Fermi Beta Decay Spectrum

Equivalent forms

\sqrt{N(E)/[p E F(Z,E)]} \propto (Q - E)

The shape of the curve betrays the neutrino's existence — long before it was detected.

Symbol	Name	SI unit	Dimension	Range
$dN/dE$	Differential rateoutput Number of betas emitted per unit energy per second	s⁻¹·J⁻¹	T⁻¹·M⁻¹·L⁻²·T²	0 – 1
$E$	Electron kinetic energy Kinetic energy of the emitted beta particle	J	M·L²·T⁻²	0 – 1e-13
$Q$	Endpoint energy Maximum available kinetic energy	J	M·L²·T⁻²	1e-15 – 1e-12

Unit systems

Symbol	SI	CGS	Natural
$dN/dE$	s⁻¹·J⁻¹	s⁻¹·erg⁻¹	MeV⁻¹
$E$	J	erg	MeV
$Q$	J	erg	MeV

Where it holds

Allowed transitions; forbidden decays require additional nuclear matrix-element factors. Neutrino mass assumed zero in the standard form.

Dimensional analysis

dN/dE \to [T^{-1}\cdot M^{-1}\cdot L^{-2}\cdot T^{2}] = [M^{-1}\cdot L^{-2}\cdot T]

G_F^{2} |M|^{2} / (\hbar ^{7} c^{5}) \times pE(Q-E)^{2}

— Fermi constant has units

[E\cdot L^{3}]

, yielding overall

[M^{-1}\cdot L^{-2}\cdot T]

Consistent.

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Discovery

Enrico Fermi · 1934

Fermi adapted Pauli's neutrino hypothesis and built the first quantum field theory of beta decay, predicting the spectral shape.

Try this

Why do beta particles emerge with a continuous range of energies instead of a single value?

Use Fermi's theory to express the differential rate dN/dE of beta electrons emitted with kinetic energy E, given endpoint Q.

Research status: stable

Real-world applications

Neutrino mass measurement (KATRIN, Project 8)
Nuclear medicine dosimetry (Y-90, Sr-90 beta sources)
Radiocarbon dating spectra (C-14)
Reactor antineutrino flux modeling for oscillation experiments

Common misconceptions

The continuous spectrum was once thought to violate energy conservation — Pauli's neutrino postulate saved it.
The endpoint shape is sensitive to neutrino mass: non-zero mass produces a kink near $E = Q$ .
F(Z,E) is essential for low-energy electrons in heavy nuclei; ignoring it distorts the spectrum substantially.

Experimental verification

Kurie plots (1936) verified the (Q-E) linearity; tritium beta endpoint experiments (KATRIN) use the spectral shape near Q to constrain neutrino mass to <

0.8\,\mathrm{eV/c}^{2}

.

Derivation

From Fermi's Golden Rule,

dW = (2\pi /\hbar )|M|^{2} d\rho

.

The two-particle phase space for electron (momentum p) and antineutrino (momentum q) with total energy fixed gives

d\rho \propto p^{2} (Q-E)^{2} dp/(2\pi ^{2}\hbar ^{3})^{2}

.

Including the Coulomb correction F(Z,E) and converting dp to dE via

E dE = p c^{2} dp

yields the stated spectrum.

Limiting cases

E \to 0

⟶

dN/dE \to 0

Low-momentum electrons have vanishing phase space (pE factor).

E \to Q

⟶

dN/dE \to 0

Neutrino phase space

(Q-E)^{2}

closes at the endpoint.

E \approx Q/3

⟶ dN/dE peaksBalance between electron momentum and neutrino phase space.

What if…

What if the neutrino had a 1 eV mass?

Spectrum endpoint truncates at $Q - m_\nu c^{2} and$ shape changes from quadratic to a sharp kink.

What if F(Z,E) were ignored for high-Z?

Low-energy region is mis-predicted by orders of magnitude due to Coulomb attraction enhancement.

What if the decay were a two-body process?

Electron would emerge with a single discrete energy $Q -$ recoil, contradicting observations.

1

Spectrum peak location for tritium

Given ·

Q:: 2.94e-15

Find · E_peak

Steps

Step 1: For low-Z, non-relativistic case, $pE \propto E^(3/2)$ .
Step 2: Maximize $E^(3/2)(Q-E)^{2}$ → derivative zero at $E = 3Q/7 \approx 0.43\,\mathrm{Q}$ .
Step 3: With $Q = 18.6\,\mathrm{keV}$ (tritium), E_peak $\approx 8\,\mathrm{keV}$ (the exact 6 keV requires F(Z,E)).

Result · Setting

d/dE[pE(Q-E)^{2}] \approx 0 (non-rel

,

p \propto \sqrt{E})

gives E_peak

\approx Q/3 \approx 9.8e-16\,\mathrm{J} \approx 6.1\,\mathrm{keV}

.

2

Endpoint shape sensitivity

Given ·

Q:: 2.94e-15
E:: 2.9e-15

Find · dN/dE

Steps

Step 1: Compute $(Q-E) = 2.94e-15 - 2.9e-15 = 4e-17\,\mathrm{J}$ .
Step 2: $(Q-E)^{2} = 1.6e-33\,\mathrm{J}^{2}$ .
Step 3: Multiplying by pE F(Z,E) gives the absolute rate near the endpoint — vanishing as expected.

Result · Near endpoint

dN/dE \propto (Q-E)^{2} \approx (4e-17)^{2} = 1.6e-33

(in

J^{2}

units before constants).

Radioactive Decay Law→Mass–Energy Equivalence→Fermi's Golden Rule→