---
title: "Fermi's Theory: Kurie Plots and ft Values"
module: Beta Decay and the Weak Interaction
moduleNumber: 6
lessonNumber: 2
order: 602
summary: >
  Fermi treated beta decay as a point-contact weak transition and read its rate
  from the golden rule. The electron spectrum then follows from phase space and
  the Coulomb Fermi function; the Kurie plot straightens it to a line whose
  intercept is the endpoint. Integrating the spectrum gives the comparative
  half-life ft, whose logarithm sorts transitions into superallowed, allowed, and
  forbidden classes governed by the Fermi and Gamow-Teller selection rules.
topics: [Beta Decay and the Weak Interaction]
draft: false
sources:
  - book: Krane
    ref: "Ch. 9 — Beta Decay; §9.2 The Fermi Theory of Beta Decay, §9.3 The Experimental Tests of the Fermi Theory"
  - book: Wong
    ref: "Ch. 5 — Nuclear Reactions and Decays; §5-3 Beta Decay"
---

Pauli's neutrino restores the conservation laws but says nothing about the decay
**rate** or the **shape** of the electron spectrum. Fermi supplied both in 1934 by
writing the weak interaction as a point contact among four fermions — the initial
nucleon, the final nucleon, the electron, and the neutrino — and feeding it
through the golden rule. The theory predicts the spectrum, linearizes it into the
Kurie plot, and packages half-lives into the comparative $ft$ value that classifies
every beta transition.[^krane-fermi]

## The golden rule for beta decay

Fermi's golden rule gives the transition rate from an initial state to a
continuum of final states,

$$
\lambda = \frac{2\pi}{\hbar}\,\lvert V_{fi}\rvert^2\,\rho(E_f),
$$

where $V_{fi} = \langle f\lvert H_\beta\rvert i\rangle$ is the matrix element of
the weak Hamiltonian and $\rho(E_f) = \d n/\d E_f$ is the density of final states.
Two features of beta decay simplify $V_{fi}$.

- **Point contact.** The range of the weak force is tiny compared with nuclear
  size, so Fermi took the interaction to act at a single point. The electron and
  neutrino are created there; their wavefunctions enter as plane waves evaluated
  at the nucleus.
- **Long wavelength.** A few-MeV electron has de Broglie wavelength of order
  $100\ \mathrm{fm}$, far larger than the nucleus, so $e^{i\vec k\cdot\vec r}
  \approx 1$ across the nuclear volume. This is the **allowed approximation**:
  the lepton wavefunctions are set to their values at the origin.

With plane-wave leptons normalized in a volume $V$, the matrix element factorizes
into a nuclear part and the lepton wavefunctions at the origin,

$$
V_{fi} = \frac{g}{V}\,M_{fi},
\qquad
M_{fi} = \int \psi_f^\ast\,\mathcal{O}\,\psi_i\,\d V,
$$

where $g$ is the weak coupling strength, $M_{fi}$ is the dimensionless **nuclear
matrix element**, and $\mathcal{O}$ is the operator (unity for Fermi transitions,
the spin operator $\vec\sigma$ for Gamow-Teller). The factors of $V$ cancel
against the phase-space normalization below.

## Phase space and the spectrum shape

The final-state density counts the ways the released energy $Q$ can be split
between the electron and the neutrino. For a particle of momentum $p$ in a volume
$V$, the number of momentum states up to $p$ is $V\,\tfrac{4}{3}\pi p^3/(2\pi\hbar)^3$,
so each lepton contributes a factor $\propto p^2\,\d p$. The electron and neutrino
momenta are independent apart from the energy constraint $E_\nu = Q - T_e$, which
fixes the neutrino momentum once the electron momentum is chosen:

$$
\d n = \frac{V\,4\pi p^2\,\d p}{(2\pi\hbar)^3}\cdot
       \frac{V\,4\pi q^2\,\d q}{(2\pi\hbar)^3},
\qquad q = \frac{E_\nu}{c} = \frac{Q - T_e}{c}.
$$

Taking the recoiling daughter to absorb momentum but negligible energy, the
density of final states per unit electron momentum is

$$
\rho(E_f) = \frac{\d n}{\d E_f}
= \frac{(4\pi)^2 V^2}{(2\pi\hbar)^6}\,p^2\,q^2\,\frac{\d q}{\d E_f}
= \frac{(4\pi)^2 V^2}{(2\pi\hbar)^6 c^3}\,p^2\,(Q - T_e)^2\,\frac{\d p}{\d E_f}.
$$

Assembling the golden rule, the $V$ factors cancel and the number of electrons
emitted with momentum between $p$ and $p + \d p$ is

$$
N(p)\,\d p = \frac{g^2\lvert M_{fi}\rvert^2}{2\pi^3\hbar^7 c^3}\,
F(Z', p)\,p^2\,(Q - T_e)^2\,\d p.
$$

