Illustrated Physics
physics, drawn
The collection

Every essay — page 38

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence.

Mechanics

Motion, force, and the quantities that refuse to change.

Every toss already decided. The outcome of a coin toss as a map of how it was launched: the upward speed in metres per second across, the spin in radians per second up, with the region where the coin shows the face it started with shaded and the region where it shows the other face left clear. A coin spinning about a diameter and caught at the height it left is in the air for 2v/g and turns 2vω/g radians, so the boundaries are the hyperbolae vω = nπg/2, one for every half-turn. Near the origin the bands are wide: a slow toss with little spin is easy to control, and a conjuror's "honest" toss lives there. Nothing in the map is random. Every point is a definite outcome, and the bands are smooth curves — there is no fractal boundary, no region where a tiny change has an unpredictable effect. What makes a toss a fair bet is only how narrow the bands become further out.

The toss that is random without chaos

A coin toss is the textbook random event and the textbook deterministic one at the same time. Its flight is a few lines of Newtonian mechanics with no chaos anywhere: two tosses that start almost alike stay almost alike, drifting apart only in proportion to time. Its outcome is fixed the moment it leaves the thumb, by a map of launch speed and spin carved into bands of heads and tails. It is a fair bet because the bands, at a real toss, are only a few per cent wide and no hand repeats itself that well — and it is not quite fair, because a coin wobbles as it flips, and the wobble keeps its starting face up a little more than half the time: 50.8 per cent, predicted in 2007 and counted in 2023 over 350,757 tosses.

5 figures · part 9 on Chaos Three throws in a spinning room. Balls thrown at 5 m/s from the floor of a station 10 m in radius, spun at 0.99 rad/s (9.5 turns a minute) so that its floor presses with one g, as seen by someone standing in it: at 45° spinward, at 45° antispinward and straight up; with the Earth's parabolas for the two 45° throws dashed. Distances along the floor and heights in metres. On the Earth both 45° throws land 2.55 m away. In the station the spinward throw lands 1.52 m away and the antispinward one 3.06 m away, and the ball thrown straight up comes down 1.30 m ahead of the thrower, spinward. Nothing acts on any of the balls in flight. Each flies straight across the station while the floor turns beneath it, and what the thrower sees as gravity bending the path is the room rotating.

The throw a spinning station bends

A space station spun fast enough presses its floor up against its crew with the force of the Earth's gravity, and from the inside it can feel like home. Throw a ball, and the illusion fails. Nothing acts on the ball once it leaves the hand: it flies in a perfectly straight line across the station while the floor turns beneath it, and what the thrower sees as a parabola is a chord of a circle seen from a turning room. In a station ten metres across, a ball thrown up a metre comes down a metre ahead of the hand, a ball thrown with the spin falls short and one thrown against it carries long, and a dropped spanner lands a metre behind the point beneath it. How big a station must be to feel like the Earth is a question projectiles answer.

5 figures · part 9 on Projectile

Waves

Oscillation, and everything that turns out to be an oscillation.

The chirp a staircase returns. The pitch of the echo of a clap from the staircase, heard by a listener 10 m from its foot, against time since the clap: the strongest frequency in successive 24 ms slices of the synthesised echo (dots), and the rate at which the reflections arrive, together with twice that rate (lines). The fundamental falls from about 767 Hz to 476 Hz over a sixth of a second, with its second harmonic an octave above — a descending chirp, which to many listeners sounds like a bird. It is Bragg reflection: the stair is a grating, the first-order echo is the frequency whose half-wavelength equals the extra path from one step to the next, and as the line of sight climbs, that extra path grows and the reflected pitch falls.

The chirp a staircase sings back

A hand clap at the foot of the great stair of El Castillo at Chichén Itzá is answered by a chirp — a falling whistle that some have compared to the call of the quetzal bird. Nothing in the stone is tuned. Each of the ninety-one steps returns its own copy of the clap, the copies arrive as a train of clicks, and the ear hears a regular train of clicks as a pitch. The pitch falls because the steps are not evenly spaced as the listener sees them: low steps are seen almost level, a tread apart, and high ones almost along the slope, a whole step's diagonal apart. It is Bragg reflection from a grating whose period, seen from one spot, changes along its length.

