The beam that stops pushing itself apart

Magnetism is electricity seen sideways showed that the force between a current-carrying wire and a moving charge is magnetic in one frame and electric in another, and that the two accounts agree to every decimal place. The wire in that essay is neutral and its electrons crawl, so the relativistic factors involved are smaller than one part in $10^{25}$ and the magnetism is the whole of a vanishingly small correction acting on an enormous number of charges. This essay takes the same bookkeeping to the opposite extreme: a beam of charges with nothing to neutralise them, moving at nearly the speed of light, where γ is ten or a thousand or a hundred thousand and the correction is the whole story.

The practical question is how a beam of like charges stays together at all. Every particle in it repels every other, and a beam in which nothing opposed that repulsion would spread until it hit the walls of its pipe. Slow beams do exactly that, and the people who build electron guns and ion sources spend much of their effort fighting it. Fast beams, it turns out, do not need nearly as much fighting. The reason is a cancellation between two forces that are both growing, and it can be told in two frames that disagree about what the forces are and agree about everything that can be measured.

Two pushes that almost cancel #

Take two equal charges moving side by side, at the same speed, along parallel lines. In the laboratory each feels two forces from the other. The first is electric. The field of a moving charge is not the Coulomb field: it is squeezed towards the plane perpendicular to the motion, weaker ahead and behind and stronger to the side, by the factor γ directly across. So the electric repulsion between the two is γ times the Coulomb force they would feel at rest — larger, not smaller, and growing without limit as the speed approaches that of light.

The second force is magnetic. Each moving charge is a current, parallel currents attract, and the magnetic field of the moving charge is β times its electric field. The magnetic force on the other charge, moving at the same speed, is β times that again: an attraction of $\gamma\beta^2$ in units of the Coulomb force. The two forces are nearly equal and opposite, and their difference is

$$\gamma - \gamma\beta^2 = \gamma(1 - \beta^2) = \frac{1}{\gamma}.$$

At nine tenths of the speed of light the push is 2.3 times Coulomb, the pull 1.9 times, and the net 0.44. At 0.99 the push is 7.1, the pull 6.9, and the net 0.14. A beam of electrons at a gigaelectronvolt, with γ near two thousand, has its particles pushing each other apart with a two-thousandth of the Coulomb force, the sum of two forces each two thousand times larger than Coulomb’s cancelling to four parts in a million of either.

And the net force is not the end of the reduction. What spreads the beam is its particles’ sideways acceleration, and a fast particle’s resistance to a sideways push is not its rest mass but γ times it — the push does not point where the body goes, and a transverse push meets inertia γm. So the sideways acceleration is the net force divided by γm, which is $1/\gamma^2$ of what the Coulomb force would give two particles at rest.

The same drift in the beam’s own frame #

Now ride with the beam. In the frame in which the two charges are at rest there is no current and no magnetic field. They repel by Coulomb’s law, full strength, and drift apart exactly as two charges at rest in a laboratory would. Nothing about their separation is contracted, because it lies across the direction of motion. The drawing computes their separation this way, in their own frame, as a function of their own time.

To say what the laboratory sees, the only thing needed is the relation between the beam’s time and the laboratory’s. The beam’s clocks run slow by the factor γ, as every moving clock must, so a history that takes a nanosecond in the beam’s frame takes γ nanoseconds in the laboratory. Over two nanoseconds of laboratory time, a pair at rest drifts from one micrometre to forty-three; at γ = 3 to thirteen and a half; at γ = 10 to under four. The laboratory, measuring accelerations, finds them reduced by $1/\gamma^2$ — one factor of γ from each of the two time derivatives — which is exactly what the net force divided by the transverse inertia gave.

The two accounts disagree about what is happening: in one there is a magnetic force nearly as large as the electric one, and in the other there is no magnetic force at all. They agree about the separation at every moment, as they must. That is the content of the field nobody can transform away put to work: the electric and magnetic fields are six numbers of one object, and what one observer calls magnetism another calls the slowness of a clock.

