Beta Decay Calculator

Beta decay is the most common mode of radioactive transformation for unstable nuclei that have too many or too few neutrons relative to protons. Unlike alpha decay, which emits a discrete particle, beta decay distributes its energy across three particles — beta particle, neutrino, and recoiling nucleus — producing a continuous energy spectrum up to the Q-value maximum. The beta decay calculator quantifies this energy release from the mass difference between parent and daughter atoms.

Why Beta Decay Happens: The N/Z Stability Valley

Stable nuclei occupy a "valley of stability" in the nuclear chart where the neutron-to-proton (N/Z) ratio is optimal for the nuclear binding forces. Nuclei above this valley (too many neutrons) undergo beta-minus decay; those below it (too many protons) undergo beta-plus decay or electron capture. The driving force is always the same: the daughter nucleus lies closer to the stability valley, representing a lower total nuclear mass and therefore lower rest-mass energy. This mass difference — the Q-value — is converted into kinetic energy of the decay products, following Einstein's mass-energy equivalence E = mc².

Q-Value Formulas: Beta-Minus vs. Beta-Plus

The Q-value uses atomic masses M (in atomic mass units, u) with the conversion factor 931.494 MeV/u:

Beta-minus (n → p + e⁻ + ν̄ₑ):

Q = [M(parent) − M(daughter)] × 931.494 MeV/u

Beta-minus is possible whenever the parent atomic mass exceeds the daughter atomic mass, even by a tiny amount.

Beta-plus (p → n + e⁺ + νₑ):

Q = [M(parent) − M(daughter) − 2mₑ] × 931.494 MeV/u

The additional 2mₑ = 2 × 0.000549 u = 1.022 MeV accounts for the positron mass that must be created from the available energy. Beta-plus decay requires Q > 1.022 MeV; below this threshold, only electron capture is possible. The radioactive decay calculator determines the remaining activity over time for any radioisotope.

Worked Examples: Common Medical and Industrial Radioisotopes

Beta decay Q-values for important applications:

• Carbon-14 → Nitrogen-14 (β⁻): Q = 0.156 MeV; the low maximum beta energy makes it a safe tracer isotope used in radiocarbon dating and biochemical labeling
• Fluorine-18 → Oxygen-18 (β⁺): Q = 0.634 MeV; the positron emitter used in PET scanning; 110-minute half-life requires on-site cyclotron production
• Iodine-131 → Xenon-131 (β⁻): Q = 0.971 MeV; combination beta and gamma emitter used for thyroid cancer treatment; concentrates naturally in thyroid tissue
• Strontium-90 → Yttrium-90 (β⁻): Q = 0.546 MeV; pure beta emitter; used in radiation therapy and industrial thickness gauges

The nuclear binding energy calculator and half-life calculator provide complementary nuclear physics analyses.

Fermi's Theory and the Beta Spectrum Shape

The continuous energy distribution of beta particles — from zero to Q_max — was the central mystery of early nuclear physics. Niels Bohr famously suggested energy might not be conserved in beta decay. Wolfgang Pauli's 1930 neutrino hypothesis resolved the paradox: the "missing" energy is carried away by an undetected neutrino (later confirmed experimentally in 1956 by Reines and Cowan). Enrico Fermi's 1934 beta decay theory correctly predicted the spectrum shape using the Fermi function and established the framework for the weak nuclear force — the interaction responsible for all beta decay processes.