5 figures · part 9 on Periodic media One note becomes a comb. The spectrum of a steady tone heard through the turning horn, level with it, against frequency offset from the tone, for tones of 250 Hz (top), 1000 Hz and 4 kHz (bottom): the power in each line as a fraction of the whole, each row on its own baseline with the height of a row standing for all of the power, computed from the received sound over one turn. A pitch that swings periodically is not a shifted line but a comb of lines spaced by the rotation rate, 6.7 Hz, with powers Jₙ(β)², where β = 2πfr/c is 0.69, 2.75 and 11.0 for the three tones; the 1000 Hz comb matches the Bessel functions line for line. The comb is narrow for the low tone, which keeps 78% of its power in the central line, and broad for the high one, whose power is spread over about ±80 Hz — so the turning horn blurs the upper partials of an organ chord into a shimmer while its fundamentals stay nearly pure.

The loudspeaker that goes round

The swirling sound of a Hammond organ comes from a loudspeaker cabinet in which the speaker's horn turns, six or seven times a second, and the swirl is the Doppler effect — the horn's mouth swings towards and away from the listener at about six metres a second, and every note it plays rises and falls by a third of a semitone. But a pitch that rises and falls periodically is not a note being shifted: it is a note split into a comb of new frequencies, spaced by the rotation rate and spread in proportion to the note itself, so the turning horn leaves the bass nearly pure and blurs the treble into a shimmer. In a room each wall returns its own copy, swinging out of step with the others, and the copies beat against each other several times a turn.

5 figures · part 10 on Doppler A pulse that comes out of the average. The cross-correlation of the two stations' records against lag, in seconds: for single sources at 0°, 45° and 90° from the line through the stations, each a copy of the noise's own ten-second wavelet shifted to that source's arrival difference — 20.0 s, 14.1 s and 0 s — and the average over all 720 sources round the ring, scaled up 9-fold to share the axes. The single-source correlations are wherever their geometry puts them. The average has peaks at ±20 s, the time a wave takes to cross 60 km at 3 km/s, one for travel from A to B and one from B to A, and little between them. That pulse is the signal a seismometer at B would have recorded had an earthquake happened at A: the Green's function between the stations, recovered from noise that never came from either.

The echo hidden in random noise

A seismometer that is not recording an earthquake is not recording silence. It hears the Earth's hum: the pounding of ocean waves on every coast, arriving from all directions as surface waves with periods near ten seconds, indistinguishable from random noise. Two seismometers sixty kilometres apart hear different noise. Yet if their records are correlated and averaged over hours, a pulse appears at a lag of twenty seconds — the time a wave takes to travel from one to the other — exactly as if an earthquake had happened under one station and been recorded at the other. Noise from everywhere contains the echo between any two points, and since 2005 seismologists have mapped the Earth's crust with it.

5 figures · part 10 on Guided waves The parabola every quartz watch follows. The frequency error of a watch's quartz fork, in parts per million, against its temperature: a downward parabola, −0.034 ppm per degree squared, about a turnover at 25 °C, where the fork runs exactly at its nominal rate; with the right-hand scale in seconds a day, since one part per million is 0.086 s a day. On a wrist, at about 31 °C, the watch runs 1.2 ppm slow, 0.11 s a day; in a drawer at 10 °C, 7.7 ppm, 0.66 s a day; at 0 °C 1.8 s a day; at 60 °C 3.6. Every departure from the turnover, hotter or colder, makes it lose. The turnover is set by the angle at which the fork is cut from the crystal, and is chosen near the temperature of the place watches are worn — the parabola is a fact about quartz, and its vertex is a fact about people.

The fork inside a watch

Every quartz watch keeps time with a tuning fork three millimetres long, cut from a crystal of quartz and ringing thirty-two thousand seven hundred and sixty-eight times a second. Each number in that sentence is a decision. The frequency is two to the fifteenth, so that fifteen halvings make one tick a second, and it sits just above the range of human hearing. The shape is a fork because two tines swinging against each other leave their base still and lose almost no energy into it. And the fork's rate drifts with temperature along a parabola whose vertex is set near twenty-five degrees, which is why a watch worn on a wrist keeps better time than one left in a cold drawer.