Three factors set the shape:

- **$p^2$** from the electron phase space, which pushes the spectrum up at low
  momentum;
- **$(Q - T_e)^2$** from the neutrino phase space, which drives it to zero at the
  endpoint $T_e = Q$;
- **$F(Z', p)$**, the **Fermi function**, correcting the plane-wave assumption for
  the Coulomb field of the daughter nucleus of charge $Z'$.

$$
% caption: The beta spectrum is the product of the rising electron phase-space
% factor p-squared and the falling neutrino factor (Q minus T) squared; their
% product peaks at intermediate energy and vanishes at the endpoint.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.2,0) node[right, black!70] {electron energy};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {rate};
  % rising p^2 factor (dashed)
  \draw[black, dashed] (0.15,0.1) .. controls (2.5,1.6) and (4.5,3.4) .. (6.2,4.0);
  \node[black, anchor=south, font=\scriptsize] at (5.5,3.5) {phase space};
  % falling neutrino factor (dotted)
  \draw[black, densely dotted] (0.15,3.9) .. controls (2.0,3.2) and (4.6,1.0) .. (6.2,0.05);
  \node[black, anchor=north, font=\scriptsize] at (1.8,3.0) {neutrino factor};
  % product (solid accent)
  \draw[acc, very thick] (0.15,0.1) .. controls (1.3,2.6) and (2.4,3.2) .. (3.2,3.15)
    .. controls (4.4,3.05) and (5.4,0.7) .. (6.2,0.05);
  \node[acc, anchor=south, font=\scriptsize] at (3.0,3.2) {spectrum};
\end{tikzpicture}
$$

The Fermi function attracts $\beta^-$ electrons toward the positive daughter
(enhancing low energies) and repels $\beta^+$ positrons (suppressing low
energies). In the nonrelativistic limit it is

$$
F(Z', p) = \frac{2\pi\eta}{1 - e^{-2\pi\eta}},
\qquad \eta = \pm\frac{Z'\alpha}{\beta},
$$

with the upper sign for $\beta^-$, the lower for $\beta^+$, and $\beta = v/c$ the
electron speed. The relativistic $F(Z', p)$ used to analyze data is tabulated
numerically.

## The Kurie plot

The three shape factors can be divided out to test the theory and read off $Q$.
Rearranging the spectrum,

$$
\sqrt{\frac{N(p)}{p^2\,F(Z', p)}}
= C\,(Q - T_e),
$$

with $C$ a constant. Plotting the left side against the electron kinetic energy
$T_e$ gives a **straight line** whose $x$-intercept is the endpoint $Q$. A linear
Kurie plot confirms the allowed shape; curvature reveals a forbidden transition or
an error in the assumed $Q$.

$$
% caption: The Kurie plot divides out the phase-space and Coulomb factors so an
% allowed beta spectrum falls on a straight line meeting the axis at Q; a massive
% neutrino bends the line down just below the endpoint.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.0,0) node[right, black!70] {electron energy};
  \draw[->, black] (0,0) -- (0,4.4)
    node[above, black!70, align=center, font=\scriptsize] {Kurie\\ordinate};
  % straight massless line
  \draw[acc, very thick] (0.3,3.9) -- (5.8,0.05);
  \node[acc, anchor=west, font=\scriptsize] at (3.4,1.9) {massless (straight)};
  \draw[black, dashed] (5.8,0) -- (5.8,0.9);
  \node[black!70, anchor=south, font=\scriptsize] at (5.8,0.85) {$Q$};
  % massive: bends down near endpoint, cut short
  \draw[black, densely dashed] (4.4,1.05) .. controls (5.0,0.5) and (5.15,0.25) .. (5.2,0.03);
  \node[black, anchor=west, font=\scriptsize] at (4.6,0.75) {massive};
\end{tikzpicture}
$$

The endpoint behavior is the tritium neutrino-mass measurement of the
[previous lesson](/nuclear-physics/beta-decay/beta-decay-energetics-neutrino): a
nonzero $m_\nu$ replaces $(Q - T_e)$ by $(Q - T_e)\bigl[(Q - T_e)^2 - m_\nu^2
c^4\bigr]^{1/4}$, bending the line down and cutting it off at $T_e = Q - m_\nu
c^2$.

