← Part II: Radioactivity & Decay
Chapter 5

Beta Decay

Beta Decay Processes

Beta decay involves the transformation of a neutron into a proton (or vice versa) inside the nucleus, mediated by the weak interaction. There are three types:

$\beta^-$ Decay

$$n \to p + e^- + \bar{\nu}_e$$

Neutron-rich nuclei. Emits electron and antineutrino.

$\beta^+$ Decay

$$p \to n + e^+ + \nu_e$$

Proton-rich nuclei. Emits positron and neutrino.

Electron Capture

$$p + e^- \to n + \nu_e$$

Captures orbital electron. Competes with $\beta^+$.

Fermi Theory of Beta Decay

Enrico Fermi (1934) developed the first successful theory of beta decay, treating it as a four-fermion point interaction. The transition rate is given by Fermi's Golden Rule:

$$\lambda = \frac{2\pi}{\hbar}|M_{fi}|^2 \rho(E_f)$$

Beta Spectrum Shape

The electron energy spectrum for allowed $\beta^-$ decay is:

$$N(T_e) \propto F(Z', E_e)\,p_e\,E_e\,(Q - T_e)^2$$

where $T_e$ is the electron kinetic energy, $E_e = T_e + m_e c^2$ is the total energy, $p_e = \sqrt{E_e^2 - m_e^2 c^4}/c$ is the momentum, and $F(Z', E_e)$ is the Fermi function that accounts for the Coulomb interaction between the emitted electron and the daughter nucleus.

The Kurie Plot

The Kurie (or Fermi-Kurie) plot is a powerful tool for analyzing beta spectra. Defining the Kurie function:

$$K(T_e) = \sqrt{\frac{N(T_e)}{F(Z', E_e)\,p_e\,E_e}} \propto (Q - T_e)$$

For allowed transitions, this is a straight line that intercepts the energy axis at $T_e = Q$. Deviations from linearity indicate forbidden transitions or a non-zero neutrino mass.

ft Values and Selection Rules

The comparative half-life (ft value) removes kinematic factors:

$$ft_{1/2} = \frac{K}{g^2|M_{fi}|^2}$$

where K is a known constant. The ft value depends only on the nuclear matrix element:

  • - Superallowed: $\log ft \approx 3.5$ (e.g., $^{14}$O $\to$ $^{14}$N)
  • - Allowed: $\log ft \approx 4-7$ ($\Delta J = 0, \pm 1$, no parity change)
  • - First forbidden: $\log ft \approx 6-9$ ($\Delta J = 0, \pm 1, \pm 2$, parity change)
  • - Second forbidden: $\log ft \approx 10-13$

Connection to Neutrino Physics

Beta decay is intimately connected to neutrino physics. The continuous electron energy spectrum was the key evidence that led Pauli to postulate the neutrino (1930). Modern beta-decay experiments probe:

  • - Neutrino mass: The endpoint of the tritium beta spectrum is sensitive to $m_{\nu_e}$. The KATRIN experiment constrains $m_{\nu_e} < 0.45$ eV.
  • - Double beta decay: Neutrinoless double beta decay ($0\nu\beta\beta$) would prove neutrinos are Majorana particles.
  • - CKM matrix: Superallowed $0^+ \to 0^+$ beta decays provide the most precise determination of $V_{ud}$.

Python Simulation: Beta Spectrum & Kurie Plot

Computes the Fermi theory beta-decay electron spectra for several nuclei and demonstrates the Kurie plot analysis technique.

Beta Decay Spectrum and Kurie Plot

Python

Fermi theory prediction for beta-minus spectra of tritium, Co-60, and P-32

script.py115 lines

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran Implementation

Calculates the Fermi function and ft values for beta-decay transitions.

Fermi Function and ft Values

Fortran

Computes statistical rate function f and comparative half-lives for beta decays

fermi_ft_values.f9071 lines

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Derivation: Fermi Theory of Beta Decay

Fermi's theory treats beta decay as a point-like four-fermion interaction. We derive the transition rate from first principles using Fermi's Golden Rule.

