Why the glow of a fast charge is blue

A charge that turns must glow found that radiation is what a charge’s field does when the charge accelerates: the field cannot rearrange itself instantly, a kink forms in it and travels outward at the speed of light, and the kink carries energy. Every later argument about radiating charges — the self-force, whether a charge on a table glows, the bremsstrahlung bill, the flash a circling charge sends once a turn, the cross-section of a shaken electron — took acceleration as the price of admission. A charge moving in a straight line at constant speed carries its field along with it, unchanged, and radiates nothing. In a vacuum that is exact, and it has to be: a charge at constant velocity is a charge at rest seen by a moving observer, and a charge at rest does not radiate.

The field that points where the charge is now noted the exception and named it. In a transparent medium the field’s disturbances travel at $c/n$ rather than $c$, and a charge moving faster than that outruns its own field and leaves a cone of light behind it, the blue glow of a reactor pool. This essay takes the exception further. The cone is half of the story; the other half is how much light there is and at what colours, and that half explains why the glow is blue, how faint it is, and what is actually doing the radiating.

A cone with no acceleration in it #

The geometry is the one the cone the source leaves behind drew for a supersonic aircraft. As the particle passes each point, the disturbance its field makes there spreads out as a wavelet at the speed of light in the medium. If the particle is faster than the wavelets, it is always ahead of every one of them, and the wavelets from all its past positions are tangent to a single cone with its tip at the particle.

Where the wavelets touch, they arrive in phase and reinforce; everywhere else they arrive at scattered times and cancel. So the light goes out along the normal to the cone, at an angle $\theta$ to the particle’s path with

$$\cos\theta = \frac{1}{n\beta},$$

where $\beta$ is the particle’s speed as a fraction of $c$. For a particle at 0.99c in water, with $n = 1.33$, the angle is 40.6 degrees. Below the threshold $\beta = 1/n$ there is no cone and no light: every wavelet runs ahead of the particle, and the field settles into the shape it would have in a vacuum, compressed and squashed but radiating nothing.

The particle never accelerates. Nothing in the construction needs it to; it needs only a speed and a medium. The difference from every earlier case is that the medium sets a speed limit lower than the particle’s.

Who is actually radiating #

The obvious question is where the light comes from, if not from an acceleration. It comes from the water. A fast charged particle passing a water molecule tugs its electrons briefly towards it and lets them go: the molecule is polarised for an instant, and a briefly polarised molecule is an oscillating dipole, which radiates. Each molecule along the path does this as the particle passes. In a vacuum there are no molecules, nothing to polarise, nothing to radiate.

In a medium, below the threshold, the molecules still radiate, but their wavelets reach any distant point at different times and cancel: the polarisation the particle drags along with it is a steady pattern moving at the particle’s speed, and a steady pattern moving slower than light in the medium cannot shed waves into it. Above the threshold the pattern moves faster than the waves it makes, and the wavelets from the dipoles along the path add up coherently along the cone. The radiation is the medium’s, organised by the particle. The electrons that do the radiating are accelerating — they are pulled and released — so the rule that radiation needs acceleration survives, applied to the right charges.

The energy comes from the particle all the same. The polarised molecules pull back on it, and the particle loses kinetic energy at exactly the rate the cone carries it away. Frank and Tamm worked out that rate in 1937, three years after Cherenkov, a graduate student in Vavilov’s laboratory in Moscow, had found the faint blue light in liquids bombarded by gamma rays and shown that it was not fluorescence: it was the same in every liquid, polarised along the direction of the gamma rays, and emitted forward. Heaviside had worked out the cone in the 1880s for a charge moving faster than light in a medium, and his result had been forgotten. The three Russians shared the 1958 Nobel Prize.

A threshold that sorts particles #

The threshold is a speed, which for a particle of known mass is a kinetic energy. In water every particle must reach a Lorentz factor of 1.517, which is 264 kiloelectronvolts of kinetic energy for an electron, 54.6 megaelectronvolts for a muon and 485 megaelectronvolts for a proton. Above it the cone opens from zero and approaches the limiting angle $\arccos(1/n) = 41.2°$ as the particle approaches the speed of light.

So at the same energy different particles behave differently. A muon of a few hundred megaelectronvolts from a cosmic-ray shower crosses a tank of water and makes a nearly full cone; a proton of the same energy makes none. The light is a speedometer, and a speedometer combined with a momentum, measured by bending the particle in a magnetic field, is a scale for mass.

