1}l_{2}j_{3})$ was missing in this equation. Reported 2012-12-31 by Yu Lin

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

Equivalently,

 34.4.3 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}={(-1)^{j_{1}+j_{3}+l_{1}+l_{3}}}\frac{\Delta(j_% {1}j_{2}j_{3})\Delta(j_{2}l_{1}l_{3})(j_{1}-j_{2}+l_{1}+l_{2})!(-j_{2}+j_{3}+l% _{2}+l_{3})!(j_{1}+j_{3}+l_{1}+l_{3}+1)!}{\Delta(j_{1}l_{2}l_{3})\Delta(j_{3}l% _{1}l_{2})(j_{1}-j_{2}+j_{3})!(-j_{2}+l_{1}+l_{3})!(j_{1}+l_{2}+l_{3}+1)!(j_{3% }+l_{1}+l_{2}+1)!}\*\mathop{{{}_{4}F_{3}}\/}\nolimits\!\left({-j_{1}+j_{2}-j_{% 3},j_{2}-l_{1}-l_{3},-j_{1}-l_{2}-l_{3}-1,-j_{3}-l_{1}-l_{2}-1\atop-j_{1}+j_{2% }-l_{1}-l_{2},j_{2}-j_{3}-l_{2}-l_{3},-j_{1}-j_{3}-l_{1}-l_{3}-1};1\right),$

where $\mathop{{{}_{4}F_{3}}\/}\nolimits$ 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 $\mathop{{{}_{4}F_{3}}\/}\nolimits$ of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).