β⁻ Decay: Electron Spectrum
In β⁻ decay a nucleus emits an electron and an electron antineutrino, sharing energy and momentum with the recoiling daughter. For an allowed transition the electron kinetic-energy spectrum is continuous from zero up to an endpoint Q (the maximum electron kinetic energy when the antineutrino carries essentially none). A standard Fermi “golden rule” phase-space cartoon gives dΓ/dE ∝ p (Q − E)² with relativistic electron momentum p(E) = √(E² + 2 m_e c² E). Coulomb distortion F(Z,E) near the nucleus is omitted here, so the curve is schematic but shows the key qualitative shape: rising from threshold, peaking, then falling to zero at E = Q. The mean kinetic energy ⟨E⟩ for the displayed shape is shown for comparison when you change Q.
Who it's for: Modern physics and introductory nuclear physics courses discussing neutrinos, lepton number, and kinematics of three-body decay.
Key terms
- Beta Decay
- Fermi Golden Rule
- Endpoint Energy
- Antineutrino
- Continuous Spectrum
How it works
Electron kinetic-energy spectrum in β⁻ decay is continuous up to an endpoint Q: the antineutrino carries the missing energy and momentum — Pauli’s 1930 hypothesis, confirmed by experiment.
Frequently asked questions
- Why is the spectrum not symmetric around Q/2?
- The (Q − E)² factor favors lower electron energies because more phase space opens when the light antineutrino takes more energy; the electron momentum factor p(E) also shapes the low-energy rise. Real nuclei add a Coulomb correction that modifies the low-energy end.
More from Chemistry
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Buffer Solution
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Gray–Scott Patterns
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Gibbs Free Energy
ΔG = ΔH − TΔS; sign vs spontaneity at constant p,T (no Q or K).
Maxwell–Boltzmann vs Eₐ
Translational energy density f(E); shaded fraction above activation energy; compare Arrhenius exp(−Eₐ/RT).
Water P–T Phase Diagram
Qualitative fusion, sublimation, vapor pressure up to critical point — probe labeled regions (pedagogical curves).