Step 1: The Interaction Hamiltonian

Fermi modeled the weak interaction as a contact interaction analogous to electromagnetic theory. The interaction Hamiltonian density is:

$$\mathcal{H}_{\rm int} = \frac{G_F}{\sqrt{2}}\,[\bar{\psi}_p \gamma^\mu \psi_n][\bar{\psi}_e \gamma_\mu \psi_\nu] + \text{h.c.}$$

where $G_F = 1.166 \times 10^{-5}$ GeV$^{-2}$ is the Fermi coupling constant. In the non-relativistic nuclear limit, this reduces to:

$$H_{\rm int} = \frac{G_F}{\sqrt{2}}\int d^3r\,\left[g_V\,\psi_e^\dagger\psi_\nu + g_A\,(\boldsymbol{\sigma}\cdot\psi_e^\dagger\boldsymbol{\sigma}\psi_\nu)\right]\,\hat{O}_{\rm nuclear}$$

where $g_V = 1$ (vector coupling, conserved vector current) and$g_A \approx -1.276$ (axial-vector coupling, renormalized by strong interactions).

Step 2: Transition Rate from Golden Rule

The differential decay rate for an electron with momentum between $p_e$ and$p_e + dp_e$ is:

$$d\lambda = \frac{2\pi}{\hbar}|M_{fi}|^2\,\frac{dn}{dE_f}$$

The matrix element factorizes into nuclear and leptonic parts:

$$|M_{fi}|^2 = \frac{G_F^2}{2}\left[g_V^2|M_F|^2 + g_A^2|M_{GT}|^2\right]$$

where $M_F = \langle f | \sum_k \tau_\pm^{(k)} | i \rangle$ is the Fermi matrix element and $M_{GT} = \langle f | \sum_k \boldsymbol{\sigma}^{(k)} \tau_\pm^{(k)} | i \rangle$ is the Gamow-Teller matrix element.

Step 3: Phase Space (Density of States)

The lepton phase space factor counts the number of final states. For the electron and neutrino confined in a normalization volume $V$:

$$dn = \frac{V\,4\pi p_e^2\,dp_e}{(2\pi\hbar)^3}\cdot\frac{V\,4\pi p_\nu^2\,dp_\nu}{(2\pi\hbar)^3}$$

Energy conservation gives $E_\nu = Q - T_e$, so $p_\nu = (Q - T_e)/c$ (for massless neutrinos). The density of final states per unit electron energy is:

$$\frac{dn}{dT_e} = \frac{16\pi^2 V^2}{(2\pi\hbar c)^6}\,p_e^2\,E_e\,(Q - T_e)^2$$

Step 4: Complete Spectrum and Total Rate

Including the Fermi function for Coulomb corrections, the electron energy spectrum is:

$$\frac{d\lambda}{dT_e} = \frac{G_F^2}{2\pi^3\hbar^7 c^6}\left[g_V^2|M_F|^2 + g_A^2|M_{GT}|^2\right]F(Z',E_e)\,p_e c\,E_e\,(Q - T_e)^2$$

The total decay rate is obtained by integrating over all electron energies:

$$\lambda = \frac{G_F^2\,m_e^5 c^4}{2\pi^3\hbar^7}\left[g_V^2|M_F|^2 + g_A^2|M_{GT}|^2\right]\,f(Z',Q)$$

where $f(Z',Q)$ is the dimensionless statistical rate function (Fermi integral):

$$f(Z',Q) = \frac{1}{(m_e c^2)^5}\int_0^{Q} F(Z',E_e)\,p_e c\,E_e\,(Q - T_e)^2\,dT_e$$

Allowed and Forbidden Transitions

Allowed Transitions (L = 0)

In allowed transitions, the leptons carry zero orbital angular momentum. The electron and neutrino wave functions are approximated as plane waves evaluated at the origin ($e^{i\mathbf{p}\cdot\mathbf{r}/\hbar} \approx 1$ inside the nucleus):

$$\text{Selection rules: } \Delta\pi = \text{no}, \quad \Delta J = 0 \text{ (Fermi) or } \Delta J = 0, \pm 1 \text{ (GT, not } 0\to0\text{)}$$

Two types of allowed transitions:

  • - Fermi transitions ($\Delta J = 0$, $\Delta T = 0$): The nuclear operator is $\hat{O}_F = \sum_k \tau_\pm^{(k)}$. The nuclear spin and isospin structure are unchanged. For $0^+ \to 0^+$ pure Fermi transitions between analog states: $|M_F|^2 = N - Z$ or $|M_F|^2 = T(T+1) - T_{3i}T_{3f}$.
  • - Gamow-Teller transitions ($\Delta J = 0, \pm 1$): The nuclear operator is $\hat{O}_{GT} = \sum_k \boldsymbol{\sigma}^{(k)}\tau_\pm^{(k)}$. These involve a spin flip of the decaying nucleon. The Ikeda sum rule constrains the total GT strength: $S_- - S_+ = 3(N - Z)$.

