Beta-Minus, Beta-Plus Decay, and Electron Capture

Beta decay is a radioactive process in which a nucleon changes into the other kind: in beta-minus decay a neutron becomes a proton and an electron is emitted (Z increases by 1), in beta-plus decay a proton becomes a neutron and a positron is emitted (Z decreases by 1), and in electron capture the nucleus captures an inner electron (Z decreases by 1). In brief: carbon-14 decays by β⁻ to nitrogen-14, fluorine-18 decays by β⁺ to oxygen-18, and in all cases the mass number A does not change.

Beta decay occurs in nuclei that have too many or too few neutrons for stability, and it is the most common mode of radioactive decay. It is responsible for carbon dating and for the tracers used in PET scans. This article describes the three processes with their equations, explains the role of the neutrino, gives a comparison table and calculates the energy released, with examples.

What is beta-minus decay?

Nuclei with too many neutrons (above the belt of stability) convert a neutron into a proton: n → p + e⁻ + ν̄_e. The electron (the beta particle, β⁻) and an electron antineutrino are emitted, and the atomic number increases by one while A stays the same: ^A_ZX → ^A_Z+1Y + e⁻ + ν̄_e. The electron is created in the process, and was not in the nucleus beforehand. Examples: ¹⁴₆C → ¹⁴₇N + e⁻ + ν̄_e and ⁶⁰₂₇Co → ⁶⁰₂₈Ni* + e⁻ + ν̄_e. A free neutron decays in the same way with a half-life of about 10 minutes.

What is beta-plus decay?

Nuclei with too many protons (below the belt of stability) can convert a proton into a neutron: p → n + e⁺ + ν_e. The positron (β⁺), the antiparticle of the electron, and an electron neutrino are emitted. The atomic number decreases by one: ^A_ZX → ^A_Z−1Y + e⁺ + ν_e. Examples: ¹⁸₉F → ¹⁸₈O + e⁺ + ν_e and ²²₁₁Na → ²²₁₀Ne + e⁺ + ν_e. A free proton cannot decay in this way, as it would need energy: the process is possible only inside a nucleus, which supplies the energy. The positron soon meets an electron and annihilates, giving two gamma photons of 511 keV each, emitted in opposite directions.

What is electron capture?

As an alternative to positron emission, a proton-rich nucleus can capture one of its inner electrons (usually from the K shell): p + e⁻ → n + ν_e. The atomic number decreases by one, A is the same: ^A_ZX + e⁻ → ^A_Z−1Y + ν_e. An example is ⁷₄Be + e⁻ → ⁷₃Li + ν_e, with a half-life of 53 days. A vacancy is left in the electron shell, which is filled by an outer electron and gives characteristic X-rays or Auger electrons. Positron emission requires an energy difference between parent and daughter atoms of at least 1.022 MeV (two electron masses); if the available energy is lower, only electron capture is possible. Nuclides with an energy above this threshold can decay by both.

Comparison of the three processes

Why are neutrinos involved?

The energies of the electrons emitted in beta decay have a continuous spectrum, from zero up to a maximum value, which at first seemed to violate the conservation of energy, since in a two-body decay the energy of the electron would be fixed. Wolfgang Pauli proposed in 1930 that a third, neutral, very light particle is emitted that carries off the rest of the energy and momentum, and Enrico Fermi called it the neutrino and developed the theory of beta decay in 1934. Neutrinos interact so weakly with matter that they were detected only in 1956, by Clyde Cowan and Frederick Reines, in a nuclear reactor. A neutrino can pass through the whole Earth with a small chance of interaction.

How much energy is released?

For β⁻ decay, the Q value can be calculated with atomic masses: Q = [m(parent) − m(daughter)] × 931.494 MeV/u, since the electron mass is included on both sides. For β⁺, Q = [m(parent) − m(daughter) − 2m_e] × 931.494 MeV/u, and for electron capture, Q = [m(parent) − m(daughter)] × 931.494 MeV/u. The energy is shared between the electron (or positron), the neutrino and (very slightly) the recoiling nucleus.

Worked examples

Example 1: writing equations

• ¹³¹₅₃I → ¹³¹₅₄Xe + e⁻ + ν̄_e (β⁻; iodine-131 has too many neutrons).
• ¹¹₆C → ¹¹₅B + e⁺ + ν_e (β⁺).
• ⁴⁰₁₉K + e⁻ → ⁴⁰₁₈Ar + ν_e (electron capture, in about 11 % of the decays of potassium-40; the main mode, in about 89 %, is β⁻ to ⁴⁰Ca).

Example 2: the energy of carbon-14 decay

m(¹⁴C) = 14.003242 u and m(¹⁴N) = 14.003074 u. Δm = 0.000168 u, so Q = 0.000168 × 931.494 = 0.157 MeV (157 keV). The beta particles of carbon-14 are therefore very low in energy (the average is about 49 keV), and are stopped by a few centimeters of air or a thin foil. This is the reason why carbon-14 is a rather safe tracer.

Example 3: predicting the mode of decay

Phosphorus-32 (Z = 15, N = 17, N/Z = 1.13) is above the belt of stability for phosphorus (the stable ³¹P has N/Z = 1.07), so it decays by β⁻ to sulfur-32 with a half-life of 14.3 days. Nitrogen-13 (Z = 7, N = 6) has too few neutrons, so it decays by β⁺ to carbon-13 (half-life 9.96 min).

Example 4: positron annihilation

The two photons from the annihilation of an electron and a positron carry the rest energy of the two particles: 2 × 0.511 = 1.022 MeV in total, or 511 keV each, with the wavelength λ = hc/E = 1.988 × 10⁻²⁵ / (511 × 10³ × 1.602 × 10⁻¹⁹) = 2.43 × 10⁻¹² m. Detecting the two photons in coincidence locates the place of the decay.

Where is beta decay used?

• PET scans: a glucose analog labeled with fluorine-18 (half-life 110 min) is injected, and positron annihilation is detected to map the activity of tissues, for example of a tumor.
• Radiocarbon dating: the β⁻ decay of carbon-14.
• Radiotherapy and tracers: iodine-131 for thyroid disease, phosphorus-32 and strontium-90 for therapy and as tracers in biology.
• Industrial gauges: beta particles passing through paper or foil are used to control the thickness during production.
• Tritium: a weak β⁻ emitter (half-life 12.3 years, 18.6 keV maximum) used in self-luminous signs and watches.

How was beta decay explained?

Henri Becquerel showed in 1900 that beta rays are electrons. The continuous energy spectrum was measured by James Chadwick in 1914 and the problem was resolved by Pauli’s neutrino hypothesis of 1930. Fermi’s theory of 1934 is the basis for the modern theory of the weak interaction. Positron emission was found by Irène Curie and Frédéric Joliot in 1934, who made the first artificial radioactive nuclide (phosphorus-30) and received the Nobel Prize in Chemistry in 1935. In 1957 Chien-Shiung Wu showed in the decay of cobalt-60 that the weak interaction does not conserve parity.

Which mistakes are common?

• Changing the mass number: A stays the same in beta decay.
• Forgetting the neutrino: it is needed to conserve energy and lepton number.
• Saying that the electron was in the nucleus: it is created in the decay.
• Mixing up β⁺ and β⁻: neutron-rich nuclei emit electrons, proton-rich ones positrons.
• Forgetting the extra 1.022 MeV for positron emission: use atomic masses with care.