The turns a muon makes at any energy

The clock that has to slow derived time dilation from a light clock: anything that keeps time, moving at speed $v$, ticks slower than the same thing at rest by $\gamma = 1/\sqrt{1 - v^2/c^2}$. The clock that does not feel the turn used muons stored on a circular orbit at CERN in 1977 to show that the slowing depends only on speed and not on the enormous acceleration of going round, a thousand million million million times gravity. Both essays treated the muon as a clock and the ring as a place to keep it.

The ring is also a clock, and that turns out to matter more than it seems. A charged particle in a magnetic field goes round at a rate set by its momentum, and relativistic momentum carries the same factor of $\gamma$ as the dilated lifetime. Followed through, the two clocks — the particle’s decay and its orbit — slow by exactly the same amount, and the consequence decides how long any unstable particle can be kept, why a muon collider is above all a problem of magnets, and why the first circular accelerators hit a wall.

Two clocks that slow together #

A charge $e$ moving at right angles to a magnetic field $B$ feels the force that does no work, at right angles to its motion, and goes round a circle whose radius is its momentum divided by $eB$. Relativistic momentum is $\gamma mv$, so a faster particle — with more momentum — goes round a larger circle. The time for one turn is the circumference over the speed:

$$T = \frac{2\pi r}{v} = \frac{2\pi \gamma m}{eB}.$$

For a slow particle $\gamma$ is one, and the period does not depend on the speed at all: a faster particle goes round a bigger circle in the same time. That is the cyclotron frequency, $eB/2\pi m$, and it is what Ernest Lawrence built the cyclotron on. For a fast one, the period grows with $\gamma$.

Meanwhile the muon’s own clock is running slow by $\gamma$: its mean life, 2.197 microseconds at rest, is $\gamma$ times longer in the laboratory. So a faster muon lives longer and takes longer to go round, by the same factor. The number of turns it makes in one mean life is

$$N = \frac{\gamma\tau}{T} = \frac{eB\tau}{2\pi m},$$

in which $\gamma$ has disappeared. In a 1.45 tesla field it is 431 turns, for a muon at three times its rest energy or at three thousand.

The cancellation is not a coincidence of formulas. In the muon’s own frame the magnetic field is not purely magnetic, and its orbit is not a circle at rest; but the count of turns is a count of events, the same in every frame, and so is the count of decays. Both the turning and the decaying are happening to the same muon, and anything that slows one of its processes in the laboratory’s description slows the other.

Why a slow orbit keeps time #

The cyclotron frequency’s independence of speed, for slow particles, deserves a second look, because it is what the relativistic version breaks. A faster particle goes round a bigger circle, and the two effects cancel exactly: double the speed, double the radius, same time per turn. The flash a circling charge sends once a turn found the consequence for radiation, a single radio frequency from slow electrons in a magnetic field whatever their energies, and the drift that does not care what the charge is used the same circles, all of one period, as the fast motion averaged away to leave a slow drift. A magnetic field is, for slow charges, a perfect clock with a rate set by the charge-to-mass ratio.

Relativity spoils the clock only by replacing the mass with $\gamma m$, which is the particle’s energy divided by $c^2$. An electron of a megaelectronvolt in a field goes round at a third of the slow-electron rate; a proton at the LHC’s energy at a seven-thousandth of the slow-proton rate. The period then measures the particle’s energy, and that is how the cancellation with the lifetime arises: both the lifetime and the period are proportional to energy, because both are $\gamma$ times something fixed.

The spin that keeps the same time #

The g − 2 rings rely on a third clock in the muon that is also untouched by its energy. A muon’s spin turns in the field too, and the spin that runs ahead of its own motion found that it turns slightly faster than the muon’s momentum does, by a fraction $a$, the anomalous part of its magnetic moment, about one part in eight hundred. The difference between the two rotations, the rate at which the spin gains on the momentum, is $aeB/m$ — once again with no $\gamma$ in it. A muon at any energy, in a 1.45 tesla field, gains one full turn of spin on its motion every 4.4 microseconds, about fifteen times in a mean life, and the experiments measure that rate to parts in ten thousand million to obtain $a$.

The momentum of 3.094 GeV/c was chosen for a different reason: at that momentum the effect of the electric fields that focus the muons on the spin’s rotation vanishes exactly, so the focusing can be electric without disturbing the measurement. That fixes $\gamma$ at 29.3 and the ring’s radius at 7.1 metres for a 1.45 tesla field. The 431 turns per mean life are then not a choice at all; they follow from the field, the muon’s mass and its lifetime, and would be the same in a ring of any size built to any energy with that field. So would the fifteen turns of the spin against the motion in each mean life: the ratio of the two clocks, decay and spin, is fixed by the muon alone, which is why the measurement’s statistical precision depends on how many muons can be stored and not on how fast they go.

