A charge that turns must glow

By 1911 the evidence for a small, heavy, positively charged nucleus with electrons around it was overwhelming. The trouble was that the picture could be shown, in about four lines of standard electromagnetism, to be impossible.

Nothing in the calculation is wrong. The acceleration is right, the radiated power is right, the energy bookkeeping is right, and the conclusion is that matter cannot exist for a nanosecond. This essay is about the law that produces that result, which is correct and useful, and about the exactly-stated place where applying it to an atom stops being legitimate.

What the law says #

An accelerating charge radiates power

$$P = \frac{q^2 a^2}{6\pi\varepsilon_0 c^3},$$

which is Larmor’s result of 1897. Three features of it decide everything below.

It contains the acceleration, not the velocity. A charge drifting at any constant speed radiates nothing, in any frame — which it must, since a constant velocity can be transformed away and radiation cannot.

It is quadratic in the acceleration. Doubling the acceleration quadruples the power, so a process that turns a charge sharply is enormously more radiative than one that turns it gently.

It carries $c^3$ in the denominator. In SI units that is $2.7\times10^{25}$, which is why an accelerating charge in an ordinary circuit radiates a negligible fraction of the energy passing through it, and why radiating deliberately requires either very many charges in step — an antenna — or very large accelerations.

A fourth feature is worth adding because it is the one that makes the formula usable in unfamiliar situations: the acceleration in it is the acceleration of the charge in its own instantaneous rest frame. For slow charges that distinction is invisible. For fast ones it is the whole of the difference between the expression above and the relativistic version, and it is where the powers of $\gamma$ later in this essay come from — not from any new physics, but from converting an acceleration between frames.

Where the radiation goes #

The total power is only half the story, and the angular distribution is the half that has consequences.

The null along the acceleration is worth pausing on, because it is a general feature rather than an accident: it is the same reason a dipole antenna radiates nothing off its own ends, and it is why the shape of an aerial decides what a transmitter covers.

The reason for the null is geometric and can be seen without any algebra. The radiation field is transverse — it has no component along the line of sight — so what an observer receives is the projection of the charge’s acceleration onto the plane perpendicular to that line. Look at the charge from a direction at right angles to its acceleration and the whole of the acceleration projects; look along the acceleration and none of it does. The sin²θ is that projection, squared because power goes as the square of the field.

The same reasoning gives the polarisation for free: the received field lies along the projected acceleration, so the light is linearly polarised in the plane containing the acceleration and the line of sight. That is an observable property, and it is how a radiating system’s geometry is inferred without ever resolving it.

Every orbit is an acceleration #

The application to atoms needs one further ingredient, which is that going round in a circle at constant speed is accelerating.

Turning is an acceleration: a velocity changes when its direction changes, not only when its magnitude does, and for circular motion that change points at the centre with magnitude $v^2/r$. An electron in a circular orbit is therefore accelerating continuously, at every instant, and the Larmor formula has no clause exempting a charge whose speed is constant. This is the step that makes the difficulty unavoidable rather than merely awkward: there is no orbit that avoids it, because an orbit is an acceleration.

For an electron at the Bohr radius the acceleration is about $9\times10^{22}$ m/s², which is a very large number, and the radiated power is around $5\times10^{-8}$ watts. That sounds tiny until it is compared with what the atom has: the binding energy is $2.2\times10^{-18}$ J, so the whole atom’s worth of energy is radiated in about $10^{-11}$ seconds.

What the collapse would actually look like #

Integrating properly gives more than a lifetime; it gives the shape of the decay and a second, independent falsification.

Since $E = -e^2/8\pi\varepsilon_0 r$ for a circular orbit, and $P = -\mathrm{d}E/\mathrm{d}t$, the radius obeys $\mathrm{d}r/\mathrm{d}t \propto -1/r^2$, so $r^3$ falls linearly and

$$r(t) = r_0\left(1 - \frac{t}{t_{\text{collapse}}}\right)^{1/3}.$$

The second falsification is spectral. As the orbit shrinks the orbital frequency rises continuously, so the emitted light would sweep smoothly upward in frequency through the whole spectrum during the collapse. Atoms emit sharp lines at fixed frequencies instead. Both facts — that atoms exist, and that their spectra are discrete — were known, and both are incompatible with this calculation.

