Theme

Only some values fit — part 2 of 2

Continuous objects producing discrete answers. A string that will sound some frequencies and not others, three permitted exponents, a speed no composition can exceed — the first sightings of quantisation, in a subject that has not reached it yet.

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.

Waves

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.

A current with no voltage, and then a voltage that is a frequency. Current against mean voltage for a Josephson junction shunted by a resistance, in units of its critical current and of the critical current times the resistance. Up to the critical current the junction carries a supercurrent at exactly zero voltage, set by the difference of the two superconductors' phases. Above it a mean voltage appears, growing as R√(I² − Ic²) and approaching the ohmic line at large current; the curve is integrated from the junction equation and checked against that result at three currents. The voltage is not steady. It is a train of pulses, each the phase slipping by one turn, at a frequency of 483.6 GHz per millivolt: for a junction with Ic = 1 mA and R = 1 Ω, one unit of the voltage axis is 1 mV and 484 GHz.

Electromagnetism

The voltage that is a frequency

Two superconductors separated by a barrier a nanometre thick carry a current with no voltage at all, set by the difference of their quantum phases. Push harder and a voltage appears — and a voltage makes that phase difference run, so the current oscillates at 483.6 gigahertz for every millivolt. Shine microwaves on the junction and the voltage locks to exact multiples of the frequency divided by a ratio of fundamental constants, with nothing about the junction in it. That is why a volt is now counted in cycles.

A swing that dies away and comes back. The envelope of the mean position of a coherent state with 9 quanta on average, in an oscillator whose levels carry a small quadratic term, Eₙ = n + n²/240, over one revival time of 240 oscillator periods. The swing collapses within about 9.1 periods, when the packet has spread round its orbit, and it stays at zero for most of the run. It returns whole at half the revival time, on the opposite side, and whole again at the full revival time; the envelope is summed from the state's energy components and checked against α·exp(−2n̄ sin²χt) to a part in a hundred million. The dashed curve is a classical ensemble started from the same distribution, each member orbiting at the frequency its own energy gives: it collapses in the same way and never returns, its swing at the half and full revival times 0.1% and 0.1% of the start.

Quantum

The return a classical cloud never makes

Put a swinging quantum packet in a well whose frequency depends a little on the amplitude and it spreads round its orbit until its swing has vanished. That part is not quantum at all: a cloud of classical oscillators does exactly the same. What no classical cloud can do is come back — and the quantum packet reassembles whole, on schedule, splitting into copies on the way, because its energies are discrete.

Two states at zero that only one side of the chain has. Every energy level of a chain of 20 cells, 40 sites, whose couplings alternate between v inside a cell and w = 1 between cells, against the ratio v/w from 0 to 2. The bulk levels fill two bands, bounded by the dashed lines ±|v − w| and ±(v + w), with a gap between them that closes at v = w. For v < w two levels sit at zero energy, in the middle of the gap, drawn in the warning colour: at v/w = 0.5 they are within a millionth of the coupling of zero. For v > w the gap is empty — at v/w = 1.5 the nearest level to zero is 0.527. Nothing about the chain's middle distinguishes the two sides of v = w; the difference is at its ends.

Waves

The end that knows how the middle was cut

A chain whose links alternate, strong and weak, has the same bands whichever kind of link is counted as inside a cell. The infinite chain cannot tell the two choices apart. A finite chain can: cut it so that a weak link is outermost and each end holds a state at exactly zero energy, in the middle of the gap; cut it the other way and it holds none. What decides is not anything at the ends but a whole number counted from the bulk — how many times a loop winds round a point.

A gap that every angle from air falls into. The stop bands of a stack of quarter-wave layers of index 4.6 and 1.6 — tellurium and polystyrene, the pair of the first such mirror — against frequency, in units of the design frequency, and the parallel index β = n₀ sin θ₀ the light brings along the layers; TE polarisation on the right, TM on the left. Shaded regions are gaps. Light from air can only have β between −1 and 1, the vertical lines; within those lines the gaps for both polarisations overlap between f = 0.848 and 1.321, the band marked across the figure, so every angle of incidence and both polarisations are reflected: a relative width of 43.6%. The TM gap narrows as β grows and closes at the internal Brewster index, 1.51, which light from air cannot reach.

Waves

The mirror that works from every direction

A stack of alternating transparent layers reflects nearly all the light of one colour arriving straight on, and less as the light tilts, because tilting moves the forbidden band. A structure that repeats in only one direction ought therefore to be a mirror for only a range of directions. It is not, if the layers differ enough: light arriving from air cannot bring enough sideways momentum to escape the forbidden band at any angle, for either polarisation, and a flat stack of plastic and tellurium reflects every angle over a band of frequencies almost half as wide as its centre.

Quantum

The field an atom calls strong

A magnetic field splits sodium's two yellow lines into ten, at spacings set by Landé's factors — until the field grows past the one the electron already feels from its own motion, when the ten reorganise into the three a theory without spin predicts. Nothing about the magnet decides which pattern appears. The atom does: the same 45 tesla is a strong field for hydrogen, a middling one for sodium and a weak one for caesium.

$The étendue, tiled. The phase space of a one-dimensional optical system: position across a twenty-micrometre aperture, against the optical direction cosine n·sinθ, for a system accepting ±0.1. The shaded rectangle is what the beam occupies, and its area — 4 micrometre-radians — is the one-dimensional étendue, the quantity no arrangement of lenses can reduce. Divided by a wavelength of 500 nanometres it is 8, and the rectangle is tiled here with exactly 8 cells of area one wavelength each. That is the whole content of the count: a cell of phase-space area λ is the smallest patch a field can be confined to, because squeezing it in position spreads it in direction by the same relation that gives a slit its diffraction pattern, so an étendue is a number of modes and its conservation is the conservation of a count. The cells are drawn four wide and two high, which is arbitrary: only a cell's area is fixed, not its shape — a beam may be confined tightly in position and loosely in angle or the other way round, and the trade is what an optical system is for.$ Optics

The invariant that is a count

Étendue is an area times a solid angle, and ray optics gives it no floor — nothing in a ray has a size. Divide it by the square of the wavelength and it becomes a number of modes, its conservation becomes the conservation of a count, and the count has a least value of one. That is where the ray bound hands over to diffraction, and the handover is the same number written two ways.