5 figures · part 12 on Resonance

Optics

Light, and the small number of rules it obeys.

The angle at which water lets the light out. The fraction of light reflected inside a windscreen of index 1.52 at its outer surface, unpolarised, against the angle of incidence, when the outer surface is dry (glass to air) and when it is wet (glass to water). Beyond 41.2° all the light is reflected from a dry surface; under water total reflection does not begin until 61.3°. Between the two angles, shaded, the surface is a perfect mirror where it is dry and a window where it is wet. A sensor working at 45° sits in that band: at a wet spot only 1.2% of its light is reflected and the rest leaves the glass into the water. The difference between a mirror and a window is the refractive index of what is touching the glass — a property of contact, not of what can be seen through it.

The windscreen that feels the rain

Behind the rear-view mirror of most modern cars a small box decides when the wipers should run, and it does so without looking at the rain. It shines infrared light into the windscreen at forty-five degrees, steep enough to be trapped inside the glass by total internal reflection, and counts how much comes back out of the far end of the glass a few centimetres away. A raindrop on the outside changes the medium the glass is touching from air to water, and water's critical angle against glass is sixty-one degrees, not forty-one: at forty-five the light is no longer trapped, and nearly all of it leaves into the drop. The sensor measures the area the rain is touching, and the rain rate from how fast that area grows.

5 figures · part 9 on Total internal reflection The focus that moves out of the eye. Parallel rays from a distant object entering a simplified eye 24 mm long, in millimetres along its axis: in air the eye's 61 dioptres bring them to a focus 21.9 mm behind its principal plane, on the retina; under water its 18 dioptres would bring them to a focus 73 mm behind, about three times the length of the eye. On the retina itself the underwater rays have converged only a third of the way, and every point of the scene is spread into a disc the size of the pupil's shadow. The eye is not out of focus by a little, as a short-sighted eye is; it is missing most of a lens.

The eye that loses its lens in water

Eyes opened under water see only a blur, and the reason is not water in the eye or salt or murk. The human eye does two-thirds of its focusing not with its lens but with the curved front of its cornea, whose power comes from the jump in refractive index between air and tissue. Water is almost as dense optically as the cornea, and the jump all but disappears: about forty-three of the eye's sixty dioptres vanish, and the focus moves to three times the eye's own length behind the retina. A mask gives the cornea back its air; a small pupil and a hard squint buy back a little; animals that hunt in both worlds have flattened their corneas and given the work to the lens.

5 figures · part 15 on Imaging

Electromagnetism

Charge, field, and the lines drawn between them.

Where the charge goes on a conducting egg. The surface charge on three conducting spheroids of the same length, a sphere and spheroids 3 and 10 times as long as they are wide, drawn as a band whose thickness is proportional to the charge per unit area at each point. On the sphere it is uniform. On an ellipsoid the density is exactly proportional to the fourth root of the surface's Gaussian curvature, and so it is 3 and 10 times as dense at the tips as at the waist — not the 9 and 100 times by which the tips' curvature exceeds the waist's, and not the ratio of their mean curvatures either. The charge does crowd to the sharp ends, but by how much is set by the whole shape, not by any property of the surface at the point.

The charge that follows no local rule

Charge on a conductor crowds where the surface curves sharply, and every high-voltage engineer rounds the edges for that reason. But curves sharply by what measure? On a conducting egg the density is exactly the fourth root of the surface's Gaussian curvature, so a tip a hundred times more curved than the waist carries only ten times the charge. On two spheres joined by a wire it is the square root, and on a flat disc the density goes without limit at the rim, where no curvature is defined. No local rule can be right, because where charge sits is decided by the whole shape at once.

5 figures · part 9 on Conductors A loop, and the butterfly it makes. Left: magnetisation against applied field for a model ferromagnet, as a fraction of saturation, with the field in units of the coercive field — the rising branch and the falling one, crossing zero magnetisation at plus and minus one. Right: the strain the same material shows along the field, as a fraction of its saturation magnetostriction, on the same two branches. Because the strain goes as the square of the magnetisation it does not care which way the magnetisation points, so it is the same at full positive and full negative saturation, and it falls to nothing at the two points where the magnetisation passes through zero — the coercive fields. A loop traced once produces a butterfly traced once: the strain goes through two full stretches for every one cycle of the field.

