Illustrated Physics
physics, drawn
Relativity

The energy above which faster arrives later

In a racing circuit a faster car laps sooner. In a proton synchrotron it does too, at first — and then, past a definite energy, the faster proton laps later. Extra momentum sends a proton round a slightly longer orbit and makes it slightly faster, and relativity decides which wins: at low energy the speed does, but near the speed of light more momentum buys almost no more speed, and the longer road takes over. At the crossing, called transition, a spread of momenta stops producing a spread of arrival times, the bunch stops oscillating in time, and the accelerating wave has to jump from one slope to the other in a few turns. The first machine to try it did so in 1959.
14 min read 5 figures Approximations that lieMade into an instrumentWhat stays the same

Assumes: The speed that stops while the energy keeps coming · The push that does not point where the body goes

The speed that stops while the energy keeps coming recounts William Bertozzi’s experiment of 1964, in which electrons given more and more energy took almost exactly the same time to cross eight and a half metres: their energy rose fifteenfold and their speed hardly at all, pinned just below that of light. It is the most direct demonstration that in special relativity momentum and speed part company. Momentum is γmv\gamma m v, and it can grow without limit while vv approaches cc; near the limit, a large change of momentum is a tiny change of speed.

In a circular accelerator that fact produces a curious reversal, with practical consequences severe enough that the machines are designed around it. A proton synchrotron accelerates bunches of protons round a ring of magnets, giving them a kick of energy each time they pass through a radio-frequency cavity. The kick has to be timed: a proton must arrive when the cavity’s oscillating field is pointing the right way. Protons in a bunch do not all have the same momentum, and a proton with a little more momentum than its neighbours arrives at the cavity a little earlier or a little later, depending on two effects that compete. Which one wins changes, abruptly, at a definite energy. That energy is called transition.

A longer road against a faster car

A proton with more momentum is stiffer against the bending magnets, and so follows an orbit of slightly larger radius, which in the ring’s focusing structure means a slightly longer path round the ring. The fractional increase in path length per fractional increase in momentum is a property of the ring’s magnetic lattice, called the momentum compaction α\alpha; for the CERN Proton Synchrotron it is about 1/371/37. At the same time, the proton with more momentum is going faster, by a fractional amount δ/γ2\delta/\gamma^2 for a fractional momentum excess δ\delta.

A longer road against a faster car. For a proton with 0.1% more momentum than the design proton in a ring 628 m round, the change in its time per turn, in picoseconds, against the Lorentz factor: from its longer orbit, which adds about 56 ps a turn at every energy; from its higher speed, which subtracts 605 ps at γ = 2 but only 5.2 ps at γ = 20, because near the speed of light more momentum buys almost no more speed; and the net, computed exactly. The speed wins below γₜ = 6.1 and the path wins above it. In special relativity a body can always be given more momentum, but its speed is capped, and that is the whole cause.
Fig. 1 For a proton with 0.1% excess momentum in a 628 m ring, the change in its time per turn against γ: the longer orbit adds about 56 ps at every energy (dashed); the higher speed saves 605 ps at γ = 2 but only 5.2 ps at γ = 20 (dashed); the exact net crosses zero at the transition γ = 6.1.

The two effects pull in opposite directions. The longer path makes the proton late, by the same fraction at every energy, because the ring’s geometry does not change. The higher speed makes it early, by a fraction that falls as 1/γ21/\gamma^2: at a Lorentz factor of 2 the extra speed is large, and the proton arrives 605 picoseconds sooner per turn, ten times more than the 56 picoseconds the longer road costs; at a Lorentz factor of 20 the extra momentum buys almost no extra speed, and the proton saves only 5 picoseconds against the same 56. In between, the net changes sign. For the PS, the crossing is at γ=6.1\gamma = 6.1, a kinetic energy of 4.8 GeV.

The quantity that combines both is the slip factor,

η=α−1γ2,\eta = \alpha - \frac{1}{\gamma^2},

which gives the fractional change in revolution time per fractional change in momentum. Its zero defines the transition Lorentz factor, γt=1/α\gamma_t = 1/\sqrt{\alpha}.

