Illustrated Physics
physics, drawn
Concept

Cross-section — where it appears

The effective area a target presents for a given process, defined as a rate divided by an incident flux and free to differ from any real size. A neutrino's is 10⁻⁴⁷ square metres and a molecule's 10⁻¹⁹, and the nineteen decades between them are the whole difference in how far each travels.

Named by 13 essays across 5 fields — each of them below, with the objects they name alongside it.

A path through a crowd. A point crossing a field of 90 scatterers, rebounding off each. The mean length of 4000 such segments is 0.2983 box widths, against the textbook form 1/2nr = 0.3086 — a departure of -3.3 per cent, from a measurement that knows nothing of the formula. It does not agree exactly and should not: the closed form is derived for a vanishingly dilute field and these discs cover 9.2 per cent of the plane. Two finite-density effects pull opposite ways — crowding shortens the path, and discs shadowing one another lengthen it — so which side of the formula a given field lands on is not something the formula can tell.

How far a molecule gets

A molecule of air travels about sixty-eight nanometres between collisions — some two hundred times its own size, and a ten-millionth of the width of a room. That ratio is the reason a gas can be treated as a continuous substance at all, and the reason it sometimes cannot.

thermodynamics · Kinetic theory Which grains the light wins. The radiation force on a spherical grain divided by the gravitational force on it, against the grain's radius, on logarithmic axes. Both forces fall as the inverse square of the distance, so the ratio does not depend on how far away the grain is — only on how big it is. Light acts on the cross-section and gravity on the volume, so the ratio goes as 1/a, and the two are equal at 287 nm for material of density 2000 kg/m³. Anything smaller than that is expelled; anything larger stays.

Light has a pressure

Sunlight pushes on a square metre with about the weight of a grain of sand, which sounds like a curiosity until the object being pushed is small enough. The demonstration in every school cupboard turns the wrong way, and the reason it does is more interesting than the effect it is supposed to show.

astrophysics · Radiation pressure The same law, across twenty-eight decades. The mean free path 1/nσ against cross-section, for a target density of 6.83·10³⁰ targets per cubic metre — solid lead. It is a straight line of slope minus one, because there is only one thing in the law. At 10⁻²⁸ m² the path is 1.46 mm; at 10⁻⁴⁷ m² the path is 1.55 light-years. Nothing about the physics changes between those ends. Only the area does.

How far a neutrino gets

A mean free path is one over the number density times the cross-section, and nothing else. Change only the cross-section — by twenty-eight powers of ten — and the same arithmetic that gives a molecule seventy nanometres in air gives a neutrino a light-year of solid lead.

astrophysics · Kinetic theory 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 Twenty-four decades of lifetime from a factor of two in energy. The half-lives of 7 alpha emitters against the reciprocal square root of the alpha's energy — the Geiger–Nuttall coordinates — with the measured values as points and a one-line tunnelling model as the open ones. The energies span a factor of 2.2 and the half-lives span 24 decades, which is what an exponent does. The model has no fitted parameter in it and reproduces every lifetime to within 0.5 decades — bad arithmetic by any ordinary standard, and a hundred-thousandth of the range it is predicting.

A wall that a factor of two makes impassable

Polonium-212 lives three tenths of a microsecond. Thorium-232 lives fourteen billion years. The alpha particles they emit differ in energy by a factor of two, and the lifetimes differ by twenty-four decades — because the quantity that decides is not the energy but an exponent built from it, and an exponent is where small differences go to become enormous.

quantum · Tunnelling Why only a sideways scattered wave takes anything away. The transmitted amplitude behind a thin scatterer, drawn as a phasor: the incident wave of unit length along the axis, plus a forward-scattered wave of length 0.12 at 0°, 60°, 90°, 150°. What a detector reads is the square of the total length. A scattered wave along the incident one lengthens or shortens the sum in proportion to itself; one at right angles changes the length only in second order, because a small perpendicular addition to a long vector barely alters its length. So a scatterer that removes energy from the beam at first order must scatter forward with a component perpendicular to the incident wave, and the size of that component is the whole extinction — which is the optical theorem.

