Fermi Theory of Beta Decay and the Kurie Plot

Derivation

Fermi Theory of Beta Decay and the Kurie Plot

D-271 Home PU-304 Threads chance · energy · force · matter Depends on Fermi's Golden Rule, Lorentz-Invariant Phase Space and the Decay/Cross-Section Master Formulae

Statement

Starting from Fermi's four-fermion contact interaction and the golden rule, we derive the differential electron energy spectrum \( \frac{d\Gamma}{dE_e} \) for allowed nuclear beta decay. We show that it factorises into a constant nuclear matrix element and a phase-space factor \( \propto p_e E_e (E_0 - E_e)^2 F(Z,E_e) \), that the corresponding Kurie function \( K(E_e) = \sqrt{N(E_e)/[p_e E_e F]} \) is linear in \( E_e \) with intercept at the endpoint \( E_0 \), and that the total rate obeys Sargent's rule \( \Gamma \propto E_0^{5} \) in the relativistic limit.

Why it matters

Beta decay was the empirical anchor for the weak interaction: the continuous electron spectrum forced Pauli to postulate the neutrino, and Fermi's 1934 contact theory turned that hypothesis into a quantitative prediction of the spectral shape. The Kurie plot linearises the spectrum so that the neutrino mass and the endpoint energy become slope-and-intercept quantities, which is exactly how tritium experiments still bound \( m_\nu \) today.

Sargent's rule, \( \Gamma \propto E_0^{5} \), is the archetypal phase-space scaling law. It explains the enormous spread of beta lifetimes across the nuclear chart from a single kinematic factor and provides the logic behind the comparative half-life \( ft \), which cleanly isolates the nuclear physics from the leptonic kinematics.

Assumptions

The interaction is a pointlike four-fermion contact couplingif dropped, a finite-range \( W \)-boson propagator \( 1/(q^2 - M_W^2) \) reintroduces momentum dependence in the matrix element and the low-energy factorisation into a constant times phase space fails at MeV/\(M_W\) order.

The decay is allowed: leptons carry away zero orbital angular momentumif dropped (forbidden transitions), the lepton wavefunctions cannot be replaced by their values at the nucleus and extra momentum powers \( (p_e^2 + p_\nu^2)^n \) enter the shape factor, curving the Kurie plot.

The nuclear matrix element is independent of lepton energy (leptons evaluated at the nucleus)if dropped, the long-wavelength approximation \( e^{i \mathbf{q}\cdot\mathbf{r}} \approx 1 \) breaks; recoil and weak-magnetism corrections of order \( E_e/M_N \) tilt the spectrum.

The neutrino is masslessif dropped, the neutrino phase space becomes \( p_\nu E_\nu \) with \( p_\nu = \sqrt{E_\nu^2 - m_\nu^2 c^4}/c \), which bends the Kurie plot sharply downward near the endpoint — the very effect used to measure \( m_\nu \).

The nucleus is infinitely heavy (no recoil kinetic energy)if dropped, a fraction \( \sim E_0/M_N c^2 \) of the released energy goes into nuclear recoil and the two-body-like endpoint \( E_0 \) shifts by a small correction.

Derivation

\[ H_{\text{int}} = \frac{G_F}{\sqrt{2}} \int d^3x \; \left[ \bar{\psi}_p \, \Gamma \, \psi_n \right] \left[ \bar{\psi}_e \, \Gamma \, \psi_\nu \right] \]

Fermi's ansatz: a Lorentz-invariant product of a hadronic current and a leptonic current at the same spacetime point, with coupling \( G_F \). \( \Gamma \) stands for the Dirac structure (V\(-\)A in the modern form). A

\[ \Gamma_{fi} = \frac{2\pi}{\hbar} \, \left| \langle f | H_{\text{int}} | i \rangle \right|^2 \, \rho(E_f) \]

Fermi's golden rule for the transition rate into a continuum, taken as a prior result. All energy dependence must come from the matrix element and the density of final states. A

\[ \psi_e(\mathbf{x}) \approx \psi_e(0), \qquad \psi_\nu(\mathbf{x}) \approx \psi_\nu(0) \]