The slip that changes sign. The slip factor η = α − 1/γ² of a synchrotron, α being its momentum compaction, whose transition is at γₜ = 6.1, as in the CERN Proton Synchrotron, against the Lorentz factor of the protons it carries: the fractional change in revolution time per fractional change in momentum. Below transition η is negative — a proton with more momentum goes round sooner — and at γ = 6.1, a kinetic energy of 4.79 GeV, it passes through zero; above, it is positive and approaches α = 0.0269 as the speed stops changing. Exactly at transition a spread of momenta produces no spread of arrival times at all, which sounds desirable and is the most dangerous moment of an acceleration cycle.
Fig. 2 The slip factor η=α−1/γ2\eta = \alpha - 1/\gamma^2 of a ring with transition at γ = 6.1 (kinetic energy 4.79 GeV): negative below, where more momentum means an earlier arrival; zero at transition; positive above, approaching α = 0.0269.

Below transition the slip factor is negative, and the machine behaves as a racing circuit does: faster laps sooner. Above transition it is positive: a proton with more momentum laps later. The reversal is purely a consequence of the relativistic relation between momentum and speed. A Newtonian proton, whose speed grows in proportion to its momentum without limit, would have η=α−1\eta = \alpha - 1, negative at every energy, and there would be no transition. The push that does not point where the body goes notes the same divorce between momentum and velocity in the direction of a force; here it is the divorce between their magnitudes, read off a stopwatch at the cavity.

The slope a bunch has to ride

The reason it matters is phase stability, the principle, found independently by Vladimir Veksler in 1944 and Edwin McMillan in 1945, that makes synchrotrons possible at all. A bunch of protons with a spread of momenta and arrival times can be held together by the accelerating wave if the kick each proton gets depends on its arrival time in the right way.

The slope a bunch has to ride. Protons' motion relative to the radio-frequency wave that accelerates them, tracked turn by turn: the phase of each proton's arrival against the wave, from 0 to π, and its fractional momentum excess, for three protons started at different phases, with the wave's voltage drawn above each panel. Below transition (left, γ = 3) a proton arriving early sits lower on the rising slope, gets a smaller kick, falls behind, and so circles the stable phase on the rising side: early-and-fast and late-and-slow trade places for ever. Above transition (right, γ = 15) more momentum makes a proton later, not earlier, and the same rising slope would push it further away; the stable phase is on the falling slope instead, 2.64 rather than 0.50 radians. Crossing transition, the accelerating system has to switch the bunch from one slope to the other in a few turns — a phase jump — or it disperses.
Fig. 3 Turn-by-turn motion of three protons relative to the accelerating wave (drawn above each panel), arrival phase against momentum excess. Below transition (γ = 3) they circle a stable phase on the wave’s rising slope, at 0.50 rad; above (γ = 15), on its falling slope, at 2.64 rad.

Below transition, put the design proton on the rising slope of the cavity’s voltage. A proton that arrives a little early meets a lower voltage and gets a smaller kick than the design proton; with less energy it goes round more slowly, since below transition less momentum means a later arrival, and next time it arrives less early. A proton that arrives late gets a larger kick, speeds up, and arrives less late. Each proton circles the design point in a plane of arrival time and momentum, and the bunch holds together. That circling is the synchrotron oscillation, and the turn-by-turn tracks on the left of the figure show it.

Above transition the logic reverses. A proton that arrives early on the rising slope gets a smaller kick, and with less momentum it now goes round faster, arriving earlier still. The rising slope has become unstable. The stable point is now on the falling slope, where an early proton gets a larger kick, which makes it late. The tracks on the right circle a point on the falling side, at a phase of 2.64 radians rather than 0.50. Every proton synchrotron that crosses transition has to move its bunches from one slope to the other, in the time it takes the energy to pass through the transition region, without losing them.

The oscillation that stops at transition

The circling below and above transition has a frequency, and the frequency depends on the slip factor.