## Comparative half-life and log ft

Integrating $N(p)$ over all momenta gives the total decay rate, hence the
half-life. The integral defines the dimensionless **Fermi integral** (or
statistical rate function)

$$
f(Z', E_0) = \frac{1}{(m_e c)^3 (m_e c^2)^2}\int_0^{p_{\max}}
F(Z', p)\,p^2\,(E_0 - E_e)^2\,\d p,
$$

with $E_0 = Q + m_e c^2$ the total endpoint energy. In terms of $f$,

$$
\frac{1}{t_{1/2}} = \frac{g^2\lvert M_{fi}\rvert^2\,m_e^5 c^4}{2\pi^3\hbar^7}\,
\frac{\ln 2}{\, }\,f(Z', E_0)
\quad\Longrightarrow\quad
f(Z', E_0)\,t_{1/2} = \frac{2\pi^3\hbar^7\ln 2}{g^2 m_e^5 c^4\,\lvert M_{fi}\rvert^2}.
$$

The product $ft \equiv f(Z', E_0)\,t_{1/2}$ removes the trivial dependence on
$Q$ (through $f$) and on the nuclear charge (through $F$), leaving a quantity set
only by the coupling $g$ and the nuclear matrix element $\lvert M_{fi}\rvert^2$.
Because $ft$ ranges over more than twenty orders of magnitude, it is quoted as
$\log_{10}(ft/\mathrm{s})$, the **log ft** value.

The dependence removed by $f$ is steep. For a light nucleus ($Z' \to 0$, so
$F \to 1$) and an endpoint large compared with the electron rest energy, the
integral is dominated by its upper limit and reduces to a power law,

$$
f(0, E_0) \approx \frac{1}{30}\left(\frac{E_0}{m_e c^2}\right)^{5}
\qquad (E_0 \gg m_e c^2),
$$

so the decay rate scales as the fifth power of the endpoint energy,
$\lambda \propto Q^5$. This is **Sargent's rule**, observed empirically before
Fermi's theory: plotting $\log\lambda$ against $\log Q$ for a decay series gives a
line of slope five. The strong $Q^5$ dependence is why a modest change in endpoint
energy swings the half-life by orders of magnitude, and why dividing it out through
$f$ is necessary before matrix elements from different nuclei can be compared.

> **Definition (Comparative half-life).** The comparative half-life is
> $ft = f(Z', E_0)\,t_{1/2}$, with $f$ the dimensionless statistical rate
> function and $t_{1/2}$ in seconds. A small $ft$ means a large matrix element and
> a fast transition; $\log ft$ classifies the transition.

The superallowed $0^+ \to 0^+$ Fermi transitions between analog states have the
largest, most nearly identical matrix elements and give the cleanest value of the
vector coupling constant.[^pdg-vud] Their measured $ft \approx 3070\ \mathrm{s}$,
combined with the muon lifetime, yields the Fermi constant

$$
\frac{G_F}{(\hbar c)^3} = 1.1664 \times 10^{-5}\ \mathrm{GeV^{-2}}.
$$

| Class | $\log ft$ | Selection rule | Example |
| --- | --- | --- | --- |
| Superallowed | $2.9$–$3.7$ | $\Delta I = 0$, no parity change, analog states | $^{14}\mathrm{O}\to\,^{14}\mathrm{N}^\ast$ |
| Allowed | $4.4$–$6.0$ | $\Delta I = 0, \pm 1$; no parity change | $^{6}\mathrm{He}\to\,^{6}\mathrm{Li}$ |
| First forbidden | $6$–$9$ | parity change; $\Delta I = 0, \pm 1, \pm 2$ | $^{111}\mathrm{Ag}\to\,^{111}\mathrm{Cd}$ |
| Higher forbidden | $> 9$ | larger $\Delta I$ | $^{40}\mathrm{K}\to\,^{40}\mathrm{Ca}$ |

## Fermi and Gamow-Teller transitions

The allowed approximation still leaves two independent operators, because the
emitted electron and neutrino can carry off their spins either antiparallel (total
lepton spin $0$) or parallel (total lepton spin $1$).

- **Fermi transitions** ($\mathcal{O} = 1$): the leptons couple to a singlet, so
  no spin is removed from the nucleus. The selection rules are $\Delta I = 0$ and
  no parity change. The nuclear operator is the total isospin raising or lowering
  operator, nonzero only between isobaric analog states.
- **Gamow-Teller transitions** ($\mathcal{O} = \vec\sigma$): the leptons couple to
  a triplet, carrying one unit of spin. The selection rules are $\Delta I = 0,
  \pm 1$ (but not $0 \to 0$) and no parity change.

