3.2 Beta decay (β-, β+, and electron capture)

Beta decay is a fascinating process where atomic nuclei transform, emitting particles and changing their identity. It comes in three flavors: beta minus, beta plus, and electron capture. Each type alters the atomic number while keeping the mass number constant.

These decay modes are governed by the weak interaction, one of nature's fundamental forces. They involve the transformation of neutrons into protons or vice versa, accompanied by the emission of electrons, positrons, and elusive neutrinos or antineutrinos.

Beta Decay Processes

Types of Beta Decay

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• Beta minus (β⁻) decay occurs when a neutron transforms into a proton, emitting an electron and an electron antineutrino
  • Increases the atomic number by one while maintaining the same mass number (A = constant, Z → Z + 1)
  • Example: Carbon-14 decays to Nitrogen-14 through β⁻ decay
• Beta plus (β⁺) decay involves a proton converting into a neutron, releasing a positron and an electron neutrino
  • Decreases the atomic number by one while keeping the mass number constant (A = constant, Z → Z - 1)
  • Example: Sodium-22 undergoes β⁺ decay to become Neon-22
• Electron capture is an alternative to β⁺ decay, where a proton captures an inner shell electron, transforming into a neutron and emitting an electron neutrino
  • Also reduces the atomic number by one while maintaining the same mass number (A = constant, Z → Z - 1)
  • Example: Potassium-40 can decay to Argon-40 through electron capture

Neutrinos and Antineutrinos

• Neutrinos and antineutrinos are nearly massless, electrically neutral particles that interact very weakly with matter
  • Electron neutrinos (νₑ) are emitted during β⁺ decay and electron capture
  • Electron antineutrinos (ν̄ₑ) are released during β⁻ decay
• The emission of neutrinos and antineutrinos allows for conservation of energy, momentum, and angular momentum in beta decay processes
• Detection of neutrinos and antineutrinos is challenging due to their weak interactions with matter, requiring large, sensitive detectors

Underlying Physics

Weak Interaction

• Beta decay processes are governed by the weak interaction, one of the four fundamental forces of nature
  • Weak interaction is responsible for the radioactive decay of subatomic particles and the nuclear processes that fuel the sun and other stars
• The weak interaction allows quarks to change flavor, enabling transitions between neutrons (composed of one up and two down quarks) and protons (two up quarks and one down quark)
  • Beta minus decay: d → u + e⁻ + ν̄ₑ
  • Beta plus decay and electron capture: u → d + e⁺ + νₑ

Fermi Theory

• Fermi theory, developed by Enrico Fermi in 1934, provides a mathematical description of beta decay
• The theory introduces the concept of the weak interaction and the existence of the neutrino
  • Fermi proposed that the neutrino is emitted during beta decay to account for the observed continuous energy spectrum of the emitted electrons
• Fermi's theory laid the groundwork for the development of the electroweak theory, which unifies the electromagnetic and weak interactions

Spectral Analysis

Energy Spectrum

• The energy spectrum of beta particles (electrons or positrons) emitted during beta decay is continuous, ranging from zero to a maximum value (Emax)
  • The continuous spectrum is due to the three-body nature of beta decay, with the available energy shared between the beta particle and the neutrino or antineutrino
• The shape of the beta spectrum depends on the specific nuclear transition and the type of beta decay (β⁻ or β⁺)
• Analysis of the beta spectrum can provide information about the parent and daughter nuclei, as well as the Q-value (energy released) of the decay

Kurie Plot

• A Kurie plot is a graphical method used to analyze the beta spectrum and determine the maximum beta energy (Emax) and the half-life of the parent nucleus
• The Kurie plot is constructed by plotting the square root of the count rate divided by the Fermi function $(N(E)/F(Z, E))^{1/2}$ against the beta energy $(E)$
  • The Fermi function $F(Z, E)$ accounts for the Coulomb interaction between the beta particle and the daughter nucleus
• The resulting plot is expected to be a straight line if the beta decay follows the allowed transition rules
  • Deviations from linearity can indicate forbidden transitions or the presence of multiple beta decay branches
• Extrapolating the straight-line portion of the Kurie plot to the energy axis yields the maximum beta energy (Emax), while the slope of the line is related to the half-life of the parent nucleus