Atomic spectra

Every element emits its own fixed barcode of sharp colours. Sixty years of measurements to five figures, not one line explained — and a classical atom that should destroy itself in sixteen picoseconds.

1814 – 191319 min

By the end you should be able to

Say what a line spectrum is and why sharp lines are so hard to explain
Explain why a classical Rutherford atom cannot exist at all
Assess Bohr’s model — what it genuinely achieved and where it is plainly a patch

In 1814 a Bavarian glassmaker named Joseph Fraunhofer was trying to make better lenses. To measure how a glass bent light he needed light of a single pure colour, so he spread sunlight through a prism to see what was available in it. What he found was the solar spectrum crossed by hundreds of narrow dark lines — places where a colour was simply missing. He mapped 574 of them and labelled the strongest with letters A through K, a notation still in use. He had no idea what they were, and he died at 39 without finding out. He was looking at the chemical composition of the Sun.

The visible emission spectrum of hydrogen: four bright lines — deep red, cyan, blue and violet — on black. — Hydrogen, photographed. Four lines and nothing in between: no continuous glow, no faint background, just these colours and darkness everywhere else. Every classical model of the atom predicts a smear. This is the measurement that has to be explained.

Forty-five years later, Gustav Kirchhoff and Robert Bunsen in Heidelberg worked out what the lines meant. Heat any element until it glows, put its light through a prism, and you do not get a rainbow. You get a handful of sharp bright lines, always at exactly the same wavelengths, and a different set for every element. Sodium gives a close pair of yellow lines; hydrogen gives four in the visible. And the dark Fraunhofer lines are the same wavelengths, in absorption: cooler gas in the Sun's outer layers removing precisely the colours it would emit if heated. The dark lines are a list of what the Sun is made of.

Switch between elements. Each has its own fixed set of lines — a fingerprint, not a family resemblance.

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Hydrogen’s lines are computed from the Rydberg formula; the rest are measured values, in air, as the reference tables give them. By the 1880s these positions were known to five significant figures — and not one of them had been explained.

The most powerful analytical tool ever handed to science

Spectroscopy did not merely raise a puzzle; it immediately became indispensable. New elements. Kirchhoff and Bunsen found lines belonging to nothing known, and thereby discovered caesium (1860) and rubidium (1861) — named for the blue and red of their lines. Elements in the Sun. In 1868, during an eclipse, a yellow line was found in the solar spectrum matching nothing on Earth. Norman Lockyer proposed a new element and named it helium, from helios. It was not found on Earth until 1895 — twenty-seven years later. We identified an element in the Sun before we found it on our own planet. And, eventually, the universe. The same lines, shifted, gave the composition of stars, the rotation of galaxies, and Hubble's expansion. All of it working perfectly, and none of it understood. In 1900, the sharpest and most precisely measured data in all of physics were completely unexplained.

The break came from an unlikely direction. In 1885 Johann Balmer — a sixty-year-old Swiss schoolteacher who taught mathematics at a girls' school in Basel, and had no research position in physics — was shown the four measured wavelengths of hydrogen's visible lines and asked whether he could find a pattern in them. He could.

While the spectra sat unexplained, the atom itself became a far worse problem. In 1911 Rutherford fired alpha particles at gold foil and found that a small fraction came straight back. As he put it, it was about as credible as firing a fifteen-inch shell at a sheet of tissue paper and having it bounce back at you. The conclusion was unavoidable: essentially all the mass and all the positive charge is concentrated in a minute nucleus, with the electrons somewhere outside it. That is the picture of the atom nearly everyone still carries around. It is also, on classical physics, completely impossible.

Watch the timer. This is the classical prediction, integrated rather than asserted.

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An accelerating charge radiates — Maxwell, from Act 1, and the operating principle of every radio transmitter. An orbiting electron accelerates continuously, so it must radiate and spiral in. Integrating the Larmor formula gives 15.6 picoseconds from the Bohr radius to the nucleus.

Classical physics does not predict the wrong atom. It predicts no atom.

This is worth stating at full strength, because it is easy to hear it as "the old model was a bit inaccurate". The lifetime is 1.6 × 10⁻¹¹ seconds. About a hundred thousand orbits, and then the electron is in the nucleus. Every atom in your body would have collapsed before you finished reading this sentence — before you started it. And the light would be wrong too. As the orbit shrinks, the orbital frequency climbs smoothly, so the emitted radiation would be a continuous sweep across all frequencies — a rising whistle terminating in silence. Not sharp lines. Not a fingerprint. A smear. So classical physics gets both halves of the problem wrong in the strongest possible way: matter should not persist, and what little light it emitted on the way out should have no structure in it whatsoever. The existence of a stable hydrogen atom is, by itself, a refutation of nineteenth-century physics.

