Pair Production Threshold

In one sentence: A very energetic particle of light can turn into matter: an electron and its twin with the opposite charge, a positron, if it has enough energy.

$E_\gamma \ge 2 m_e c^2 \approx 1.022 \text{ MeV}$

Read it out loud: “E gamma is greater than or equal to 2 times m e c squared, which is about 1.022 MeV. E gamma is the energy of the photon, a particle of light, sometimes called a gamma ray. m e is the mass of one electron. c is the speed of light, and c squared means c multiplied by itself. The symbol for greater than or equal to means at least this big. MeV means million electron volts.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Photons and gamma rays: A photon is one packet of light or other electromagnetic radiation. Visible light photons carry only a few electron volts each. A gamma ray is a photon with very high energy, from radioactive nuclei and from space. Gamma rays carry about a million electron volts or more per photon, hundreds of thousands of times the energy of a visible photon.
Powers of ten, like $10^{6}$ and $10^{-13}$: Particle energies are written with powers of ten. The prefix mega, M, means $10^{6}$ , a million, so 1 MeV is one million electron volts. A negative power means divide by ten that many times. $10^{-13}$ is a tenth of a trillionth, which is written 0.0000000000001. So 1.6 times $10^{-13}$ joules is 0.00000000000016 joules, with twelve zeros after the decimal point. When multiplying powers of ten, add their exponents.
Electrons, positrons and antiparticles: An electron is a tiny particle with a negative electric charge. A positron is its antiparticle: it has the same mass as the electron and the same size of charge, but the charge is positive. When an electron and a positron meet, they can annihilate each other and turn back into photons. Pair production is the reverse of that process.
Rest energy and $mc^2$: Einstein’s rule $E = mc^2$ says a particle standing still holds an energy equal to its mass times the speed of light squared. The rest energy of an electron is 0.51099895 MeV. Making an electron and a positron from energy therefore costs at least the rest energy of both, which is $2m_e c^2$ .
Momentum and its conservation: Momentum measures how much motion a thing carries. A photon carries momentum even though it has no mass, with $p = E/c$ . Momentum must be conserved, which means the total momentum before a reaction equals the total momentum after it. A reaction that breaks this rule cannot happen.
Conservation of energy: Energy cannot appear from nothing or disappear. In pair production the photon’s energy must equal the rest energy of the new electron and positron, plus whatever motion energy they carry. If the photon has less energy than the rest energy of the pair, the reaction cannot happen.
Atomic nucleus: An atomic nucleus is the dense center of an atom, made of protons and neutrons. It is far heavier than an electron. Because it is so heavy, a nucleus can take up momentum from a photon while picking up almost no energy, so it recoils only slowly.

What each symbol means

Symbol	Meaning
$E_\gamma$	The energy of the photon, also called a gamma ray, in MeV (million electron volts) or in joules (J). The threshold of about 1.022 MeV is the smallest photon energy that can make an electron and a positron.
$m_e$	The mass of one electron, $9.1093837 \times 10^{-31}$ kg. The positron has exactly the same mass. The negative power of ten shows that the number is very small: the decimal point is followed by 30 zeros before the digits 91 begin.
$c$	The speed of light in vacuum, $299792458$ m/s, exact by definition. In this formula it turns the rest mass $m_e$ into an energy, through $m_e c^2$ .

The formula piece by piece

$m_e$: The mass of one electron. Each particle that the photon makes, the electron and the positron, has this same mass, so the mass is counted twice in the threshold.
$m_e c^2$: The rest energy of one electron, $0.51099895$ MeV. It is the energy an electron holds while standing still, and the photon must supply this much energy to make one electron.
$2 m_e c^2$: The rest energy of an electron and a positron together. Since both have the same mass, the total is twice the rest energy of one, which is $1.0219979$ MeV. This is the threshold energy.
$E_\gamma \ge 2 m_e c^2$: The photon energy must be at least the threshold. Any extra energy above the threshold turns into motion energy of the pair.
$E_\gamma \ge 2 m_e c^2 \approx 1.022 \text{ MeV}$: The whole formula in one sentence: a photon can make an electron and a positron only if its energy is at least twice the rest energy of an electron, which is about 1.022 MeV. Any photon with less energy cannot make the pair, however many such photons arrive.

What the formula is doing, and why

What goes in and what comes out. A photon with energy $E_\gamma$ travels near a heavy nucleus. If its energy is large enough, the photon disappears and two particles appear: an electron and a positron. The formula gives the smallest photon energy that can do this, which is $2 m_e c^2$ . Both particles are made from energy, and each one needs its rest energy $m_e c^2$ .

Why is the threshold twice the rest energy? The electron and the positron have the same mass, so the photon must pay the rest energy of both. The electron’s rest energy is $m_e c^2 = 0.51099895$ MeV, so the threshold is $2 \times 0.51099895 = 1.0219979$ MeV, which is about 1.022 MeV. Any energy above that threshold becomes motion energy of the electron and positron.

Why is a nucleus needed nearby? A photon in empty space cannot make a pair, even with enough energy. Momentum must also be conserved, and a photon alone cannot share its energy and momentum in the right way. A nearby nucleus takes up the extra momentum and recoils. Because the nucleus is very heavy, it takes almost no energy, so the energy still works out.

