The light a charge makes by changing medium

A charge that turns must glow put acceleration at the centre of radiation: a charge that changes its velocity leaves a kink in its field, and the kink runs off as light. Why the glow of a fast charge is blue found the first exception. A charge in uniform motion through water radiates if it outruns the speed of light in water, and what radiates is not the charge but the water’s electrons, polarised in sequence by the field sweeping past them.

That exception needs the particle to be fast compared with light in the medium. There is a second exception that needs nothing of the kind. A charge crossing a boundary between two media radiates at the boundary, at any speed, though it does not accelerate and does not outrun anything. Vitaly Ginzburg and Ilya Frank predicted it in 1945 and called it transition radiation. For a slow particle it is a feeble curiosity. For a very fast one it turns out to carry an energy proportional to the particle’s Lorentz factor, which makes it one of the very few ways of measuring how relativistic a particle is once its speed has stopped changing at all.

A field that has to be rebuilt #

The simplest way to see why there must be radiation is to take the boundary between vacuum and a perfect conductor. A charge approaching the metal induces charges on its surface that pull towards it, and the field outside the metal is exactly what it would be if the metal were replaced by an equal and opposite charge at the mirror-image point inside it — the image that the inside of a conductor makes necessary. Seen from far away, charge and image form an electric dipole whose length is twice the charge’s distance from the surface, and that length shrinks to zero at a steady rate as the charge arrives. When the charge enters the metal, the dipole vanishes.

A dipole whose strength changes radiates, and one that collapses from a finite size to nothing at a constant rate makes a pulse of light when the collapse ends. The charge itself kept its speed throughout. What changed was the arrangement of the field, which in vacuum spreads out in every direction and inside the metal is cancelled within a short distance by the electrons crowding round. The energy that the outside field carried has partly to be left behind as a free wave.

For a real material the same thing happens more gently. A fast charge in vacuum carries its field squeezed into a thin disc at right angles to its motion, the flash of light a passing charge is for any observer it sweeps past, with a transverse reach of about γ times c/ω for each frequency ω in the flash. Inside a medium the electrons respond to the field and alter it. They follow the field almost freely at frequencies far above their binding, and there the medium’s dielectric constant is $\varepsilon = 1 - \omega_p^2/\omega^2$, with $\omega_p$ the plasma frequency set by the density of electrons, about 21 electronvolts for polypropylene. The field the charge carries in the medium is therefore not the field it carried in vacuum. At the boundary one has to turn into the other, and the difference is radiated.

A hollow cone at one over gamma #

The pattern for an ultrarelativistic charge was worked out from Maxwell’s equations by matching the fields across the boundary. The energy per unit frequency and solid angle is

$$\frac{d^2W}{d\omega\,d\Omega} = \frac{\alpha\hbar}{\pi^2}\,\theta^2\left[\frac{1}{\gamma^{-2}+\theta^2} - \frac{1}{\gamma^{-2}+\theta^2+\omega_p^2/\omega^2}\right]^2,$$

where α is the fine-structure constant and θ the angle to the particle’s path. The two terms inside the bracket are the vacuum field and the medium’s field, and the radiation is their mismatch squared. Exactly along the path the factor $\theta^2$ kills it, so the light comes out in a hollow cone, and the brightest ring lies close to $\theta = 1/\gamma$: at 0.98, 0.88 and 0.68 in units of 1/γ for the three frequencies drawn. That angle is the signature of every radiation from an ultrarelativistic source, the same one-over-gamma cone into which a circling electron concentrates its synchrotron light. For a 2 GeV electron it is a quarter of a milliradian, so the X-rays travel on almost exactly along the particle’s track.

The bracket also shows where the radiation stops. When ω is much larger than $\gamma\omega_p$, the third term in the second denominator is negligible against the first, the two fractions become equal, and the mismatch vanishes: at such frequencies the medium’s electrons cannot alter the field quickly enough over the length that matters, and the charge carries the same field on both sides. When ω is much smaller than $\gamma\omega_p$, the second fraction is small and the vacuum field is radiated more or less entirely. The light is produced in the band below $\gamma\omega_p$, and the bigger γ, the wider that band.

Energy that counts gamma #

Integrating over angle gives a spectrum with a single shape,

$$\frac{dW}{d\omega} = \frac{\alpha\hbar}{\pi}\left[(1+2\nu^2)\ln\!\left(1+\frac{1}{\nu^2}\right)-2\right],\qquad \nu = \frac{\omega}{\gamma\omega_p},$$

and integrating that over frequency gives the total:

$$W = \tfrac{1}{3}\,\alpha\,\gamma\,\hbar\omega_p.$$

The total is proportional to γ. A pion and an electron of the same momentum, two GeV per unit c, move at speeds that differ by about one part in a thousand, and every effect that depends on speed — the Cherenkov angle, the time of flight, the cone a decay cannot leave — can barely tell them apart. Their Lorentz factors differ by the ratio of their masses, 273, and the energy each radiates at a surface differs by the same ratio. A 2 GeV electron, γ about four thousand, sheds two hundred electronvolts at one boundary with polypropylene; the pion sheds less than one.

