Beta decay and Fermi's theory

Mass tables list neutral atoms, not nuclei, so do the electron bookkeeping once and reuse it. In β⁻ the parent has Z electrons and the daughter atom needs Z + 1; the emitted electron supplies exactly the missing one, the Z-dependence cancels, and Q(β⁻) = [M(X) − M(Y)]c² with atomic masses used straight. In β⁺ the daughter needs only Z − 1, so one of the parent's electrons is left over — still present, but missing from the tabulated daughter mass — and the decay must also create a positron: two debits of 0.511 MeV, 1.022 MeV in all. Electron capture consumes one of the parent's own electrons, so the counts already match on both sides and Q(EC) is the full mass difference less the binding energy of the shell left vacant in the daughter atom, 8.33 keV for a K-shell hole in nickel. Take ⁶⁴Cu, which exceeds ⁶⁴Zn by 6.220 × 10⁻⁴ u and ⁶⁴Ni by 1.798 × 10⁻³ u. Those two differences give Q(β⁻) = 0.579 MeV, Q(EC) = 1.666 MeV and Q(β⁺) = 0.653 MeV. All three channels are open, and ⁶⁴Cu duly uses all three.

If β⁻ were a two-body decay, momentum conservation in the parent's rest frame would fix the electron's energy exactly and the spectrum would be a line. It is not: counts run continuously from zero to a sharp endpoint at Q, and the mean electron energy is only about Q/3. Spin makes the same point independently — n → p + e⁻ would put one spin-½ particle on the left and two on the right, which cannot balance. Pauli's 1930 letter proposed a light, neutral, spin-½ particle to carry off the balance of energy, momentum and angular momentum; Fermi named it the neutrino and built the rate around it. Lepton number then sorts the versions: β⁻ emits an antineutrino, while β⁺ and capture emit a neutrino. Reines and Cowan confirmed the particle in 1956 through the inverse reaction ν̄ + p → n + e⁺, tagged by a delayed coincidence between two 0.511 MeV annihilation photons and a neutron-capture gamma. The cross-section is of order 10⁻⁴³ cm², a mean free path in water of hundreds of light-years, which is why confirmation took twenty-six years.

Fermi wrote the rate with the golden rule, λ = (2π/ħ)|M_fi|² dn/dE_f. The interaction he used is a contact one — all four fermions meeting at a single point with strength G_F — and we now know why that works: the exchanged W weighs 80.4 GeV, so its range is ħc/M_Wc² = 197.3/80400 fm = 2.5 × 10⁻³ fm, utterly unresolved by a nucleus. The second simplification is the allowed approximation. Expand each outgoing lepton wave as e^(ik⋅r) ≈ 1 + ik⋅r; for a 1 MeV electron pc = 1.42 MeV, so ħc/pc = 139 fm against a nuclear radius near 5 fm, and kR ≈ 0.04. Keep only the 1, evaluate both lepton wavefunctions at the origin, and |M_fi|² becomes a constant. Everything left is counting. The electron contributes 4πp² dp states, the neutrino 4πp_ν² dp_ν with p_νc = Q − Tₑ and dp_ν/dE = 1/c, so N(p) dp ∝ p²(Q − Tₑ)² dp. The shape of a beta spectrum is not dynamics at all: it is the number of ways two particles can share Q.

The electron leaves through the daughter's Coulomb field, so its wavefunction at the nucleus is not a plane wave's. The repair is the Fermi function, F(Z, E) = 2πη/(1 − e^(−2πη)) in its non-relativistic form, with η = Zα/β for β⁻ and −Zα/β for β⁺. Attraction pulls slow electrons in and lifts the low-energy end: for the ³²P daughter, Z = 16, F runs from 1.81 at T = 0.1 MeV down to 1.43 at 1.2 MeV. Repulsion does the reverse, which is why positron spectra from the same Q look visibly harder. Kurie's move is to divide the measurement by everything the theory already knows and take a square root: K = √[N(p)/(p²F)] should be proportional to Q − T, a straight line meeting the axis at the endpoint. Two payoffs follow. Q comes from extrapolating a line fitted where counts are plentiful, rather than from chasing the last few counts. And departures are diagnostic: a low-energy bow means a missing F or self-absorption in the source, a high-energy break means an unresolved second branch or a forbidden shape factor.

Beta half-lives run from milliseconds to 10¹⁶ years, but much of that range is phase space rather than nuclear physics: integrating p²(Q − T)² over the spectrum makes the rate climb roughly as Q⁵. To compare nuclei you divide it out. The Fermi integral f = ∫₁^(W₀) F(Z, W)(W₀ − W)² W √(W² − 1) dW, with W = E/m_ec² and W₀ = 1 + Q/m_ec², collects all of it; ft½ in seconds is the comparative half-life, and it is quoted as log₁₀(ft) because even after the division the spread is enormous. The bands are the useful part. Superallowed 0⁺ → 0⁺ decays cluster at log ft ≈ 3.5 — ¹⁴O gives 3.49; ordinary allowed transitions sit between 4.5 and 6; first-forbidden ones between 6 and 9; and ¹¹⁵In, fourth-forbidden with a half-life of 4.4 × 10¹⁴ years, reaches log ft ≈ 22.5. Each step of forbiddenness costs about (kR)² ≈ 10⁻³ in rate, three to four units of log ft, which is exactly what the ladder is measuring.

Because the leptons leave with ℓ = 0, an allowed transition cannot change the nuclear parity. Their spins couple two ways, and each way defines an operator. Antiparallel, S = 0, is Fermi: the operator is the identity, and it demands ΔJ = 0 with no parity change. Parallel, S = 1, is Gamow-Teller: the operator is σ, and it permits ΔJ = 0 or 1 but forbids 0 → 0, since a spin-1 lepton pair carries away angular momentum the nucleus must supply. The pure cases are the clean tests. ¹⁴O(0⁺) → ¹⁴N*(0⁺) can only be Fermi; ⁶He(0⁺) → ⁶Li(1⁺) can only be Gamow-Teller; the free neutron and the mirror decays mix the two. When both operators return zero — because ΔJ ≥ 2, or because the parity changes — the leading term is gone and the ik⋅r term has to be kept. That is not a small correction. The rate falls by roughly (kR)², and the shape factor is no longer flat, so the Kurie plot itself curves until that factor is divided out too.