Two hundred photons a centimetre #

The cone says where the light goes. Frank and Tamm’s formula says how much of it there is. For a particle with a single charge, the number of photons emitted per unit length of path and per unit wavelength is

$$\frac{d^2N}{dx\,d\lambda} = \frac{2\pi\alpha}{\lambda^2}\left(1 - \frac{1}{\beta^2 n^2}\right),$$

with $\alpha$ the fine-structure constant, about 1/137. The bracket is $\sin^2\theta$, zero at threshold and largest for the fastest particles. Everything else is a pure number divided by the square of the wavelength.

Integrated between 400 and 700 nanometres, the visible band, the formula gives about 214 photons per centimetre of water for a particle near the speed of light. It is a faint light. A fast particle crossing water loses about two megaelectronvolts per centimetre to ionising the molecules it passes, and its Cherenkov light carries a few hundred electronvolts — about a four-thousandth of its energy loss. But a muon crossing the forty-metre tank of a large neutrino detector makes most of a million photons, and the glow of a reactor pool is the light of billions of electrons from fission products, each making its few hundred.

The count also explains a detail of how the light rises from threshold. Just above it the cone is narrow, $\sin^2\theta$ is small, and almost nothing is emitted; an electron reaches half the limiting count at 459 kiloelectronvolts, 195 above its threshold, and nine-tenths only at about a megaelectronvolt. Detectors therefore see most clearly the particles well above threshold, and the softest particles that just cross it are nearly invisible.

Why the light is blue #

The colour follows from the $1/\lambda^2$. Equal intervals of wavelength contain more photons at the short end than at the long: between 400 and 500 nanometres there are 101 photons per centimetre, between 600 and 700 only 48. Measured as energy, which adds another factor of $1/\lambda$, the spectrum is weighted to the blue more steeply still. The eye, which sees nothing below about 400 nanometres, is shown the long-wavelength tail of a spectrum that is strongest in the ultraviolet, and the tail is blue.

That the formula rises without limit towards short wavelengths is the clue to where it stops. It does not stop because of anything the particle does; it stops because the medium changes. Water’s index rises a little towards the ultraviolet, which only strengthens the light, and then the water begins to absorb, below about 200 nanometres, and its index falls through one. Above the frequencies at which a medium’s electrons can respond, its index is less than one — the index that falls below one found it slightly below one for X-rays in every material — and with $n < 1$ no particle, however fast, can outrun the light, and the spectrum ends. Cherenkov light exists only in the band where the medium is transparent and slows light, which is why it is ultraviolet, visible and infrared, and never X-ray.

The contrast with the sky is worth noticing. Why the sky is blue and the sunset is not found that air scatters blue light more than red, as the fourth power of frequency, because the molecules respond more strongly to faster shaking. Cherenkov light is blue for a different and simpler reason: the number of modes a cone can fill in a given wavelength band grows as the frequency, and each carries a share of energy that grows as the frequency again. Two blues, one from a cross-section and one from a count.

The glow of a pool, and what it is used to check #

The electrons that light a reactor pool come from two sources. Fission products are unstable and decay by emitting electrons with energies of up to several megaelectronvolts, far above the 264-kiloelectronvolt threshold; and the gamma rays from fission and decay knock electrons out of the water’s molecules by Compton scattering, giving them similar energies. Both kinds slow down within a few millimetres, so the light comes from a thin layer of water around everything radioactive, and its brightness follows the activity. A running reactor’s core glows strongly; a fuel assembly removed from it goes on glowing for years, dimming as its fission products decay.

That dimming is used. Inspectors who verify that spent nuclear fuel in a storage pond has not been removed or replaced with dummies photograph the Cherenkov glow from above with image-intensifying viewers: an assembly with real fuel glows with the pattern of its rods, a dummy of steel does not glow at all, and an assembly with some rods missing glows with gaps. The glow can be checked without touching or moving anything, through metres of water, because the light is made in the water itself.

The same light is made inside people. Patients receiving radiation therapy to the head have long reported flashes of light with their eyes closed, and part of the explanation, confirmed in 2019 by imaging the eye during treatment, is Cherenkov light made by fast electrons in the vitreous humour, the clear gel that fills the eyeball: the eye is a small Cherenkov detector with its own photoreceptors attached. Astronauts outside the Earth’s protective field report similar flashes, from cosmic rays crossing the eye, and Cherenkov light is thought to be one of the mechanisms there too. Radiotherapy now also uses a camera aimed at the patient’s skin to image the Cherenkov light a treatment beam makes as it enters the body, a direct picture of where the dose is going.