That linear growth is unusual and worth setting beside the energy loss that does saturate. A fast charged particle crossing matter loses energy mainly by knocking electrons out of atoms, and the rate rises slowly with energy as the particle’s flattened field reaches out to more distant atoms. Then it stops rising. Enrico Fermi explained the plateau in 1940: the medium’s own polarisation screens the far-reaching part of the field beyond a distance of about $c/\omega_p$, so a faster particle gains nothing more from reaching further. That is the same plasma frequency, setting the same screening, that makes the field inside the medium differ from the field outside. Ionisation stops measuring γ exactly where the mismatch at a boundary starts to measure it.

The X-rays no cone can make #

The comparison with Cherenkov light is sharper than it looks. Cherenkov radiation needs the particle to outrun light in the medium, $\beta n > 1$, and at the frequencies where transition radiation is strongest that is impossible for any particle at all. The plasma formula gives an index $n = \sqrt{1-\omega_p^2/\omega^2}$, a little less than one, which is the result the index that falls below one found for every material at X-ray frequencies: light there runs faster than c in phase, and no particle can be faster than it. A water tank shines blue under a beam of electrons and gives no Cherenkov X-rays whatever their energy.

So in the X-ray band the medium cannot radiate by being outrun. It can only radiate by being entered or left, and the energy available to that process grows without limit as γ does, while the Cherenkov yield per centimetre, where it exists, stops growing once β is close to one. The two exceptions to “only acceleration radiates” divide the spectrum between them. Below the material’s electronic resonances, where $n > 1$, a fast particle drags a cone; above the plasma frequency, where $n < 1$, it can only leave a flash at each boundary, and the flash carries the particle’s γ. The plasma frequency that separates them is the same one that sets the frequency below which nothing gets in to a metal or an ionosphere: below it the electrons screen a field out, above it they let it through changed.

Photons at the rate of alpha #

Two hundred electronvolts is not much, and it is not delivered as one photon. Dividing the spectrum by ħω to count photons gives a number of order α per surface. The figure counts only X-rays above 4 keV, which are the ones energetic enough to escape the plastic and be caught in a gas.

The electron’s curve rises from below one photon per hundred crossings at 1 GeV/c to several at a hundred; the pion’s is flat at nothing until its own γ reaches the thousands, near a hundred GeV/c, and only at a TeV does it give something comparable to the electron’s yield at a few GeV. The muon, two hundred times heavier than the electron, joins between them. The separation is clean because the spectrum’s edge, γħω_p, falls on one side or the other of the X-ray band: for a pion at 2 GeV/c it lies at three hundred electronvolts, where the photons are absorbed in the material that made them, and for an electron at eighty keV.

So the signal is weak but nearly binary, and the obvious remedy is to make the particle cross many surfaces. That brings in the one piece of physics the single-surface formula leaves out.

The length a photon takes to be made #

A surface cannot radiate by itself, since there is no surface without two media. A foil has two surfaces, entry and exit, and their mismatches have opposite signs: going in, the field changes from vacuum to medium; coming out, it changes back. If the two wavelets arrived together, they would cancel, as the two reflections from a quarter-wave coating are arranged to cancel — here by accident rather than design.

They do not arrive together, because the charge and the light it has emitted travel at slightly different speeds along the path. Over a distance $Z$ the charge falls behind its own wavelet, or the wavelet behind the charge, by a phase of one radian, and that distance is the formation zone:

$$Z = \frac{2c}{\omega}\,\frac{1}{\gamma^{-2}+\theta^2+\omega_p^2/\omega^2}.$$

In vacuum, without the last term, it is γ²c/ω at the angle 1/γ, hundreds of micrometres for 10 keV X-rays from a 2 GeV electron. In polypropylene the plasma term dominates and it shrinks to about nine micrometres, growing with photon energy because the medium matters less at higher frequency: two micrometres at 2 keV, twenty-one at 30 keV.

A foil much thinner than the formation zone does nothing: its two surfaces radiate in antiphase and cancel. The 5 micrometre foil in the figure is thick enough for the soft X-rays, where the zone is short, and too thin for the hard ones, which it cuts off. A foil comparable with the zone does better than two independent surfaces, because the phase slip across it can be close to half a cycle, which turns cancellation into reinforcement: the 15 micrometre foil gives 1.8 times two free surfaces near 6 keV. A much thicker foil oscillates about the dashed curve as the phase slip winds through whole cycles, and averages to it. The same reasoning applies to the gaps between foils, with the longer vacuum formation zone, which is why a radiator stacks foils about fifteen micrometres thick a few hundred micrometres apart.

The formation zone is the honest answer to a question that sounds empty: where, exactly, is the light emitted? Not at the surface. It is emitted over a distance $Z$, and inside that distance the light and the charge are not yet distinct objects. A charge that meets a second surface before the first one’s light has separated from it radiates according to both together.