Allowed approximation: the lepton de Broglie wavelengths (\( \sim 10^{2}\,\text{fm} \) at MeV energies) greatly exceed the nuclear radius (\( \sim \text{few fm} \)), so \( e^{i\mathbf{q}\cdot\mathbf{r}} \approx 1 \) and the plane waves are pulled outside the nuclear integral. B

\[ \langle f | H_{\text{int}} | i \rangle = \frac{G_F}{\sqrt{2}} \, M_{\text{nucl}} \; u_e^{\dagger}(0)\, u_\nu(0) \]

The nuclear matrix element \( M_{\text{nucl}} = \int \bar{\psi}_p \Gamma \psi_n \, d^3x \) separates from the leptonic plane-wave factors evaluated at the origin; \( M_{\text{nucl}} \) no longer depends on the sharing of energy between the leptons. B

\[ \sum_{\text{spins}} \left| u_e^{\dagger}(0) u_\nu(0) \right|^2 \equiv C \, \overline{|M_{\text{nucl}}|^2} \]

Summing over final and averaging over initial spins for an allowed transition gives an energy-independent constant \( C \) (containing \( |M_F|^2 \) and \( |M_{GT}|^2 \)); no lepton momenta survive the spin sum at leading order. C

\[ \rho(E_f)\, dE_f = \frac{V\, 4\pi p_e^2 \, dp_e}{(2\pi\hbar)^3} \cdot \frac{V\, 4\pi p_\nu^2 \, dp_\nu}{(2\pi\hbar)^3} \]

Three-body final state (recoil nucleus fixes momentum conservation, its kinetic energy neglected), so the phase space is the product of the electron and neutrino momentum-space volumes — the relativistic phase-space factor taken as a prior result. B

\[ E_0 = E_e + E_\nu \quad\Rightarrow\quad \frac{dp_\nu}{dE_f}\bigg|_{E_e} = \frac{1}{c}\,\frac{dE_\nu}{dE_f} = \frac{1}{c} \]

At fixed electron energy the total released energy \( E_0 \) is shared with the neutrino, so \( dE_\nu = dE_f \) and, for a massless neutrino \( E_\nu = p_\nu c \), the density of final states is evaluated at \( p_\nu = (E_0 - E_e)/c \). C

\[ p_\nu^2 \frac{dp_\nu}{dE_f} = \frac{(E_0 - E_e)^2}{c^3} \]

Substituting the massless-neutrino kinematics gives the characteristic \( (E_0 - E_e)^2 \) tail: all neutrino energy dependence is now expressed through the electron energy. B

\[ \frac{d\Gamma}{dp_e} = \frac{2\pi}{\hbar}\,\frac{G_F^2}{2}\, C\,\overline{|M_{\text{nucl}}|^2}\; \frac{(4\pi)^2 V^2}{(2\pi\hbar)^6}\; p_e^2 \, \frac{(E_0 - E_e)^2}{c^3} \]

Assemble golden rule, squared matrix element and phase space; the box volumes \( V \) cancel against the wavefunction normalisation \( 1/\sqrt{V} \) per lepton, leaving a finite rate. B

\[ \frac{d\Gamma}{dp_e} = \frac{G_F^2\,\overline{|M_{\text{nucl}}|^2}}{2\pi^3 \hbar^7 c^3}\; F(Z,E_e)\; p_e^2 \, (E_0 - E_e)^2 \]

Collect all constants into one prefactor and insert the Fermi (Coulomb) function \( F(Z,E_e) \), which corrects the outgoing-electron plane wave for the daughter nucleus's Coulomb field — a distortion of \( \psi_e(0) \) omitted by the free-plane-wave step. C

\[ \frac{d\Gamma}{dE_e} = \frac{d\Gamma}{dp_e}\,\frac{dp_e}{dE_e} = \frac{d\Gamma}{dp_e}\,\frac{E_e}{p_e c^2} \]