Forbidden Transitions

When the allowed matrix elements vanish (due to selection rules), higher-order terms in the expansion of the lepton wave functions contribute. The $n$-th forbidden transition involves the $n$-th term in the plane wave expansion:

$$e^{i\mathbf{p}\cdot\mathbf{r}/\hbar} = 1 + \frac{i\mathbf{p}\cdot\mathbf{r}}{\hbar} + \frac{1}{2}\left(\frac{i\mathbf{p}\cdot\mathbf{r}}{\hbar}\right)^2 + \cdots$$

Each order of forbiddenness introduces a factor of $(pR/\hbar)^2 \sim (R/\lambda_e)^2 \approx 10^{-4}$suppression, where $R \sim 5$ fm and $\lambda_e \sim 400$ fm for ~1 MeV electrons:

Order$\Delta\pi$$\Delta J$$\log ft$Suppression
AllowedNo0, 13-71
1st forbiddenYes0, 1, 26-10$\sim 10^{-3}$
2nd forbiddenNo2, 310-14$\sim 10^{-6}$
3rd forbiddenYes3, 414-20$\sim 10^{-9}$
4th forbiddenNo4, 520-24$\sim 10^{-12}$

Derivation: ft Values and Comparative Half-Lives

The ft Value

The comparative half-life removes the kinematic (phase space) dependence, isolating the nuclear matrix element. From the total decay rate:

$$\lambda = \frac{\ln 2}{t_{1/2}} = \frac{G_F^2 m_e^5 c^4}{2\pi^3 \hbar^7}\left[g_V^2|M_F|^2 + g_A^2|M_{GT}|^2\right]f$$

Therefore:

$$ft_{1/2} = \frac{2\pi^3 \hbar^7 \ln 2}{G_F^2 m_e^5 c^4}\cdot\frac{1}{g_V^2|M_F|^2 + g_A^2|M_{GT}|^2} = \frac{K}{g_V^2|M_F|^2 + g_A^2|M_{GT}|^2}$$

where the constant $K$ has the value:

$$K = \frac{2\pi^3\hbar\ln 2}{G_F^2 m_e^5 c^4 / \hbar^6} = 6146 \pm 6 \text{ s}$$

Superallowed $0^+ \to 0^+$ Transitions

For pure Fermi transitions between $0^+$ analog states ($M_{GT} = 0$), the ft value directly determines $G_F$ (or equivalently $V_{ud}$):

$$\mathcal{F}t = ft(1 + \delta_R')(1 + \delta_{NS} - \delta_C) = \frac{K}{2G_V^2(1 + \Delta_R^V)}$$

where $\delta_R'$, $\delta_{NS}$ are radiative corrections, $\delta_C$ is the isospin-symmetry breaking correction, and $\Delta_R^V$ is the transition-independent radiative correction. From 14 precisely measured superallowed transitions:

$$\overline{\mathcal{F}t} = 3072.1 \pm 0.7 \text{ s}$$

This yields $|V_{ud}| = 0.97373 \pm 0.00031$, the most precise determination of this CKM matrix element and a cornerstone of the unitarity test of the first row:$|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 1$.

Electron Capture in Detail

Electron capture (EC) is the process where a proton-rich nucleus captures an inner orbital electron and converts a proton to a neutron:

$$p + e^- \to n + \nu_e$$

EC Decay Rate

The EC rate depends on the electron wave function at the nucleus. For K-shell capture (the dominant mode):

$$\lambda_{\rm EC} = \frac{G_F^2}{2\pi^3\hbar}\left[g_V^2|M_F|^2 + g_A^2|M_{GT}|^2\right]|\psi_e(0)|^2\,E_\nu^2$$

The electron density at the nucleus for a K-shell electron in a hydrogen-like atom is:

$$|\psi_{1s}(0)|^2 = \frac{1}{\pi}\left(\frac{Z}{a_0}\right)^3$$

The neutrino energy is monoenergetic (two-body final state):

$$E_\nu = Q_{\rm EC} - B_e \approx (M_P - M_D)c^2 - B_e$$

where $B_e$ is the binding energy of the captured electron. EC competes with$\beta^+$ decay but has no threshold (unlike $\beta^+$ which requires $Q > 2m_e c^2 = 1.022$ MeV). For low-Q decays, EC is the only allowed mode.