Where it applies. One photon with less energy than about 1.022 MeV can never make a pair near a nucleus. Above this energy, pair production becomes one of the main ways gamma rays lose energy in dense materials such as lead. That is why heavy materials are used to shield against gamma rays.

An everyday picture

Think of a club where two guests can enter only if they pay the full price of two tickets, which is $2 m_e c^2$ . A guest with 0.8 MeV is short of the price and cannot enter. A guest with more than the full price can enter, and any spare money is handed to the new pair as motion energy, the energy they use to move apart.

Worked example: Gamma rays at 0.8 MeV and 1.5 MeV

Two gamma-ray photons travel toward a lead nucleus. One has energy 0.8 MeV, and the other has energy 1.5 MeV. Find out which one can make an electron and a positron. Use the electron rest energy $m_e c^2 = 0.51099895$ MeV, so the threshold is $2 m_e c^2$ .

Find the threshold. The threshold is the rest energy of the two particles added together. Doubling the electron rest energy gives $2 \times 0.51099895 = 1.0219979$ MeV, which we round to 1.022 MeV. Each photon must be compared with this number, and any photon below 1.022 MeV is not enough to make the pair.
Test the 0.8 MeV photon. Subtract the threshold from its energy: $0.8 - 1.022 = -0.222$ MeV. The negative result means the photon is 0.222 MeV short of the price of the pair. So this photon cannot make an electron and a positron, however close it passes to the nucleus.
Test the 1.5 MeV photon. Subtract the threshold again: $1.5 - 1.022 = 0.478$ MeV. The positive result means the photon can pay for the pair and still has 0.478 MeV left over. So this photon can make a pair, provided a nucleus is nearby to take up the extra momentum.
Share the leftover energy. The 0.478 MeV left over becomes motion energy of the electron and the positron. The nucleus is so heavy that it takes almost none of this energy, so the electron and the positron share it between themselves. The two particles then fly apart, each carrying part of that energy.
Convert the threshold to joules to see the scale. One MeV is $10^{6}$ eV, and one eV is $1.602176634 \times 10^{-19}$ J. Multiplying the powers of ten adds their exponents: $10^{6} \times 10^{-19} = 10^{-13}$ , so 1 MeV is $1.602176634 \times 10^{-13}$ J. The threshold of 1.022 MeV is therefore about $1.637 \times 10^{-13}$ J, a very small amount of energy for one photon.

Answer: The 0.8 MeV gamma ray cannot create an electron and a positron. The 1.5 MeV gamma ray can, and about 0.478 MeV is left over as motion energy of the pair. The threshold is about 1.022 MeV, or about $1.637 \times 10^{-13}$ J, for every nucleus to a very good approximation.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Expecting a lone photon to make a pair in empty space. Momentum must be conserved, so a nearby nucleus is needed to absorb the extra momentum.
Using the proton mass. The pair here is an electron and a positron, so the electron mass sets the threshold. A proton is about 1,836 times heavier than an electron, which would give a threshold far larger than 1.022 MeV.
Thinking the energy above the threshold is lost. It becomes the kinetic energy of the electron and the positron, so none of it is wasted.
Comparing joules with MeV directly. Convert first, using 1 MeV $= 1.602176634 \times 10^{-13}$ J, before comparing two energies.

Where you will meet it

Gamma-ray shielding design, where pair production decides how strongly high-energy gamma rays are absorbed by dense materials such as lead.
Particle physics experiments, which create electron and positron pairs from high-energy photons to study matter and antimatter.
Medical and nuclear physics, where pair production is one of the ways high-energy gamma rays deposit their energy in matter and in detectors.

Questions people ask

What is a positron?

A positron is the antiparticle of the electron. It has the same mass as an electron and the same size of charge, but the charge is positive. When a positron meets an electron, the two can annihilate and turn back into photons.

Why does the photon need a nucleus nearby?

Momentum must be conserved. A photon has momentum $E/c$ but no rest mass, so a lone photon cannot give the pair both its energy and its momentum. A nearby nucleus takes up the extra momentum while taking almost none of the energy.

Does the threshold change for different nuclei?

To a very good approximation, no. The threshold is $2 m_e c^2$ for any nucleus. The nucleus affects how likely the process is, but not the minimum energy needed. The true threshold is a tiny bit above $2 m_e c^2$ , because the nucleus takes a small share of the energy as it recoils.

Energy-Momentum Relation: One equation ties together a particle’s total energy, its momentum (the amount of motion it carries) and its mass, and it works at every speed.
Compton Scattering: When an X-ray bounces off an electron, it gives away some of its energy, so its wavelength gets longer, and the size of that change depends only on the angle it bounces through.
Photon Momentum: A photon has no mass, yet it still carries a small push, so light can push on objects, and light with a shorter wavelength pushes harder for each photon.
Photoelectric Effect: Light can knock electrons out of a metal, but only if each tiny packet of light, called a photon, carries enough energy to pay the cost of leaving, and any extra energy becomes the electron’s speed.