Thermodynamics

The condition three modes never meet

Whether two modes of a chain can hand energy to a third is arithmetic on the dispersion relation, and for a chain of masses the answer is never: a sine is concave, so the sum of two frequencies always exceeds the frequency of their sum. The smallest shortfall anywhere on a chain of N masses is π³/4(N+1)³ — never zero, and never far from it — and a shortfall turns a transfer into a beat.

How much a block knows about the rest. The entanglement between a block of a one-dimensional chain and everything outside it, against how long the block is, for three states of the same number of particles. The straight line is a randomly chosen state, whose entanglement is the block's length times the logarithm of two — a volume law, and what almost every state in Hilbert space does. The flat curve is the ground state of a chain with a gap: it saturates, varying by less than a twentieth of a per cent from a block of eight to one of forty, because the boundary of a one-dimensional block is two points however long the block is. The middle curve is the ground state of a gapless chain, which grows as the logarithm of the size with a coefficient measured here as 0.333 against the third that conformal field theory gives. Both ground states are enormously less entangled than a random state, and that is not a detail about chains: it is why a ground state can be written down at all.

Quantum

The corner of Hilbert space that is ever visited

Monogamy between three parties says how much of a correlation a pair may hold. Read across a boundary in a many-body system it says something much stronger: the entanglement between a region and the rest scales with the boundary rather than the volume, for the ground state of anything with local interactions. That is why such a state can be written down at all — and why almost every state in Hilbert space is one that nothing ever prepares.

The count that works for everything with a mass. How many beams a Stern-Gerlach analyser splits a particle into, for four spins, with the photon on the bottom row. For anything with a mass the answer is 2j+1 — go to the particle's rest frame, where its spin can point in any direction, and count the projections along whichever axis the magnet defines. A spin-one particle gives three: up, down, and a middle beam that is not deflected at all. The photon has spin one and gives two. The middle state does not exist, and it is not that it is hard to produce or weakly coupled — there is no such state of the electromagnetic field. A light wave has two polarisations and the third one, in which the field would oscillate along the direction of travel, is not a solution of Maxwell's equations at all.

Quantum

Two states where the counting says three

A particle of spin one has three states, and the photon has two. The missing one is not rare or weakly coupled — there is no such state of the electromagnetic field. What removes it is that a massless particle has no rest frame, so the rotations that would turn one projection into another are not available; and the state comes back the moment the particle acquires a mass, which is what a photon does in a plasma.

Three identical photons in a three-way splitter: some outcomes are forbidden. One photon enters each input of a symmetric three-way splitter, which sends each photon to each output with equal probability. Across: the ten ways three photons can leave, written as how many leave by each output. Bars, for each outcome: photons that can be told apart, identical photons, and identical fermions. Distinguishable photons spread over all ten, as independent coins would. Identical photons never produce 210, 201, 120, 102, 021, 012 — the 6 outcomes whose output labels do not add to a multiple of three — and pile into the rest: 300 with 0.222, 111 with 0.333, 030 with 0.222, 003 with 0.222. Identical fermions leave one per output every time (probability 1.000). Each probability is a sum of amplitudes over the distinct ways to reach the outcome, and each set sums to one.

Quantum

The outcomes identical photons refuse

Two identical photons meeting at a beam splitter always leave together, and that one fact carries three more. No classical light can empty the coincidence dip more than halfway, so the depth is a test of what light is; the depth measures how identical two photons are, however they differ; and with three photons in a three-way splitter whole classes of outcome become impossible — the first case of a sum over paths that no known algorithm can evaluate quickly as the photons multiply.

A hump that is not a soliton comes apart into solitons. A single smooth hump of height 6, shaped as the square of a hyperbolic secant, released into the Korteweg–de Vries equation and followed by a pseudo-spectral integration, drawn at times 0.00, 0.15, 0.35, 0.60, each snapshot raised above the last. The hump is too tall for its width to be a soliton, and it separates: by the last time there are 2 crests, of heights 8.00 and 2.00, running apart at different speeds, with a small ripple left behind. Read as a potential well, the same hump holds 2 bound states, at κ = 2.000 and 1.000, and a soliton of height 2κ² belongs to each: 8.00 and 2.00. The integration conserved the hump's area to 3.1·10⁻¹⁵.

Waves

The solitons a hump already contains

A soliton is one height for one width. Release a hump of any other shape and it does not keep that shape or simply spread — it comes apart into a fixed number of solitons of fixed heights, running off in order of size, with a ripple left behind. The number and the heights can be read off before anything moves, by treating the hump upside down as a well and counting the levels it holds.

A pendulum with unequal steps. The potential −EJ cos φ of the junction against its phase, at EJ/EC = 50, with the lowest 4 levels of the circuit drawn across the well between their classical turning points, in units of the charging energy. The transitions are 18.94, 17.79, 16.50, each smaller than the one below it; a harmonic well of the same curvature would space them all at the square root of 8·EJ·EC, 20.00. The spacing shrinks because the cosine is flatter than a parabola away from its bottom, and the shrinking is what lets a microwave pulse tuned to the lowest transition leave the next one alone.

Electromagnetism

The circuit that forgets its charge

A tiny superconducting island joined to its surroundings through a Josephson junction has discrete energy levels, and two of them make a quantum bit. The first such circuits were ruined by stray charges on nearby surfaces, which moved their levels and scrambled any superposition within a nanosecond. The cure was to make the junction's energy fifty times the charging energy. That makes the levels exponentially insensitive to charge while costing only a power-law loss in the unequal spacing that lets one transition be driven alone.

The area each quantum state is allowed. Phase-space orbits of the four lowest quantum levels, computed as eigenvalues of the discretised Schrödinger equation, for a harmonic well (left) and a box with hard walls (right), in units ħ = m = 1. Each orbit encloses an area that is a whole number of Planck constants plus a fixed offset: for the harmonic well 0.500, 1.500, 2.500, 3.500 h — n + ½ — and for the box 1.000, 2.000, 3.000, 4.000 h — n + 1. The harmonic orbit turns round smoothly at both ends and loses a quarter of a cycle at each; the box orbit is reflected sharply by each wall and loses half a cycle there.