Switch series and pick transitions. Every line in hydrogen’s spectrum is one jump between two rungs.

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Bohr’s ladder. Allowed energies are −13.6/n² eV, so the levels crowd together as they rise towards the ionisation limit. Jumps landing on n = 2 are visible — those are Balmer’s four lines. Jumps to n = 1 are ultraviolet (Lyman), to n = 3 infrared (Paschen); both were predicted by this picture and subsequently found.

The experiment

Deriving the Rydberg constant from e, m and h

Niels Bohr · 1913 · Manchester (Rutherford’s laboratory), and Copenhagen

The question: Balmer’s formula contains a constant that had to be measured from the spectrum itself. Can a physical model of the atom predict that constant from quantities measured in completely unrelated experiments?
The apparatus: None — this is a calculation checked against sixty years of accumulated spectroscopy. The inputs are the electron mass from J.J. Thomson’s deflection experiments, the elementary charge from Millikan’s oil drops, and Planck’s constant from the blackbody spectrum. Not one of them was measured using light from hydrogen.

Theory predicted

If the model is merely a restatement of Balmer, the constant remains an adjustable parameter. If the model is right, R = me⁴/8ε₀²h³c must come out equal to the spectroscopically measured Rydberg constant, with nothing to tune.

They measured

The computed value agreed with the measured Rydberg constant to within the experimental uncertainty of the constants going in — a fraction of a per cent. With modern values the agreement is exact: 1.0973732 × 10⁷ m⁻¹.

How sure could they be? Limited entirely by how well e, m and h were then known, which was a few tenths of a per cent. Bohr also correctly attributed a set of lines then assigned to hydrogen in stellar spectra to ionised helium instead, by noting the model required a factor of four — a prediction confirmed within months.

Why it mattered

This is what separated Bohr from Balmer. A formula that fits is a fit; a formula whose constant is forced by unrelated measurements is an explanation. Einstein, on hearing of the helium result, called it "an enormous achievement" and said the theory must be right. It was also, as Bohr knew, built on a postulate he could not justify.

You might think

Bohr explained the atom — electrons orbit the nucleus in fixed shells, like planets.

Actually

The model gets hydrogen’s spectrum exactly right and is still wrong in almost every particular. It is not derived from anything: the claim that an orbiting electron does not radiate is an assertion inserted to prevent collapse, contradicting Maxwell with no argument offered. It only works for one electron. Applied to helium — two electrons, the second-simplest atom in the universe — it fails outright, and a decade of attempts to extend it failed too. And the orbits are not real. Electrons do not follow paths around nuclei; there are no shells in the planetary sense, and the ground state has zero orbital angular momentum, which Bohr’s rule forbids. What survives is the part Bohr got right for the wrong reasons: atoms have discrete energy levels, and light is emitted in jumps between them.

1814Fraunhofer maps 574 dark lines in the solar spectrum, and cannot explain any of them.
1859Kirchhoff and Bunsen establish that each element has a unique line spectrum.
1868Helium is identified in the Sun — 27 years before it is found on Earth.
1885Balmer, a schoolteacher, finds a formula fitting hydrogen’s four visible lines.
1888Rydberg generalises it, predicting whole series not yet observed.
1911Rutherford finds the nucleus — and with it an atom that should collapse in 16 ps.
1913Bohr postulates non-radiating orbits and derives the Rydberg constant.
1925Heisenberg and Schrödinger replace the model entirely.

Problem

Find a line nobody showed you

Bohr’s formula gives the wavelength of every hydrogen line, not just the ones in the picture. Use it to predict the H-beta line — the transition from n = 4 down to n = 2 — and give the answer as a wavelength in air, in nanometres.

Rydberg constant: R_H = 1.09678 × 10⁷ m⁻¹
Lower level: n = 2
Upper level: n = 4

Where this leaves us

The position in 1913 is a strange one. There is a model that reproduces hydrogen exactly, predicts a fundamental constant from unrelated quantities, correctly reassigns a set of stellar lines to ionised helium — and rests on a postulate its author admitted he could not defend, while failing on the second-simplest atom in existence. It is a signpost rather than a theory. What it says correctly is: atoms have discrete energy levels, and light is emitted in jumps between them. What it cannot say is why. And that raises an obvious and testable question. All the evidence for these energy levels so far comes from light — from spectra. If the levels are physically real, they should show up in something that has nothing to do with light at all. In 1914, two physicists in Berlin fired a beam of electrons through mercury vapour and watched what happened to their energy.