The cross-section that forgets the colour

Light passing an electron shakes it. A shaken charge radiates, as a charge that turns must glow establishes, and the radiation goes off in every direction except along the line of the shaking. Some of the light that was heading one way is now heading every other way. That is scattering, and its strength is summarised by a cross-section: the power scattered divided by the intensity of the light arriving, an area, the size of the target the electron presents to the beam.

Why the sky is blue and the sunset is not uses this for molecules in air and gets the famous fourth power: blue light, with about 1.8 times the frequency of red, is scattered about ten times more strongly. The fourth power is usually quoted as though it were a law of scattering. It is a law of one regime of it — the regime in which the light is shaking the charge much more slowly than the charge would oscillate by itself. The same electron, shaken at its own frequency or much faster, obeys two quite different laws, and the second of them contains no frequency at all.

One line, three laws #

The model is an electron held near its equilibrium by a restoring force — a charge on a spring, with a natural frequency $\omega_0$ — and damped by nothing except the energy it radiates. That damping is the radiation reaction of the force a charge exerts on itself, with its time constant $\tau = 6.27\times10^{-24}$ seconds. Driven by a wave of frequency $\omega$, the charge oscillates with an amplitude that depends on how close $\omega$ is to $\omega_0$, radiates at Larmor’s rate, and the cross-section comes out as

$$\sigma(\omega) = \sigma_T\,\frac{\omega^4}{(\omega_0^2-\omega^2)^2 + \tau^2\omega^6}, \qquad \sigma_T = \frac{8\pi}{3}\left(\frac{e^2}{4\pi\varepsilon_0 m_e c^2}\right)^2 = 6.652\times10^{-29}\ \text{m}^2.$$

$\sigma_T$ is the Thomson cross-section, and the quantity in brackets is the classical electron radius, 2.82 femtometres — the radius at which a sphere of the electron’s charge would have electrostatic energy equal to $m_ec^2$. Nothing in the problem is that size. It is simply the combination of charge, mass and the speed of light that a length can be made from.

The shaded region on the left is the sky. There the spring dominates the denominator, the charge moves as the force pushes it, and the cross-section climbs as the fourth power of the frequency: at a hundredth of the resonant frequency it is a hundred-millionth of the Thomson value, and the slope measured off the curve is 4.001. Air molecules resonate in the far ultraviolet, so all of visible light is in this region, and so is the blue sky.

At the resonance the charge is driven at its own frequency and its amplitude is limited only by the damping — here, only by its own radiation. The cross-section spikes by twenty-three decades above where it was a hundredth of the way down, to $2.5\times10^{15}$ times the Thomson value, over a band of frequencies only two parts in a hundred million wide.

Above the resonance the charge can no longer keep up with the spring at all. Driven too fast for the restoring force to matter, it responds as though it were free: its acceleration is simply the electric force divided by its mass, whatever the frequency, and the power it re-radiates is a fixed fraction of the power passing it. The cross-section stops depending on the colour. From 4.6 times the resonant frequency — 9.8 electronvolts, in the far ultraviolet — to 29 kiloelectronvolts, three and a half decades, it stays within ten per cent of $\sigma_T$. A photon anywhere in that span, from vacuum ultraviolet to hard X-ray, sees the same target.

That flatness is what makes the plateau usable as a constant of astrophysics. The brightness a mass cannot exceed balances the gravity on a proton against the push of a star’s light on the electron beside it, and the push is $\sigma_T$ times the flux divided by the speed of light — with no need to know the star’s spectrum, because in a fully ionised gas every colour pushes equally. The same number sets how far a photon travels between scatterings in the Sun’s core, a fraction of a millimetre, and so how long light takes to random-walk out, which the light that takes a hundred thousand years to leave computes; there other kinds of opacity add to it, and it is the floor rather than the whole.

The plateau ends at the right, where the photon’s energy becomes comparable to the electron’s rest energy and the classical picture of a charge shaken by a smooth wave stops applying. That last stretch is the quantum correction of Klein and Nishina, and it has its own section below.

A peak that has forgotten the electron #

The height of the resonant spike contains a cancellation that is easy to miss and hard to believe. With radiative damping, the damping rate is $\tau\omega_0^2$, and the peak cross-section is $\sigma_T/(\tau\omega_0)^2$. Both $\sigma_T$ and $\tau^2$ are proportional to the fourth power of the charge over the square of the mass. They cancel, and what is left is

$$\sigma_{\text{res}} = \frac{6\pi c^2}{\omega_0^2} = \frac{3\lambda^2}{2\pi}.$$

The resonant cross-section is set by the wavelength and by nothing else. For yellow light at 589 nanometres it is $1.66\times10^{-13}$ square metres — the area of a disc 460 nanometres across, around a charge whose “size” in the formula was 2.82 femtometres.