Mechanics

The quarter cycle a turning point costs

The old quantum rule said an orbit's action is a whole number of Planck constants. It gets the harmonic oscillator wrong by exactly half a unit, the box by exactly one, and a neutron bouncing on a mirror by three quarters. Each is the same correction: a wave loses a quarter of a cycle every time it turns round smoothly, and half a cycle every time it hits a wall.

Why a flip of the spin takes eleven million years. Spontaneous emission rates against frequency on logarithmic axes. The two lines are the rates an electric dipole of e a₀ and a magnetic dipole of one Bohr magneton would give, both rising as the cube of the frequency; they differ by α²/4, 1.3 × 10⁻⁵, the square of the ratio of the two moments. Measured lines sit near them: Lyman α at 6.3 × 10⁸ s⁻¹, Hα and sodium's D line near 10⁸, and hydrogen's 21 cm line at 2.87 × 10⁻¹⁵ s⁻¹ — a lifetime of 11.0 million years. Its slowness is two factors multiplied: a magnetic moment rather than an electric one, and a frequency nearly a million times lower, cubed.

Quantum

The line that takes eleven million years

A hydrogen atom in its ground state can flip its electron's spin relative to the proton's and emit a photon of 21 centimetres. Left alone, one atom takes eleven million years to do it. The Galaxy is nonetheless bright at that wavelength, because there is so much hydrogen, and because the flip is so slow that the line's brightness counts atoms without caring how hot they are.

A smooth swing that sharpens into pulses. The shape of a string whose far end moves back and forth by 1% of its length at 2 times its fundamental frequency, started swinging in its fundamental, after 0, 10, 20, 30, 40 round trips of a wave. The smooth arch gathers into two ever-narrower pulses running in opposite directions, drawn at the moments they cross in the middle, with their height unchanged and their sides ever steeper; each reaches the moving end just as it moves inwards fastest. The energy has become 1.00, 1.09, 3.15, 10.9, 38.3 times the starting value. Every shape is computed exactly from the moving end, with no approximation to the wall's motion.

Waves

The string whose end will not keep still

Move one end of a vibrating string slowly and each mode simply retunes. Move it quickly, at twice the note the string is sounding, and something else happens: the smooth swing gathers into pulses, the energy grows by the Doppler factor of the moving end on every round trip, and it climbs through the modes without stopping — because the modes of a string are evenly spaced, and a wall that couples one pair couples them all.

Each unit of angular momentum costs a factor of tens of thousands. Weisskopf's estimates of the half-life of a gamma-ray transition in a nucleus of mass number 100, against the photon's energy, for electric multipoles of order 1 to 5 — the photon carrying away 1 to 5 units of angular momentum. At 1 MeV the estimates are 10⁻¹⁵ s for E1, 10⁻¹¹ s for E2, 2.0 µs for E3, 292.5 ms for E4, 17.3 h for E5. Each extra unit multiplies the half-life at 1 MeV by between 60,000 and 210,000: about a thousand of that is (λ/2πR)², the square of the photon's reduced wavelength over the nucleus's radius, which grows as the energy falls, and the rest is a numerical factor that grows with the order. A transition that must carry away five units at 100 keV has an estimated half-life longer than the age of the universe; the same energy carried off with one unit takes less than a nanosecond.

Quantum

The spin a photon has to carry away

An excited nucleus sheds its energy as a gamma ray in a millionth of a millionth of a nanosecond — or in minutes, or centuries, or never within the age of the universe. The energy available hardly matters. What matters is how much angular momentum the photon must carry off, because a source much smaller than its wavelength is very bad at launching a wave that twists, and each extra unit of twist costs a factor of tens of thousands in time. Selection rules are usually stated as what cannot happen; they are really a price list, and it spans forty decades.

The heat an eccentricity buys. The power tides dissipate inside Io against the eccentricity of its orbit, from (21/2)(k₂/Q) G M² R⁵ n e²/a⁶ with k₂/Q = 0.015: it grows as the square of the eccentricity. At Io's forced eccentricity, 0.0041, it is 9.3 × 10¹³ W, against about 1.0 × 10¹⁴ W measured from Io's infrared glow — the same to within the uncertainty of both — and against 4.7 × 10¹³ W for the whole of the Earth's internal heat, from a body forty-three times Io's volume. Halve the eccentricity and the heating falls fourfold; take it away and Io would freeze.

Astrophysics

The moon kept molten by its neighbours

Io is the most volcanic body known, with hundreds of active vents and lava hotter than any erupting on the Earth, on a moon smaller than the Earth's own and far too small to have kept any heat from its formation. The heat is made continuously, by tides: Jupiter stretches Io, and because Io's orbit is slightly eccentric the stretch changes every day and a half, flexing the rock and warming it. Tides should have rounded the orbit off in a hundred thousand years and switched the heating off. They cannot, because two other moons keep tugging the orbit back into its oval, and the heat Io radiates is paid for by Jupiter's spin.

Electromagnetism

The number of dimensions a planet can orbit in

Gauss's law says that a point source's field is its strength spread over the surface around it, and the size of that surface depends on how many dimensions space has. In three it grows as the square of the distance, and the field falls as the inverse square. Count differently and the field falls differently — and above three dimensions it falls so steeply that no planet could hold a stable orbit and no atom could hold its electron. The inverse-square law is not a fact about gravity or electricity. It is a fact about space, and it is one of the conditions for there being anything in space to orbit.

Five images of one source, and the curves where they are made. The sky around a lens with a core of 0.10 and an external shear of 0.15, in Einstein radii, with its two critical curves: an outer one near the Einstein ring and a small inner one round the core. A source at (0.06, 0.04) in the source plane — inset, with the caustics the critical curves map to — lies inside the diamond-shaped caustic and has 5 images. Their magnifications are +4.99, −3.76, +3.42, −2.63, +0.01 — a minus sign for an image turned over — and the parities sum to +1. Four sit near the outer curve and carry nearly all of the 14.8-fold total; the fifth, at the centre, is magnified by only 0.013. A source further out, at (0.55, 0.1), has 3.

Astrophysics

The fifth image a galaxy hides

A single star bends light into two images of whatever lies behind it. A galaxy, whose mass is spread out, has to make an odd number — one, three or five — because the map from the sky to the source is smooth and has nowhere to put an even count. The famous cross of four quasar images is therefore five images, with the fifth thousands of times too faint to see, sitting at the centre of the galaxy that made it. Images are born and destroyed only in pairs, and only on the curves where the map folds.

