Pair Production Threshold Energy

Equivalent forms

hf_min = 2 m_e c²

The simplest expression of mass-energy equivalence in action: 1.022 MeV in, electron + positron out.

Symbol	Name	SI unit	Dimension	Range
$E_γ^min$	Threshold photon energyoutput Minimum gamma-ray energy for pair production	J	M·L²·T⁻²	0 – 1e-12
$m_e$	Electron rest mass Invariant mass of electron / positron	kg	M	9.1093837015e-31 – 9.1093837015e-31
$c$	Speed of light Speed of light in vacuum	m/s	L·T⁻¹	299792458 – 299792458

Unit systems

Symbol	SI	CGS	Natural
$E_γ^min$	J	erg	MeV
$m_e$	kg	g	MeV
$c$	m/s	cm/s	1

Where it holds

Strictly valid for photon-induced pair production. For other reactions, replace

2\,\mathrm{m}_e c^{2}

with the sum of produced rest energies.

Dimensional analysis

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

2\,\mathrm{m}_e c^{2} \to [M][L\cdot T^{-1}]^{2} = [M\cdot L^{2}\cdot T^{-2}]

Both sides are energy.

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Discovery

Paul Dirac & Carl Anderson · 1933

Dirac's 1928 equation predicted antiparticles. Anderson discovered the positron in 1932 cosmic-ray cloud chamber tracks, and pair production from gamma rays was confirmed shortly after.

Try this

What's the minimum photon energy needed to make matter from light?

What is the minimum photon energy to produce an electron-positron pair in the field of a heavy nucleus?

Research status: stable

Real-world applications

PET medical imaging (reverse process: positron-electron annihilation)
Gamma-ray attenuation in shielding (dominant above $\sim 10\,\mathrm{MeV})$
Particle-accelerator physics (electron-positron colliders)
Cosmic-ray cascades in the atmosphere

Common misconceptions

Pair production in vacuum is forbidden by momentum conservation — a third body is required.
The 'pair' can be any particle-antiparticle pair, but electron-positron has the lowest threshold.
Just above threshold, the cross section is very small — significant pair production requires E ≫ $2\,\mathrm{m}_e c^{2}$ .

Experimental verification

Bubble-chamber photographs show characteristic V-shaped electron-positron tracks from high-energy gammas hitting nuclei. The 1.022 MeV threshold is exploited in PET imaging (detecting back-to-back 511 keV annihilation photons).

Derivation

Conservation of 4-momentum: in the center-of-mass frame, the produced pair must be at rest at threshold, with total energy

2\,\mathrm{m}_e c^{2}

.

For a photon hitting a heavy nucleus, the lab-frame threshold equals

2\,\mathrm{m}_e c^{2}

because the heavy nucleus absorbs the recoil momentum at negligible energy cost.

Limiting cases

Free photon in vacuum⟶ ForbiddenA photon alone cannot produce a pair — momentum conservation requires a third body (nucleus or another photon).

Nuclear field⟶

E \ge 2\,\mathrm{m}_e c^{2}

Standard threshold for heavy nuclei (recoil negligible).

Electron field⟶

E \ge 4\,\mathrm{m}_e c^{2}

Atomic-electron 'triplet production' requires higher threshold due to lighter recoil partner.

What if…

What if we wanted to produce a muon pair instead?

Threshold becomes $2\,\mathrm{m}_\mu c^{2} \approx 211\,\mathrm{MeV}$ — $207\times$ higher because muons are $207\times$ heavier.

What if the photon had only 0.5 MeV?

Pair production is impossible; the photon will instead Compton-scatter or photoelectrically eject electrons.

1

Threshold energy in MeV

Given ·

m e:: 9.1093837015e-31
c:: 299792458

Find ·

E_\gamma ^\min

Steps

Step 1: $m_e c^{2} = 9.109e-31 \times 8.988e16 = 8.187 \times 10^{-14} J = 0.511\,\mathrm{MeV}$ .
Step 2: Multiply by 2: $E_\gamma ^\min = 1.637 \times 10^{-13} J = 1.022\,\mathrm{MeV}$ .
Step 3: Below 1.022 MeV, pair production is impossible regardless of field strength.

Result ·

E_\gamma ^\min = 2 \times 9.109e-31 \times (3e_{8})^{2} = 1.637 \times 10^{-13} J = 1.022\,\mathrm{MeV}

Mass–Energy Equivalence→klein nishina cross sectionCompton Scattering Wavelength Shift→