The barcode of hydrogen
Pass the light of a glowing hydrogen tube through a prism and instead of a rainbow you see a few sharp colored lines: red, blue-green, violet. In 1885 the Swiss schoolteacher Johann Balmer found a formula that matched their wavelengths. In 1888 the Swedish physicist Johannes Rydberg generalized it to all of hydrogen’s lines, and in 1913 Niels Bohr explained it: each line is a photon emitted when the electron drops from one energy level to a lower one.
The formula
1/λ = R × Z² × (1/n₁² − 1/n₂²)
- λ is the wavelength in vacuum.
- n₁ is the lower level and n₂ the upper level, with n₂ > n₁.
- Z is the nuclear charge: 1 for hydrogen and deuterium, 2 for He⁺, 3 for Li²⁺.
- R is the Rydberg constant corrected for the mass of the nucleus. For an infinitely heavy nucleus, R∞ = 10,973,731.568 m⁻¹; for hydrogen it is about 10,967,758 m⁻¹.
The photon energy is E = hc ÷ λ, which is 1239.84 ÷ λ in nanometers, in electronvolts.
Worked example: H-alpha
For hydrogen, n₂ = 3 to n₁ = 2:
- 1/λ = 10,967,758 × (1/4 − 1/9) = 1,523,300 m⁻¹.
- Wavelength: 656.47 nm in vacuum, a deep red.
- Photon energy: 1.889 eV; frequency 456.67 THz.
- The Balmer series converges to a limit of 364.71 nm as n₂ grows.
- Ionizing the atom from level 2 takes 3.40 eV.
The series
| Lower level | Series | Region | First line |
|---|---|---|---|
| 1 | Lyman | Ultraviolet | 121.6 nm |
| 2 | Balmer | Visible and near UV | 656.5 nm |
| 3 | Paschen | Infrared | 1,875 nm |
| 4 | Brackett | Infrared | 4,051 nm |
| 5 | Pfund | Infrared | 7,460 nm |
The four visible Balmer lines are H-alpha (656 nm, red), H-beta (486 nm, blue-green), H-gamma (434 nm, violet) and H-delta (410 nm, violet).
Why it matters
The red glow of nebulae photographed by amateur astronomers is H-alpha light. Astronomers measure the shift of hydrogen lines to find how fast galaxies move. The tiny difference between hydrogen and deuterium lines, about 0.18 nm for H-alpha, is how Harold Urey discovered deuterium in 1931.
Limits
The formula is exact only for one-electron systems. For atoms with several electrons, the inner electrons screen the nucleus and the levels shift. Very fine effects, such as the fine structure and the Lamb shift, split each line into components a fraction of a picometer apart. Tables often list wavelengths in air, which are about 0.03% shorter than the vacuum values given here.