The triangle of collisions that survive. Every collision of a particle ε above a filled Fermi sea with a partner from the sea, at zero temperature: the partner's energy across, one product's energy up, both from the Fermi level, the other product's fixed by energy conservation. Without exclusion every point in the plotted square would be an allowed collision. With it, the partner must come from below the Fermi level, and both products must land above it: the partner no deeper than ε below, the first product above zero, the second product above zero too. What survives is the shaded triangle, whose area is half of ε squared, and the collision rate is proportional to it. Halve the particle's excess energy and the triangle shrinks to a quarter; at the Fermi level itself it vanishes, and a particle there cannot scatter at all.

Quantum

The collisions the exclusion principle forbids

A copper wire holds one free electron per atom, packed a quarter of a nanometre apart and repelling one another with the full Coulomb force. It seems impossible that an electron could travel any distance through that crowd without colliding. It travels micrometres — thousands of atoms — before another electron deflects it. The collisions are not weak. They are forbidden: to collide, both electrons need empty states to move into, and in a sea filled to the brim almost none exist. The same rule makes cold helium-3 thicken as the square of how cold it gets.

A sharp angle costs a spread of turning. Left: the probability of finding a rotor at each angle, for three states. Right: the probability of each value of its angular momentum, in units of ħ, for the same states. A state bunched to a circular spread of 0.025 has an angular-momentum spread of 2.21ħ; a state bunched to a circular spread of 0.302 has an angular-momentum spread of 0.59ħ; a state with no preferred angle has an angular-momentum spread of 0.00ħ. The angular momentum can only take whole-number values, so its probabilities are bars; the angle is continuous. The state with no preferred angle has exactly one value of angular momentum, and its angle is spread evenly round the circle — where the naive rule would say that its angle's spread times ħ/2 should be infinite, it is π/√3.

Quantum

The angle the uncertainty principle cannot be written for

Position and momentum trade exactly: squeeze one and the other spreads, with a product never below ħ/2. Angle and angular momentum look like the same pair and are not. A rotor in a state of definite angular momentum has zero spread in angular momentum and a finite spread in angle, π/√3, and their product is zero. Nothing is wrong with the rotor. The relation was written with a commutator that does not hold, because an angle lives on a circle and a line has no ends to wrap round. The relation that does hold measures the angle's spread by how far its average position on the circle falls short of the rim.

$What correlation is worth, to a helium atom and to H⁻. The energy needed to remove the second electron, by three descriptions, as a fraction of the measured value, for the negative hydrogen ion and for helium. Helium: one shared orbital 23.07 eV, one electron in and one out 23.83 eV, measured 24.59 eV — the simple picture is 94 per cent right. H⁻: −0.74, 0.362 and 0.755 eV — the simple picture is not merely inaccurate but has the wrong sign. The energy the simple picture misses — the electrons avoiding each other — is about 1.5 eV in both; in helium it is a small correction to 24.6 eV, and in H⁻ it is everything.$ Quantum

The ion that exists because its electrons keep apart

Add a second electron to a hydrogen atom and it holds on, by three-quarters of an electronvolt. The simplest respectable calculation says it should not — that the two electrons, sharing one orbital, repel each other more than the proton can hold them, and the ion falls apart. The same calculation gets helium's energy right to two per cent. The difference is not a failure of the method but a measure of what it leaves out: the two electrons avoiding each other. In helium that is a small correction. In the negative hydrogen ion it is the whole of the binding, and that ion is what makes the Sun opaque.

Three identical quarks in one state, and the label that rescues them. The Δ⁺⁺ particle, mass 1,232 MeV and charge +2, drawn as the quark model describes it: three up quarks, all in the lowest orbital state, all with their spins pointing the same way, giving spin 3/2. Left: with nothing else to tell them apart, the state is unchanged by swapping any two quarks — symmetric in where they are, in their spins and in their flavour — which the exclusion principle forbids for identical fermions. Right: give each quark a label that takes three values, and put the three in the one combination of the three values that changes sign under every swap. The whole state is then antisymmetric, as it must be, and the particle is allowed. The label is called colour, and three is the smallest number of values that can do this for three quarks.

Quantum

The label three identical quarks forced

In 1964 the quark model had one embarrassment it could not explain away. A particle called the Δ⁺⁺, well measured and entirely ordinary, had to be three identical up quarks in the same state with their spins pointing the same way — which the exclusion principle forbids outright. The way out was to give quarks a hidden property that takes three values, so that the three could be different after all. Nobody could see the property, and for years it looked like a bookkeeping trick. Then three independent measurements counted it, and every one came out at three. The exclusion principle, which keeps electrons in shells, had predicted the colour of the strong force.

$Only a slow grating sends everything one way. The largest fraction of particles, arriving at Bragg's angle, that a standing light wave can put into the first diffracted order, choosing the wave's depth freely, against how long it acts, in units of ħ divided by the two-kick recoil energy, on a logarithmic axis. A brief pulse acts as a thin phase grating and can put at most 33.9 per cent into that order (dashed), the Raman–Nath limit, spreading the rest over the others. A pulse lasting longer than a few units has time to notice that only one exchange conserves energy, and can put nearly all of it there: 99.9 per cent at the longest time drawn. The same distinction between thin and thick gratings decides whether sound in a crystal diffracts light into many orders or one.$ Quantum

The grating made of light

A diffraction grating is ordinarily made of matter and diffracts light. In 1933 Pyotr Kapitza and Paul Dirac proposed the opposite: a standing wave of light, whose bright and dark bands repeat every half wavelength, should diffract a beam of electrons. It took fifty-three years to see the effect with atoms and sixty-eight to see it with electrons, because a free electron barely notices light. The grating works by momentum: it can only change a particle's motion in whole pairs of photon kicks. A brief flash scatters particles into a spread of orders in proportions fixed by Bessel functions; a slow, gentle one sends every particle one way, which is how laser light became the mirrors and beam splitters of atom interferometers.