A long beam in its own frame #

The same factors turn up for a whole beam, and following them through the beam’s own frame is a useful check on the bookkeeping. A beam carrying a given charge per metre in the laboratory is, in its own frame, longer by γ — the laboratory sees it contracted along its motion, so its rest length is γ times greater — and so it carries γ times less charge per metre. Its electric field at the edge, and the Coulomb push on each particle there, is smaller by γ than a laboratory observer who ignored relativity would calculate. The particles’ sideways acceleration in their own frame is that push divided by their rest mass. And the laboratory, whose clocks run γ times faster than the beam’s, sees that acceleration reduced by a further $\gamma^2$. Altogether, $1/\gamma^3$.

In the laboratory the same $1/\gamma^3$ is assembled differently: the magnetic pull cancels all but $1/\gamma^2$ of the electric push, and the transverse inertia γm divides by one more γ. Neither account is more correct. One uses length contraction and time dilation and has no magnetism in it; the other uses the laboratory’s charge density and has magnetism instead. That charge and current are one thing, seen from different frames, is exactly the statement that these two sets of factors must multiply to the same answer.

How far a beam goes before it doubles #

For a whole beam the same reduction appears in the equation for its radius. A round beam carrying a current $I$ is pushed outward by its own charge with a strength measured by a single number, the generalised perveance, $K = 2I/(I_0\beta^3\gamma^3)$, where $I_0$ is a natural current built from the particle’s charge and mass — seventeen thousand amperes for electrons. The $\gamma^3$ in the denominator is two factors of γ from the magnetic pull cancelling the push of the beam’s charge per metre, and one from the transverse inertia; the $\beta^3$ comes from turning a current into a charge per metre, which divides by the speed, and a time into a distance along the beam, which divides by it twice more. The drawing solves the equation for the distance over which a beam that starts parallel doubles its width.

An electron beam of one ampere at ten thousand electronvolts, the energy of an old television tube, doubles its width in about a centimetre. At a million electronvolts it takes more than half a metre; at a thousand million, about twelve kilometres. The violent part of an accelerator’s life, as far as its own charge is concerned, is its first few metres, where the particles are still slow, and electron guns are built with electrodes shaped to hold the beam together against its own charge until it has been accelerated past the point where the cancellation takes over. Protons at the same kinetic energy are far slower, and a one-ampere proton beam at a million electronvolts doubles in six centimetres; the low-energy stages of proton accelerators are where their designers fight space charge hardest.

Guns built against their own charge #

The low-energy end of the curve is where the engineering happened first. In the 1930s and 1940s the designers of the electron guns in television tubes, microwave amplifiers and early accelerators found that beams of useful current spread within centimetres of the cathode, and that no arrangement of magnetic lenses downstream could undo what space charge did in the first few millimetres. John Pierce’s solution, in 1940, was to shape the electrodes around the cathode so that the electric field they produced exactly balanced the beam’s own outward push at its edge, making a beam whose particles moved in parallel lines as if they did not repel each other at all. Pierce electrodes, with their characteristic angle of 67.5 degrees to the beam’s edge, are still how guns are built.

The same space charge limits how much current a gun can draw in the first place. Near the cathode, where the electrons are slowest, their charge depresses the accelerating field, and the current a gap can carry rises only as the three-halves power of the voltage — the Child–Langmuir law. A beam emerges from its gun already shaped by the fight against its own charge, and it is only after acceleration, as γ grows and the cancellation sets in, that it can be handed to magnets that treat it as a stream of independent particles.

A few ions and the beam pinches #

The cancellation leaves a beam balanced on a knife edge, and a small thumb on the scale tips it. A beam travelling through a gas ionises some of it, and the slow positive ions it leaves behind sit in the beam’s path. They cancel some of its charge, and so some of its electric push, but being almost stationary they carry no current and add nothing to the magnetic pull. For a slow beam that barely matters: the push is far larger than the pull and needs most of its charge cancelled before the balance changes. For a fast beam the push and pull are almost equal, and cancelling a fraction $1/\gamma^2$ of the charge is enough to leave the pull ahead. At γ = 10, one ion for every hundred electrons makes the beam focus itself.

