Forbidden beta decay (${\pi }_f/{\pi }_i=(-1{)}^l$)

A beta decay transition for which the leading $l=0$ lepton orbital term vanishes by the angular momentum or parity selection rules. The least contributing orbital order $l=n\ge 1$ gives the name $n$th forbidden. First-forbidden transitions change parity and can carry total lepton angular momentum $0,1,2$; second-forbidden transitions preserve parity and can carry $1,2,3$, with Fermi-type $S=0$ contributions restricted to $J=l$. Forbidden is suppression by higher powers of lepton momentum times nuclear size, rather than an absolute prohibition.

An $l$th-forbidden beta decay transition with the maximal leading angular-momentum transfer $l+1$. Only the spin-dependent $S=1$ coupling can attain that transfer at this orbital order. Thus a $0^{+}\to 2^{-}$ transition is first-forbidden unique, whereas a same-parity $\Delta J=2$ transition is second-forbidden nonunique. Unique refers to the leading weak multipole structure, not a claim that every correction to the decay vanishes.

In a paired nuclear shell model with the post-28 orbital order $2p_{3/2},1f_{5/2},2p_{1/2}$, proton counts 32 and 38 fill $2p_{3/2}$ and $1f_{5/2}$, so the odd 39th proton occupies $2p_{1/2}$. With paired valence neutrons this gives a $1/2^{-}$ yttrium-91 ground-state prediction. The daughter has paired 40 protons and one neutron beyond the 50-neutron closure in $2d_{5/2}$, giving $5/2^{+}$. The inferred ground-state beta transition is therefore first-forbidden unique. This is a stated paired-shell model deduction; an endpoint diagram alone does not determine the spin assignment or an evaluated isotope mass.

A first-forbidden unique beta decay has $|\Delta J|=2$ with a parity change. Its leading lepton shape factor contains two additional powers of momentum relative to an allowed beta decay. In the massless, no-Coulomb limit, ${\int }_0^Q{E}^2(Q-E{)}^2[E^2+(Q-E{)}^2]dE=2Q^7/105$. Thus its microscopic reduced nuclear matrix element multiplies a seventh-power phase-space factor, rather than the fifth power in the Sargent rule. A fifth-power comparison across different forbiddenness classes instead extracts convention-dependent effective strengths.