Change the differential variable using the relativistic relation \( E_e^2 = (p_e c)^2 + (m_e c^2)^2 \Rightarrow p_e\,dp_e = E_e\,dE_e/c^2 \), converting a momentum spectrum into the measured energy spectrum. B

\[ N(E_e) \equiv \frac{d\Gamma}{dE_e} = \frac{G_F^2\,\overline{|M_{\text{nucl}}|^2}}{2\pi^3 \hbar^7 c^5}\; F(Z,E_e)\; p_e E_e \,(E_0 - E_e)^2 \]

The final allowed-spectrum shape: a single kinematic envelope multiplying an energy-independent nuclear factor. A

\[ K(E_e) \equiv \sqrt{\frac{N(E_e)}{p_e E_e F(Z,E_e)}} = \sqrt{\frac{G_F^2\,\overline{|M_{\text{nucl}}|^2}}{2\pi^3\hbar^7 c^5}}\;(E_0 - E_e) \]

Divide out the analytically known factors \( p_e E_e F \) and take the square root: the Kurie function is exactly linear in \( E_e \), crossing zero at the endpoint \( E_0 \). Curvature in a real plot signals forbiddenness or a non-zero \( m_\nu \). A

\[ \Gamma = \int_{m_e c^2}^{E_0} N(E_e)\, dE_e \;\propto\; \int_{m_e c^2}^{E_0} p_e E_e (E_0 - E_e)^2 \, dE_e \]

Integrate over all electron energies (ignoring \( F \approx 1 \) and setting \( m_e \to 0 \) in the relativistic limit) to obtain the total rate. B

\[ \Gamma \;\propto\; \int_0^{E_0} E_e^2 (E_0 - E_e)^2 \, dE_e = \frac{E_0^{5}}{30} \]

In the extreme-relativistic limit \( p_e c \to E_e \); the dimensional integral of \( E_e^2(E_0-E_e)^2 \) scales as the fifth power of the endpoint. This is Sargent's rule. B

Result

\[ N(E_e) = \frac{G_F^2\,\overline{|M_{\text{nucl}}|^2}}{2\pi^3 \hbar^7 c^5}\; F(Z,E_e)\; p_e E_e (E_0 - E_e)^2, \qquad K(E_e) \propto (E_0 - E_e), \qquad \Gamma \propto E_0^{5} \]

Reading. The electron spectrum is a product of three energy-dependent pieces: the outgoing-electron momentum-space weight \( p_e E_e \), the neutrino phase-space tail \( (E_0 - E_e)^2 \), and the Coulomb factor \( F(Z,E_e) \) that pulls electrons toward low energy (\( \beta^- \), attractive daughter) or pushes them up (\( \beta^+ \), repulsive). Everything nuclear collapses into the single constant \( \overline{|M_{\text{nucl}}|^2} \). The Kurie transform strips the kinematics so a straight line falls to zero at \( E_0 \); the total rate scales as the fifth power of the energy release, so a decay with twice the \( Q \)-value is roughly thirty times faster.

Units check. \( G_F^2 \) has SI dimensions \( (\text{J}\,\text{m}^3)^2 = \text{J}^2\,\text{m}^6 \). Then \( G_F^2/(\hbar^7 c^5) \) carries \( \text{J}^2\text{m}^6 / [(\text{J s})^7 (\text{m/s})^5] = \text{J}^2\text{m}^6 \, \text{J}^{-7}\text{s}^{-7}\,\text{m}^{-5}\text{s}^{5} = \text{J}^{-5}\,\text{m}\,\text{s}^{-2} \). Multiplying by \( p_e E_e (E_0-E_e)^2 \) with \( [p_e]=\text{kg m/s}= \text{J s/m} \) and \( [E]^3 = \text{J}^3 \) gives \( \text{J}^{-5}\,\text{m}\,\text{s}^{-2} \cdot \text{J s m}^{-1} \cdot \text{J}^3 = \text{J}^{-1}\text{s}^{-1} \), i.e. \( N(E_e)\,dE_e \) has units \( \text{s}^{-1} \), a rate. Good.

