The partner that sets the price of a pair

Mass is a form of energy drew the distinction between energy and a source of energy, and the test of mass and energy that needs a ruler showed how closely $E = mc^2$ has been checked, with a nucleus emitting a gamma ray. The reverse process, turning energy into matter, looks equally simple and carries a number everyone knows. An electron has a rest energy of 0.511 MeV, so making an electron and a positron from light takes 1.022 MeV. Carl Anderson’s discovery of the positron in 1932, and Blackett and Occhialini’s cloud-chamber pictures of pairs a year later, gave that figure its status in textbooks, and it is the threshold used in gamma-ray shielding calculations and in nuclear physics everywhere.

The number is correct, and it is the right answer to a slightly different question from the one it appears to answer. It is a good approximation for one special case. A photon alone, with no other particle nearby, cannot make a pair at any energy at all. With another particle nearby, the threshold depends on that particle’s mass, and it varies by a factor of two across the particles a photon could encounter in ordinary matter. The fact has observable results, including a third track in old bubble-chamber photographs.

Why a photon alone cannot do it #

An energetic photon has energy $E$ and momentum $E/c$. An electron–positron pair with zero relative motion, at some total speed, has total energy $E' = \gamma 2mc^2$ and momentum $\gamma 2m\beta c$. Their relation is the familiar one for any object of rest mass $2m$:

$$E'^2 = (p c)^2 + (2mc^2)^2.$$

Put the two on a graph, and the difficulty shows itself. For any given momentum, a pair has more energy than a photon with the same momentum, by an amount that shrinks at high energies and never reaches zero. So no photon can become a pair while conserving both energy and momentum: it has too much momentum for its energy. It would be conserving energy while losing momentum, and that has no place to go. Looked at in another way, the photon has zero invariant mass, and a pair at rest has invariant mass $2m$, and no frame or energy can change the invariant mass of a single photon. The invariant that survives a boost showed why that quantity cannot change.

Something else has to be present to take up the difference. The field of a nucleus works, and so does an electron, or a second photon, the last of which the gamma ray its own kind cannot absorb treats in its astronomical setting. In each case, the invariant mass of the system with two objects can be large, and the only question is what that requires.

One formula, from the invariant mass #

Let a photon of energy $E$ strike a partner of mass $M$ at rest. Before the collision the total energy is $E + M$ and the total momentum is $E$, in units with $c = 1$. The invariant mass squared is therefore

$$s = (E + M)^2 - E^2 = M^2 + 2EM.$$

After the reaction, at threshold, the partner and the pair must all be at rest together in the frame where total momentum vanishes. Anything else would use up more energy than is needed. So the invariant mass of the final state is simply $M + 2m$. Setting these equal gives

$$E_{\text{th}} = \frac{(M + 2m)^2 - M^2}{2M} = 2m\left(1 + \frac{m}{M}\right).$$

The first factor is the rest energy of the pair, and the second factor is the price of moving things. When $M$ is much larger than $m$, as for any nucleus, the second factor is effectively 1, and the textbook’s value is correct: lead’s correction is less than a part in a hundred thousand. For a proton, the correction is 0.054 per cent. For a free electron, $M = m$ and the threshold is exactly $4m$, twice the textbook figure, or 2.044 MeV. As $M$ falls further, the threshold rises without limit, and for the limit of a massless partner at rest the formula has no solution, in agreement with the discussion above, since such a thing cannot be at rest.

It seems at first paradoxical that a small partner should demand more. An electron has a smaller rest energy than a proton and so seems like it should need less energy to set in motion. The opposite occurs because the question is one of momentum.

Momentum for free, energy at a price #

At threshold everything moves together at the speed of the centre of momentum. With total momentum $E$ and total energy $E + M$, that speed is $\beta = E/(E + M)$. An electron partner at threshold therefore moves at $\beta = 4/5$, while a lead nucleus moves at a tiny fraction of a per cent of $c$. Now compare what each costs.

Momentum is proportional to speed and kinetic energy to its square. An object of large mass can therefore absorb a given momentum at a very low speed, and so for almost no kinetic energy: $T = p^2/2M$ in the non-relativistic case. A lead nucleus takes up the 1.02 MeV/c of a threshold photon and moves away with about 2.7 eV, less than a typical chemical bond and a very small amount compared with the MeV energies present. That is what an effectively infinitely massive partner does: it is a momentum sink. The push that needs nothing to push against discussed the same asymmetry between energy and momentum at much lower speeds.