The oscillation that stops at transition. The frequency at which protons in a bunch oscillate back and forth in arrival time — the synchrotron frequency — as a fraction of the revolution frequency, in thousandths, against the Lorentz factor, for a ring with transition at γ = 6.1, harmonic number 8 and a 200 kV accelerating voltage. At γ = 3 the bunch oscillates once every 365 turns; within one per cent of transition, once every 6897; at transition itself it does not oscillate at all, because nothing restores a proton's arrival time when arrival no longer depends on momentum. For the few milliseconds around transition the bunch's longitudinal motion is frozen while the energy keeps rising, the oscillation cannot follow the changing conditions adiabatically, and the bunch's length and momentum spread swing to extremes: short and intense enough to excite instabilities, as the first machines found.
Fig. 4 Synchrotron frequency as a fraction of the revolution frequency, against γ, for a ring with transition at 6.1, harmonic number 8 and 200 kV of accelerating voltage: once every 365 turns at γ = 3, once every 6,897 within one per cent of transition, and zero at transition itself.

The restoring action is a product of two things: how much a proton’s kick depends on its arrival time, set by the voltage and the slope, and how much its arrival time depends on its momentum, which is the slip factor. When the slip factor vanishes the second factor is zero and nothing restores a proton’s phase. The synchrotron frequency, proportional to the square root of ∣η∣|\eta|, falls to zero. In the model ring the bunch oscillates once every 365 turns at γ=3\gamma = 3, once every seven thousand within a per cent of transition, and not at all at transition.

That is why transition is dangerous rather than convenient. Away from it the oscillation is fast compared with the rate at which the machine’s conditions change, so the bunch adjusts smoothly: its length and momentum spread change slowly and predictably as the energy rises, the adiabatic behaviour of any slowly perturbed oscillator. Near transition the oscillation is slower than the changes, the bunch cannot follow, and its shape freezes while the conditions under it pass through zero. The bunch emerges short and with a large momentum spread, and while it is short its charge density is high, and the fields it induces in the walls of the beam pipe act back on it. In the first machines this produced instabilities and losses at every crossing.

There is a further twist above transition. A proton that gains energy and laps later behaves, in the longitudinal direction, as if it had a negative mass: push it forward and it falls back. Two protons that repel each other along the bunch are therefore drawn together, which makes the charge distribution of an intense bunch above transition prone to clumping — the negative-mass instability, predicted in 1959 and observed soon after. The relativistic speed limit does not only reverse the sense of phase stability; it reverses the sense of the bunch’s own space-charge forces.

Crossing it in 1959

CERN’s Proton Synchrotron first accelerated protons through transition on 24 November 1959, and the night it did so is part of the laboratory’s history: after weeks of beam lost at transition, a reconfiguration of the radio-frequency phase control got the beam through and on to its design energy of 24 GeV, briefly the highest in the world. Brookhaven’s Alternating Gradient Synchrotron followed in 1960. Both relied on a phase jump: at the moment the energy passes transition, the accelerating system shifts the cavity’s phase so that the bunches, which had been sitting on the rising slope, find themselves on the falling one.

Later machines added a second technique, the transition jump: a set of fast quadrupole magnets that briefly change the ring’s focusing, and so its momentum compaction, so that the transition energy itself moves and the beam crosses it in a fraction of the time the acceleration alone would take. The PS still uses such a scheme. The less time the bunch spends in the non-adiabatic region, the less its shape is distorted.

Which machines have to cross

Since transition is set by the momentum compaction, which is set by the focusing, a machine’s designer decides whether it will have to cross. Rings with strong focusing have small momentum compaction and high transition energies; the transition Lorentz factor is roughly the ring’s horizontal tune, the number of betatron oscillations per turn.

Which machines have to cross. The range of Lorentz factors through which five proton and ion synchrotrons accelerate their beams, on a logarithmic scale, with each ring's transition γₜ marked as a tick: the CERN PS Booster (γ from 1.17 to 3.13, transition at about 4.1), the Proton Synchrotron (3.1 to 27.7, transition 6.1), the Super Proton Synchrotron (27.7 to 480, transition about 18), RHIC with gold ions (about 10.5 to 107, transition about 23) and the LHC (480 to 7460, transition 55.7). The Booster stays below its transition and the SPS and LHC above theirs; the PS and RHIC must cross, and both carry the hardware to do it — a fast jump of the accelerating phase, and in the PS a quadrupole system that briefly moves γₜ itself so the beam crosses it faster. A ring's transition is set by its focusing, roughly its horizontal tune, so the designer chooses whether a machine will have to cross; injecting above transition is one reason the LHC is fed by the SPS rather than straight from the PS.
Fig. 5 Acceleration ranges in γ (bars, logarithmic) and transition energies (ticks) for the CERN PS Booster (1.17–3.13, transition about 4.1), PS (3.1–27.7, 6.1), SPS (27.7–480, about 18), RHIC with gold ions (about 10.5–107, about 23) and the LHC (480–7460, 55.7). Only the PS and RHIC cross.