The ring that measures by turning #

The muon storage rings of the g − 2 experiments, at CERN in the 1970s, at Brookhaven around 2000 and at Fermilab in the 2010s and 2020s, all used 1.45 tesla fields, 7.1 metres in radius, with muons at 3.094 GeV/c, a Lorentz factor of 29.3 chosen for reasons to do with how the muons’ spins precess. A muon there goes round in 149 nanoseconds, lives 64.4 microseconds, and makes 431 turns in its mean life. The decays are counted turn by turn for six or seven hundred microseconds, ten mean lives, and the experiments measure the muon’s magnetic moment from a wiggle in the count; the exponential beneath the wiggle is the dilated lifetime.

The figure shows the cancellation as a measurement would. Plotted against time, the three populations decay at very different rates; plotted against turns, they decay identically, and the dots marking every hundred turns line up across the curves. Counting turns is counting the muon’s own proper time in units fixed by the field. That is one way to see why the 1977 experiment could test time dilation to a part in a thousand: the ring’s revolution frequency, which the experiment measured to high precision, already contains $\gamma$, and comparing the decay rate with it compares two clocks that relativity says must slow alike.

The particles that cannot be kept #

Because the number of turns is fixed by the field and by the particle’s own mass and lifetime, a particle’s suitability for storage is a property of the particle. A muon makes about three hundred turns per tesla. A charged pion, living a hundredth as long and a third heavier, makes under three; even in the strongest magnets built, around twenty tesla, a pion would make fifty turns before most of it had gone. A kaon makes a third of a turn per tesla. No energy, however high, improves this. Of all the unstable charged particles, only the muon lives long enough, relative to its mass, to be circulated, which is part of why the muon is the only one whose magnetic moment has been measured in a storage ring to parts in ten thousand million.

The same arithmetic governs a proposed muon collider. Muons are attractive for colliding because, unlike electrons, they are heavy enough not to lose their energy to synchrotron radiation in a ring, and, unlike protons, they are point particles that put all their energy into a collision. But they decay, and a collider must keep them circulating long enough to collide many times. Raising the energy, which would help any other machine, gains nothing here: the muons live longer and take longer to go round, in proportion. The only remedy is a stronger average field around the ring. Design studies for a collider at ten teraelectronvolts assume superconducting dipoles of ten to sixteen tesla, and with the space taken by straight sections and focusing magnets the muons make about a thousand turns in a mean life. The machine’s luminosity is set by its magnets, not its energy.

The cyclotron that fell out of step #

The revolution’s slowing has a history of its own, and it came first.

Lawrence’s cyclotron, built in Berkeley from 1931, accelerated protons in a flat cylindrical chamber between the poles of a large magnet. Two hollow D-shaped electrodes with a gap between them carried an alternating voltage at the cyclotron frequency; a proton crossing the gap was pushed, spiralled round half a turn in the field, and arrived back at the gap just as the voltage had reversed to push it again. Because the slow proton’s revolution time does not depend on its speed, a single fixed frequency kept every proton in step as it spiralled outward and sped up. The design was so simple that it gave the highest-energy beams in the world for a decade.

It had a built-in limit, which Hans Bethe and Morris Rose pointed out in 1937. As the proton’s energy approached a few per cent of its rest energy, $\gamma$ grew past one, the revolution slowed, and the proton began to arrive at the gap late. Over the hundreds of turns needed to reach high energy, a frequency deficit of one or two per cent accumulated into a phase slip of half a cycle, at which point the voltage decelerated the proton instead. The classical cyclotron could not get protons much beyond twenty megaelectronvolts — the time dilation of a circular orbit, built into the timing of the machine.

Two cures followed, and both are corrections for relativity built into iron and electronics. Edwin McMillan and Vladimir Veksler showed in 1944 and 1945 that bunches of protons stay stable if the voltage’s frequency is lowered as they speed up — the synchrocyclotron — and the synchrotron extended the idea by raising the field as well. Llewellyn Thomas had shown in 1938 that a field which grows with radius, shaped by sectors to keep the beam focused, can make the revolution period the same at every energy; isochronous cyclotrons built on his idea accelerate protons continuously to hundreds of megaelectronvolts, the largest to 590 at the Paul Scherrer Institute, and supply the beams used in proton cancer therapy.

A hundred metres of decay #

Particles that cannot be stored can still be sent somewhere, and there the same factor of $\gamma$ decides how far they get.

