A nuclear beta decay connects initial and final quantum states through vector and axial-vector parts of the weak charged current. At leading order in the lepton wavelength expansion, Fermi transitions carry lepton-pair spin , while Gamow-Teller transitions carry . If the lepton pair carries orbital angular momentum , its total angular momentum is for , or for . The nuclear states must obey , and their parities satisfy in this leading orbital classification. The least permitted labels the degree of forbiddenness. Real forbidden transitions involve several weak multipole operators; the elementary spin classification does not assert that only one operator or nuclear matrix element exists.
The product of a dimensionless beta-decay statistical phase-space factor and the partial half-life for the transition. The tabulated normally means . Factoring out this approximate phase space dependence allows comparisons of nuclear weak matrix elements. Strong superallowed Fermi transitions have small ; forbidden beta decays generally have larger values, with nuclear-structure exceptions and overlaps between categories.
A beta decay transition for which the leading lepton orbital term vanishes by the angular momentum or parity selection rules. The least contributing orbital order gives the name th forbidden. First-forbidden transitions change parity and can carry total lepton angular momentum ; second-forbidden transitions preserve parity and can carry , with Fermi-type contributions restricted to . Forbidden is suppression by higher powers of lepton momentum times nuclear size, rather than an absolute prohibition.
An th-forbidden beta decay transition with the maximal leading angular-momentum transfer . Only the spin-dependent coupling can attain that transfer at this orbital order. Thus a transition is first-forbidden unique, whereas a same-parity 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 , proton counts 32 and 38 fill and , so the odd 39th proton occupies . With paired valence neutrons this gives a yttrium-91 ground-state prediction. The daughter has paired 40 protons and one neutron beyond the 50-neutron closure in , giving . 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 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, . 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.
A beta decay transition whose leading lepton orbital term has . Fermi transitions allow and Gamow-Teller transitions allow , excluding a Gamow-Teller transition. Both preserve nuclear parity. Allowed describes the selection rules; cancellations in the nuclear matrix element can still make an allowed transition slow.
The approximate fifth-power scaling of an allowed beta decay rate with available energy when electron mass and Coulomb distortion are negligible and the nuclear matrix element is held fixed. It follows from . Different nuclear matrix elements or nonrelativistic electron kinematics spoil an exact universal law.
With a momentum-independent allowed nuclear matrix element, negligible nuclear recoil and negligible neutrino mass, the electron spectrum in natural units is , where . Transforming to an energy spectrum requires the Jacobian determinant .
The electron-nucleus Coulomb-distortion factor multiplying an otherwise plane-wave beta decay spectrum. It depends on daughter charge, lepton charge and lepton energy. Its energy dependence means that a pure fifth-power phase-space rule is not generally exact.
An especially strong allowed beta decay, conventionally including pure Fermi transitions between isobaric analog states. Those states have closely related spatial and spin structure, so the isospin ladder operator has a large overlap. A assignment alone fixes the allowed Fermi character, but does not establish analog-state overlap or a large rate; a small comparative half-life supplies additional evidence.
The spin-dependent axial-vector contribution to an allowed beta decay. Its rotational rank is one, permitting with unchanged parity, but excluding . Here acts on nucleon spin and on isospin. Higher-orbital spin-dependent contributions are conventionally described as Gamow-Teller-type forbidden transitions.
The spin-independent vector contribution to an allowed beta decay. The isospin operator changes a neutron into a proton or the reverse; the sign convention depends on the assigned nucleon isospin projection. Its rotational rank is zero, so and the parity is unchanged. The phrase Fermi-type forbidden contribution extends the classification to higher orbital angular momentum.
The Fermi transition operator is an isospin ladder operator. Acting between normalized members of one isospin multiplet gives the displayed squared matrix element. An edge transition in a multiplet has Fermi strength three, whereas the transition between the two mirror members has strength one. This fixes the vector contribution; nonzero-spin analogue transitions can also have a Gamow-Teller transition contribution that is not fixed by this counting.