In CERN’s chain the PS Booster stays below its transition throughout; the PS crosses; the SPS receives protons from the PS already above its own transition and accelerates them to 450 GeV; and the LHC receives those, far above its transition of 55.7. Avoiding transition is one of the reasons for a chain of machines rather than a single ring, though not the only one. The Relativistic Heavy Ion Collider at Brookhaven, which accelerates gold ions from a Lorentz factor of about ten, must cross its transition at about 23, and does so with a transition jump. Electron machines never face the problem: an electron at a few hundred MeV is already at a Lorentz factor of several hundred, far above any ring’s transition, and is above it from the moment it is injected.

Why the machines are chained

The reason an accelerator complex is a chain of rings rather than one is mostly a matter of range. A ring’s magnets can be ramped over a limited span of fields, typically a factor of ten or twenty, and the radio-frequency system over a limited span of frequencies. A proton injected at low energy changes its speed substantially in the first rings — from a third of the speed of light out of a linear accelerator to nearly all of it within a few GeV — and the cavities must follow; at higher energies the speed barely changes and the frequency is almost fixed, as the ship that never arrives at c would lead one to expect. Transition falls naturally into this structure. It occurs where the speed is still changing appreciably but no longer dominates, a few GeV for a ring of moderate focusing, and a chain lets each ring live entirely on one side of it except the one designed to cross.

The energies at the far end of the chain are the point of the whole arrangement. A 7 TeV proton in the LHC carries a Lorentz factor of 7,460, and since its speed differs from light’s by less than one part in a hundred million, its revolution time is set by the ring’s circumference to the same precision. That is what lets two beams of such protons be kept colliding at fixed points for hours, and what makes the collision that wastes most of the energy of a fixed-target experiment avoidable: colliding beams of equal energy put all of it into the collision. The relativistic speed limit that creates transition at low energy is what makes the timing of a collider trivial at high energy.

The same crossing in other places

The competition between a longer path and a faster speed is not peculiar to protons. A light pulse in an optical fibre has the same structure, with wavelength playing the part of momentum. Different wavelengths travel at different group speeds through glass, and the dependence changes sign at one wavelength: below about 1.3 micrometres in standard fibre, longer wavelengths arrive first, and above it they arrive last. At the crossing a pulse’s spread of wavelengths produces no spread of arrival times to first order, and the wavelength a fibre does not smear shows why the first long-distance fibre systems were built to operate there. The zero-dispersion wavelength of a fibre is its transition energy.

In electron machines the momentum compaction is put to the opposite use. A short bunch of electrons is needed to drive a free-electron laser, and the bunch is compressed by giving it an energy spread correlated with position — the tail faster than the head — and then sending it through a chicane of four bending magnets in which the higher-energy electrons take a shorter path. Electrons at a Lorentz factor of thousands all travel at essentially the speed of light, so only the path length matters, and the tail catches up with the head. The compressed bunch then radiates coherently in the undulator of the magnet period that comes out as an X-ray. The bunch compressor is a machine operated deep above transition, where the speed term has vanished and only the path term is left.

Why the cyclotron could not have this problem

The predecessor of the synchrotron, the cyclotron, accelerated particles in a fixed magnetic field along a spiral, and its revolution frequency qB/2πγmqB/2\pi\gamma m — the subject of the force that does no work — fell as the particles’ energy grew. At non-relativistic energies the frequency was constant, independent of the orbit’s size, and the fixed-frequency cyclotron worked. Relativity spoiled that above a few tens of MeV for protons, because γ\gamma in the denominator grew and the particles fell out of step with the fixed accelerating voltage. The synchrocyclotron fixed it by lowering the frequency during the acceleration, and the isochronous cyclotron by raising the field with radius in proportion to γ\gamma.