The frequency of a mass on a spring that weighs something. The angular frequency of a mass M on a spring of stiffness k and mass mₛ, in units of √(k/M), against the ratio mₛ/M on a logarithmic axis: exact, from β tan β = mₛ/M (solid); Rayleigh's estimate √(k/(M + mₛ/3)) (dashed); and the textbook answer that ignores the spring's mass, 1 (dotted). For a spring a tenth of the mass the exact value is 0.9836, Rayleigh's 0.9837 and the textbook's 1 — 1.7 per cent too fast. With spring and mass equal the exact frequency is 0.8603, Rayleigh's 0.8660: the third is good to 0.66 per cent where the textbook answer is 16 per cent out.

Mechanics

The third of itself a spring carries

Every textbook gives the frequency of a mass on a spring as the square root of stiffness over mass, and every real spring has mass of its own, which the formula leaves out. The repair is famous and oddly specific: add a third of the spring's mass to the load. The third is not a fudge. It is what the spring's kinetic energy comes to if each coil moves in proportion to its distance from the fixed end, and it is good to two-thirds of a per cent even when the spring weighs as much as the mass. Where it fails, it fails because the spring has stopped being a spring and become a column of waves, with notes of its own above the bounce — the notes that make the valve springs of an engine lose control.

Five square logs of different densities, as they float. Cross-sections of five long square logs in their stable floating positions, with relative densities 0.1, 0.25, 0.5, 0.75 and 0.9, each drawn with its water line and its submerged part shaded, and a dot at its centre. The lightest floats flat and high; at 0.25 it has tilted to about 27°; at 0.5 it floats on its corner with a diagonal vertical and the water line across the middle; at 0.75 it tilts to the same angle as at 0.25, and at 0.9 it floats flat again, low in the water. The heavy logs are the light ones turned upside down, with air and water exchanged.

Fluids

The log that floats on its corner

A long square beam of very light wood floats flat, one face up, as anyone would expect. A beam of wood half as dense as water does not: it rolls onto a corner and floats with a diagonal vertical, like a diamond. Between the two, at a quarter of water's density, it floats tilted at twenty-seven degrees, neither flat nor on edge. Heavier beams repeat the sequence in reverse, and the whole pattern is symmetric: a beam of density s floats at exactly the angle of one of density 1 − s. None of it depends on anything but a square, a density and Archimedes, and the same arithmetic decides which way an iceberg rolls when it breaks.

$Light from one guide, spreading through a row of them. The power in each of a row of identical coupled waveguides after light launched into the central one has travelled 1.5, 3, 5 and 7 coupling lengths (1/C) along the row, drawn as stacked bars with the distance increasing upward: |Jₙ(2Cz)|², where n counts guides from the centre. The light does not spread into a single widening hump as it would in a uniform slab. It forms two beams that run outward at a constant rate, carrying most of the power in their leading edges near n = ±2Cz, and leaves the centre nearly dark: after 7 coupling lengths the central guide holds 2.9 per cent and the brightest guides, at ±12, hold 8.1 per cent each. This is discrete diffraction: the spreading of light that can only step between neighbouring sites.$ Waves

The light that spreads in two beams

Shine light into one of a row of identical optical fibres laid side by side and close enough to leak into each other, and it does not spread out into a widening glow as it would in a slab of glass. It splits into two beams that run outward at a fixed speed, bright at their leading edges, leaving the middle almost dark. The reason is that light hopping between neighbours has a fastest possible sideways speed, and piles up at it. The same periodicity gives the row two tricks no uniform medium has: a direction in which a beam travels without spreading at all, and a direction in which diffraction runs backwards, so that a spread beam can be gathered up again.

Quantum

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.

The heights a neutron is allowed to bounce to. A neutron above a horizontal mirror in the Earth's gravity: the potential energy mgz (straight line), the first four allowed energies and, drawn on each, the probability density of the neutron's height. The natural length is z₀ = (ħ²/2m²g)^(1/3) = 5.87 μm and the natural energy mgz₀ = 0.602 peV. The allowed energies are 1.41, 2.46, 3.32, 4.08 peV — pico-electronvolts — and a classical neutron with each energy would rise to 13.7, 24.0, 32.4, 39.8 μm. Each density is an Airy function, with n − 1 nodes, leaking a little above the classical turning height and vanishing at the mirror. The lowest state never touches the mirror and never rises above about 25 μm.

Quantum

The bounce that comes in fixed heights

A ball dropped on a floor can bounce to any height. A neutron cannot. Slow enough, and resting on a flat mirror in the Earth's gravity, it has a lowest state about fifteen micrometres high, with an energy of 1.41 pico-electronvolts, and a staircase of higher states above it — the first quantum states ever seen in a gravitational field, measured in 2002 with a slit narrower than a hair. Shaking the mirror at 255 hertz drives a neutron from the first state to the second. The staircase is now a laboratory for gravity at distances where nothing else can test it.

$How a solute moves both ends of the liquid range. The molar chemical potential of water against temperature near its freezing point (left) and its boiling point (right): liquid (blue), ice (green), vapour (red), each a line whose slope is minus its entropy, drawn relative to the liquid at the transition. A solute at mole fraction 0.05 — about 2.9 mol/kg — lowers the liquid's line by RTx (dashed) and leaves the ice and the vapour unchanged. The crossings move: the freezing point down by RTx divided by the entropy of melting, 5.16 K, the boiling point up by RTx divided by the entropy of boiling, 1.42 K. The same vertical shift moves the freezing crossing 3.6 times as far, because ice's line is nearly parallel to the liquid's — melting changes the entropy by 22.0 J/mol K — while the vapour's line crosses steeply, with 109 J/mol K between them. Lowering the liquid makes it the winner over a wider range of temperature at both ends: the solution is liquid from below 0 °C to above 100 °C.$ Thermodynamics

The degree a spoonful of solute buys

Dissolve anything in water and the water freezes lower and boils higher. The two shifts come from one cause — the solute lowers the liquid water's chemical potential, so the liquid wins against ice and against vapour over a wider range — but they are not the same size. For every mole of particles per kilogram the freezing point falls 1.86 degrees and the boiling point rises only 0.51, three and a half times less, and the reason is in the entropies: ice and water are so alike that a small push moves their crossing a long way. The same arithmetic says salting pasta water does nothing a cook could notice, weighed molecules for a century of chemistry, and in 1887 showed that salts in water are made of ions.