The hum at twice the mains

A transformer hums, and the note is not the frequency of the mains but twice it: a hundred hertz in Europe, a hundred and twenty in America. The reason is that iron changes shape when it is magnetised, by a few parts in a million, and the change does not care which way the magnetisation points. A core driven at fifty cycles a second therefore stretches a hundred times a second. The same evenness makes the strain trace a butterfly against the field, puts harmonics at every even multiple of the mains and none at the odd ones, and makes the hum climb steeply as the steel nears saturation.

5 figures · part 8 on Magnetisation The resistance a pan offers the coil. The surface resistance — the square root of πfμρ — of five pan materials against frequency, on logarithmic axes, with the band where induction hobs work, 20 to 100 kHz, shaded. All five rise as the square root of the frequency. At 25 kHz carbon steel offers 9.9·10⁻⁴ Ω and ferritic stainless 0.0024 Ω, while austenitic stainless, which is not magnetic, offers 2.7·10⁻⁴ Ω, aluminium 5.1·10⁻⁵ Ω and copper 4.1·10⁻⁵ Ω. For the same coil current the heating goes as this resistance, so steel heats 19 times as hard as aluminium. Matching steel with aluminium by frequency alone would need 377 times the frequency — 9 MHz, where the hob's own electronics and coil would be the problem.

The pan a better conductor will not heat

An induction hob heats a pan with currents the pan carries itself, driven by a field oscillating twenty-five thousand times a second, and it will heat a cast-iron skillet furiously while an aluminium saucepan, which conducts electricity six times better, barely warms. The better conductor is the worse load. What a pan presents to the coil is a surface resistance that goes as the square root of permeability times resistivity, and on that measure iron beats aluminium twenty times over. The same arithmetic explains why kitchen foil on the same hob can catch fire, and why a steel tip heated this way switches itself off at a fixed temperature.

5 figures · part 9 on Induction

Thermodynamics

Heat, disorder, and the one law with a direction in it.

An attraction found by counting. How often two large discs are found at each gap between their surfaces, relative to how often they are found far apart, in a direct simulation of two discs among 270 small particles that pass through each other freely and do not enter the discs (dots), and the prediction from counting the small particles' available area (line); small particles a quarter of the large radius. Nothing in the simulation attracts anything: every move that does not put a small particle inside a large one is accepted. Yet the discs are found touching 2.7 times as often as apart (2.6 predicted), because touching returns the overlap of their exclusion zones to the small particles, and the probability of a separation is proportional to the number of arrangements the small particles have at it. Beyond a gap of twice the small radius the zones no longer overlap and the effect vanishes. Written as a potential, −kT times the logarithm of that ratio, it is an attraction about 1.0 kT deep.

The attraction that is the small ones' room

Put large particles into a crowd of small ones that cannot get inside them, and the large ones begin to stick together — though nothing in the mixture attracts anything. Each large particle keeps the small ones' centres out of a thin shell around it. When two large particles touch, their shells overlap, and the overlap is handed back to the small particles as room to move in. More room is more arrangements, more arrangements is more entropy, and the large particles are pushed together by the pressure of the small ones on their outer sides. Fumio Oosawa and Sho Asakura worked it out in 1954. The force is now used to purify DNA, sort colloids onto patterned surfaces and explain why the inside of a cell behaves as though its proteins were stickier than they are.

5 figures · part 11 on Entropy Why dissolving beats evaporating near absolute zero. Cooling power against temperature on logarithmic axes, for pumping on liquid helium-4, pumping on liquid helium-3 — both with a pump of 100 litres a second at room temperature — and a dilution refrigerator circulating 100 micromoles of helium-3 a second (the T² law, dashed above 0.1 K where it no longer holds). Evaporative cooling is proportional to the vapour pressure, which collapses as e^(−L/T): helium-4 gives 7·10⁻¹¹ W at 0.3 K, helium-3 10⁻⁴ W at 0.3 K and 2·10⁻²⁴ W at 50 mK. The dilution refrigerator's "vapour" is the dilute phase, whose density of helium-3 stays at 6.6% however cold it gets, so its cooling falls only as T²: 2·10⁻⁵ W at 50 mK. Below about 0.3 K nothing that evaporates can compete, and most experiments that must stay continuously below a tenth of a kelvin — superconducting quantum computers, bolometers for the cosmic microwave background, searches for dark matter — run on dilution.

