The Balmer lines: ratios in light

The light of hydrogen has four visible colours. In 1885 Johann Jakob Balmer found their wavelengths as simple fractions of square numbers, among them 4 : 3 and 9 : 8, the fourth and the whole tone.

The Pythagoreans found number in the notes of a string. Two and a half thousand years later it was found in the colours of light, in the light of the simplest of all atoms, hydrogen.

Lines in the spectrum

Split light with a prism and you get a band of colours, the spectrum. In 1814 Joseph von Fraunhofer found fine dark lines in it. William Wollaston had noticed some of them in 1802, but Fraunhofer was the first to measure them precisely. In 1859 Gustav Kirchhoff and Robert Bunsen showed that each chemical element has its own pattern of such lines, like a fingerprint. Glowing hydrogen does not shine in all colours, only in four sharp lines in the visible range: red, blue-green, blue and violet. The Swede Anders Ångström had measured their wavelengths very precisely.

Coloured plate: a band of colour from red through yellow and green to violet, crossed by many dark vertical lines, above it a brightness curve — Joseph von Fraunhofer’s own drawing of the solar spectrum with the dark lines that now bear his name (1814/15). The curve above shows how bright the spectrum appeared to him at each point. The letters he used to label the strongest lines are still in use today.

Balmer’s fractions

Johann Jakob Balmer taught mathematics at a girls’ school in Basel and was 60 years old when a physicist at the university drew his attention to these four numbers. Balmer had no physical theory. He was only looking for a ratio. In 1885 he found it: divide the four wavelengths by a common base length B = 364.56 nm and you get the fractions

\frac{9}{5}, \quad \frac{4}{3}, \quad \frac{25}{21}, \quad \frac{9}{8}

At first this looks disordered. But write 4/3 as 16/12 and 9/8 as 36/32, and the pattern appears: on top are the square numbers 9, 16, 25, 36, below each the same square number minus 4.

\lambda = B \cdot \frac{n^2}{n^2 - 4}, \qquad n = 3, 4, 5, 6

Balmer predicted further lines for $n = 7$ , 8 and so on, crowding ever closer to the limit B. They had already been measured in the light of hot stars and fitted very well.

Top: the spectrum of hydrogen with the Balmer lines, crowding towards the limit at 364.56 nm. Bottom: the energy levels of the atom. Each line arises when the electron falls from level n to level 2. The buttons select one of the four visible lines.

Two of the four fractions are familiar intervals, the fourth 4 : 3 and the whole tone 9 : 8, and 9 : 5 is a minor seventh. Whether this means more than a pretty coincidence cannot be said. What is certain is that the colours of hydrogen are fixed by whole numbers, just like the notes of a string.

From Balmer to Bohr

In 1888 Johannes Rydberg wrote Balmer’s formula more generally. Instead of 4, which is $2^{2}$ , any square number may stand. With $1^{2}$ you get a series in the ultraviolet, found by Theodore Lyman in 1906, with $3^{2}$ one in the infrared, found by Friedrich Paschen in 1908. All the light of hydrogen follows a single law of squares of whole numbers.

Niels Bohr explained why in 1913. The electron in the hydrogen atom can only occupy certain levels, numbered $n = 1$ , 2, 3 … Their energy is inversely proportional to $n^{2}$ . When the electron falls from a higher level to a lower one, it emits light whose colour matches the gap between the levels. The Balmer lines are the jumps down to level 2. In 1926 the new quantum mechanics derived these levels from its basic equations, first Wolfgang Pauli with matrix mechanics, then Erwin Schrödinger from the wave nature of the electron.

The same number `n`

The level number n of the Balmer formula is the same one that counts the shells of the atoms. The n-th shell holds $2 \cdot n^{2}$ electrons, twice a square number (see Atomic shells and magic numbers). The colours of light and the build of atoms follow the same square numbers.

Balmer was occupied all his life with numbers and proportions, in architecture as well, and so he looked for a ratio where others saw only measurements. That made him a late Pythagorean, and the physics of the 20th century proved him right: at the beginning of the road to quantum physics there also stood a ratio, found by a teacher who was looking for harmony in light.

Sources and further reading

Sources

Joseph von Fraunhofer, Bestimmung des Brechungs- und Farbenzerstreuungs-Vermögens verschiedener Glasarten, Denkschriften der Akademie der Wissenschaften zu München 5, 1814 to 1815
Gustav Kirchhoff and Robert Bunsen, Chemische Analyse durch Spectralbeobachtungen, Annalen der Physik 110, 1860
Johann Jakob Balmer, Notiz über die Spectrallinien des Wasserstoffs, Annalen der Physik 25, 1885
Niels Bohr, On the Constitution of Atoms and Molecules, Philosophical Magazine 26, 1913