There is a third objection, less often made and rather harder to dismiss, about how the collapse would end. As the radius falls the speed rises — a circular orbit has $v^2 = e^2/4\pi\varepsilon_0 m r$ — so the electron reaches an appreciable fraction of the speed of light in the last stages, and the non-relativistic Larmor formula stops applying before the collapse finishes. The correction makes matters worse rather than better: the relativistic form radiates more. The lifetime computed here is therefore an over-estimate, and the honest statement is that classical physics gives an atom at most sixteen picoseconds.

It is worth noticing what the failure did not show. Nothing here suggests that electromagnetism is wrong, or that the nucleus is not there, or that the electron is not charged — all of which were live possibilities in 1912, and all of which the calculation is often loosely said to have ruled out. What it shows is narrower and much sharper: a charged point particle following a classical trajectory around a nucleus is not a possible description of an atom. Every assumption in that sentence except the last was eventually kept.

What replaced it, and what did not #

The resolution is not that the Larmor formula is wrong. It is that an electron in an atom is not a small charged ball following a trajectory.

What is there instead is a stationary distribution of charge — a probability density that does not change with time. The distinction matters more than it first appears. A charge distribution that does not change has no time-varying dipole moment, and the radiation formula acts on a time-varying moment and on nothing else, so there is simply no source term. The ground state is not stable because the radiation was switched off, or because some new force balances the old one. It is stable because the quantity that would have to be oscillating is not oscillating, and Larmor’s result is left exactly as true as it was.

The stability is worth stating in the form that makes it a physical answer rather than a definition. What sets the size of the ground state is a competition: confining the electron to a smaller region raises its kinetic energy, by the uncertainty relation, faster than it lowers the electrostatic energy. Minimising the sum gives a radius, and that radius is the Bohr radius. Nothing about that argument mentions radiation, which is the point — the atom has a size for a reason unconnected with the process that would otherwise destroy it.

Two things survive intact into the quantum treatment. The first is that transitions do radiate, and the rates are computed from a matrix element that is recognisably the same dipole quantity — with the classical acceleration replaced by an oscillation between two states. The second is the angular pattern: light emitted in an atomic transition comes out with the same sin²θ distribution about the relevant axis, which is measurable and measured.

What does not survive is the picture of an orbit. It is worth being precise about which part failed: not the electromagnetism, not the mechanics, but the assumption that a bound electron has a trajectory at all.

The searchlight #

The other end of this subject is a device rather than a paradox. Accelerate a charge to a speed near $c$ and the radiation pattern is transformed by aberration, sweeping the lobes forward into a narrow cone. The apparatus that does it is a magnetic field, which bends a charged particle’s path without changing its speed and so supplies pure centripetal acceleration — the one arrangement that gets a charge to radiate hard without doing any work on it.

The consequences compound. In the relativistic form of the Larmor formula, for a charge moving in a circle, the power carries a factor of $\gamma^4$. So a highly relativistic electron in a magnet radiates enormously, in a beam whose angular width is about $1/\gamma$ — for a 1 GeV electron, $\gamma$ is about 2000 and the cone is half a milliradian.

It is worth separating the two relativistic effects, because they are often run together and they are different things. The first is the aberration that redirects the pattern, which changes where the light goes and not how much there is. The second is the transformation of the acceleration between frames, which changes how much there is. A charge that is merely moving fast in a straight line experiences the first and not the second, and radiates nothing at all; a charge being turned experiences both.

The $\gamma^4$ has a hard engineering consequence. Because $\gamma$ is energy over rest energy, and a proton is 1836 times heavier than an electron, a proton at the same energy has a $\gamma$ smaller by that factor and radiates $1836^4 \approx 10^{13}$ times less. That single number is why the highest-energy circular colliders accelerate protons and why an electron machine at comparable energy has to be built in a straight line.