A pion at rest lives 26 nanoseconds, in which light travels 7.8 metres. In a laboratory beam its decay length is $\gamma\beta c\tau$, which for a fast pion is its momentum in units of its rest mass times 7.8 metres: 56 metres at one GeV/c, 560 at ten. Over a hundred-metre beamline, 17 per cent of one-gigaelectronvolt pions survive and 84 per cent of ten-gigaelectronvolt ones. Without time dilation, essentially none would. Every secondary beam at an accelerator — the pions, kaons and muons made when protons hit a target and are then steered to experiments — is designed with these curves, and the experiments sit as close to their targets as shielding allows.

The atmosphere runs the same experiment without any apparatus. Cosmic rays striking the upper atmosphere make pions, the pions decay within a few hundred metres to muons, and the muons, typically at a few gigaelectronvolts, must cross fifteen kilometres of air to reach the ground. At rest a muon travels at most 660 metres in its mean life; at four gigaelectronvolts its decay length is twenty-five kilometres, and more than half of them arrive. Measurements of the muon flux on mountains and at sea level in the 1940s were among the first direct evidence of time dilation, as the clock that has to slow recounted; here they are one more point on the beamline curve, with a fifteen-kilometre beamline and no magnets.

Kaons make the separation by decay length into a tool. Neutral kaons come in a short-lived form, whose decay length at a few gigaelectronvolts is some tens of centimetres, and a long-lived form, whose decay length is about a hundred metres. A beam of neutral kaons seventeen metres from its target has lost almost all of its short-lived component, and in 1964 James Cronin and Val Fitch looked in exactly such a beam for decays that only the short-lived kaons should make — two pions — and found a few, the first evidence that nature distinguishes matter from antimatter. Time dilation chose which kaons were left in the beam, and a nucleus with no clock is the reason the choice was sharp: decay is exponential, and tens of decay lengths remove a population completely.

Neutrino beams use the same numbers the other way. A long-baseline neutrino experiment makes pions and lets them decay in flight along a tunnel, a kilometre at CERN for the beam to Gran Sasso and 675 metres at Fermilab for the beam to Minnesota; the tunnel’s length is chosen so that most pions of the useful energies decay before reaching the end, which, at several gigaelectronvolts, means hundreds of metres. The tunnels’ lengths are time dilation written in concrete.

What the uniform ring leaves out #

The figures treat a ring of uniform magnetic field all the way round. Real storage rings and colliders have straight sections, focusing magnets and injection gaps, so the average field is lower than the dipoles’ field, and the number of turns per mean life is set by the average. Particles in a real ring lose energy — muons to ionisation in residual gas and to synchrotron radiation, electrons far more — and a particle’s energy, and so its $\gamma$, changes slowly during storage, which the figures ignore. The cyclotron figure shows the revolution frequency of a single proton; how a beam stays in step depends on the spread of phases and energies within a bunch, which is what phase stability is about.

The beamline figure uses a single momentum for each curve, while real secondary beams have a spread, and ignores the particles’ passage through matter and their loss at apertures, usually larger than the loss to decay. None of these affects the central result, which holds for any particle in any field: the turning and the decaying are both clocks of the same particle, slowed by the same $\gamma$.

The domain of the argument is a charged unstable particle in a magnetic field, at any energy, with its lifetime set by its proper time — the clock hypothesis that the earlier argument tested — and its orbit by the Lorentz force. Within it, the turns per mean life depend on the field and on nothing about the particle’s energy.

Still open: whether muons can be cooled fast enough to collide #

A muon collider would make muons by smashing protons into a target and collecting the pions, letting the pions decay to muons, and then compressing the resulting diffuse, hot cloud of muons into a beam fine enough to collide — all within a few muon lifetimes. The compression, called cooling, must be done by passing the muons through absorbers that slow them in every direction and then re-accelerating them only forwards, which takes metres of apparatus and microseconds of the muons’ dilated lives. A demonstration of this ionisation cooling at the Rutherford Appleton Laboratory reported its first success in 2020, reducing a beam’s spread by a small but measurable amount. Whether cooling can be pushed far enough and fast enough, before the muons decay, and whether the radiation from the neutrinos their decays produce can be kept within safe limits at the surface above a ring at teraelectronvolt energies, are the questions on which a muon collider rests.

The arithmetic of keeping one is already settled. A charged particle circling a field B goes round in 2πγm/eB and lives γτ, so the γ cancels and it makes eBτ/2πm turns per mean life at any energy — 431 for a muon at 1.45 T, under three per tesla for a pion — and the same slowing of the revolution put the first cyclotrons out of step near 20 MeV. In a ring, a particle’s lifetime is measured in turns, and the turns are set by the magnet.