Illustrated Physics
physics, drawn
Concept

Fine structure constant — where it appears

The dimensionless number α ≈ 1/137 that measures the strength of the electromagnetic interaction between charged particles. In an atom it is the speed of the innermost electron in units of c, so fine-structure splittings are a fraction α² of the level spacing.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

One level, split by the speed of the electron in it. Hydrogen's n = 2 level on the left at the scale of the whole spectrum, and the same level on the right at a scale ten thousand times finer. The gross structure puts it 3.4014 electronvolts below the ionisation limit; the relativistic and spin–orbit corrections then split it into a lower pair, j = ½, and an upper, j = 3/2, separated by 45.3 microelectronvolts. That gap is 1.331e-5 of the level's own binding energy, which is exactly α²/4 — so the size of the splitting is a measurement of how fast the electron is going, α being its speed in units of c on the innermost orbit. The same physics in sodium is larger by four orders of magnitude: its two D lines at 588.995 and 589.5924 nanometres differ by 2.133 millielectronvolts, which is 1.01e-3 of the transition energy, because the outer electron of sodium penetrates to where the nuclear charge is far from screened and the correction goes as the fourth power of the charge it sees.

The line that is really two

Sodium's yellow line is two lines six-tenths of a nanometre apart, and the gap is not a property of the light. It is a splitting of the atom's own level, caused by the electron's magnetic moment sitting in the field it sees because it is moving — and its size, relative to the level it splits, is exactly α²/4.

quantum · Atomic spectra The levels that refuse to cross. The six energies of sodium 3p against an applied field from zero to 4 crossover fields, 147 T, from diagonalising spin–orbit coupling and the field together. At zero field there are two levels, 515.5 GHz apart. The two states with the largest |mⱼ| are straight lines at every field, because nothing else shares their mⱼ. The other four bend. The two states with mⱼ = −½ approach within 486.0 GHz near 12.3 T and then separate again — √2 ζ exactly, since the coupling between them sets how close they may come — while the mⱼ = −3/2 state falls straight through the lower mⱼ = +½ state at 16.6 T as if it were not there. States that the field cannot mix cross freely; states it can mix exchange character instead, and that exchange is the passage from one pattern to the other.

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.

quantum · Atomic spectra Both clocks move, and their ratio does not. Two years of a clock's fractional frequency against a distant clock (upper panel), and of the ratio of two unlike clocks kept side by side (lower panel). Above, both clocks slow as the Earth nears the Sun, with an amplitude of 1.65·10⁻¹⁰, and if the redshift is universal the two curves are one curve. Below, the ratio: the flat line is what universality predicts, and the sinusoid is what a clock responding to the potential 10⁻⁶ more strongly than the other would produce — an annual term of 1.65·10⁻¹⁶, a million times smaller than the shift both clocks share and within reach of clocks that compare to parts in 10¹⁷.

The clocks that must all slow together

Every clock on the Earth runs slower in January than in July, by three parts in ten thousand million, because the orbit carries the planet deeper into the Sun's potential at perihelion. No clock on the Earth can see this, and that invisibility is the claim worth testing. If the redshift is a property of time rather than of clocks, two clocks built on different physics must slow by exactly the same fraction, and their ratio must not move with the seasons. A ratio that did move would mean the constants of nature depend on where they are measured.

astrophysics · Gravitational redshift A rosette: the orbit of a body whose inertia is its energy. An orbit of eccentricity 0.45 under inertia from energy, Newton's pull, integrated for 7 revolutions from the closest approach of a body with k/Lc = 0.35. The dashed curve is the Newtonian ellipse from the same starting position and momentum, which closes on itself. The integrated orbit does not: its closest approach moves forward by 24.3° each revolution, against the exact 24.3° of 2π(1/Γ − 1) with Γ = √(1 − k²/L²c²). The dots are the successive closest approaches.

The orbit special relativity cannot close

An inverse-square orbit closes on itself because of a conserved vector that nothing else has. Give the orbiting body an inertia that grows with its speed, as special relativity requires, and the vector turns — the orbit becomes a rosette, advancing by a sixth of Mercury's famous 43 arcseconds a century. A different theory that respects special relativity just as well turns the rosette backwards by the same amount. The 43 is not special relativity plus a correction; it is a measurement of what gravity pulls on.

astrophysics · Orbit stability The innermost electron of an atom with Z = 79, with and without relativity. The probability of finding the innermost electron at each distance from a nucleus of charge 79 (Au), with the distance in units of a₀/Z so that the non-relativistic curve is the same for every element. The dashed curve is Schrödinger's, r² e^(−2Zr/a₀). The solid curve is Dirac's exact ground state, r^(2γ) e^(−2Zr/a₀) with γ = √(1 − (Zα)²) = 0.817. It is pulled in towards the nucleus: its mean radius is 0.878 of Schrödinger's, a contraction of 12 per cent, and it has a sharper peak close in. Nothing about the nucleus has changed; the electron, moving at an appreciable fraction of the speed of light near it, simply behaves as a heavier particle, and a heavier particle is bound more tightly and more closely.

The metal that is yellow because it is heavy

Gold sits directly below silver in the periodic table, with the same arrangement of outer electrons, and by every non-relativistic calculation it should look like silver. It is yellow because its innermost electrons move at more than half the speed of light. The mass that motion adds pulls the s orbitals in, the tightened s level drops towards the d band beneath it, and the gap an absorbed photon must bridge shrinks from the ultraviolet into the blue. The same contraction is why mercury is a liquid.

quantum · Atomic structure 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.

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.

quantum · Atomic spectra 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.

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.

quantum · Atomic structure The innermost level dives towards the sea. The energy of an atom's innermost (1s) electron, in units of the electron's rest energy mc², against the nuclear charge Z: dashed, Dirac's formula for a point nucleus, √(1 − (Zα)²); solid, the same equation solved for a nucleus of realistic size. Above +mc² lie the free electrons; below −mc², the filled sea of negative-energy states that Dirac's theory requires. For a point nucleus the level stops at Z = 137, where the formula fails. A real nucleus keeps it: it crosses zero near Z = 148 and reaches −mc² at Z ≈ 170, binding the electron by twice its own rest energy. Beyond that the level has nowhere left to go but into the sea.

The nucleus that would make positrons from nothing

Dirac's equation gives the innermost electron of an atom an energy that falls as the nuclear charge rises, and at 137 protons the formula fails outright. Real nuclei have a size, and with that correction the level does not stop: it falls through zero and on towards minus the electron's rest energy, the edge of the sea of states that Dirac's theory says are already full. It gets there at about 170 protons. Past that point an empty innermost level could be filled from the sea, and the vacuum round the nucleus would emit positrons of its own accord. No nucleus that large exists — but two uranium nuclei, for an instant, come close enough to be one.

quantum · Atomic spectra

Named alongside it

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

Energy levelsSpinAngular momentumAtomic spectraDegeneracyGeneral relativityThe Lorentz factorMagnetic momentScreeningSelection rulesSpectral lineAbsorption

All concepts