A decay with both $\Delta I = 0$ and neither state $0^+$ proceeds by a mixture, so
the rate involves both matrix elements and their couplings,

$$
\frac{1}{ft} \propto g_V^2\lvert M_F\rvert^2 + g_A^2\lvert M_{GT}\rvert^2,
$$

where $g_V$ and $g_A$ are the vector and axial-vector coupling constants. The pure
Fermi decay $^{14}\mathrm{O}\to\,^{14}\mathrm{N}^\ast$ ($0^+ \to 0^+$) isolates
$g_V$; the pure Gamow-Teller decay $^{6}\mathrm{He}\to\,^{6}\mathrm{Li}$
($0^+ \to 1^+$) isolates $g_A$. Their ratio measured in free-neutron decay,

$$
\frac{g_A}{g_V} = -1.27,
$$

is a fundamental parameter of the weak interaction and the input to the
[V minus A current](/nuclear-physics/beta-decay/weak-interaction-parity-violation)
of the next lesson.[^pdg-vud]

$$
% caption: In a Fermi transition the electron and antineutrino spins are
% antiparallel and remove no nuclear spin; in a Gamow-Teller transition they are
% parallel and remove one unit, allowing a spin change of one.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % ---- Fermi (antiparallel) ----
  \begin{scope}
    \node[black!70, anchor=south, font=\scriptsize] at (1.2,2.6) {Fermi};
    \draw[->, very thick] (0.6,0.4) -- (0.6,2.0);
    \draw[->, very thick] (1.8,2.0) -- (1.8,0.4);
    \node[black, anchor=north, font=\scriptsize] at (0.6,0.35) {electron};
    \node[black, anchor=north, font=\scriptsize] at (1.8,0.35) {neutrino};
    \node[black, anchor=south, font=\scriptsize] at (1.2,2.1) {sum spin zero};
  \end{scope}
  % ---- Gamow-Teller (parallel) ----
  \begin{scope}[xshift=4.4cm]
    \node[black!70, anchor=south, font=\scriptsize] at (1.2,2.6) {Gamow-Teller};
    \draw[->, very thick] (0.6,0.4) -- (0.6,2.0);
    \draw[->, very thick] (1.8,0.4) -- (1.8,2.0);
    \node[black, anchor=north, font=\scriptsize] at (0.6,0.35) {electron};
    \node[black, anchor=north, font=\scriptsize] at (1.8,0.35) {neutrino};
    \node[black, anchor=south, font=\scriptsize] at (1.2,2.1) {sum spin one};
  \end{scope}
\end{tikzpicture}
$$

$$
% caption: log ft sorts transitions by matrix-element strength, from the
% superallowed 0-to-0 analog decays near 3.5 up through allowed and forbidden
% groups spanning more than five decades.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % axis
  \draw[->, black] (0,0) -- (0,5.2) node[above, black!70, font=\scriptsize] {$\log ft$};
  \foreach \y/\lab in {0.5/3, 1.7/5, 2.9/7, 4.1/9, 4.9/11} {
    \draw[black] (-0.08,\y) -- (0.08,\y);
    \node[anchor=east, black, font=\scriptsize] at (-0.12,\y) {\lab};
  }
  % bands
  \draw[acc, very thick] (0.6,0.3) -- (2.4,0.3);
  \node[acc, anchor=west, font=\scriptsize] at (2.5,0.3) {superallowed};
  \draw[black, very thick] (0.6,1.5) -- (2.4,1.5);
  \node[black, anchor=west, font=\scriptsize] at (2.5,1.5) {allowed};
  \draw[black, very thick, dashed] (0.6,3.0) -- (2.4,3.0);
  \node[black, anchor=west, font=\scriptsize] at (2.5,3.0) {1st forbidden};
  \draw[black, very thick, densely dotted] (0.6,4.5) -- (2.4,4.5);
  \node[black, anchor=west, font=\scriptsize] at (2.5,4.5) {higher forbidden};
\end{tikzpicture}
$$

Forbidden transitions occur when both Fermi and Gamow-Teller matrix elements
vanish in the allowed approximation, forcing the lepton wavefunction expansion to
higher order in $\vec k\cdot\vec r$. Each additional order suppresses the rate by
about $(kR)^2 \sim 10^{-3}$ to $10^{-4}$, raising $\log ft$ by three to four
units per degree of forbiddenness and allowing larger spin changes and a parity
flip.

[^krane-fermi]: **Krane**, _Introductory Nuclear Physics_, §9.2 (The Fermi Theory of Beta Decay) and §9.3 (Experimental Tests): the golden-rule derivation of the spectrum, the Fermi function, the Kurie plot, the $ft$ value, and the Fermi/Gamow-Teller selection rules. Representative $\log ft$ ranges follow Krane's Table 9.2 grouping.
[^pdg-vud]: **Particle Data Group**, _Review of Particle Physics_, "$V_{ud}$, $V_{us}$, the Cabibbo Angle" and the electroweak review: superallowed $0^+\to0^+$ $ft$ values, the Fermi constant $G_F$, and the axial-to-vector ratio $g_A/g_V$ from free-neutron decay, [pdg.lbl.gov](https://pdg.lbl.gov/). The Fermi constant value is the CODATA/PDG figure, [physics.nist.gov](https://physics.nist.gov/cuu/Constants/).