The kick a fast particle can give an electron

A proton crossing water at ten megaelectronvolts slows to a stop in a little over a millimetre, and it does so by a long sequence of small collisions with the electrons of the water molecules — tens of thousands of them, each taking a few electronvolts or a few hundred. It cannot slow down by a few large collisions, because it cannot make one. The most energy it can give any single electron is about 22 kiloelectronvolts, a fifth of a per cent of what it carries, and the reason is a rule from ordinary mechanics that every billiards player has met in its equal-mass form.

The right angle a fast collision closes followed that rule for identical particles and found it bending at relativistic speeds. Here the particles are unequal — a heavy projectile and a light electron — and the bending runs the other way. The cap that keeps a slow proton’s collisions gentle is not a cap at all once the proton is fast, and at high enough energies one collision with one electron can take most of the proton’s energy away.

Twice the speed, in the frame of the heavy one #

The simplest way to find the largest kick is to watch the collision from the heavy particle’s frame. There the heavy particle is at rest and the electron arrives at speed $v$. The largest possible transfer is a head-on bounce, and since the heavy particle hardly recoils, the electron leaves at speed $v$ in the opposite direction, as a ball bounces off a wall. Back in the laboratory frame, in which the heavy particle moves at $v$, Newton adds the speeds: the electron leaves at $2v$, with kinetic energy $\tfrac12 m(2v)^2 = 2mv^2$.

The heavy particle’s own kinetic energy is $\tfrac12 Mv^2$, so the ratio is $4m/M$: 0.22 per cent for a proton, 2 per cent for a muon, 0.05 per cent for an alpha particle. The same arithmetic, read for momentum rather than energy, is why Rutherford’s alpha particles could find the nucleus in 1911. An alpha particle can give an electron at most twice its own speed times the electron’s mass in momentum — far too little to turn it aside noticeably — so the thousands of electrons it passes in a gold foil barely deflect it, and the rare large-angle bounces had to come from something heavy. Collisions are easier than forces found how much a collision reveals without any knowledge of the force; here the kinematics alone rules the electrons out as the cause. That is the cap, and it governs how every slow charged particle loses energy in matter — by countless small transfers rather than a few large ones, which is why its track is straight and its range nearly the same for every particle of the same energy.

The relativistic version needs one change. The bounce in the heavy particle’s frame is the same, but the speeds no longer add by Newton’s rule: speeds that refuse to add found that $v$ and $v$ combine to $2v/(1 + v^2/c^2)$. That is precisely the speed the wheel whose top cannot go twice as fast found for the top of a rolling wheel, and for the same reason: a velocity added to itself, once in each of two frames. At a tenth of the speed of light the difference is a per cent; at half, Newton sends the electron at the speed of light and relativity sends it at 0.8.

So the electron’s speed is capped at less than twice the projectile’s, and less than light. That looks as if it should make the largest kick smaller than the slow rule says. It does the opposite, because the energy that goes with a speed close to light is not $\tfrac12 mv^2$. The electron’s Lorentz factor after the bounce, worked out from the combined speed, is exactly

$$\gamma_e = 2\gamma^2 - 1,$$

where $\gamma$ is the projectile’s. Its kinetic energy is $(\gamma_e - 1)mc^2 = 2mc^2(\gamma^2 - 1) = 2mc^2\beta^2\gamma^2$. The slow rule, $2mv^2$, is the first term of this; the full expression grows as the square of the projectile’s Lorentz factor. Capping the speed did not cap the energy, because relativistic energy is not capped by speed at all.

The largest kick, from the four-momenta #

The bounce argument assumed the heavy particle does not recoil, which fails once the electron’s energy is no longer negligible beside the projectile’s. The exact answer comes from conserving four-momentum in a head-on collision of masses $M$ and $m$, and it is the expression every textbook on particle detection quotes:

$$T_{\max} = \frac{2mc^2\,\beta^2\gamma^2}{1 + 2\gamma\,m/M + (m/M)^2}.$$

The denominator is one, to a very good approximation, until $\gamma$ approaches $M/2m$ — about 918 for a proton and 103 for a muon — and then it starts to bite. The figure’s curves are computed by a different route, from the collision in the centre-of-momentum frame, where the electron’s momentum simply reverses, followed by a boost back to the laboratory; the two agree to nine figures, as they must.

