34 $\mathrm{3j},\mathrm{6j},\mathrm{9j}$ SymbolsProperties34.9 Graphical Method34.11 Higher-Order $3nj$ Symbols

In a $\mathrm{3j}$ 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 $\mathrm{3j}$ symbol is
zero. Similarly the $\mathrm{6j}$ symbol (34.4.1) vanishes when the
triangle conditions are not satisfied by any of the four $\mathrm{3j}$ symbols in the
summation. Such zeros are called *trivial zeros*. However, the $\mathrm{3j}$ and
$\mathrm{6j}$ symbols may vanish for certain combinations of the angular momenta and
projective quantum numbers even when the triangle conditions are fulfilled.
Such zeros are called *nontrivial zeros*.

For further information, including examples of nontrivial zeros and extensions to $\mathrm{9j}$ symbols, see Srinivasa Rao and Rajeswari (1993, pp. 133–215, 294–295, 299–310).