The instrument that came out of the nuisance #

For twenty years the radiation from a bending magnet was a problem: it was where the energy in an electron machine went, and the machines were built for particle physics rather than for light. It was first seen directly in 1947, at General Electric’s 70 MeV synchrotron, through a transparent section of the vacuum chamber, and named after the machine that produced it.

The properties that made it a nuisance are the ones that make it valuable. The beam is naturally collimated to about $1/\gamma$, so the brightness — power per unit area per unit solid angle — is enormous without any focusing optics. The spectrum is broad and continuous, extending to a critical frequency that goes as $\gamma^3$, so one machine covers infrared to hard X-rays and a monochromator selects. The output is pulsed at the ring’s own bunch frequency, which gives sub-nanosecond timing for free. And it is polarised, in the plane of the orbit, because that is where the acceleration is.

Dedicated machines followed, then insertion devices — arrays of alternating magnets that make the electron wiggle many times and add the contributions in phase. The modern versions are brighter than a laboratory X-ray tube by ten orders of magnitude and are the routine instrument for determining what a crystal’s structure is. All of it comes from an electron being made to turn, which is the same sentence that killed the classical atom.

Why an antenna has to be big #

The $c^3$ in the denominator was noted as the reason a charge radiates so feebly. Turned into engineering it becomes a hard constraint on how small a transmitter can be, and the constraint has no way round it.

An antenna is a device for making charges accelerate in step. Summing the Larmor contribution over a short wire carrying a current gives a radiated power proportional to the square of the current times the square of the wire’s length divided by the wavelength — and expressing that as an equivalent resistance, so it can be compared with the wire’s own losses, gives a radiation resistance of about $790\,(L/\lambda)^2$ ohms for a short dipole.

The square is what does the damage. An antenna a hundredth of a wavelength long has a radiation resistance of eight hundredths of an ohm, which is comparable with or smaller than the ohmic resistance of the wire and the losses in the ground beneath it — so most of the power supplied goes into heat and only a small fraction is radiated.

Make the antenna half a wavelength long and the radiation resistance is 73 ohms, which is a comfortable match to an ordinary feed line and larger than any loss in the system. That is why the half-wave dipole is the standard element of the whole subject: it is the shortest length at which an antenna stops being mostly a heater.

The consequence bites hardest where the wavelength is long. Communication with a submerged submarine requires a wave that penetrates seawater, which means very low frequency: at twenty kilohertz the wavelength is fifteen kilometres. A three-hundred-metre mast is then a fiftieth of a wavelength, and its radiation resistance is a fraction of an ohm. The stations built to do it are among the largest structures ever erected for a single purpose — arrays of towers spanning kilometres, carrying a canopy of wire to force more current into the vertical section — and they consume megawatts to radiate a few hundred kilowatts.

At extremely low frequency it becomes absurd, and instructively so. A system operating near eighty hertz has a wavelength of four thousand kilometres, so an antenna eighty kilometres long is a fiftieth of a per cent of one. Its radiation resistance is measured in milliohms. Such a system was built, used two enormous buried ground conductors, drew megawatts, radiated a handful of watts, and carried a few characters per minute.

Every one of those numbers comes out of the same formula that killed the classical atom, and the direction is the same in both cases: the constant is small, so getting anything out of it takes either enormous acceleration or an enormous number of charges moving together.

The acceleration that costs nothing #

The essay asserts that an electron machine at high energy must be built in a straight line. The arithmetic behind that is worth doing, because the contrast between the two geometries is not a factor of a few.

For a charge accelerated along its own direction of motion — which is what a linear accelerator does — the radiated power depends on the rate at which the energy is being given, and the fraction of the supplied energy that is lost to radiation works out as the accelerating gradient divided by a characteristic gradient: the electron’s rest energy divided by its classical radius, which is about $2\times10^{20}$ volts per metre.

Achievable gradients are a hundred megavolts per metre in a conventional structure and perhaps a few gigavolts per metre in the most aggressive plasma schemes. Dividing gives a radiated fraction of order $10^{-12}$. A linear accelerator loses essentially nothing to radiation, at any final energy, because the loss depends on the gradient and not on how long the machine is.