Astrophysics

The circling that comes in quanta

An electron circling a magnetic field radiates at the frequency of its circling, and classically that is all: a continuous glow at whatever energy the circling has. On a neutron star the field is a hundred million tesla, the quantum of circling is a hard X-ray, and an electron lifted one step up falls back in a quarter of a femtosecond. So every electron there sits on the bottom step, and the X-rays that match the step are scattered out of the beam. The gap they leave is a line, and its energy is the field.

Three waves, and the points where the light is zero. The overlap of three equal plane waves travelling 120° apart: brightness (shading) and lines of equal phase every 60° (curves). Lines of every phase meet at isolated points where the brightness is exactly zero (dots); elsewhere only branches of a single phase cross, at the bright spots, where the phase is momentarily level. Round each such point the phase changes by one full turn, clockwise (red, charge +1) or anticlockwise (green, charge −1); in this window there are 18 of one and 18 of the other, arranged on a lattice in which every neighbour has the opposite sign. No adjustment of the phases removes them; shifting a wave only slides the lattice.

Optics

The dark lines a wave cannot avoid

Overlap three waves of light and there are places where they cancel exactly — not along fringes, as two waves cancel, but at isolated points, round which the phase of the light turns through a full circle. They cannot be removed by any small change, they can only be created or destroyed two at a time, and random light is full of them, about three to every square wavelength. A grating with a fork in it writes one into a beam on purpose, and the beam that comes out has a dark core and carries angular momentum in its twist.

Waves

The tide a lake makes by itself

On Lake Geneva the water at Geneva rises and falls by a few centimetres every seventy-three minutes, and the water at the far end does the opposite, as though the whole lake were a bath slopping back and forth. It is a standing wave, set ringing by wind and air pressure, and its period follows from the lake's length and depth alone. A harbour rings the same way with its mouth as a node, and it does something odder: narrowing its entrance, the obvious way to keep waves out, makes its resonance stronger.

Absorption below and above the gap. The light absorbed by a direct-gap crystal against photon energy measured from the gap, in units of the exciton's Rydberg, from Elliott's formula with each line broadened to a half-width of 0.06 Rydberg (solid), beside the absorption the same crystal would have if electron and hole did not attract (dashed), which rises from zero as the square root of the energy above the gap. Below the gap the attraction adds a series of lines at −1, −1/4, −1/9 … Rydberg, the hydrogen series scaled down, with strengths falling as 1/n³, crowding towards the gap. Above it the continuum is lifted: the attraction pulls each free pair together, and at the gap itself the absorption is a finite step rather than zero.

Quantum

The hydrogen atom inside a crystal

Light absorbed by a semiconductor lifts an electron across the gap and leaves a hole behind, and the two attract. In an ordinary crystal they bind into a copy of the hydrogen atom, with the same series of levels and the same 1/n² — but the crystal screens their attraction and lends them light masses, so the copy is a few thousand times shallower and several hundred times larger. That rescaling is the whole theory, it predicts lines below the gap and a finite step at it where no attraction would give a smooth rise from zero, and it says which crystals keep their excitons at room temperature.

$Two lifetimes a thousand times apart. The fraction of positronium atoms surviving in the ground state against time since formation, on a logarithmic time axis, for the spin singlet, para-positronium (red), and the spin triplet, ortho-positronium (blue), in vacuum. Para lives 125.1 ps and ortho 142.1 ns, a ratio of 1135. The lowest-order formulas give 124.5 ps, from α⁵mc²/2ħ, and 138.7 ns, from 2(π² − 9)/9π · α⁶mc²/ħ; the ratio is almost entirely one extra power of the fine-structure constant, the price of a third photon.$ Quantum

The atom with no nucleus

An electron and a positron can bind into an atom whose structure is hydrogen's with every level halved and every size doubled, because neither partner sits still at the centre. The atom then destroys itself, and how fast depends on one symmetry. With the two spins opposed it vanishes in 125 picoseconds as two photons; with them aligned it must make three, and lives 142 nanoseconds — over a thousand times longer — for no reason but that a rule about charge conjugation forbids the shorter route.

A heat capacity that goes up and comes down. The heat capacity per system, in units of Boltzmann's constant, against temperature in units of the level spacing over k: for a system with two levels (red), Schottky's hump x²eˣ/(1 + eˣ)², x = Δ/kT, and for a harmonic oscillator with the same spacing (blue), which climbs to one k and stays there. Both are frozen out when kT is well below the gap. The oscillator then fills its unending stack of levels and reaches equipartition's value. The two-level system peaks at 0.439k when kT = 0.417Δ, and then, with both levels nearly equally occupied and nothing higher to go to, it stops absorbing heat: its capacity falls back towards zero as 1/T².

Thermodynamics

The heat capacity that goes up and comes down

Equipartition promises every way a system can move a fixed share of heat, and for a harmonic oscillator the promise is kept: once the temperature is high enough, it holds one k per oscillator and goes on holding it. A system with only two levels cannot keep it. It absorbs heat while its upper level is filling, then stops — the two levels equally full and nowhere higher to go — so its heat capacity rises to a peak and falls back to zero. The hump, named after Walter Schottky, is the signature of a degree of freedom with a ceiling, and the area under it counts how many states were there.

Waves

The speed below which nothing on water leaves a wake

Every wave on clean water travels at 23 centimetres a second or faster: long waves are fast because gravity drives them, short ones because surface tension does, and in between there is a slowest wave. A boat, a duck or a fishing line can only make waves that keep pace with it, so anything crossing the water more slowly than 23 centimetres a second makes no waves at all. Faster, it makes two kinds at once — short ripples that run ahead of it and long waves that trail behind — and the threshold between nothing and both is set by gravity, surface tension and density alone.

The cavities that hold a beam. Every two-mirror cavity as a point (g1, g2), with g = 1 − L/R for each mirror. The shaded region, between the axes and the hyperbolas g1 g2 = 1, holds a ray for ever; everywhere else a ray is walked out. The commonest designs sit on the boundary: two flat mirrors (1, 1), the confocal cavity with the mirrors' foci together (0, 0), the concentric cavity with their centres together (−1, −1), and a flat mirror facing a curved one at its own radius (1, 0). A boundary cavity is stable in principle and marginal in practice, which is why the confocal one, sitting at the corner where the two shaded regions touch, is usually built a little inside it.