A proton on a spring of the same frequency has a Thomson cross-section 1,836 squared times smaller, and it reaches the same peak. So does a caesium ion, a quarter of a million times heavier than an electron. What the charge and the mass do decide is how long the peak lasts in frequency: the heavier the charge, the more slowly it radiates away its energy, the longer it rings, and the narrower its resonance. The electron rings for 15.6 nanoseconds and the caesium ion for 3.8 milliseconds, and the widths stand in the same inverse ratio — the relation the width that is a lifetime establishes between how long a resonator rings and how sharply it answers. A narrow peak holds exactly as much area-times-width as its heavier and lighter neighbours; it simply spends its height over less of the spectrum.

The electron’s number is worth checking against an atom. A classical electron bound at 589 nanometres radiates its energy away in 15.6 nanoseconds. The upper level of sodium’s yellow line, measured, lives for 16.2. The classical model is not a theory of the atom — the quantum treatment attaches an oscillator strength to each transition and an atom has many — but for the strongest line of the simplest kind of atom the spring-and-charge picture gets the lifetime within four per cent. It is also why a resonant atomic vapour is so opaque: a cloud of sodium a few centimetres long at a pressure of a millionth of an atmosphere blocks its own yellow line completely, each atom presenting a target of $\lambda^2$ rather than of an atom.

The cancellation is not a coincidence of this model. Everything a scatterer removes derives the optical theorem, which ties the total cross-section of any scatterer to what it does to the light straight ahead, and a resonant dipole that loses energy only by radiating is the case where scattering and extinction are the same thing. The limit $3\lambda^2/2\pi$ is the largest cross-section a single dipole can present at a single frequency, reached only when nothing but its own radiation damps it. Collisions, a spread of velocities or any other loss broaden the resonance and lower its peak.

Every electron counts the same #

The plateau has a consequence far from the sky. An atom has several electrons, bound with resonances ranging from a few electronvolts for the outermost to thousands for the innermost of a heavy atom. X-rays of ten or twenty kiloelectronvolts lie above nearly all of them, so every electron in the atom scatters as though it were free, with the same Thomson amplitude. Scattered straight ahead, their waves are in step, so the amplitudes add: an atom with $Z$ electrons scatters $Z$ times one electron’s amplitude and $Z^2$ times its intensity.

The nucleus is in the formula too. It has a charge $Ze$ and scatters with an amplitude proportional to its charge squared over its mass, the same combination that sits inside $\sigma_T$. Its charge is only $Z$ times an electron’s and its mass thousands of times larger, and the result on the logarithmic axis is decisive: a nucleus contributes between six parts in a hundred million and three parts in ten million of what the electrons around it do.

So an X-ray diffraction pattern is a map of electron density, and the positions of atoms are inferred from where their electrons crowd. That is usually a good proxy and occasionally a misleading one. A hydrogen atom has one electron, often pulled partly towards the atom it is bonded to, so it scatters 64 times less than an oxygen atom and its apparent position sits noticeably short of its nucleus; the hydrogen positions in a protein structure are the least certain thing in it, and are found reliably only by diffracting neutrons, which scatter off nuclei and cannot see electrons at all. The image that is a diffraction pattern twice describes how a pattern of scattered waves becomes an image; the plateau is what decides that the image is of electrons.

The same counting sets the refractive index of every material at X-ray wavelengths. Above the resonances, the electrons respond as free charges, a little behind the driving field, and the index comes out slightly less than one by an amount proportional to the number of electrons per unit volume — the high-frequency side of the plasma behaviour of the long-range force that does not reach. X-ray mirrors work by grazing reflection off that deficit, and it is the same count of electrons again.

Where colour-blindness ends #

The plateau is classical, and it ends where the classical picture ends. A photon with a momentum derives the Compton shift: a photon scattered by a free electron comes away with less energy, because it has given the electron a kick, and the loss depends on the angle. In the wave picture of the plateau the scattered light has exactly the colour of the incident light. Both statements are true in their own regimes, and the boundary between them is where the photon’s momentum becomes comparable to $m_ec$.

The two curves go together. Below a few kiloelectronvolts a photon scattered through a right angle loses a fifth of a per cent of its energy or less, and the cross-section is Thomson’s to within one per cent. By 511 kiloelectronvolts, the electron’s own rest energy, a photon scattered through a right angle keeps only half its energy, and the cross-section has fallen to 0.43 of Thomson’s. By ten megaelectronvolts it is under a tenth. Klein and Nishina derived the curve in 1929 from Dirac’s then-new equation for the electron, and its agreement with measurements of gamma-ray absorption was among the first confirmations that Dirac’s theory described something real.

The plateau and the unchanged colour end at the same energy, and for the same reason. Thomson’s picture is an electron that is shaken and does not recoil; Compton’s is an electron that recoils; the crossover is the energy at which one photon can move the electron appreciably. It is the same crossover that the bath that pushes back meets from the other side, where an electron moving fast through a bath of low-energy photons scatters them up in energy and loses its own.

The corona, whose light has no lines #

The cleanest demonstration of the plateau is in the sky, and it is a thousand kilometres above the Sun’s surface.