Limiting cases

Low electron energy \( E_e \to m_e c^2 \): \( p_e \to 0 \), so \( N \to 0 \) unless the Coulomb factor \( F \) diverges; for \( \beta^- \) the attractive \( F \) keeps a finite non-zero count at threshold, for \( \beta^+ \) the repulsive \( F \) drives the count to zero.
Endpoint \( E_e \to E_0 \): \( N \propto (E_0 - E_e)^2 \to 0 \) quadratically; the Kurie plot reaches the axis linearly, which is why the endpoint is read from the straight-line intercept, not the (noisy) spectrum tail.
Massive neutrino: replace \( (E_0 - E_e)^2 \to (E_0 - E_e)\sqrt{(E_0 - E_e)^2 - m_\nu^2 c^4} \); the Kurie line bends down and meets the axis vertically at \( E_0 - m_\nu c^2 \).
Extreme-relativistic endpoint \( E_0 \gg m_e c^2 \): \( p_e c \approx E_e \) throughout, so \( \Gamma \propto E_0^5 \) exactly (Sargent).
Non-relativistic endpoint \( E_0 - m_e c^2 \ll m_e c^2 \): \( p_e \approx \sqrt{2 m_e (E_e - m_e c^2)} \) and the rate scales instead as \( \Gamma \propto (E_0 - m_e c^2)^{7/2} \).

Breaks when

Forbidden transitions. When the leading matrix element vanishes by angular-momentum or parity selection, the \( e^{i\mathbf{q}\cdot\mathbf{r}}\approx 1 \) step fails and a momentum-dependent shape factor \( S(p_e,p_\nu) \) multiplies the spectrum, curving the Kurie plot; the pure \( E_0^5 \) scaling is replaced by higher powers.
Near the endpoint with finite neutrino mass. The massless-neutrino phase space \( (E_0 - E_e)^2 \) is wrong within \( \sim m_\nu c^2 \) of \( E_0 \); the derivation's linear Kurie plot is exactly the quantity that breaks, and that break is the signal used to bound \( m_\nu \).
High momentum transfer \( q \sim M_W c \). The contact approximation collapses once the exchanged energy approaches the \( W \) mass; the propagator \( 1/(q^2 - M_W^2 c^2) \) can no longer be treated as the constant \( -1/M_W^2 c^2 \) that defines \( G_F \).
Heavy-nucleus, low-energy electrons. When \( Z\alpha/\beta \gtrsim 1 \) the non-relativistic Fermi function is inadequate; relativistic and screening corrections to \( F(Z,E_e) \) are required or the extracted \( M_{\text{nucl}} \) is biased.

Failure modes

Plotting \( \sqrt{N} \) instead of \( \sqrt{N/(p_e E_e F)} \). Forgetting to divide out the phase-space and Coulomb factors leaves a curved plot and a wrong endpoint; the Kurie function is defined precisely to remove them.
Using momentum \( p_e \) where energy \( E_e \) is meant. The extra Jacobian \( dp_e/dE_e = E_e/(p_e c^2) \) is dropped, turning a \( p_e^2(E_0-E_e)^2 \) momentum spectrum into a mislabelled energy spectrum.
Writing \( (E_0 - E_e) \) linear instead of squared. Confusing the phase-space power: the neutrino contributes \( p_\nu^2 \, dp_\nu/dE \propto (E_0-E_e)^2 \), not the first power.
Treating \( E_0 \) as the electron kinetic-energy endpoint vs total energy. Mixing \( E_e \) (total, includes \( m_e c^2 \)) with kinetic \( T_e = E_e - m_e c^2 \) shifts the endpoint by \( 511\,\text{keV} \).
Applying Sargent's \( E_0^5 \) to a non-relativistic decay. The fifth-power law needs \( E_0 \gg m_e c^2 \); for low \( Q \) the correct scaling is nearer \( E_0^{7/2} \) to \( E_0^{4} \).
Assuming \( \overline{|M_{\text{nucl}}|^2} \) is energy dependent. Reintroducing lepton momenta into an allowed matrix element double-counts phase space.

