Fermi's Theory: Kurie Plots and ft Values

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.¹

The golden rule for beta decay

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

where $V_{fi}=⟨f|H_{\beta }|i⟩$ is the matrix element of the weak Hamiltonian and $\rho (E_f)=\mathbf{d}n/\mathbf{d}{E}_f$ 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,

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 $V\frac{4}{3}\pi p^3/(2\pi \hslash {)}^3$, so each lepton contributes a factor $\propto p^2\mathbf{d}p$. 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:

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

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

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 $T_e=Q$;
• $F(Z^{\prime },p)$, the Fermi function, correcting the plane-wave assumption for the Coulomb field of the daughter nucleus of charge $Z^{\prime }$.

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

with the upper sign for ${\beta }^{-}$, the lower for ${\beta }^{+}$, and $\beta =v/c$ the electron speed. The relativistic $F(Z^{\prime },p)$ 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,

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 $Q$.

The endpoint behavior is the tritium neutrino-mass measurement of the previous lesson: a nonzero $m_{\nu }$ replaces $(Q-T_e)$ by $(Q-T_e)[(Q-T_e{)}^2-m_{\nu }^2{c}^4{]}^{1/4}$, 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)

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

The product $ft\equiv f(Z^{\prime },E_0)t_{1/2}$ 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 $|M_{fi}{|}^2$. 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 ($Z^{\prime }\to 0$, 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,

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.

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.² Their measured $ft\approx 3070\mathrm{s}$, combined with the muon lifetime, yields the Fermi constant

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,

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,

is a fundamental parameter of the weak interaction and the input to the V minus A current of the next lesson.²

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.