Optics

The two mirrors that hold a beam

A laser is two mirrors facing each other, and the light in it makes thousands of round trips before it leaves. Whether a ray stays between the mirrors for ever or is walked out after a few bounces depends on a single number, built from their spacing and their curvatures, and the answer has a sharp edge: two flat mirrors sit exactly on it, which is why the first lasers were so hard to align. The criterion is the one that decides whether a pumped swing grows, whether a wave crosses a crystal, and whether a particle beam stays in an accelerator.

The square root of an X-ray's energy, one step per element. The square root of the energy of each element's Kα X-ray line, in units of the Rydberg energy, against its atomic number, for 24 elements from aluminium to lead (dots), with Moseley's law, √(3/4)·(Z − 1), drawn as a line. The points march up in equal steps, one per element, because each step adds one proton to the nucleus and one unit to the charge the falling electron feels. When Moseley first did this, in 1913–14, the line had gaps where no known element fell: the numbers marked, 43, 61, 72 and 75, were found later as technetium, promethium, hafnium and rhenium. At high atomic number the points rise above the line, by 6 per cent for lead, because the innermost electrons there move fast enough for relativity to deepen their binding.

Quantum

The X-ray line that counts the protons

Strike a metal with fast electrons and it gives off X-rays at a few sharp energies characteristic of the metal. In 1913 Henry Moseley measured the strongest of them for element after element and found that the square root of its energy rose in equal steps, one step per element — as if something inside each atom were being counted. It was the charge on the nucleus. The periodic table had been ordered by weight; Moseley's line ordered it by protons, fixed the elements it was missing, and showed that an electron deep inside a heavy atom behaves like the electron in hydrogen, with the charge turned up.

Quantum

The electrons that pay in fixed amounts

Fire electrons through mercury vapour and raise the voltage that drives them, and the current they carry should simply rise. In 1914 James Franck and Gustav Hertz found it rising, then collapsing, then rising again, over and over, every 4.9 volts. Below that energy an electron bounces off a mercury atom and keeps almost everything; above it, it can hand the atom exactly 4.9 electronvolts and no other amount. The atom will accept energy only in the size of the gap between its levels — shown here by collisions rather than by light, and misread by its discoverers as something else entirely.

A resistance counted in whole numbers. The Hall resistance of a gallium-arsenide layer with 3 × 10¹⁵ electrons per square metre, against the perpendicular field (solid), beside the classical B/ne (dotted); and the resistance along the current, on an arbitrary scale (red). Classically the Hall resistance rises in proportion to the field. In the layer it climbs in steps and stays on flat plateaus at h/νe² — 25813, 12906, 8604, 6453 ohms — each centred where the classical line crosses it, at integer filling. On each plateau the resistance along the current falls to zero: the current flows with no dissipation at all. Between plateaus both change. The model drawn here gives each Landau level a width 0.18 of the level spacing and lets only the middle 0.16 of it carry current across the sample.

Quantum

The resistance counted in whole numbers

Pass a current along a thin layer of electrons in a strong magnetic field and measure the voltage across it. Classically the ratio rises steadily with the field. In fact it climbs in steps, and on each step it sits at the fundamental constant h/e² divided by a whole number — 25,812.807 ohms, then half that, then a third — the same to better than a part in a thousand million in every sample, every material and every laboratory, however dirty the layer and however irregular its shape. The steps exist because of the dirt, and they are exact because of a count that disorder cannot change.

Quantum

The condensate a trap makes

A gas of atoms held in a magnetic trap and cooled to a few hundred billionths of a degree does something no gas in a box does at any temperature it can reach: a large share of its atoms falls into the trap's single lowest state, and a sharp peak rises out of the cloud. The reason is counting. The trap's excited states can hold only so many atoms at a given temperature, and the number grows as the cube of the temperature — so below one temperature the atoms run out of room, and every atom that does not fit has only one place to go.

How a hanging chain can swing. The shapes of the first four ways a uniform hanging chain can swing, side by side, against height from its free end: J₀(j₀,ₙ √(y/L)), fixed at the top and largest at the free bottom end. For a 2-metre chain they sound at 0.424, 0.973, 1.525, 2.078 hertz. The first swings as a whole, like a pendulum, with its bottom moving most. The higher modes have nodes, and the nodes are not evenly spaced: they crowd towards the bottom, where the tension is least and waves travel slowest, so a stretch near the free end holds more of each wave than a stretch of the same length near the top.

Waves

The chain that swings in Bessel's notes

Hang a chain from a hook and set it swinging. It can swing as a whole, like a pendulum, or in shapes with still points along it, as a guitar string does — but its higher notes are not two, three and four times its lowest. They are 2.295, 3.598 and 4.903 times it, the zeros of a function that Daniel Bernoulli met for the first time in exactly this problem in 1732, a century before Bessel's name was attached to it. The reason is that a hanging chain is a string whose tension grows from nothing at its free end to its whole weight at the top, and a wave crossing it changes speed all the way.

The diamonds where the island's charge is fixed. The stability diagram of a single-electron island with identical junctions: bias voltage (units of e/CΣ) against the charge the gate induces (units of e). Inside each shaded diamond, labelled with the number of extra electrons the island holds, every possible tunnelling step costs energy and no current flows; outside, at least one step pays for itself and electrons pass one at a time. The diamonds repeat every time the gate charge rises by one electron, and touch at gate charges of e/2, 3e/2, … where two charge states cost the same and the island conducts at any bias. Their height is e/CΣ in each direction. Probing the model at three points outside the diamonds gives currents of 0.60, 0.25, 0.53; inside them, nothing.

Electromagnetism

The island that takes electrons one at a time

A capacitor holds charge in proportion to its voltage, and the proportion is a property of its shape — on any piece of metal large enough to see. Make the metal small enough and cold enough, and joined to the world only by tunnel junctions, and the proportion breaks into steps: one electron's charge costs more energy than the heat or the battery can supply, so no current flows at all until it can, and then the electrons cross one by one. A charge of a single electron becomes something a voltmeter can see.