Discussion

The physical content of the result is a clean separation of scales. The nuclear structure, encoded in the Fermi and Gamow-Teller matrix elements \( |M_F|^2 \) and \( |M_{GT}|^2 \), is frozen at the nuclear energy scale (tens of MeV), while the emitted leptons live at the \( Q \)-value scale (a few MeV). Because the lepton wavelengths dwarf the nucleus, the leptons see only a point source and the spectrum shape is pure kinematics. This is why the comparative half-life \( ft \) — the measured half-life times the phase-space integral \( f(Z,E_0) \) — is the quantity nuclear physicists tabulate: it divides out the \( E_0^5 \) kinematics and exposes \( |M_{\text{nucl}}|^2 \) directly, isolating superallowed \( 0^+ \to 0^+ \) transitions that fix \( G_F \) and test CKM unitarity through \( V_{ud} \).

The Kurie plot is the experimental workhorse that follows from step 13. By transforming the raw counts \( N(E_e) \) into a quantity that is linear in \( E_e \), it converts endpoint determination and neutrino-mass searches into a slope-and-intercept measurement, where statistical leverage is far better than fitting a curved tail. A non-zero neutrino mass distorts the last few electronvolts below \( E_0 \); the KATRIN tritium experiment exploits exactly this, and the shape derived here — with the \( (E_0-E_e)\sqrt{(E_0-E_e)^2 - m_\nu^2 c^4} \) replacement — is the fit function it uses.

Sargent's rule connects to a much broader principle: decay rates are governed by available phase space, and for a three-body leptonic final state that means the fifth power of the energy release. The same \( E_0^5 \) logic reappears in muon decay \( \mu \to e \nu \bar\nu \), where \( \Gamma \propto m_\mu^5 G_F^2 \), and it is precisely this muon lifetime that provides the most accurate value of \( G_F \). The universality of \( G_F \) across nuclear beta decay and muon decay was the first quantitative hint of a single weak coupling.

At a deeper level the contact interaction is the low-energy limit of \( W \)-boson exchange: \( G_F/\sqrt{2} = g_w^2/(8 M_W^2 c^2) \). Fermi's dimensionful coupling, with its \( [\text{energy}]^{-2} \) character, is the fingerprint of an integrated-out heavy propagator, and its dimensionfulness is exactly what makes the theory non-renormalisable and drives the cross-section to violate unitarity near \( \sqrt{s} \sim M_W \). The modern V\(-\)A structure \( \Gamma = \gamma^\mu(1-\gamma_5) \) further encodes maximal parity violation, absent from Fermi's original vector ansatz and only forced by the 1957 experiments; it modifies the spin sum in step 5 but leaves the energy spectrum shape untouched, which is why Fermi's 1934 shape prediction survived the parity revolution intact.

Common misconceptions. The continuous spectrum is not evidence that energy conservation fails in beta decay — it is evidence of a third, unseen body (the neutrino) sharing the energy; the fixed endpoint \( E_0 \) is precisely the two-body \( Q \)-value the electron would carry if the neutrino took nothing. And the Coulomb function \( F(Z,E_e) \) is not a small correction: for medium and heavy nuclei it dramatically reshapes the low-energy end of the spectrum and must be divided out before any Kurie analysis.

Worked examples

Example A — Sargent-rule lifetime scaling. Two allowed decays have endpoints \( E_0^{(1)} = 1.5\,\text{MeV} \) and \( E_0^{(2)} = 4.5\,\text{MeV} \) with comparable nuclear matrix elements. Estimate the ratio of their partial half-lives, assuming the relativistic limit. A

\[ \Gamma \propto E_0^{5} \quad\Rightarrow\quad \frac{\Gamma_2}{\Gamma_1} = \left(\frac{E_0^{(2)}}{E_0^{(1)}}\right)^{5} \]

Sargent's rule with equal \( |M_{\text{nucl}}|^2 \); symbols first. A

\[ \frac{\Gamma_2}{\Gamma_1} = \left(\frac{4.5}{1.5}\right)^{5} = 3^{5} = 243 \]

Insert the numbers (dimensionless ratio, units cancel). A

\[ \frac{t_{1/2}^{(2)}}{t_{1/2}^{(1)}} = \frac{\Gamma_1}{\Gamma_2} = \frac{1}{243} \approx 4.1\times 10^{-3} \]

Half-life is inverse to rate. A

\[ \frac{t_{1/2}^{(2)}}{t_{1/2}^{(1)}} \approx \frac{1}{243} \]

Reading. Trebling the energy release makes the decay about 243 times faster. This single kinematic factor spans the observed range of beta half-lives from milliseconds to \( 10^{9} \) years. Units check. A pure ratio of energies raised to a power; dimensionless, as a lifetime ratio must be.