Hundreds of foils for one photon #

Each foil gives about three X-rays per hundred crossings. A stack of a hundred gives about two per electron, and the curve bends as it grows, because X-rays made at the front of the stack must pass through the rest of it, and the plastic absorbs the softest of them. Beyond a few hundred foils the gain is small. The X-rays then travel on, within a quarter of a milliradian of the track, into a gas of heavy atoms — xenon is the usual choice, since it absorbs X-rays of ten keV in about a centimetre — where each one makes a burst of ionisation larger than anything the particle’s own passage leaves in the same few millimetres of gas.

That is the transition radiation detector. ATLAS at the Large Hadron Collider wrapped its inner tracker in polypropylene and threaded it with three hundred thousand thin xenon-filled tubes, so that every track was tested for the large clusters an electron’s X-rays leave and a pion’s do not. Detectors flown in space use the same principle to pick out cosmic-ray electrons and positrons among a hundred times as many protons of the same momentum. In each case what is measured is γ, through a light that is emitted only because a boundary was crossed.

The same physics with far fewer photons runs the other way in an accelerator. A thin polished foil placed in a beam emits optical transition radiation as each bunch passes through it, and a camera aimed at the foil’s surface sees an image of the beam’s cross-section, made at the instant of crossing, with a resolution of micrometres. The light’s ring of radius 1/γ, imaged at infinity instead, gives the beam’s energy. Optical transition radiation screens are among the commonest instruments along the beamlines of electron accelerators, because the foil is thin enough to disturb the beam hardly at all.

The same light, with the boundary in time #

There is a second way to look at the crossing. In the charge’s own frame nothing is moving except the medium, whose surface rushes towards it at nearly c and engulfs it. From the charge’s point of view the medium around it changes in an instant, from vacuum to plastic, everywhere it can feel. That is a change of a medium in time, which the reflection that needs no surface found splits any wave already present into a forward and a backward part. The field the charge carries is such a wave, or a sum of them, and a sudden change of medium has to shed part of it.

The two pictures, a boundary in space crossed at speed and a change in time seen from rest, are the same physics in two frames, and transition radiation is the clearest place to see that a radiating charge does not need to be accelerated. It needs its field to stop being a solution of the equations it is carrying it under. Acceleration does that, as the Larmor formula says; a medium that outruns the field’s travel does it, as the Cherenkov cone does; a change of medium does it too. In all three, what is radiated is the part of the field that no longer fits.

What the formula leaves out #

The spectrum drawn here assumes three things that real radiators relax. It treats the medium as a plasma of free electrons, $\varepsilon = 1 - \omega_p^2/\omega^2$, which holds for X-rays far above the binding of the material’s electrons and fails in the visible and ultraviolet, where optical transition radiation obeys a different formula with the material’s actual dielectric constant. It treats the foils as independent beyond their own two surfaces, adding the yields of successive foils instead of their amplitudes, which neglects the interference across the gaps that a regular stack shows as a further modulation of the spectrum and a sharper threshold. And it scales the plastic’s X-ray absorption crudely, as the inverse cube of the photon energy from a single value at 10 keV, which is enough to show why the curve saturates and not enough to give a detector’s exact yield.

The pion is drawn as giving nothing, and in that idealised sense it gives nothing. In a working detector the pion still ionises the gas, and occasionally knocks out an energetic electron that deposits as much as an X-ray would. Those rare large clusters, not transition radiation, are what limit how well a real detector rejects pions, usually to about one in a hundred while keeping nine electrons in ten.

The domain of the formula is a particle with γ much larger than one, photons far above the plasma frequency, and smooth boundaries at least a formation zone apart. A slow charge crossing a boundary still radiates, but weakly and mostly in the visible, with an energy that does not grow with γ.

Still open: how far the signal can be pushed into the TeV #

At the highest energies the electron’s signal saturates, because a regular radiator’s formation zones cap the useful spectrum at some value of γ, while the pion’s signal keeps rising until it too saturates. Above a few hundred GeV the two become hard to tell apart, which matters for cosmic-ray experiments that want to separate electrons from protons at TeV energies, where the γ of a proton is itself in the thousands. Radiators built from foils of different thicknesses, irregular foams, or multilayer structures tuned to move the threshold upward are being designed and tested to stretch the range, and how far a transition radiation detector can be pushed before it stops telling masses apart, and what the cleanest radiator for TeV particles looks like, are being settled by simulation and beam tests rather than by any closed formula.

The habit worth carrying away is to ask what a measurement is sensitive to once the obvious variable has saturated. Above a few GeV every particle’s speed is c to a part in a thousand, but its Lorentz factor still carries its mass, and a boundary crossed at that speed radiates αγħω_p/3 — two hundred electronvolts for an electron at 2 GeV, about one for a pion. The charge does not accelerate; its field is rebuilt, and the rebuilding is counted in X-rays.