During a total eclipse the Sun’s corona shines with a pearly white light. Most of that light near the Sun is sunlight scattered off free electrons in the corona, and it has the colour of sunlight exactly — which on the plateau it must, since every visible wavelength is scattered by the same amount. What it lacks is the Sun’s dark lines. The photosphere’s spectrum is cut by thousands of narrow absorption lines; the corona’s scattered light has almost none.

The electrons do not remove the lines by absorbing anything. They are moving, and each one Doppler-shifts the light it scatters by an amount set by its velocity. At a million kelvin an electron’s typical speed is several thousand kilometres a second, so each wavelength is smeared by about two per cent — ten nanometres in the blue, against lines a nanometre wide or less. The narrow lines are spread across a band ten times their width and fade into the continuum; only the broad pair of calcium lines survives as a shallow dip. A colder scatterer would leave them: dust in the outer corona scatters the same sunlight with the lines intact, because a grain of dust moves at tens of kilometres a second, not thousands.

Walter Grotrian made exactly this argument in 1931. The absence of lines required scatterers moving at thousands of kilometres a second, and only electrons at a temperature of the order of a million kelvin would do. It was the first sign that the corona was vastly hotter than the surface beneath it — a decade before the corona’s bright emission lines were identified as iron stripped of a dozen electrons, which only a million-kelvin gas could produce. The colour-blindness of the plateau is what made the lines’ disappearance a thermometer: the electrons change nothing about the light except to smear it by their speed, so the smear measures the speed.

Where the charge-on-a-spring stops #

One electron, one resonance. A real atom has many resonances with different strengths, and a molecule has vibrational and rotational ones as well. Between and below them the cross-section is a sum of terms like this one, with oscillator strengths that only quantum mechanics supplies. The three regimes survive — the $\omega^4$ law below all resonances is why the sky is blue, and the plateau above all of them is Thomson’s — but the middle of the curve for a real atom is a forest, not a single spike.

Radiative damping only. The peak of $3\lambda^2/2\pi$ is the ceiling, reached only when nothing but radiation damps the oscillator. In a gas at room temperature the atoms’ thermal motion Doppler-broadens a sodium line about a hundred times beyond its radiative width and lowers its peak cross-section by roughly the same factor. The cancellation of charge and mass survives; the number it gives is an upper bound.

Independent scatterers. Everything here is one charge. Many charges scatter coherently in the forward direction and wherever their spacing makes the waves add, which is the whole of diffraction; they scatter incoherently otherwise. The electron density in a corona is low enough, and the electrons disordered enough, that the incoherent sum is right. In a metal the conduction electrons act collectively, reflect visible light outright and are not a gas of independent Thomson scatterers at all.

Low photon momentum. The plateau needs the electron not to recoil, and the Klein–Nishina curve replaces it above a few kiloelectronvolts. At the other end, the classical radiation reaction in the damping term becomes unphysical at frequencies approaching $1/\tau$, far above anywhere the quantum correction has already taken over.

Where the scattered light goes, and how it is polarised #

Each figure here reduces scattering to one number per frequency. What that number omits is where the scattered light goes. A charge shaken along a line radiates nothing along that line, so light scattered through a right angle is completely polarised, perpendicular to the plane containing the two beams — true on the plateau and below it alike, and the reason the white corona is strongly polarised and the blue sky partly so. None of the figures draws that pattern.

The atoms figure shows only forward scattering, where every electron’s wave is in step. At larger angles the waves from different parts of an atom’s electron cloud begin to cancel, the atom scatters less than $Z^2$ and the fall-off with angle encodes the size of the cloud; that fall-off, the atomic form factor, is what crystallographers tabulate and is not drawn. The corona figure is a model with seven lines and one geometry. The real K-corona’s spectrum is a superposition over the whole path through the corona, with electrons of different densities and temperatures along it, and its measured lines are shallower and broader than any single-temperature curve.

Still open: how the corona got so hot #

Grotrian’s smeared lines, and the iron lines that followed, established that the corona sits at one to two million kelvin above a surface at under six thousand. Why is not settled. The energy has to come from the churning of the photosphere and must be carried up by the magnetic field, and the two main candidates have been argued over for decades: waves in the field that travel up and dissipate, and small, frequent releases of energy as the tangled field reconnects. Spacecraft passing through the inner corona since 2021 have measured both kinds of structure in the solar wind, and both appear to be present; what share of the heating each supplies, and whether the answer changes between the quiet Sun and active regions, is still being measured.

The habit worth carrying away is to ask which side of a resonance a problem sits on. A response to a periodic push is set by the push’s frequency relative to the system’s own, and the same object obeys three different laws depending on which is faster. The electron that makes the sky blue, the electron that blocks a sodium lamp’s light completely, and the electron that keeps the corona white while erasing its lines are the same electron. Only the colour of the light changed, and on the plateau, even that stops mattering.