Example B — Kurie endpoint of tritium. Tritium decays \( ^3\text{H} \to \, ^3\text{He} + e^- + \bar\nu_e \) with kinetic-energy endpoint \( T_0 = 18.6\,\text{keV} \). For an electron of kinetic energy \( T_e = 15.0\,\text{keV} \), compute (i) the electron momentum term \( p_e c \) and (ii) the ratio \( N(15.0)/N(10.0) \) of spectrum heights at \( 15.0 \) vs \( 10.0\,\text{keV} \), neglecting the slowly varying \( F \). Use \( m_e c^2 = 511\,\text{keV} \). B

\[ E_e = T_e + m_e c^2, \qquad p_e c = \sqrt{E_e^2 - (m_e c^2)^2} \]

Relativistic energy-momentum relation; symbolic form. A

\[ E_e = 15.0 + 511 = 526.0\,\text{keV}, \quad p_e c = \sqrt{526.0^2 - 511^2} = \sqrt{15558}\;\text{keV} \approx 124.7\,\text{keV} \]

Numbers with units; \( 526^2 - 511^2 = (526-511)(526+511) = 15\times 1037 = 15555 \) keV\(^2\). A

\[ N(T_e) \propto p_e E_e (T_0 - T_e)^2 \]

Allowed spectrum with \( F \approx \text{const} \); here \( E_0 - E_e = T_0 - T_e \) since the \( m_e c^2 \) cancels in the difference. A

\[ p_e(10)c = \sqrt{521^2 - 511^2} = \sqrt{10320}\approx 101.6\,\text{keV}, \quad E_e(10)=521\,\text{keV} \]

Second point, same relation. A

\[ \frac{N(15)}{N(10)} = \frac{124.7 \times 526.0 \times (18.6-15.0)^2}{101.6 \times 521.0 \times (18.6-10.0)^2} = \frac{124.7\times526.0\times12.96}{101.6\times521.0\times73.96} \]

Insert all three factors; \( (3.6)^2=12.96 \), \( (8.6)^2=73.96 \). A

\[ \frac{N(15)}{N(10)} = \frac{8.50\times 10^{5}}{3.915\times 10^{6}} \approx 0.217 \]

Evaluate: numerator \( \approx 8.50\times10^5 \), denominator \( \approx 3.92\times10^6 \). A

\[ p_e c \approx 124.7\,\text{keV}, \qquad \frac{N(15.0)}{N(10.0)} \approx 0.22 \]

Reading. The spectrum falls steeply toward the endpoint because the \( (T_0 - T_e)^2 \) neutrino factor collapses; only \( 3.6\,\text{keV} \) of phase space remains at \( T_e = 15\,\text{keV} \). This vanishing count near \( T_0 \) is exactly why tritium's tiny \( Q \)-value makes it the premier neutrino-mass isotope: a large fraction of decays land close to the endpoint. Units check. \( p_e c \) in keV (momentum \(\times c\) as energy); the count ratio is dimensionless.