The synchrotron’s transition is the same relativistic fact in a different machine. In both, the trouble arises because a particle’s speed stops tracking its momentum, so that a timing tuned for one stops working for the other. The cyclotron’s answer was to follow the particles’ slowing revolution frequency; the synchrotron’s is to follow the change in the sign of their response. And in a muon storage ring the same mathematics appears once more, in the turns a muon makes at any energy, where a “magic” Lorentz factor of 29.3 makes the muon’s spin precession independent of the electric focusing fields, a special energy chosen for the opposite reason: to make a dependence vanish rather than to survive one.

A ring reduced to one compaction

The figures use the PS’s circumference and transition energy, a single accelerating harmonic and an illustrative voltage, and treat the momentum compaction as one number, ignoring the higher-order terms that make it depend on the momentum excess itself and that slightly separate the transition energies of different protons in a bunch — a real effect, which smears the crossing. The longitudinal tracking uses a simple turn-by-turn map with constant energy, so the tracks are closed orbits; real acceleration makes them shrink or grow slowly. The proton starting phases are chosen to show the motion clearly, and the resulting momentum spreads are larger than a real bunch’s. Space charge, the impedance of the beam pipe and the negative-mass instability are described in words and omitted from the figures. The machine parameters are approximate and have changed over the decades as the rings were upgraded.

Still open: crossing transition with intense beams

The intensity of the proton beams the LHC’s injectors deliver is limited partly by what the PS can carry through transition without the bunch’s own fields driving it unstable. Upgrades for the high-luminosity LHC have pushed the PS to bunch intensities at which transition crossing again requires careful control of the bunch’s shape, and proposals for rings that avoid transition entirely — by designing lattices with an imaginary transition Lorentz factor, whose momentum compaction is negative, so that the slip factor never changes sign — have been built in some smaller machines. Whether such lattices can be made with the other properties a large proton synchrotron needs, at a cost worth paying, is one of the questions shaping the designs of the next generation of proton machines.

The habit worth carrying away is to ask which way a small difference grows. A proton with a little more momentum goes round a longer road and goes slightly faster, and below a Lorentz factor of 6.1 in the PS the speed wins and it laps sooner, while above it the speed has nowhere left to go and the road wins — so the stable phase flips to the other slope of the accelerating wave, and at the crossing the bunch’s timing oscillation stops dead. Relativity’s speed limit, which Bertozzi measured on a stopwatch, is designed into the hardware of every proton accelerator that has to cross it.

Part 17 of 18

This essay is one argument about Relativistic dynamics. The others:

Thrown across a uniform field, a fast charge does not draw a parabola. The paths of a charge launched across a uniform electric field (which points up the page) at 0.30, 0.60, 0.90, 0.99 of the speed of light, from the exact relativistic motion (solid), beside the parabolas Newton's law gives for the same launches (dashed); lengths are in units of mc²/qE, the distance over which the field does work equal to the charge's rest energy. Near its launch each exact path is a parabola, but one with the inertia γ₀m in place of m, so it is deflected less than Newton says — 1.05, 1.25, 2.29, 7.09 times less for the four. Further out each turns steeply along the field: one unit across, the 0.3c charge is 11.6 units along where Newton says 5.6, because its sideways speed has fallen while the field kept pushing. Each exact path is a hyperbolic cosine. The field adds to the momentum along it while the sideways momentum stays fixed, and the charge's growing energy is a growing inertia that slows its sideways progress. The throw a uniform field bends into a chain Part 16 — Throw a stone sideways across the Earth's gravity and it draws a parabola. Throw a charge sideways across a uniform electric field and, at low speed, it does the same. At high speed it draws a different curve, a hyperbolic cosine — the shape of a chain hanging between two posts, stretched — and the reason is that nothing pushes it sideways yet its sideways speed falls. Its sideways momentum is fixed, the field keeps adding energy, and energy is inertia. The charge gets only logarithmically far across the field however long it flies, and its sideways distance turns out to be a reading of its own clock. The peak no pion spectrum can move. The photon spectrum from a population of neutral pions, summed from their boxes, for pion spectra falling as three different powers of the energy. Whatever the pion spectrum, each curve peaks at the same 67.5 MeV and is symmetric on the logarithmic axis, since every box contains that point and a box's height falls as it widens; checked here at the point and its mirror, the ratio is 1.000. This is the shape, known as the pion bump, that Floyd Stecker worked out in 1971. Gamma-ray telescopes found it in 2013 in two supernova remnants, which was taken as direct evidence that such remnants accelerate protons, because the electrons that produce most other gamma rays have no reason to make a peak here. The flat spectrum a fast pion must paint Part 18 — A neutral pion falls apart into two photons, each with half its rest energy, in a hundred-million-billionth of a second. Seen in flight, those photons could have any energy between two edges, and every energy in that range is equally likely, whatever the pion's speed. Add up a whole population of pions and something unexpected appears: a peak at exactly the 67.5 MeV that no individual decay shows. Astronomers took it as evidence of where cosmic rays are made.