Everything a scatterer removes, from one direction

How much light a particle takes out of a beam — by scattering it anywhere at all, and by absorbing it — is fixed entirely by what it does in the forward direction, where its scattered wave cannot be told apart from the incident one. The mechanism is interference, and it also gives the refractive index.

optics · Scattering An absorption that goes up when the photon gets harder. Mass attenuation coefficients for 4 materials against photon energy, both logarithmic, from tabulated measurements interpolated between. Away from a threshold every curve falls steeply — a harder photon is less easily absorbed, which is what makes X-rays penetrating. At a K edge the curve jumps upward: the photon has become able to eject an innermost electron, a whole new population becomes available, and the absorption multiplies by 4.4 for iodine at 33.17 keV and 4.0 for lead at 88 keV. That is a discontinuity in a material property as a function of frequency, and nothing in the classical description of absorption produces one. It exists because absorption at these energies is a transition of a bound electron rather than a loss in a medium.

The steps in an absorption curve

Every absorption law up to this point is smooth — a fixed loss per cycle, a relaxation, a power of frequency. Take the photon energy up to where it can eject an electron from an atom and the curve acquires steps, and they go the wrong way: the material becomes suddenly more opaque to a harder photon. Iodine absorbs four times as strongly at 33.4 keV as at 33.0, and that discontinuity is why it is injected into people.

waves · Attenuation How far the vacuum is from being a nonlinear medium. The size of the vacuum's departure from linearity, as a fraction, against the electric field it is subjected to — thirteen decades of field and twenty-eight of correction, both logarithmic. The scale is the Schwinger field, computed here from the electron's mass and the fundamental constants as 1.32e+18 volts per metre: the field at which a pair gains its own rest energy over a Compton wavelength, and therefore the field at which the vacuum stops being a passive backdrop. The marks are the strongest fields that exist. a laboratory magnet, 10 T is 2.3e-9 of it; a hydrogen atom's own field is 3.9e-7 of it; a 10²² W/cm² laser focus is 2.1e-4 of it; the Schwinger field is 1.0e+0 of it; a magnetar, 10¹¹ T is 2.3e+1 of it. So a laboratory is twenty-eight decades from making the effect large, and a magnetar's field is above the critical one — which is why the only places the vacuum's nonlinearity has been seen are the two where the fields are not human: the ultraperipheral collision of two heavy nuclei, and the surface of a neutron star.

The one medium that was supposed to add exactly

Superposition holds because an equation is linear, and every material stops being linear at some amplitude. Empty space was the exception: Maxwell's equations are linear exactly, and two beams cross with no interaction of any kind. Quantum electrodynamics says otherwise — light scatters light, and a strong field makes the vacuum birefringent — at a field of 1.3 × 10¹⁸ volts per metre, which no laboratory has come within four decades of.

waves · Superposition Four bodies sent past one mass at 30 masses, at four speeds. Paths traced from the exact geodesic of a non-rotating mass, every one aimed at the same impact parameter of 30 masses and differing only in speed. The slower the body, the harder it is turned: 0.4c by 32.9°, 0.6c by 16.6°, 0.8c by 11.1°, light by 8.5°. The weak-field rule (2GM/bv²)(1 + v²/c²) gives 27.7°, 14.4°, 9.8°, 7.6° — close for the fast bodies and increasingly short for the slow ones, which pass near enough to feel the field's strong part. The angles are the true ones: near a compact mass nothing has to be exaggerated.