The mixture that stays mixed at absolute zero

The third law says a mixture's entropy must vanish at absolute zero, so every mixture must either separate completely or order into a compound as it is cooled. Liquid helium-3 dissolved in helium-4 does neither. Below 0.87 kelvin it splits into two liquids, but the one that is mostly helium-4 keeps 6.6 per cent helium-3 however cold it gets — and the third law permits it, because the dissolved atoms are a degenerate Fermi gas with a single ground state and no entropy of mixing to lose. Pushing helium-3 across that boundary absorbs heat the way evaporation does, from a 'vapour' that never thins out. That is the dilution refrigerator, the machine that holds every superconducting quantum computer at ten millikelvin.

5 figures · part 9 on Third law Two metals mixing at unequal speeds. The fraction of the fast-diffusing species A against position across a welded couple, A on the left and B on the right, at three times in the ratio 1 : 4 : 16, when A diffuses 3 times as fast as B, from Darken's equations, in units in which B's diffusivity and the middle time are both one; each profile's marker — an inert wire laid at the original weld — as a dot on it. The profiles spread as the square root of time, and they are lopsided: A pours into B's side faster than B into A's, so the mixed zone extends further into B. The markers do not stay at the weld. They move into A's side — by 0.39, 0.80 and 1.63 units at the three times — because the lattice planes of A's side are being emptied faster than they are filled. That drift, first seen in copper and brass in 1947, is the Kirkendall effect.

The markers that move because atoms are unequal

In 1947 Ernest Kirkendall and Alice Smigelskas wound fine molybdenum wires around a bar of brass, plated it thickly with copper, and heated it for fifty-six days. The wires, which do not dissolve in either metal, ended closer together: the brass inside them had shrunk. Zinc had diffused out of the brass faster than copper diffused in, and the lattice planes of the brass had been taken away along with the zinc. If atoms in a metal moved by swapping places with each other, the two flows would have had to be equal. They were not, and the only way the books balance is if atoms move by jumping into vacancies — empty lattice sites — that flow the other way. The wires were the first direct evidence that diffusion in metals runs on holes.

5 figures · part 16 on Diffusion Newcomen's dilemma. Two consequences of how cold a Newcomen engine's injection spray leaves its cylinder, against that temperature: the pressure difference the atmosphere then pushes the piston down with, one atmosphere less the vapour pressure of the water left in the cylinder; and the steam the next stroke wastes reheating the cylinder's iron skin, as a fraction of the most it wastes at 30 °C (8.2 cylinder-fulls). A cylinder cooled to 40 °C gives a push of 0.93 atmospheres and wastes 7.0 cylinder-fulls of steam a stroke; one left at 70 °C wastes half as much but pushes with only 0.69. Every choice was bad, because the one vessel had to be both hot, to receive the steam, and cold, to make the vacuum. Newcomen's engineers ran it in between.

The cylinder that must not be cooled

The first steam engines that did useful work, Thomas Newcomen's pumps of 1712, made a vacuum by spraying cold water into a cylinder full of steam. It worked, and on every stroke it cooled the cylinder, so that the next charge of steam condensed on the cold iron until the walls were hot again — several cylinders' worth of steam spent heating metal that the next spray would cool. James Watt saw in 1765 that the vessel was being asked to be hot and cold at once, and moved the cold to a separate condenser. The improvement was a factor of three or four in coal. A century later the reason had a name: heat passing from hot to cold across a large difference, doing nothing on the way, is work thrown away.

5 figures · part 12 on Heat engines The pipe that out-conducts any solid. The temperature difference needed to carry 30 W along a rod 300 mm long and 6 mm across, on a logarithmic scale: for solid aluminium 1343 K, for copper 796 K, for diamond, the best conductor of heat at room temperature, 159 K; and for a water heat pipe of the same size 0.8 K, almost all of it across the thin wet wicks where the heat goes in and comes out. Solid metal could not do it at all: the hot end of the copper rod would have to be hundreds of degrees above the cold. The heat pipe carries the heat as latent heat in moving vapour, and the vapour needs almost no temperature difference to flow. As an effective conductivity that is 1000 times copper's — which is why every laptop and many satellites move their heat in pipes rather than through metal.