Rings that name a particle #

The angle’s dependence on speed makes the light an instrument for identifying particles. A dense medium like water has a low threshold and a wide cone and suits particles of modest energy; for particles of tens of gigaelectronvolts, whose speeds differ from $c$ by parts in a thousand, a gas with an index barely above one spreads those small differences of speed into large differences of angle. In a gas of index 1.0014, a pion of ten gigaelectronvolts per $c$ makes a cone of 51 milliradians and a kaon of the same momentum one of 19. A spherical mirror focuses each cone into a ring on a plane of photon detectors, and the radius of the ring names the particle. Ring-imaging detectors of this kind are how the experiments that study B mesons tell a decay into kaons from a decay into pions, which differ in the physics they test.

The largest Cherenkov detectors are vast tanks of water. Super-Kamiokande holds fifty thousand tonnes of pure water in a cavern under a Japanese mountain, watched by eleven thousand photomultiplier tubes. A neutrino, which crosses almost anything, occasionally strikes an electron or a nucleus in the water and makes a charged particle, which makes a ring: sharp-edged for a muon, which travels straight; fuzzy for an electron, which scatters and showers as it goes. Counting sharp rings and fuzzy ones, from neutrinos made in the atmosphere on the far side of the Earth and directly overhead, showed in 1998 that muon neutrinos change into another kind on the way through the planet — the discovery that neutrinos have mass. IceCube does the same in a cubic kilometre of Antarctic ice, and on the surface, telescopes watch for the faint flashes of Cherenkov light that a high-energy gamma ray’s shower of particles makes in the air, at an angle of about a degree.

What the medium remembers #

Cherenkov radiation has a sibling that makes the same point more starkly. A charge crossing the boundary between two media with different indices radiates even if it is slower than light in both, because its field must rearrange as the medium changes. That transition radiation, predicted by Frank and Ginzburg in 1945, is again radiation without acceleration, and again what changes is the medium rather than the charge. Both are reminders that “an accelerating charge radiates” is a statement about a charge in a vacuum. In matter, a charge in uniform motion is surrounded by a pattern of polarisation, and whenever that pattern cannot keep up with the charge — because the charge is too fast, or because the medium suddenly changes — the pattern sheds light.

The momentum light carries into glass met the same theme from the other side: light in a medium is partly field and partly the medium’s response, and which share of its momentum belongs to which is a question about how the two are divided. Cherenkov light is born in that division. It is radiated by the medium, organised by the charge, and paid for out of the charge’s kinetic energy.

What the pictures cannot show #

The figures use a constant index of 1.33 for the angles and counts and a simple fit to water’s measured index for the spectrum; real water’s index varies by about one per cent across the visible band, which shifts the angle by a few tenths of a degree from blue to red. They treat the particle as moving straight at constant speed, while a real electron in water scatters and slows within centimetres, so its cone wanders and narrows; a real muon goes straight for metres. They count photons emitted, not photons detected, and the fraction a detector records depends on absorption in the water, the reflectivity of its walls and the efficiency of its photomultipliers, which together are often a few tens of per cent. The gas radiator is drawn with a fixed index; real ring-imaging detectors correct for the gas’s dispersion and temperature.

Still open: the radio pulse of a cosmic ray in the ice #

At wavelengths much longer than the size of a particle shower, the Cherenkov emission of all its charges adds coherently, and because a shower in dense matter collects electrons from the medium and loses positrons to annihilation, it carries a net negative charge of about a fifth of its particles. Askaryan predicted in 1962 that this excess makes the shower glow in radio waves, with a power that grows as the square of the shower’s energy rather than in proportion to it. The effect has been measured in sand, salt and ice at accelerators. Detectors that would use it to catch the rarest, highest-energy neutrinos from beyond the Galaxy, through their radio flashes in hundreds of cubic kilometres of Antarctic or Greenland ice, are being built; whether such neutrinos arrive in numbers they can see is not yet known.

The habit worth carrying away is to ask what a rule assumes about its surroundings. “Only an accelerating charge radiates” is a rule for a charge in a vacuum; in a medium a charge in uniform motion radiates whenever it outruns light there, the medium’s electrons doing the radiating, at $\cos\theta = 1/n\beta$ and in numbers proportional to $\sin^2\theta/\lambda^2$ — about 214 visible photons per centimetre of water, most of them blue. The glow is the medium’s, and the colour is a count.