Beta decay

Beta decay is a type of radioactive decay in which an unstable atomic nucleus transforms by emitting a beta particle (an electron or a positron) and an associated neutrino or antineutrino, thereby altering its atomic number by one unit while keeping the mass number unchanged.

Introduction

Beta decay is one of the three main classes of radioactive decay (alongside alpha decay and gamma decay). It is a fundamental process governed by the weak nuclear force, one of the four fundamental interactions in nature. Unlike alpha decay, which involves the emission of a helium nucleus, beta decay changes the identity of the nucleus itself: a neutron converts into a proton, or a proton converts into a neutron, with the release of a beta particle (a high-energy electron or positron) and a nearly massless, weakly interacting neutral particle called a neutrino (or antineutrino). The decay process was first observed in the early 20th century, and its theoretical explanation by Enrico Fermi in 1934 marked a milestone in particle physics.

Types of Beta Decay

Beta-minus (β⁻) decay

In β⁻ decay, a neutron inside the nucleus transforms into a proton, an electron (the beta particle), and an electron antineutrino. The emitted electron carries a negative charge and is ejected from the nucleus with a continuous spectrum of kinetic energy. The general equation for β⁻ decay of a nucleus X with atomic number Z and mass number A is:
\[
^{A}_{Z}\mathrm{X} \rightarrow \;^{A}_{Z+1}\mathrm{Y} + e^{-} + \bar{\nu}_e
\]
A common example is the decay of carbon-14:
\[
^{14}_{6}\mathrm{C} \rightarrow \;^{14}_{7}\mathrm{N} + e^{-} + \bar{\nu}_e
\]

Beta-plus (β⁺) decay

In β⁺ decay, a proton inside the nucleus converts into a neutron, a positron (the antimatter counterpart of the electron), and an electron neutrino. The positron is a positively charged beta particle. This process occurs only in nuclei that have an excess of protons. The general equation is:
\[
^{A}_{Z}\mathrm{X} \rightarrow \;^{A}_{Z-1}\mathrm{Y} + e^{+} + \nu_e
\]
An example is the decay of fluorine-18:
\[
^{18}_{9}\mathrm{F} \rightarrow \;^{18}_{8}\mathrm{O} + e^{+} + \nu_e
\]

Electron capture (EC)

Electron capture is an alternative decay mode to β⁺ decay. In this process, an inner atomic electron (usually from the K or L shell) is captured by a proton in the nucleus, transforming the proton into a neutron and emitting an electron neutrino. No positron is emitted; instead, the capture leaves a vacancy in the electron shell, which fills by emitting characteristic X-rays or Auger electrons. The equation is:
\[
^{A}_{Z}\mathrm{X} + e^{-} \rightarrow \;^{A}_{Z-1}\mathrm{Y} + \nu_e
\]
For example, potassium-40 can undergo electron capture to form argon-40.

Mechanism and the Weak Interaction

Beta decay is mediated by the weak nuclear force, which operates at subatomic distances. The fundamental process involves the transformation of a down-type quark into an up-type quark (in β⁻ decay) or vice versa (in β⁺ decay), via the exchange of a W boson. Specifically:

• In β⁻ decay, a down quark (d) inside a neutron turns into an up quark (u), converting the neutron into a proton. This changes the quark composition from udd to uud. The process emits a virtual \(W^{-}\) boson, which subsequently decays into an electron and an antineutrino.
• In β⁺ decay, an up quark inside a proton turns into a down quark, converting the proton into a neutron (uud → udd), emitting a virtual \(W^{+}\) boson that decays into a positron and a neutrino.

The weak interaction is the only force that can change the flavor of quarks, making beta decay a unique probe into the electroweak theory.

The Continuous Beta Spectrum and the Neutrino

A perplexing observation in early beta decay experiments was that the emitted electrons did not have a discrete energy (as in alpha decay) but instead exhibited a continuous energy spectrum from zero up to a maximum value, the Q-value of the decay. This violated the law of conservation of energy, leading Wolfgang Pauli in 1930 to postulate the existence of a new, nearly undetectable particle—the neutrino (or antineutrino)—that carried away the missing energy and momentum. Enrico Fermi later developed a quantitative theory of beta decay (Fermi's theory) that incorporated the neutrino, successfully explaining the continuous spectrum. The neutrino was first detected experimentally in 1956 by Clyde Cowan and Frederick Reines.

The Q-value of beta decay is the difference in rest-mass energy between the parent and daughter nuclei, and it determines the maximum kinetic energy of the beta particle. Because the emitted neutrino also carries variable energy, the beta particle's energy can range from zero to the Q-value.

Fermi Theory and the Beta Decay Spectrum

Fermi's golden rule, applied to beta decay, yields the transition rate (decay constant) as proportional to the square of a nuclear matrix element and the density of final states. The energy distribution of beta particles is given by the Fermi–Kurie plot: the number of beta particles emitted at a given kinetic energy depends on the phase space factor and the nuclear matrix element. For allowed transitions, the shape factor is nearly constant, and the spectrum is approximated by the equation:
\[
N(E) \, dE \propto p E (E_0 - E)^2 F(Z, E) \, dE
\]
where \(p\) is the electron momentum, \(E\) its total energy, \(E_0\) the maximum energy, and \(F(Z, E)\) the Fermi function that accounts for the Coulomb effect of the nucleus on the emitted beta particle. This theoretical prediction was confirmed experimentally and remains a cornerstone of weak interaction physics.

History and Significance

• 1896: Henri Becquerel discovered radioactivity, including beta radiation.
• 1900: Ernest Rutherford identified beta particles as electrons.
• 1914: James Chadwick observed the continuous energy spectrum of beta decay.
• 1930: Wolfgang Pauli proposed the neutrino (initially called "neutron") to resolve the energy crisis.
• 1934: Enrico Fermi published his theory of beta decay, introducing the term "neutrino" and the concept of a weak interaction.
• 1956: Clyde Cowan and Frederick Reines detected the antineutrino, confirming its existence.
• 1968: The double beta decay process was observed, providing insights into neutrino properties.
• Later: Precision studies of beta decay have tested the Standard Model, led to the discovery of neutrino oscillations, and constrained beyond-Standard-Model physics.

Applications

Beta decay has numerous practical uses:

• Medical imaging and therapy: Positron emitters (e.g., fluorine-18) are used in positron emission tomography (PET) scans. β⁻ emitters (e.g., iodine-131, strontium-89) are used for radioisotope therapy.
• Dating: Carbon-14 beta decay is the basis of radiocarbon dating for archaeological and geological samples.
• Energy production: Beta decay from fission products contributes to decay heat in nuclear reactors.
• Nuclear structure research: Beta decay measurements yield information about nuclear masses, shapes, and weak interaction coupling constants.
• Neutrino physics: Double beta decay experiments (e.g., search for neutrinoless double beta decay) aim to determine whether neutrinos are Majorana particles.

See also

• Weak interaction
• Neutrino
• Positron emission
• Electron capture
• Fermi's golden rule
• Double beta decay

References

• Fermi, E. (1934). "Versuch einer Theorie der β-Strahlen". Zeitschrift für Physik.
• Konopinski, E. J., & Rose, M. E. (1952). "The Theory of Beta Decay". Handbuch der Physik.
• Wu, C. S. (1957). "Beta Decay". Encyclopedia of Physics.
• Griffiths, D. (2008). Introduction to Elementary Particles (2nd ed.). Wiley-VCH.