An electron cannot do this. It takes 0.341 MeV as its share of kinetic energy, and the pair it stays with, moving at the same speed, has a further 0.681 MeV, because they are carried along with it. Those two together make up the additional 1.022 MeV that doubles the threshold. Each number in the bar chart was checked to satisfy energy conservation: partner plus pair, kinetic plus rest, comes to the photon’s energy in every row.

The physical result here has nothing specific to do with light. It is the same reason why a tennis ball rebounds from a wall with all its energy, and the wall has absorbed all the momentum for no energy at all. A collider’s advantage comes from the same fact, in reverse: in the collision that wastes most of the energy the target at rest is forced to carry momentum, and most of the beam’s energy goes to moving the debris forward. Rearrange the same expression, with a heavier and heavier beam energy, and it gives the square-root law that drove accelerator builders to collide head-on.

The third track #

The electron threshold has been directly seen. A photon striking an atomic electron in a bubble chamber produces a pair, and the struck electron receives enough energy to show as a third track from the same point. It is called triplet production for the three tracks. Pair production on the nucleus shows only two, as a neat V, since a nucleus takes up too little energy to leave a mark. Bubble-chamber physicists used the triplet as a check on photon energies at about the time that the first accelerator-produced gamma-ray beams became available. The minimum of 2.04 MeV is sharp: triplets do not appear below it, even though pairs on the nucleus already are common by that energy.

A reason for it being comparatively rare appears in the cross-section. Pair production uses the field of the partner, and a rate is the square of the amplitude, so the cross-section on a nucleus of charge $Z$ goes as $Z^2$. Each of the $Z$ atomic electrons gives a contribution proportional to 1 in the same units, so together they contribute $Z$ and not $Z^2$. In lead the nucleus therefore dominates by about 82 to 1. The expensive partner is also the rare one. In hydrogen the two are nearly equal, with the single electron contributing about as much as the single proton does at high energies. That makes hydrogen an interesting case for high-energy physics, since it allows the two to be separated, and a hydrogen bubble chamber was where the formula was most directly tested.

At very high energy, the rate stops rising. Pairs are then created at such a distance from the nucleus that the atom’s other electrons screen its charge, an effect that Bethe and Heitler included in their calculation in 1934. This limit is expressed as the radiation length, the distance over which a very high energy electron loses most of its energy. It governs how showers grow in any calorimeter. Near a strong enough nucleus, a further limit occurs in which no real photon at all is needed: the nucleus that would make positrons from nothing treats it.

One formula for many prices #

Nothing above is special to photons or electrons. For any target at rest with mass $M$, and final-state rest masses that sum to $\Sigma m$ including the target, a massless projectile’s threshold energy is $E_{\text{th}} = ((\Sigma m)^2 - M^2)/2M$. A massive projectile has a slightly different version because its own mass enters the invariant.

The light partner is the extreme case throughout. A photon making a muon pair costs 211 MeV next to a lead nucleus. Next to an electron, it costs 43.9 GeV: the threshold formula gives $2m_\mu(1 + m_\mu/m_e)$, with the ratio $m_\mu/m_e = 207$. Nothing like this high energy is necessary to supply the muons’ rest energy. Almost all of it goes to moving the very light electron, which must carry a momentum that it could never hold at low speed. That is the same physics that makes reactions on electrons kinematically forbidding for almost anything heavy.

The most famous case in the history of accelerators is the antiproton. Conservation of baryon number means that a proton–antiproton pair must be made, since an antiproton cannot be made by itself. So $p + p \to p + p + p + \bar{p}$. In a proton beam on a stationary proton, the threshold is a kinetic energy of $6m_pc^2$, or 5.63 GeV, for a total of seven proton rest energies in the projectile. The Bevatron at Berkeley, named for its energy in billions of electronvolts, was designed in the late 1940s with that number in mind. Owen Chamberlain, Emilio Segrè and their collaborators identified the antiproton there in 1955 by time of flight and magnetic deflection, at an energy not far above threshold. This is why the machine’s specification had nothing to do with the antiproton’s rest energy of about 0.94 GeV alone.

The photoproduction of a pion on a proton is another exact case, at 144.7 MeV, slightly above the pion’s own mass of 135 MeV by just the factor predicted. Its reversal in space, where cosmic-ray protons of very high energy strike microwave-background photons, is a similar invariant-mass condition seen from a different frame. What is the same in all these cases is that nothing else is needed: only the invariant mass and a partner’s mass.

Thresholds in thermal matter #

An electron-positron pair can also be made in a hot gas, by colliding photons, in which case neither photon is fixed and the invariant mass depends on the angle between them. The heat that turns into matter deals with that. The threshold on two photons is $E_1E_2(1 - \cos\theta) \geq 2m^2$, the same expression that gives the pion’s mass. It is a fine illustration of what a threshold is: the condition for reaching a fixed invariant, and how much energy this requires depends on how the other particle moves.