Where this goes next

The speed that stops while the energy keeps coming. The speed of an electron, as a fraction of the speed of light, against the kinetic energy given to it, from special relativity (solid) and from Newton's ½mv² (dashed); dots at the five energies Bertozzi used in 1964. Newton's speed passes the speed of light at a quarter of a megaelectronvolt and is 7.7 times it at 15 MeV. Relativity's speed is 0.863 c at 0.5 MeV, 0.967 c at 1.5 MeV, 0.9948 c at 4.5 MeV and 0.99946 c at 15 MeV. Between 1.5 and 15 MeV the energy grows tenfold and the speed by three and a half per cent. The energy does not stop arriving; the speed stops answering it. The speed that stops while the energy keeps coming In 1964 William Bertozzi gave electrons energies from half a megaelectronvolt to fifteen, timed them over eight and a half metres, and caught them in an aluminium disc to measure the heat they brought. By Newton the fastest should have crossed in under four nanoseconds, seven times faster than light. They took twenty-eight, and so did the slower ones, more or less. The heat, meanwhile, rose in proportion to the energy put in. The experiment is famous for showing that nothing outruns light, but its sharper result is the second half: the energy did arrive, all of it, carried at very nearly the same speed. Two clocks that slow together. For a muon circling a 1.45 T magnetic field, its mean life as the laboratory measures it and the time it takes to go round once, against its Lorentz factor γ, on logarithmic scales. Time dilation stretches the life to γ × 2.197 μs: 6.6 μs at γ = 3, 64.4 μs at γ = 29.3, the muon g − 2 experiments' value. The orbit grows with the momentum, so the revolution takes γ times longer too: 15.3 ns at γ = 3, 149 ns at γ = 29.3. The two lines are parallel. Their ratio, the number of turns in one mean life, is 431 at every energy — set by the field and the muon's own properties, eBτ/2πm, with γ cancelled out. The turns a muon makes at any energy A muon lives two microseconds, and in a storage ring it lives longer the faster it goes, because its clock runs slow. Its orbit runs slow too: a faster muon carries more momentum, bends less in the same field and takes longer to go round. The two slowings are the same factor of gamma, so they cancel. A muon in a 1.45 tesla ring makes 431 turns in a mean life whether it is moving at nine-tenths of the speed of light or at nine nines, and the only way to more turns is a stronger magnet. The same cancellation is why the first cyclotrons stopped working near twenty million electronvolts. Same field, same charge, three momenta. Electrons entering a 10 mT field at right angles to it, at 1, 4, 9 keV, each drawn for a quarter of its turn. The radius is mv/qB — 10.7 mm, 21.3 mm, 32.0 mm — so measuring the curvature of a track measures the momentum of whatever made it, which is how every particle detector since the cloud chamber has worked. The time to go once round is 2πm/qB = 3.57 ns for all three: the faster particle travels a proportionally longer way round and arrives at the same moment. The force that does no work A magnetic field can turn a moving charge through any angle at all and cannot add a joule to it. Everything a magnet is good for follows from that one prohibition — including the fact that a bent track is a reading of momentum, and that a machine built on it stops working at five kilovolts for an electron.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

The Lorentz factorMomentum compactionParticle acceleratorPhase stabilityRelativistic momentumSlip factorSynchrotronTransition energy