Nuclear gamma-ray multipole selection rules ($|J_i-J_f|\le L\le J_i+J_f,L\ge 1$)

A real photon emitted in gamma decay carries a radiation multipole of integer angular momentum $L\ge 1$. The triangle rule gives the displayed range. An electric multipole $EL$ requires ${\pi }_f/{\pi }_i=(-1{)}^L$, while a magnetic multipole $ML$ requires ${\pi }_f/{\pi }_i=(-1{)}^{L+1}$. There is no real single-photon $E0$ decay; an electric monopole transition can instead proceed through internal conversion or, above threshold, pair creation. In the long-wavelength regime $kR\ll 1$, increasing $L$ usually strongly suppresses the rate, but nuclear transition matrix elements and collective enhancement can make an allowed lower multipole weaker than a higher one.

An absent otherwise-fast gamma transition can support a spin assignment that forbids it. It is not an absolute selection rule: a nuclear reduced matrix element may be small even when angular momentum and parity allow the mode. Meaningful limits require detection thresholds, branching normalization and competing-channel information. Assuming ordinary unhindered strengths can yield a useful conventional assignment, but that assumption must be distinguished from strict observed-line constraints.

If a cascade determines only absolute spin differences relative to a known reference spin, reflection of all inferred spins around that reference can preserve the same multipole pattern. Nonnegative-spin constraints, independently known spins, angular correlations or polarization measurements may break the ambiguity. A set of line energies and lowest-multipole labels need not do so.