Moving partners change the threshold, since a moving target contributes to the invariant mass. If an electron is moving toward the photon, it offers more invariant mass for the same photon energy, and the threshold falls with the Doppler factor. That is the reason why nuclear reactions that are forbidden on stationary targets can occur in a hot gas or a star, with the fast tail of the distribution supplying the needed extra. It is also why the threshold is so sharp only for a target at rest in the lab. A nucleus’s bound electrons have an orbital motion of their own, so the observed threshold in matter is smeared slightly compared with what the formula shows. The binding energy of the electrons is itself a further small correction, and it means an electron’s threshold in matter is close to, but not precisely, 2.044 MeV.

Where the positrons come from in a hospital #

A gamma ray of a few MeV in a patient or a block of lead is unremarkable, and pair production is one of three processes competing for it, with their shares set by energy and atomic number. At low energy the photoelectric effect dominates, then Compton scattering takes over at around an MeV; above about 5 MeV in lead and well above 20 MeV in water, pairs win. Radiation therapy machines deliberately use beams with energies of 6 to 18 MeV. For them the threshold has a practical meaning. Above 1.02 MeV, each pair’s positron soon annihilates with an electron and produces two 511 keV photons, and those secondary photons deposit energy elsewhere in the patient, so the planning calculation must include them. Bremsstrahlung from the same fast particles adds to this.

Positron emission tomography runs on the reverse process: an injected isotope emits a positron, which annihilates, and two back-to-back 511 keV photons identify the line along which it happened. Near threshold pair production has an analogue here too, for a photon in a detector crystal may make a pair, and when one of the annihilation photons then escapes from the crystal, a line appears 511 keV below the full energy, and a second escape gives another line at 1,022 keV below it. Gamma-ray spectroscopists call these the single-escape and double-escape peaks, and anyone who has calibrated a germanium detector has seen them. They are as direct a measurement of the pair’s rest energy as can be had, on a laboratory bench, and they follow exactly the kinematics above since a germanium nucleus is a nearly infinite momentum sink.

What the formula does not contain #

A threshold is a condition from kinematics, and it says nothing about how probable a reaction is. Near threshold, a cross-section is typically vanishingly small, since the outgoing particles have no phase space, and rises slowly from zero. The energy at which a process becomes significant can be much larger than the threshold. Pair production in lead becomes important compared with Compton scattering only at about 5 MeV and dominates beyond about 10 MeV. Thresholds also assume that final particles can be free. Near a nucleus, the positron feels repulsion and the electron attraction, and this Coulomb distortion changes the spectrum near threshold in a way not contained in the formula.

The cross-section figure uses a high-energy approximation, with a rough adjustment for the electrons’ contribution, and it is unreliable below about 20 times the electron’s rest energy. Both curves have been drawn only above that point. The picture of a threshold as a sharp line is, finally, an idealisation for free particles at zero temperature, and a thermal or bound target smears the edge.

Still open: whether a photon can make matter out of nothing but other light #

The purest version of this process has only recently been tested. The Breit–Wheeler reaction, two real photons producing a pair, was calculated in 1934, but the available beams of photons have far too low a density to make it observable by any direct experiment. In 2021 the STAR collaboration at Brookhaven reported pairs from the intense photon clouds that surround colliding gold nuclei, and argued they were real photons in the relevant sense, though there is continuing discussion about how far the photons in those fields are really on their mass shell. A cleaner test, with a laser beam against a gamma-ray beam, would produce a pure light-on-light reaction, and several facilities are preparing it.

A related question is what happens in a very strong laser field. An electron in a field near the critical value of $10^{18}$ V/m may absorb many photons at once and produce a pair: the effective mass becomes field-dependent, so that the threshold itself shifts. The SLAC E144 experiment saw this in 1997 at a weak level, using many laser photons on a 46.6 GeV beam of electrons. Whether a pure vacuum can be induced to break down into pairs by a sufficiently focused laser, without anything else present, has not been observed, and it needs field strengths that no laser has so far reached.

A photon’s momentum is too large for its energy, so it cannot become a pair on its own. A partner must carry the difference, with the threshold $2mc^2(1 + m/M)$. A lead nucleus does this for 2.7 eV and sets the textbook’s 1.02 MeV. An electron, with the same mass as the particles being made, must be flung away with the pair, and the price is exactly double. The formula sets the price of every reaction, including the one the Bevatron was built for.