EC/Beta-Plus Ratio

The ratio of EC to $\beta^+$ rates depends on $Z$ and $Q$:

$$\frac{\lambda_{\rm EC}}{\lambda_{\beta^+}} = \frac{2\pi^2|\psi_e(0)|^2 E_\nu^2}{f_{\beta^+}} \propto \frac{Z^3}{f_{\beta^+}}$$

EC dominates over $\beta^+$ for: (a) high-Z nuclei (larger electron density at nucleus), (b) low-Q transitions (small $\beta^+$ phase space), and (c) is the only option when$Q < 2m_e c^2$. Observable signatures include characteristic X-rays from the filling of the K-shell vacancy and Auger electrons.

Double Beta Decay

Two-Neutrino Double Beta Decay ($2\nu\beta\beta$)

When single beta decay is energetically forbidden but double beta decay is allowed, the nucleus can undergo simultaneous conversion of two neutrons:

$$(A,Z) \to (A,Z+2) + 2e^- + 2\bar{\nu}_e$$

This is a second-order weak process with an extremely long half-life:

$$\left(t_{1/2}^{2\nu}\right)^{-1} = G^{2\nu}(Q,Z)\,g_A^4\,|M^{2\nu}_{GT}|^2$$

Observed in ~12 nuclides with half-lives of $10^{18}\text{-}10^{24}$ years. Examples: $^{76}$Ge ($t_{1/2} = 1.9 \times 10^{21}$ y),$^{136}$Xe ($t_{1/2} = 2.2 \times 10^{21}$ y).

Neutrinoless Double Beta Decay ($0\nu\beta\beta$)

If neutrinos are Majorana particles (i.e., their own antiparticles), a lepton-number violating mode becomes possible:

$$(A,Z) \to (A,Z+2) + 2e^-$$

The rate is proportional to the square of the effective Majorana mass:

$$\left(t_{1/2}^{0\nu}\right)^{-1} = G^{0\nu}(Q,Z)\,g_A^4\,|M^{0\nu}|^2\,\left(\frac{\langle m_{\beta\beta}\rangle}{m_e}\right)^2$$

where the effective Majorana mass is:

$$\langle m_{\beta\beta} \rangle = \left|\sum_i U_{ei}^2\,m_i\right|$$

Current experiments (KamLAND-Zen, GERDA/LEGEND, CUORE) set limits of$\langle m_{\beta\beta} \rangle < 36\text{-}156$ meV, probing the inverted hierarchy region. Discovery of $0\nu\beta\beta$ would prove that neutrinos are Majorana particles and lepton number is violated, with profound implications for baryogenesis.

Distinguishing $2\nu$ from $0\nu$ Modes

The key experimental signature is the summed electron energy spectrum:

  • - $2\nu\beta\beta$: Continuous spectrum with endpoint at $Q_{\beta\beta}$, because energy is shared with two neutrinos. The shape is $\propto (Q - T)^5 T^2$.
  • - $0\nu\beta\beta$: Sharp peak at $T_{\rm total} = Q_{\beta\beta}$, since all energy goes to the two electrons (no neutrinos). This is a monoenergetic line broadened only by detector resolution.

Neutrino Mass from Beta Decay Endpoint

Effect of Neutrino Mass on the Spectrum

If the neutrino has mass $m_\nu$, the beta spectrum is modified near the endpoint. The spectrum becomes:

$$N(T_e) \propto F(Z',E_e)\,p_e\,E_e\,(Q - T_e)\sqrt{(Q - T_e)^2 - m_\nu^2 c^4}\,\Theta(Q - T_e - m_\nu c^2)$$

The key differences from the massless case:

  • - The endpoint shifts from $T_e^{\rm max} = Q$ to $T_e^{\rm max} = Q - m_\nu c^2$
  • - The spectrum vanishes more quickly near the endpoint (the Kurie plot curves downward)
  • - The effect is only significant in the last ~$m_\nu c^2$ of the spectrum

The KATRIN Experiment

KATRIN (Karlsruhe Tritium Neutrino Experiment) uses molecular tritium ($T_2$) as a source and a MAC-E (Magnetic Adiabatic Collimation with Electrostatic) filter to measure the tritium beta spectrum near its endpoint ($Q = 18.574$ keV):

$$^3\text{H} \to\, ^3\text{He} + e^- + \bar{\nu}_e, \quad Q = 18.574 \text{ keV}$$

Tritium is ideal because of its low endpoint (maximizing the fraction of events near the endpoint), simple nuclear structure ($|M_{GT}|^2 = 3$), and manageable half-life ($t_{1/2} = 12.3$ years).

The current KATRIN upper limit is $m_\nu < 0.45$ eV (90% C.L.), with a design sensitivity of 0.2 eV. The effective electron neutrino mass measured is:$m_\nu^2 = \sum_i |U_{ei}|^2 m_i^2$.

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