Overview#

• Beta decay changes a neutron into a proton (or vice versa) within the nucleus, keeping \(A\) constant but changing \(Z\). Three processes: beta-minus (\(n \to p\)), beta-plus (\(p \to n\)), and electron capture (\(p + e^- \to n\)).

• A neutrino or antineutrino is emitted in each process. It is required for energy, momentum, and lepton number conservation — and it explains the continuous energy spectrum of the beta particle.

• The Q-value determines whether a given beta decay is energetically allowed; mass tables (or atomic masses) are used to compute it.

Beta-Minus Decay (\(\beta^-\))#

In beta-minus decay, a neutron inside the nucleus converts to a proton, emitting an electron and an antineutrino:

At the nuclear level: \(_Z^A\text{X} \to _{Z+1}^{A}\text{Y} + e^- + \bar{\nu}_e\). The mass number \(A\) is unchanged; the atomic number increases by 1. Example: \(^{14}\text{C} \to ^{14}\text{N} + e^- + \bar{\nu}_e\).

Beta-Plus Decay (\(\beta^+\))#

In beta-plus decay, a proton converts to a neutron, emitting a positron (\(e^+\)) and a neutrino:

At the nuclear level: \(_Z^A\text{X} \to _{Z-1}^{A}\text{Y} + e^+ + \nu_e\). Example: \(^{11}\text{C} \to ^{11}\text{B} + e^+ + \nu_e\).

Electron Capture (EC)#

In electron capture, a proton captures an atomic electron and converts to a neutron, emitting a neutrino:

At the nuclear level: \(_Z^A\text{X} + e^- \to _{Z-1}^{A}\text{Y} + \nu_e\). The captured electron usually comes from the innermost shell (K-shell); the process is sometimes written as K-capture.

Why the Neutrino?#

The neutrino (\(\nu_e\)) and antineutrino (\(\bar{\nu}_e\)) are electrically neutral, nearly massless particles that carry energy, momentum, and lepton number.

Why they are required:

1. Energy spectrum: Without a third particle, the electron (or positron) would have a unique energy (two-body final state). Experimentally, the beta particle has a continuous spectrum from 0 up to a maximum \(E_{\max} = Q\). The “missing” energy is carried by the neutrino.

2. Momentum: A two-body decay would fix the electron’s momentum. The three-body decay allows a range of momentum sharing.

3. Lepton number: The electron has lepton number \(L = +1\); the antineutrino has \(L = -1\). The decay \(n \to p + e^-\) would violate lepton number; adding \(\bar{\nu}_e\) conserves it.

Q-Value for Beta Decay#

The decay energy \(Q\) is the total kinetic energy shared by the products. For beta-minus (using atomic masses; the \(Z\) electrons cancel):

For beta-plus, the daughter has one fewer electron than the parent; creating the positron requires \(2m_e c^2\):

For electron capture, the parent atom has one more electron than the daughter nucleus needs:

where \(E_B\) is the binding energy of the captured electron (usually small compared to \(Q\)).

Summary#

• \(\beta^-\): \(n \to p + e^- + \bar{\nu}_e\); \(Z \to Z+1\), \(A\) unchanged.

• \(\beta^+\): \(p \to n + e^+ + \nu_e\); \(Z \to Z-1\).

• EC: \(p + e^- \to n + \nu_e\); \(Z \to Z-1\); competes with \(\beta^+\) when \(Q_{\beta^+}\) is small.

• Neutrino/antineutrino: conserves energy, momentum, lepton number; explains continuous beta spectrum.

• \(Q > 0\) required for each process; use atomic masses with care for \(\beta^+\) and EC.