Bending is entirely different. A charge turned round a radius $R$ has an acceleration of $v^2/R$ in the laboratory, which transforms into its own frame with two extra powers of $\gamma$, and the radiated power then carries $\gamma^4$. The energy lost per turn goes as $\gamma^4/R$, so it climbs as the fourth power of the energy at a fixed ring size.

That is the whole difference. Straight-line acceleration is free of radiation because the acceleration is along the motion and the geometry supplies no additional factors; circular acceleration is expensive because the acceleration is transverse and the relativistic transformation piles four powers of $\gamma$ onto it.

Put numbers on the electron case and the conclusion is forced. At a hundred billion electronvolts an electron in a ring of a few kilometres’ radius radiates a substantial fraction of its energy every turn, and the machine spends most of its power replacing what the bending removed. The same ring with protons in it loses $1836^4$ — about $10^{13}$ — times less, which is why every high-energy circular collider built has been a proton machine and why the proposals for electron colliders beyond a few hundred billion electronvolts are all tens of kilometres long and perfectly straight.

What it costs, and where the model stops #

Radiation reaction is not a solved part of classical electromagnetism. If a charge radiates, it loses momentum, so the radiation must react back on the charge. The classical expression for that force — the Abraham–Lorentz force — is proportional to the derivative of the acceleration, and a differential equation of that form has solutions in which a charge accelerates before the force is applied, and others in which it accelerates for ever without a force. Both are unphysical, both are exact solutions, and no fully satisfactory classical resolution exists. What is used in practice is an approximation that discards the offending term.

The self-energy diverges. A point charge’s own field carries an energy that goes as $1/r$ as the radius goes to zero, so a genuinely point-like electron has infinite electromagnetic mass. This is not a subtlety at the edge; it is the reason the classical theory of a charged particle cannot be completed, and the reason the problem was handed to quantum field theory.

A uniformly accelerating charge is genuinely confusing. By the Larmor formula it radiates. By the equivalence principle, a charge sitting on a table in a gravitational field is accelerating in exactly the same sense — and does not appear to radiate. The resolution turns on the fact that whether a given observer detects radiation depends on that observer’s own acceleration, and that the radiation zone of a uniformly accelerating charge lies beyond a horizon for a co-accelerating observer. It is a genuinely subtle problem and it is still written about.

The formula is the leading term of a multipole expansion. It is the electric dipole term. Higher terms — quadrupole, magnetic dipole — are smaller by powers of the source size over the wavelength, and dominate exactly where the dipole term vanishes by symmetry. That is the same expansion gravitational radiation uses, with the important difference that gravity’s first two terms are forbidden outright.

The shape of the whole subject, in one comparison #

Everything on this page is one formula read in three regimes, and the comparison is worth keeping.

At low speed and small acceleration it is a nuisance term: an antenna’s efficiency, a circuit’s stray emission, a correction nobody computes. At high acceleration and low speed it is a diagnostic: the sin²θ pattern and its polarisation say which way a source is being shaken, which is how a radiating system’s geometry is inferred without resolving it. At high speed it is an industry: a beamed, broadband, polarised, pulsed source of X-rays, and the dominant energy loss in any circular electron machine.

One expression, three uses, and the parameter that moves between them is $\gamma$. That is the sense in which this ladder’s rungs are a single subject rather than a list.

The ladder from here #

Later rungs on this anchor: the retarded potentials from which the Larmor formula is derived, and the separation of the near field from the radiation field; the synchrotron spectrum, whose critical frequency goes as $\gamma^3$ and which spans decades from one machine; bremsstrahlung, where the acceleration comes from a collision rather than from a magnet; Thomson and Compton scattering as an accelerating-charge problem; and radiation reaction, taken seriously enough to state exactly which approximations make it usable.

The neighbouring ladders are the quantum treatment of the atom, which is what the collapse computed here forced into existence, and gravitational radiation, where the same multipole machinery loses its first two terms to conservation laws.