Problems

(Grade 1) An allowed beta emitter has endpoint \( E_0 = 2.0\,\text{MeV} \). By what factor does its decay rate change if a nuclear-structure change leaves \( |M|^2 \) fixed but the endpoint rises to \( 3.0\,\text{MeV} \)?
Solution
By Sargent's rule \( \Gamma \propto E_0^5 \). Ratio \( = (3.0/2.0)^5 = 1.5^5 = 7.59 \). The rate increases by about a factor of \( 7.6 \).
(Grade 2) Show that the electron energy at which the momentum spectrum \( d\Gamma/dp_e \propto p_e^2 (E_0-E_e)^2 \) peaks satisfies \( 2(E_0 - E_e)\,\frac{dE_e}{dp_e} = \frac{2(E_0-E_e)^2}{p_e} \) at the maximum, and hence find the peak condition in the ultra-relativistic limit \( E_e \approx p_e c \).
Solution
With \( E_e \approx p_e c \), write \( f(p_e) = p_e^2 (E_0 - p_e c)^2 \). Then \( f'(p_e) = 2 p_e (E_0 - p_e c)^2 + p_e^2 \cdot 2(E_0 - p_e c)(-c) = 2p_e(E_0 - p_e c)\left[(E_0 - p_e c) - p_e c\right] \). Setting \( f'=0 \) (non-trivial root) gives \( E_0 - p_e c - p_e c = 0 \Rightarrow p_e c = E_0/2 \), i.e. \( E_e = E_0/2 \). The momentum spectrum peaks at half the endpoint energy in the relativistic limit.
(Grade 3) Derive the total-rate integral \( I = \int_0^{E_0} E_e^2 (E_0 - E_e)^2 \, dE_e \) and confirm the \( E_0^5 \) scaling.
Solution
Substitute \( x = E_e/E_0 \), \( dE_e = E_0\,dx \): \( I = E_0^5 \int_0^1 x^2(1-x)^2\,dx \). The Beta integral \( \int_0^1 x^2(1-x)^2 dx = B(3,3) = \frac{\Gamma(3)\Gamma(3)}{\Gamma(6)} = \frac{2!\,2!}{5!} = \frac{4}{120} = \frac{1}{30} \). Thus \( I = E_0^5/30 \propto E_0^5 \), confirming Sargent's rule with the numerical coefficient \( 1/30 \).
(Grade 4) A Kurie plot of an allowed decay is straight and hits the axis at \( E_0 = 1.000\,\text{MeV} \) total energy. A second isotope's Kurie plot is straight down to \( 0.995\,\text{MeV} \) but then bends and meets the axis vertically at \( 0.9995\,\text{MeV} \). Interpret the bend quantitatively and estimate the neutrino mass.
Solution
A massive neutrino replaces \( (E_0 - E_e)^2 \to (E_0-E_e)\sqrt{(E_0-E_e)^2 - m_\nu^2 c^4} \). The Kurie plot then deviates from the straight line within \( \sim m_\nu c^2 \) of the true endpoint and meets the axis vertically at \( E_e = E_0 - m_\nu c^2 \). The near-endpoint distortion sets in around \( 0.995\,\text{MeV} \), i.e. \( \sim 5\,\text{keV} \) below \( E_0 \). The vertical intercept at \( 0.9995\,\text{MeV} \) gives \( m_\nu c^2 = E_0 - 0.9995\,\text{MeV} = 0.5\,\text{keV} \). (Real tritium limits are \( < 0.8\,\text{eV} \); this exaggerated value is illustrative.)
(Grade 5) For a superallowed \( 0^+ \to 0^+ \) transition only the Fermi matrix element contributes, \( |M_F|^2 = 2 \) (for \( T=1 \)). Given the phase-space integral \( f = 3000 \) and a measured half-life \( t_{1/2} = 3.0\,\text{s} \), estimate \( ft \) and comment on how it constrains \( G_F \).
Solution
The comparative half-life is \( ft = f \cdot t_{1/2} = 3000 \times 3.0\,\text{s} = 9.0\times10^3\,\text{s} \), close to the canonical superallowed value \( ft \approx 3070\,\text{s} \) when \( f \) and \( t \) are defined consistently (this simplified \( f \) is illustrative). The relation \( ft = \frac{K}{G_F^2\,|M_F|^2} \) with \( K = 2\pi^3\hbar^7 \ln 2 /(m_e^5 c^4) \) shows \( ft \) is independent of the isotope for pure Fermi decays: measuring several superallowed \( ft \) values and averaging fixes \( G_F |V_{ud}| \). Because \( |M_F|^2 = 2 \) is known exactly from isospin, the extracted \( G_F \) tests CKM unitarity via \( |V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 1 \). A constant \( ft \) across nuclei is the experimental signature that \( G_F \) is universal.