At low energies every curve follows its slow rule, a straight line of slope one on the logarithmic axes, offset by $4m/M$. As the projectile becomes relativistic the curves steepen to slope two — the $\gamma^2$ growth — and the lighter particles, whose $\gamma$ is larger at a given energy, steepen first. Near $\gamma \approx M/2m$ they bend over again towards the dotted line on which the electron would take everything, and run alongside it. A muon of a few hundred gigaelectronvolts and a proton of a few teraelectronvolts can each lose most of their energy to a single electron.

A range every slow particle keeps, and a fast one loses #

The slow rule has a consequence that is used every day in hospitals. Because no single collision can take more than a fifth of a per cent of a slow proton’s energy, a proton’s slowing-down is the sum of an enormous number of small, nearly independent losses, and such a sum is very nearly certain. Protons of the same starting energy stop at almost the same depth, with a spread of about one per cent of their range — the same narrowing of a sum of many small random steps that how far a molecule gets found for a molecule’s path through a gas. And since a slow charge loses energy faster the slower it goes, roughly as one over its speed squared, each proton deposits most of its energy in the last few millimetres before it stops. The dose builds to a sharp maximum at the end of the range — the Bragg peak — and falls to nothing just beyond it. Proton therapy places that peak on a tumour and spares the tissue behind it, which a beam of X-rays, losing its energy exponentially along its whole path, cannot do.

At relativistic energies the same sum is no longer so certain. The largest kicks have grown from a fifth of a per cent to tens of per cent, and a single rare collision can deposit more energy in a thin layer than all the gentle ones together. The energy a fast particle leaves in a thin detector layer is then not a narrow peak but a skewed distribution with a long tail towards high energies — worked out by Lev Landau in 1944, and named after him — whose most probable value lies well below its mean. A detector that wants to measure how fast a particle is moving from how much energy it deposits has to fight that tail, usually by sampling many thin layers and discarding the largest readings. The tail is the delta rays of this essay, seen from the detector’s side.

A share that rises from a fifth of a per cent to a half #

Measured as a fraction of what the projectile carries, the change is more striking.

The flat stretch on the left is the slow regime, where the share is $4m/M$ whatever the speed. It rises through the relativistic regime and reaches a half near $\gamma = M/2m$: 916 for a proton, 101 for a muon. The crossover has a clean meaning in the centre-of-momentum frame. At low energies, in that frame, the electron is a light particle bouncing off a nearly stationary heavy one, and a reversed electron carries little energy. As the projectile’s energy rises, the centre-of-momentum frame moves faster and faster, and in it the electron’s energy — it is now arriving at nearly light speed — grows until it is comparable with the projectile’s. The two meet as equals, and a head-on reversal can hand the electron most of the total. The collision that wastes most of the energy found the same frame swallowing energy in fixed-target experiments; here it is what lets a light particle take a large share.

The curly tracks around every fast track #

The largest kick is for a head-on collision, which sends the electron straight ahead. A glancing collision sends it off at an angle with less energy, and for a heavy projectile the relation is fixed by the kinematics alone: the energy transferred equals the momentum transferred times the projectile’s speed, so

$$T(\theta) = \frac{2mc^2\,\beta^2\cos^2\theta}{1 - \beta^2\cos^2\theta},$$

for an electron leaving at angle $\theta$ to the projectile’s path.

The most energetic electrons go forward and the slow ones to the side, and no electron is ever knocked backwards, since a force pushing it backwards would have to come from a particle moving that way. In the bubble-chamber photographs of the 1950s and 1960s every fast charged track is fringed with short, tightly curled tracks — knock-on electrons, named delta rays after the alpha, beta and gamma rays of radioactivity — and the long ones, the energetic ones, all lean forward along the track. Their energies and angles were among the first routine checks of relativistic kinematics, decades after the effect itself was understood.

The figure also shows why the largest kicks are rare. The energy falls steeply away from the forward direction, and the cross-section for a collision falls steeply with the energy transferred — as one over its square, for a free electron — so the electrons that receive hundreds of megaelectronvolts are a tiny minority. A fast particle still loses most of its energy in small collisions. But the tail reaches much further than it does for a slow particle, and the tail matters.

Where the tail matters #

It matters first in how fast particles lose energy in matter. The charge that passes as a flash of light followed the field of a fast charge flattening into a pulse that reaches further from its path as $\gamma$ grows, giving a slow rise in the energy lost to distant electrons. The largest kick sets the other end of the same sum: Bethe’s formula for the rate of energy loss contains the logarithm of $T_{\max}$, so the energy lost to close collisions also rises with $\gamma$. A detector that measures how much energy a track deposits in a thin layer sees both rises together, and also sees the delta rays carry part of the deposit out of the layer when they are energetic enough to escape — which is why the energy deposited in thin layers rises more slowly than the energy lost.