This self-pinching of partly neutralised relativistic beams was worked out by Gersh Budker in the 1950s and is the beam version of the pinch that holds a current-carrying plasma together. It has been used on purpose to transport intense beams through gas-filled channels without magnets, and it has had to be suppressed where it was not wanted, because a beam that pinches too hard becomes unstable. In a plasma rather than a gas the effect is extreme: a relativistic bunch entering a plasma expels its electrons and leaves a channel of bare ions that focuses it hard, which is part of how a plasma wave can accelerate electrons while holding them together. The drawing shows why it needs so little help at high energy: the magnetic attraction is already nearly as strong as the electric repulsion, and anything that weakens the repulsion without touching the attraction wins.

A beam that meets another beam #

The cancellation depends on the two currents being parallel. Reverse one of them and the magnetic force reverses: two like charges moving in opposite directions are antiparallel currents, which repel, so the magnetic force adds to the electric one instead of cancelling it. The drawing sets the two cases side by side. Moving together, the sideways force falls as $1/\gamma$. Moving towards each other, it grows as $\gamma(1 + \beta^2)$, nearly twice γ.

That asymmetry sets one of the central limits of particle colliders. A collider brings two beams into collision head on, and the rate of collisions grows with how densely the particles are packed at the crossing point. Each beam barely disturbs itself, for the reason drawn here, but at the crossing point it acts on the oncoming beam with a force hundreds or thousands of times stronger than Coulomb’s — focusing oncoming particles of opposite charge, defocusing those of like charge, and in either case deflecting them. This beam–beam force, not the beam’s own space charge, limits how densely a high-energy collider can pack its bunches; in the electron–positron colliders it also makes the particles radiate as they are bent by the other beam, a loss called beamstrahlung that grows with energy. A passing charge’s field is almost a pulse of light, and a bunch crossing another is two flashes passing through each other.

Where the two-charge picture stops #

Uniform, steady beams. The forces drawn are between two charges moving at exactly the same velocity. Real beams have a spread of velocities, both sideways and along the beam, and the cancellation is exact only for the component the charges share. The sideways spread — the beam’s emittance — adds a pressure of its own that competes with space charge, and at high energy, where space charge has been cancelled away, it is emittance that sets how small a beam can be focused.

No walls. A beam in a metal pipe induces image charges and currents in the walls, and in a pipe that is not perfectly conducting or not perfectly smooth those images lag behind, producing forces that can grow into instabilities. Much of accelerator physics at high current is the physics of the beam’s interaction with its surroundings rather than with itself.

Radiation. Charges forced to change direction radiate. For a beam going straight the effect is negligible, but in a ring a circling charge sends out a flash once a turn, and the collective version of that radiation, from a beam whose particles are bunched, is what free-electron lasers amplify.

What the forces do not show #

The drawings show forces and distances at the instant they are computed. They do not show that the cancellation is also a statement about energy: a relativistic beam carries almost as much energy in its magnetic field as in its electric field, and the two field energies grow together. Nor do they show what happens at the beam’s ends. A bunch of finite length has a field that is not purely transverse at its head and tail, and there the longitudinal field pushes particles forward and back, lengthening the bunch — an effect that, like the transverse one, falls away as the bunch gets faster, because the field of each particle is flattened into a pancake that barely reaches its neighbours ahead and behind.

Still open: beams dense enough to break the cancellation #

The newest accelerator concepts pack charge into bunches micrometres long and wide, dense enough that the particles’ own fields are comparable with the fields of the accelerating structure. At such densities the simple picture of a slight residual repulsion gives way to collective effects — the bunch radiating coherently as it moves, the radiation acting back on the bunch’s own tail, the beam’s field driving the plasma around it. How to predict and control a beam in that regime, where the cancellation that makes ordinary beams easy is no longer the dominant fact, is being worked out with large simulations and with experiments at facilities built for the purpose, and the answers decide how bright the next generation of X-ray lasers can be made.

The habit worth carrying away is to look for a cancellation before accepting a small result. When a force comes out weak at high speed, ask whether it is weak or the difference of two strong ones — because a difference balanced on a knife edge can be tipped by a fraction $1/\gamma^2$ of anything that touches only one side, and a beam that barely pushes itself apart can still pinch itself or wreck the beam it collides with.