The period of a ball bouncing on a gas. The period of a 16 mm steel ball (16.8 g) oscillating in a vertical tube that fits it on top of a flask, against the flask's volume on a logarithmic axis, for helium, air and carbon dioxide at room temperature, if every compression is adiabatic: T = 2π√(mV/γPA²). On a 10 litre flask the ball swings with a period of 0.983 s on helium, 1.072 s on air and 1.117 s on carbon dioxide. The gas is the spring, and its stiffness is γ times the pressure: the gas whose molecules have fewest ways to store energy is the stiffest, because less of the work done in compressing it goes into motions that do not push on the walls.

Thermodynamics

The ball that weighs how a gas stores heat

Drop a steel ball into a glass tube that fits it closely, on top of a flask of gas, and it does not fall to the bottom. It bounces, up and down, on the gas trapped beneath it, with a period of about a second. The gas is a spring, and its stiffness is the pressure multiplied by a number that counts how many ways the gas's molecules can hold energy. So a stopwatch on a bouncing ball measures something that otherwise needs a calorimeter: whether the molecules can rotate, whether they can vibrate, and how far a vibration has woken up at the temperature of the room.

The two lines a vibration adds to scattered light. The light scattered by silicon's optical vibration at 520 cm⁻¹, against its shift from the 532 nm laser line in cm⁻¹ (negative, lost to the crystal; positive, gained from it), on a logarithmic intensity axis, at 100 K, 300 K and 600 K; the strong unshifted line and its wings are cut out at the centre. The Stokes line, where the light gives up a quantum of vibration, barely changes: 1.00, 1.09 and 1.40 in relative strength. The anti-Stokes line, where it takes one, needs a vibration already excited and grows from 7·10⁻⁴ of the Stokes line at 100 K to 0.103 at 300 K and 0.36 at 600 K. The light comes back bluer only as often as the crystal is warm enough to pay for it.

Optics

The light that comes back bluer when it is warm

Shine a green laser on a crystal and a tiny fraction of the light comes back with a different colour: a little redder, having given a quantum of energy to one of the crystal's vibrations, or a little bluer, having taken one. Losing energy is always possible. Gaining it needs a vibration already excited, so the bluer light is fainter, and fainter still the colder the crystal. The ratio of the two is a Boltzmann factor, fixed by quantum statistics and nothing else. It is a thermometer that needs no contact and no calibration — and in a long glass fibre, it is thousands of thermometers at once.

Where shaking makes the surface unstable. For water 1 cm deep shaken vertically at 50 Hz, the regions of wavenumber and shaking acceleration in which a surface wave grows rather than decays, computed mode by mode from the Floquet multipliers of its damped Mathieu equation (inside the curves, unstable). The largest tongue reaches down to 0.80 m/s² — 0.081 g — at k = 633 m⁻¹, the wave whose own frequency is exactly half the drive's: a wavelength of 9.9 mm. A second, narrower tongue sits at k = 1066 m⁻¹, where the wave's frequency equals the drive's, and needs much harder shaking. Without damping both tongues would reach down to zero; viscosity lifts their tips, most for the shortest waves.

Waves

The ripples that answer at half the shaking

Shake a dish of water up and down fifty times a second and nothing happens — until the shaking passes a threshold, when the whole surface erupts into a pattern of standing ripples, squares or stripes, each crest rising and falling twenty-five times a second, at exactly half the frequency of the shaking. Michael Faraday saw it in 1831. The size of the ripples is not set by the dish. It is chosen by the water itself, as the one wavelength whose natural frequency is half the drive's, and the threshold is set by how quickly the water's own viscosity would kill that wave.

Relativity

The parabola that decides which way a nucleus decays

A weak decay turns a neutron into a proton or a proton into a neutron and leaves the number of nucleons alone, so it moves a nucleus sideways among its isobars. Whether it can happen, and in which direction, is settled by weighing: a decay goes wherever the atom ends up lighter, by any amount at all, except that emitting a positron needs two electron masses more. Laid out against proton number, the masses of a family of isobars fall on a parabola — or, for an even number of nucleons, on two — and the shape of that parabola is why potassium-40 decays both ways and germanium-76 can only take two steps at once.

$Two measures of how uncertain a split state is. For the two-peaked state, against half the separation of the peaks in units of their width: Heisenberg's product σₓσₚ divided by its minimum ħ/2 (solid), and the entropic sum Hₓ + Hₚ minus its minimum ln(πeħ) (dashed), both computed from the state's distributions. As the peaks separate the product grows without limit, 8.1 times the minimum at the right of the figure, because the variance counts the empty space between the peaks as uncertainty. The entropic excess levels off at 2 ln 2 − 1 = 0.386 (0.386 at the right): the position's entropy gains ln 2, one bit, for not knowing which peak, and the momentum's gives back 1 − ln 2 because its fringes pack it into narrow bands. The entropic relation says what is true of the state: it is a fraction of a coin toss away from a minimum-uncertainty packet, however far apart the two places are.$ Quantum

The sharpness a variance cannot see

Heisenberg's inequality is a statement about standard deviations, and a standard deviation is a poor measure of how sharply something is known. A particle that is certainly in one of two narrow places a metre apart has a position spread of half a metre, and satisfies the inequality with room to spare while being sharply localised. Measure uncertainty by information instead — by the Shannon entropy of each distribution — and the relation tightens: the sum of the two entropies has a floor that the split state cannot get under either, and the floor survives in places where the variance form says nothing at all.

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.

Thermodynamics

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.

Quantum

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.

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

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.

Where the heat would land. The average field strength squared on a horizontal plane half-way up the oven from 3 of the in-band modes taken together, 2,3,2; 3,2,2; 2,4,1, darker meaning stronger. The bright patches are spaced by about half a wavelength of the mode along each wall, some 7 to 15 cm for these indices, and between them are nodes where the heating falls to almost nothing. A plate of cheese on a stopped turntable shows this as melted spots, and a measurement of their spacing against the frequency gives the speed of light to within several per cent, a popular school experiment. Since many modes can be involved and the load disturbs them, the method is not as clean as it looks, and here the load's own effect is not drawn.

Waves

The oven that cooks in spots

A microwave oven is a metal box, and a metal box has only so many ways of ringing at a given frequency. At 2.45 GHz a kitchen oven allows a handful, and the field inside is the sum of those few standing patterns, with hot patches and nodes several centimetres apart. Food on a stopped plate shows them as a pattern of scorch and frost. A turntable can smear this only into rings, and frozen food has a further trap, because ice absorbs thousands of times less than water.

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