Illustrated Physics
physics, drawn
Concept

Rayleigh scattering — where it appears

Scattering by particles much smaller than the wavelength, going as the inverse fourth power of it, and the reason the sky is blue. The same fourth power reddens a sunset, since what is scattered out of a long path is what is missing from the light that arrives.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

How much each colour is scattered. Scattering strength against wavelength, as the inverse fourth power, normalised to one at 550 nanometres. Light at 450 nanometres is scattered 4.35 times as strongly as light at 650 nanometres — which is the whole reason the sky is the colour it is.

Why the sky is blue and the sunset is not, from one exponent

Scattering goes as the inverse fourth power of wavelength, and that single number produces a blue sky and a red sun without any second explanation. The two facts look opposite and are the same arithmetic.

optics · Scattering Where the sky's blue goes. How many times more strongly a sphere scatters 450 nm light than 650 nm light, against its radius, with the radius on a logarithmic axis running from a couple of nanometres to twenty micrometres. On the left the ratio sits at 4.35, which is the fourth power of the wavelength ratio and the whole reason the daytime sky is blue. It does not stay there. By a radius of 245 nm the preference has halved, and by a micrometre it has essentially gone: a particle comparable with the wavelength scatters every visible colour within a few per cent of equally, which is why a cloud is white, why fog is white, why milk is white and why the exhaust of a cold diesel is white while the smoke of a cigarette — whose particles are ten times smaller — is blue. Nothing about the material changed between one end of this axis and the other; only the size did.

When the particle is the size of the wave

The sky is blue because small things scatter short wavelengths far more strongly. A cloud is made of the same water and scatters every colour alike. Nothing about the material changed — only the size, and one dimensionless number crossing one.

optics · Scattering The blow-out band has two edges, not one. The ratio of radiation force to gravity against grain radius, with the radiation-pressure efficiency included: a grain much smaller than the wavelength of the light barely interacts with it — the efficiency falls as the fourth power of the size, which is Rayleigh's law — so the ratio stops rising and turns over. The dashed line is the same ratio with the efficiency taken as one, which is the usual drawing and is right only to the right of the turnover at 115 nm. The consequence is that a grain can be too small to be blown out as well as too large. Taking the threshold at a half — the value at which a grain released from a circular orbit is unbound — the band runs from 48.2 nm to 574 nm, and the largest ratio any grain of this material reaches is 1.87. Everything outside that band stays, and what stays does not stay put: it spirals.

The size the light cannot blow away

Radiation pressure and gravity both fall as the inverse square of distance, so their ratio is a property of the grain and not of where it is. What follows is a band of sizes that get blown out — with a lower edge as well as an upper one — and a drag, on everything else, that is the same pressure read one order further in v/c.

astrophysics · Radiation pressure N when the phases are random, N² when they are not. Scattered intensity against the number of scatterers, both logarithmic, for two ways of adding the same amplitudes. The lower curve averages 400 draws of N unit amplitudes with independent random phases and grows as N^1.000; the upper one adds them in phase and grows as N². At 3000 scatterers the two differ by a factor of 2921. Nothing about the scatterers is different between the two — same number, same strength, same wavelength. Only the arrangement is, and it is worth three decades here.

Why a litre of water is not blue for the reason the sky is

The same molecules that make the sky blue also make the refractive index of air, and the two numbers agree because the sideways sum has random phases and the forward one does not. Condense those molecules into a liquid and the sideways sum collapses by a factor of sixteen — and what is left is thirty-four times smaller than the absorption that actually colours the water.

optics · Scattering A force that points across the beam, not along it. The intensity of a beam focused to a waist of 0.5 µm, and the force it exerts on a 60 nm polystyrene sphere in water, across the beam. The force is not a pressure and does not point along the light: it is the pull on an induced dipole sitting in a non-uniform field, proportional to the gradient of the intensity rather than to the intensity, and it points up the gradient — towards the bright axis from either side. It vanishes exactly on the axis, which is what makes the axis a trap rather than a place. The well is 47 times kT deep at 100 milliwatts, which is why the particle stays: thermal motion explores it and does not escape it. The stiffness near the bottom is 3.070 piconewtons per micrometre, and a particle in a spring that stiff wanders 36 nanometres from centre.

The light that pulls rather than pushes

Radiation pressure is momentum arriving, and it points along the beam. A gradient of intensity does something else entirely — it polarises a particle and then pulls the induced dipole up the gradient — so a focused beam holds a particle at its waist against the push, and the force that does it is not a pressure at all.

astrophysics · Radiation pressure The pattern the whole sky is written in. The sky as a disc — zenith at the centre, horizon at the rim, equal angles at equal distances — with the sun 30° above the horizon. Each short line is the direction the electric field vibrates in at that point, and its length and darkness are how polarised the light there is. The directions are perpendicular to the plane containing the sun, the observer and the point, which puts them tangent to circles centred on the sun. The heavy arc is the locus 90° from the sun, where the polarisation is strongest — 74 per cent here — and it is a great circle rather than a patch: a band across the sky, not a region near the horizon. This is what a polarising filter on a camera acts on, and it is why turning one darkens a band of sky and leaves the rest almost untouched, and why the effect is strongest when the sun is off to one side and absent when it is behind the photographer.