The pipe that carries heat as vapour

A copper rod six millimetres across and thirty centimetres long, asked to carry thirty watts from a laptop's processor to its fan, would need its hot end eight hundred degrees above its cold end. The flattened copper tube that actually does the job manages it with less than a degree. It is a heat pipe: sealed, emptied of air, and holding a few drops of water and their vapour. The water boils at the hot end, the vapour rushes to the cold end and condenses, and a wick of sintered copper powder drags the liquid back by capillarity. The heat goes as latent heat, and vapour needs almost no temperature difference to flow — which makes the pipe a thousand times better than copper, as long as its wick can keep up.

5 figures · part 16 on Phase change

Relativity

Space and time, drawn on the same axes.

A longer road against a faster car. For a proton with 0.1% more momentum than the design proton in a ring 628 m round, the change in its time per turn, in picoseconds, against the Lorentz factor: from its longer orbit, which adds about 56 ps a turn at every energy; from its higher speed, which subtracts 605 ps at γ = 2 but only 5.2 ps at γ = 20, because near the speed of light more momentum buys almost no more speed; and the net, computed exactly. The speed wins below γₜ = 6.1 and the path wins above it. In special relativity a body can always be given more momentum, but its speed is capped, and that is the whole cause.

The energy above which faster arrives later

In a racing circuit a faster car laps sooner. In a proton synchrotron it does too, at first — and then, past a definite energy, the faster proton laps later. Extra momentum sends a proton round a slightly longer orbit and makes it slightly faster, and relativity decides which wins: at low energy the speed does, but near the speed of light more momentum buys almost no more speed, and the longer road takes over. At the crossing, called transition, a spread of momenta stops producing a spread of arrival times, the bunch stops oscillating in time, and the accelerating wave has to jump from one slope to the other in a few turns. The first machine to try it did so in 1959.

5 figures · part 17 on Relativistic dynamics A mass that goes missing as light. For six captures of a slow neutron — ¹H to ²H, ¹²C to ¹³C, ⁵⁶Fe to ⁵⁷Fe, ²⁸Si to ²⁹Si, ³⁵Cl to ³⁶Cl and ³²S to ³³S — the mass that disappears, the atom plus the neutron less the atom formed, in thousandths of an atomic mass unit, against the energy released as gamma rays, in MeV, both computed from the tabulated atomic masses. All six lie on the line Δm = E/c², whose slope is one atomic mass unit per 931.494 MeV: ¹H → ²H loses 2.388 mu and releases 2.225 MeV; ¹²C → ¹³C loses 5.310 mu and releases 4.946 MeV; ⁵⁶Fe → ⁵⁷Fe loses 8.208 mu and releases 7.646 MeV; ²⁸Si → ²⁹Si loses 9.097 mu and releases 8.474 MeV; ³⁵Cl → ³⁶Cl loses 9.211 mu and releases 8.580 MeV; ³²S → ³³S loses 9.277 mu and releases 8.642 MeV. Drawn this way the line is a definition — the energies were computed from the masses. The experiment of 2005 measured the two quantities for silicon and sulphur independently, the masses as frequencies and the energies as angles, and asked whether they still fell on it.

The test of mass and energy that needs a ruler

The most direct test of E = mc² weighed two silicon atoms and measured a gamma ray. A silicon-28 nucleus that captures a slow neutron becomes silicon-29 and loses three ten-thousandths of its mass as gamma rays; weigh both atoms in a trap where they circle a magnetic field, measure the gamma rays by the angle at which a silicon crystal diffracts them, and the two sides agree to four parts in ten million. What the comparison needed is the interesting part: the recoil of the atom, a correction four hundred times larger than the precision, computed from the momentum relativity gives to light; angles of a few hundred picoradians; and a chain of constants tying a frequency and a ruler to the same number.