The half of the bend a slow body never feels

Light passing the Sun is bent by twice Newton's angle, and the factor of two is usually read as a fact about light. It is the end point of a curve every body lies on. A body at speed v is bent by (2GM/bv²)(1 + v²/c²) — the 1 from the curvature of time, which a slow body feels in full, and the v²/c² from the curvature of space, which is the same absolute angle for a comet as for a photon and is simply swamped when the body is slow.

astrophysics · Light deflection 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 A target that grows as the traveller slows. The capture cross-section of a non-rotating black hole for particles arriving with speed v, in units of the horizon's own cross-sectional area π(2GM/c²)², against v/c, both on logarithmic axes (solid). For light it is 27/4 = 6.75 times the horizon's area. It grows as the particle slows, following 4c²/v² (dashed) — 404 at a tenth of light speed and 7.43·10⁶ at the 220 km/s of the Galaxy's dark matter. The dotted curve is a Newtonian ball of radius 8GM/c² — four horizon radii — capturing by ordinary gravitational focusing, and for slow particles the hole and the ball cannot be told apart.

The target that grows as the traveller slows

Light aimed at a black hole is swallowed if it passes within about five of the hole's gravitational radii, a disc six and three-quarter times the horizon's own. A particle arriving slowly is swallowed from much further out, and the disc grows as the inverse square of its speed: dark matter drifting at 220 kilometres a second sees a target seven million times the horizon. The reason is that slow matter carries almost no angular momentum for a given aim, and angular momentum is the only thing that keeps anything out. For slow traffic the hole behaves exactly like an ordinary ball four times its own size.

astrophysics · Horizons Fringes that identical nuclei add. The cross-section for two identical spinless nuclei repelling each other, divided by the classical answer — the Rutherford counts for the nucleus hit and the nucleus thrown, added — against the scattering angle in the centre-of-mass frame, for Sommerfeld parameters η = 1, 4 and 16 (η is half the closest approach in units of the wavelength over 2π). Because the detector cannot tell which nucleus it caught, the two amplitudes add before squaring, and the cross term oscillates in angle: at 90° the two paths are mirror images and the rate is exactly twice the classical one at every energy. Between 30° and 90° the curve completes about 2 oscillations for η = 4 and 7 for η = 16 — their number grows in proportion to η, which is to say in proportion to 1/ħ.

The scattering quantum mechanics returns unchanged

Rutherford worked out how alpha particles bounce off a nucleus by treating them as tiny planets on hyperbolic orbits, two years before Bohr's atom and fifteen before quantum mechanics. He had no right to be exactly correct, and he was: the full quantum calculation returns his formula without a single correction, at every energy. Coulomb's force is the one force for which that happens. Make the two nuclei identical, though, and quantum mechanics adds fringes no orbit can produce — doubling the rate at right angles for one kind of particle and halving it for the other.

quantum · Correspondence How much of a wave of gravity a bar can catch. A resonant bar's cross-section for gravitational waves, integrated over its resonance, (8/π)(GM/c)(v/c)², against its mass, on logarithmic axes, for sound speeds of 3.0 km/s, 5.4 km/s, 10.0 km/s (aluminium is about 5.4). It is proportional to the mass, which is why the bars weighed tonnes, and to the square of the speed of sound over the speed of light — the bar answers a strain only by being stretched against its own stiffness, and (v/c)² is about 3 × 10⁻¹⁰. Weber's first bar caught 2.3·10⁻²⁵ m² Hz: to absorb a single joule, a burst must carry 4.4·10²⁴ J per square metre per hertz at the bar's frequency. Dots: Weber's bar, and the two cryogenic aluminium bars that ran longest, ALLEGRO in Louisiana and NAUTILUS near Rome.

The bar a wave of gravity barely touches

The first gravitational-wave detectors were tonnes of aluminium, hung in a vacuum and listened to for ringing. A passing wave stretches and squeezes the bar, and the bar can answer only by vibrating against its own stiffness, so the fraction of the wave it catches is set by the speed of sound over the speed of light, squared: about three parts in ten billion. That one ratio explains why the bars had to be enormous and cold, why Joseph Weber's reported detections of 1969 would have required the Galaxy to be converting dozens of suns a day into gravitational waves, and why the waves were finally found by light instead.

astrophysics · Gravitational waves

Named alongside it

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

ScatteringAttenuationMean free pathAbsorptionEffective potentialGeneral relativityImpact parameterInterferenceKinetic theoryNumber densityRadiation pressureRayleigh scattering

All concepts