The pattern the sky is written in

Scattered sunlight is polarised, so the whole sky carries a direction of vibration at every point — arranged in circles about the sun, strongest on the great circle ninety degrees away from it, and vanishing at points that were found by looking before anyone could explain them. Bees navigate by it and a camera filter reads one band of it.

optics · Polarisation Every colour a bound electron can meet. The light-scattering cross-section of one electron bound with a resonance at 2.105 eV (589 nm) and damped only by its own radiation, in units of the Thomson cross-section, against the frequency of the light in units of the resonance, on logarithmic axes spanning nine decades of frequency and twenty-five of cross-section. Far below the resonance it rises with slope 4.001, Rayleigh's fourth power: at a hundredth of the resonance it is 1.0 × 10⁻⁸ times the Thomson value. At the resonance it spikes to 2.5 × 10¹⁵ times the Thomson value, which is 3λ²/2π = 1.66 × 10⁻¹³ m² exactly, over a width of 2.0 × 10⁻⁸ of the resonant frequency. Above it the cross-section settles to the Thomson value, 6.652 × 10⁻²⁹ m² — 1.0002 times it at a hundred times the resonance — and stays within 10 per cent of it from 4.6 times the resonant frequency, 9.8 eV, to 29 keV, 3.5 decades, until the photon energy approaches the electron's rest energy, where the Klein–Nishina factor for a free electron takes it down to 0.431 of the Thomson value at 511 keV and 0.077 at 10 MeV.

The cross-section that forgets the colour

An electron on a spring, shaken by light, scatters it — weakly and very blue below its resonance, enormously at it, and above it by exactly the same amount whatever the colour. That flat stretch is the Thomson cross-section, 6.65 × 10⁻²⁹ m², and it is why an X-ray picture of a crystal is a map of electrons and why the corona is white.

astrophysics · Radiating charge Which atoms in a Fermi sea are allowed to take a photon's kick. A slice through the occupied momentum states of a gas of identical fermions at zero temperature — a filled disc whose radius is the Fermi momentum — with each state marked by whether it can absorb a recoil of 0.8 of the Fermi momentum to the right, the kick a scattered photon gives. A state can scatter only if its kicked momentum lands outside the sea, in an empty state. The kicked sea is the second circle; the atoms in the overlap of the two (grey) land on occupied states and cannot scatter, and only those in the crescent (blue) can. In this slice 50 per cent of the states are allowed; over the whole sphere the allowed share is S(q) = 3q/4 − q³/16 = 0.568. A kick larger than the sea's diameter, twice the Fermi momentum, is never blocked, which is why the effect needs a sea whose Fermi momentum is comparable with the photon's.

The light a full sea will not scatter

An atom that scatters a photon recoils, and the recoil has to put it into a new state of motion. In a cold, dense gas of identical fermions most of the states it could recoil into are already taken, and the exclusion principle forbids the scattering outright. Nothing about the atom has changed — not its electrons, not its transition, not its cross-section — yet the gas becomes dimmer than the same number of atoms scattered one at a time. The effect was predicted in the 1990s and seen in 2021, and it turns transparency into a measurement of how full a quantum gas is.

quantum · Exclusion How a dark ridge fades into the sky. The apparent contrast against the horizon sky of a black ridge at a given distance, seen in green light (550 nm) along a level path at sea level, on a logarithmic scale: e^(−βx), the light from the ridge scattered out of the path and the sky's light scattered in. The dashed line is the eye's threshold, a contrast of 2%. Clean air: β = 0.0116 per km, visible to 337 km; light haze, 0.05 per km: β = 0.06 per km, visible to 64 km; haze, 0.3 per km: β = 0.31 per km, visible to 13 km. Pure air sets a limit of a few hundred kilometres to how far anything can be seen; in practice haze shortens it to tens.

How far the air lets a mountain be seen

A dark mountain on the horizon fades not because its light grows too faint to see but because the air between it and the eye glows. Sunlight scattered into the line of sight fills the dark ridge in until it matches the sky behind it, and its contrast falls as a simple exponential of the distance. With the eye's threshold at two per cent, that gives a visual range of 3.912 divided by the air's scattering coefficient: about 337 kilometres for perfectly clean air in green light, far less in the hazes real air carries. The same arithmetic makes near ridges blue and far ones pale.

optics · Scattering

Named alongside it

The objects these essays reach for when they reach for this one.

PolarisationScatteringCross-sectionDipole radiationExtinctionMie scatteringMultiple scatteringPolarisabilityRadiation pressureRefractive indexAberrationAbsorption

All concepts