5 figures · part 12 on Mass-energy

Quantum

Where the continuous picture runs out, and what replaces it.

The cloud that stops shrinking. The rms size of a cloud of 10⁵ ideal atoms in a harmonic trap, in units of the Fermi radius, against temperature in units of the Fermi temperature, for fermions, for bosons and for a classical gas. The classical cloud shrinks as the square root of half the temperature ratio, all the way to nothing. The bosons follow it down to their condensation temperature, 0.52 of the Fermi temperature, and then collapse into the trap's ground state, whose rms size is 0.054 Fermi radii. The fermions part from the classical line near the Fermi temperature and level off at one over the square root of eight, 0.354 of the Fermi radius: at a quarter of the Fermi temperature they are 1.17 times the classical size, and cooling further barely changes them. Mixed together in one trap, as lithium-7 and lithium-6 were in 2001, the two clouds start the same size and end 6.5 times apart in width for ideal atoms of these numbers — further still in volume. The fermions are held up by nothing but the rule that no two of them may share a state.

The cloud that stops shrinking

Two kinds of lithium atom, cooled together in one magnetic trap to a few hundred billionths of a degree, start as clouds of the same size and end very differently. The lithium-7 atoms, which are bosons, shrink with the temperature and then collapse into a condensate. The lithium-6 atoms, which are fermions, stop shrinking at about a quarter of their Fermi temperature and refuse to get any smaller, however much colder they become. Nothing pushes them apart but the rule that no two identical fermions may share a state. The photographs taken in 2001 were the first time Fermi pressure — the pressure that holds up white dwarfs — was seen as a picture.

5 figures · part 7 on Exclusion The push that makes an electron swing. An electron in a band of width 4J, pushed by a constant force F = 0.25 J per lattice spacing, over two Bloch periods h/Fa. Above: its crystal momentum, which grows steadily as Newton's law says it should, and is folded back each time it reaches the edge of the Brillouin zone, ±π/a. Below: its position, in lattice spacings, which does not grow steadily at all. The electron's velocity is the slope of the band, and the slope reverses at the zone edge, so it moves forward, slows, stops 16 lattice spacings along — 4J/F — and comes back, oscillating instead of accelerating. A free particle with the band-bottom effective mass, pushed by the same force, is dashed: the two agree only for the first fraction of a period. The electron is not stopped by anything; it is turned round by the lattice, which takes the momentum the force gives it.

The push that makes an electron swing

Push a free electron with a steady force and it accelerates for as long as the force lasts. Push an electron in a crystal with the same force and, according to the band theory every transistor depends on, it should do something else entirely: move forward, slow, stop, come back and do it again, oscillating on the spot at a frequency set by the force and the lattice spacing. Felix Bloch's oscillation was predicted in 1928 and not seen for sixty-four years, because in a real metal the electron is knocked off course ten thousand times before it completes one swing. It was found in crystals built with a thirty-times-larger spacing, and in lattices made of light where nothing scatters at all.

5 figures · part 8 on Bands

Fluids

Matter that will not hold a shape, and the forces that act in it anyway.

The film that thins too slowly. The thickness of the air film under a silicone-oil drop of radius 0.38 mm pressed onto its pool, against time, on logarithmic axes, by Reynolds's law: dh/dt ∝ −h³, so the film thins quickly at first and then ever more slowly, its thickness falling only as the inverse square root of time once it is thin. The shaded band is one bounce, 4.1 ms, in which the film falls from 10 µm to 1240 nm; the dashed line is the 100 nm at which van der Waals forces pull the surfaces together, reached after 0.65 s. The air does not have to be held up; it has only to be slow to leave, and the cube of the thickness makes it slower and slower the thinner it gets. A vibrating bath that lifts the drop off every few milliseconds refills the film each time.

The drop that will not merge

A drop of oil let fall onto a pool of the same oil ought to vanish into it at once: merging releases surface energy, and nothing holds the two apart. Instead it bounces. A film of air is trapped between drop and pool, and air, though it offers no strength at all, is slow to leave a gap a few micrometres thick: the film takes the better part of a second to drain to the hundred nanometres at which the surfaces can touch, while a bounce lasts four milliseconds. Shake the pool and the drop can bounce for ever, in rhythms that double as the shaking grows — and, near the pool's own instability, it starts to walk across the surface on the waves of its own making.

