Illustrated Physics
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Atomic spectra — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Hydrogen's emission lines, where they are actually seen. Every transition down to level 1, 2, 3 in hydrogen, drawn at the wavelength it emits, on a logarithmic axis in nanometres. The Lyman series begins at 121.5 nm and crowds toward its limit at 91.1 nm; The Balmer series begins at 656.1 nm and crowds toward its limit at 364.5 nm; The Paschen series begins at 1874.6 nm and crowds toward its limit at 820.1 nm. Only the Balmer series has lines in the visible band, which is why it was the one found first.

    The spectrum is a subtraction, not a list of values

    An atom emits a handful of sharp wavelengths and nothing in between. They are not the atom's energies — they are the differences between them, which is why the lines come in families that crowd onto a limit.

    part 1 · quantum
  2. 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.

    part 2 · quantum
  3. 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.

    part 3 · quantum
  4. 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.

    part 4 · quantum
  5. An atom made of powers of n. Five properties of a hydrogen atom in a level of principal quantum number n, each divided by its value in the ground state, against n from 1 to 300 on logarithmic axes. The radius of a circular orbit rises as n², to 0.53 µm at n = 100; the orbital period as n³, to 1.5 × 10⁻¹⁰ s; the radiative lifetime of the circular state, from the Larmor loss on its orbit, as n⁵ — slope 5.000 — to 0.9 s. The binding energy falls as n⁻², to 1.36 meV, and the electric field that pulls the electron away falls as n⁻⁴, to 3.2 V/cm. Across n = 1 to 300 the lifetime changes by 2 × 10¹² and the ionising field by 8 × 10⁹.

    The atom the size of a bacterium

    Excite hydrogen's electron to the hundredth level and the atom is a micrometre across, bound by a thousandth of an electronvolt, pulled apart by the field of a torch battery, and — in its most circular state — able to live for nearly a second. Every one of those numbers is a power of n, and together they make the most exaggerated atom there is into one of the most useful.

    part 5 · quantum
  6. The hyperfine structure, read off a room-temperature cell. The difference between the probe's absorption with the pump off and on, which removes the Doppler hump and leaves only the Doppler-free features, against detuning. The three transitions, labelled by the excited level F′, sit at −423.6, −266.65 and 0 MHz, and the three crossovers midway between them at −345.1, −211.8, −133.3 MHz. In this model, which empties the ground level only by exciting it, the crossover between F′ = 2 and 3 is the second strongest feature and the weak transitions to F′ = 1 and 2 are small. In real rubidium spectra the crossovers are often the strongest of all, because excited atoms can fall to the other ground level and be lost to the transition altogether — optical pumping, which the model leaves out and which deepens every hole the pump burns. The positions do not depend on any of that: the separations between the features are the hyperfine splittings of rubidium's excited state, read to a fraction of a megahertz in a glass cell on a bench.

    The lines a warm vapour hides

    Shine a tunable laser through a cell of rubidium vapour and the absorption is one smooth hump half a gigahertz wide, though the atoms' own lines are eighty times narrower and three of them lie inside it. The atoms are moving, each sees the light Doppler-shifted by its own speed, and their lines are smeared together. Send a second, stronger beam back the other way and narrow dips appear in the hump at exactly the frequencies of the hidden lines — because only atoms standing still can be talked to by both beams at once. The vapour stays at room temperature and the motion is not removed; it is selected against.

    part 6 · quantum
  7. 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.

    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.

    part 7 · quantum
  8. 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.

    part 8 · quantum

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