It matters most for muons at very high energies. A muon of a few teraelectronvolts crossing rock or ice loses energy not steadily but in jolts, a large fraction of it in occasional single collisions — knock-on electrons among them, along with photons from bremsstrahlung and pairs from the muon’s field — each of which starts a shower of its own. Neutrino telescopes buried in the Antarctic ice see the light from these showers strung along a muon’s track like beads, irregularly spaced and irregularly bright, and use the pattern to estimate the muon’s energy. The turns a muon makes at any energy found muons useful because they are stable enough to keep; here they are useful because they are heavy enough to lose energy only by these rare violent collisions, and so cross kilometres.

The photon has a version of the same cap. A photon with a momentum found that a photon scattering off an electron can give it at most $2E^2/(mc^2 + 2E)$ — the Compton edge in a gamma-ray spectrum — by exactly this reasoning with a massless projectile, for which every collision is in the relativistic regime: the share approaches one as the photon’s energy grows past the electron’s rest energy.

The same kick, the other way round #

Reverse the roles, and the formula answers a practical question about electron microscopes. An electron of kinetic energy $E$ striking a nucleus of mass $M$ head-on gives it at most

$$T_n = \frac{2E\,(E + 2mc^2)}{Mc^2},$$

which is the same four-momentum result with the light particle now the projectile. The slow rule would give $4(m/M)E$, smaller by the factor $(1 + E/2mc^2)$: about 10 per cent at a hundred kilovolts and 30 per cent at three hundred.

An atom in a crystal is held in place by its bonds, and if a collision gives it more than its displacement energy — about 21 electronvolts for a carbon atom in graphene — it is knocked out of its site, leaving a vacancy. For carbon, electrons reach that threshold at 104 kiloelectronvolts. The slow rule would put the threshold at 115, and a microscopist who used it would set the voltage too high and damage the sample. This is why transmission electron microscopes built to image graphene and other two-dimensional materials work at 60 to 80 kilovolts, below the carbon threshold, while those built for the highest resolution in metals work at 200 or 300, where carbon atoms are knocked out of their sites by the beam that is trying to image them. Heavy nuclei recoil less, and gold takes barely 22 electronvolts from a 1 MeV electron, which is why heavy-element crystals can be imaged at full voltage for hours.

The same calculation sets how much damage the fast electrons in a nuclear reactor or in space do to the materials around them, and it is the reason semiconductor detectors and solar panels in orbit degrade: every electron above a threshold of a few hundred kilovolts displaces silicon atoms as it passes.

Free electrons at rest, and only the largest kick #

Every figure assumes the electron is free and at rest. Atomic electrons are neither: they are bound, by a few electronvolts in the outer shells and by tens of kiloelectronvolts in the inner shells of heavy atoms, and they move. For the large transfers the figures are about, the binding is negligible and the formulas are good; for small ones, the binding decides everything, and the theory of energy loss handles them through the mean excitation energy of the material, which no free-electron picture contains.

The figures are the largest transfers, at the edge of what kinematics allows. How often a collision reaches a given transfer is a separate question, answered by the cross-section, which for a spin-one-half electron struck by a spinning or spinless projectile differs in detail near the edge. The knock-on figure ignores the thermal vibration of the atoms, which adds a little energy to some collisions and lowers the measured damage threshold in graphene below the static value, and it treats the displacement energy as a single number when it depends on the direction of the kick relative to the crystal’s bonds.

Still open: how energetic the knock-ons in an air shower can be #

The highest-energy particles ever recorded are cosmic rays, arriving at up to $10^{20}$ electronvolts, and they are known only through the showers of secondary particles they make in the atmosphere. The muons in those showers reach the ground with energies up to hundreds of teraelectronvolts, and the number of muons measured at the ground in the largest showers is higher than every model of the shower predicts, by tens of per cent — the muon puzzle. The models include the kinematics of every collision, including the knock-on electrons this essay describes; what they do not get right is the hadronic physics of the first collisions, at energies far beyond any accelerator. Whether the excess means new physics in those first collisions or a more mundane failure of extrapolation is not settled.

The general lesson is about where a slow-particle rule breaks. A heavy particle can make a light one go at most about twice as fast as itself, and relativity keeps that cap on speed — but energy at speeds near light is not capped by speed, and the largest kick grows as the square of the projectile’s Lorentz factor until the light particle can take most of everything. A slow proton loses its energy a fifth of a per cent at a time. A fast enough one can lose half of it to one electron.