5 figures · part 12 on Surface tension The tube that is stiff because it is full. The bending moment a thin-walled tube 1 mm in radius can carry when inflated, in newton-millimetres, against its internal pressure in megapascals: the moment at which the side in compression goes slack and wrinkles, πPr³/2, and the moment at which the whole section folds, πPr³. Both are proportional to the pressure and contain nothing about the wall's own stiffness: a membrane that cannot resist compression at all carries bending only by being pre-stretched, and the pressure is the pre-stretch. At 0.74 MPa the tube wrinkles at 1.17 N·mm; with no pressure, at nothing. That is the whole of how a lettuce leaf, a petal or a young shoot keeps its shape. It is also how an inflatable boat, a bouncy castle or the boom of an inflatable antenna holds up a load, and how they fail: by wrinkling on the compressed side first.

The stem that stands by pressure

A lettuce leaf has no bones, and a young sunflower stem has hardly any wood, yet both hold their shape against gravity and wind. They do it with pressure. Each cell holds more dissolved solute than the water around it, water flows in until the cell wall pushes back, and the wall ends up stretched taut by a pressure of seven or eight atmospheres — three times a car tyre's. A tube of membrane under tension resists bending in exact proportion to its pressure, whatever the membrane is made of, so a soft plant is stiff exactly as long as its cells are full. Lose a few per cent of their water and the pressure is gone, and the plant wilts.

5 figures · part 10 on Osmosis

Astrophysics

Gravity read as geometry, and the laws carried where no laboratory can follow.

The deeper the line, the closer to gravity. The shift of a solar line in sunlight seen from the Earth, against the line's depth, in the two-stream model with the convective velocities dying away above the photosphere over 150 km and a deeper line taken to form higher (600 km × depth²). Weak lines, formed low among the fastest flows, show 411 m/s — about two-thirds of the gravitational value; the deepest lines, whose cores form in the upper photosphere where the granules' motion has nearly stopped, show 627 m/s and are still climbing towards the 633 m/s that gravity alone gives, drawn as the dashed line. That is the measurement made with integrated sunlight reflected from the Moon and calibrated with a laser frequency comb: plot the shift against line depth, and read the gravitational redshift off where the strongest lines are heading. Its published value, 638 ± 6 m/s, agrees with general relativity.

The redshift the Sun's boiling hides

Light leaving the Sun climbs out of a gravitational well and should arrive redder by 633 metres a second in velocity terms — a prediction Einstein made in 1907, before he had general relativity. It took half a century to confirm, and the reason was not the size of the effect but the Sun's surface, which boils. Bright hot gas rising in granules and fainter cool gas sinking between them blueshift every line by a few hundred metres a second, by an amount that depends on the line and on where on the disc it is seen. The gravitational shift was found by treating that nuisance as a second measurement: deeper lines and lines near the limb both converge on the same number.

5 figures · part 8 on Gravitational redshift The bend that depends on when the planet was there. The difference, in microarcseconds, between the quasar's deflection computed from Jupiter's position when the light passed it — its position as seen from the Earth, 53 minutes earlier — and from its position at the moment of observation, over the day of the 2002 pass. In those 53 minutes Jupiter moves 40998 km, 0.57 of its own radius, and the deflection, a vector of 1.14 mas pointing away from the planet, turns and shrinks by up to 46 µas — about one twenty-fifth of itself. That small term was what the experiment was designed to see. The VLBA measured 51 ± 9 µas, the value general relativity predicts for a deflection pointing away from the position light reveals.

The bend a moving planet gives

Jupiter bends starlight by sixteen thousandths of an arcsecond at its edge, a hundredth of what the Sun manages, and in 2002 radio telescopes spread across a continent measured the bend on a quasar three and a half arcminutes from the planet. The experiment was designed to see something smaller still: the fact that Jupiter moves, by half its own width in the time light takes to reach the Earth, so that the bend depends on where the planet was when the light went past. The measured term was announced as the speed of gravity, and the argument over whether it was anything of the kind is the clearest lesson there is in what an experiment's number is a number of.

5 figures · part 8 on Light deflection

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