Atomic spectra and quantised energy levels

A spectrometer measures wavelength; the physics lives in energy. A photon of wavelength λ carries E = hf = hc/λ, and with E in eV and λ in nm, hc = 1239.84 eV nm, so hydrogen's red line at 656.47 nm carries 1.8886 eV. The atom that emitted it did not possess 1.8886 eV in any useful sense: it fell by 1.8886 eV from one level to another. A hot, dilute gas gives emission lines because collisions keep lifting atoms into upper levels; a cooler gas in front of a continuous source gives dark absorption lines where photons match an upward jump from a level that is actually occupied. Quote vacuum wavelengths. Air has a refractive index near 1.00028 in the visible, so a line reads about 3 parts in 10⁴ shorter in air: Hα, 656.47 nm in vacuum, is listed near 656.28 nm in air-wavelength tables. That correction is about ten times larger than the fine-structure splitting this unit meets later.

In 1885 Balmer found that the four visible hydrogen lines fit λ = B n²/(n² − 4) with n = 3, 4, 5, 6 and B = 364.56 nm. The rule is opaque in wavelength and transparent in wavenumber σ = 1/λ, where it becomes σ = (4/B)(1/2² − 1/n²), a difference of two terms of the same shape. Rydberg generalised it to σ = R_H(1/n₁² − 1/n₂²), and σ is the natural variable because it is proportional to energy, E = hcσ. The generalisation made predictions. With n₁ = 1 it puts a series in the ultraviolet from 121.57 nm down to 91.18 nm, which Lyman found from 1906; with n₁ = 3 it puts one in the infrared from 1875.6 nm down to 820.6 nm, which Paschen found in 1908. Each series is the family of lines sharing one lower level. In hydrogen the families occupy separate bands until n₁ = 4, whose Brackett lines, 4052 nm to 1459 nm, overlap the long-wavelength end of the Paschen series.

Ritz saw in 1908 that the content is not the 1/n² but the structure underneath it: every line of an atom has a wavenumber equal to the difference of two entries in one list of terms, T(n). Lines that share a term therefore combine into a third. The cascade 3 → 2 → 1 emits Hα at 15 233 cm⁻¹ and then Lyman α at 82 258 cm⁻¹, and their sum, 97 491 cm⁻¹, is Lyman β at 102.57 nm, observed on its own. Two lines that share an upper level subtract instead: σ(3 → 1) − σ(3 → 2) = σ(2 → 1). Multiply every term by hc and the principle is energy conservation for an atom with a fixed set of levels. It also marks what lines cannot fix. Add the same constant to every term and every difference, and so every line, is unchanged: a spectrum measures the spacing of the ladder, never the height of its floor.

As n₂ → ∞ the lines of a series crowd toward σₗᵢₘ = R_H/n₁², and adjacent lines close up as 2R_H/(n₂ + ½)³, the derivative of the term. Past the limit absorption does not stop; it turns continuous, because a photon above the limit frees the electron and the surplus becomes kinetic energy, which is not quantised. So the limit photon is the one that just ionises the atom from the lower level, and its energy is that level's binding energy: the Balmer limit at 364.71 nm gives 3.400 eV for n = 2, and the Lyman limit at 91.18 nm gives 13.598 eV for n = 1. Put the zero at the threshold and every level sits at Eₙ = −13.598 eV/n². Electron-impact experiments need the same 13.6 eV to ionise hydrogen, an independent check that the ladder is real. The crowding is the correspondence check: for large n the gap to the next level is about 2/n of the binding energy, so near the threshold the energy becomes effectively continuous, as classical physics expects.

An absorption line needs atoms waiting in its lower level. In thermal equilibrium the populations follow a Boltzmann factor, Nₙ/N₁ = (gₙ/g₁)e^(−ΔE/kT), where ΔE = Eₙ − E₁, k is Boltzmann's constant and gₙ = 2n² counts the states in level n. For n = 2 the gap is 10.199 eV while kT = 0.0259 eV at 300 K, so N₂/N₁ ≈ 2 × 10⁻¹⁷¹: a laboratory cell of atomic hydrogen absorbs only from n = 1, in the Lyman series and the continuum beyond its limit, all of it in the vacuum ultraviolet, and is transparent to every Balmer line. At 9500 K, kT = 0.819 eV and N₂/N₁ = 4e^(−12.46) ≈ 1.6 × 10⁻⁵. That fraction is small, but a stellar atmosphere holds so many atoms along the line of sight that Balmer absorption lines are the strongest features in the spectra of stars near that temperature. Hotter still, most of the hydrogen is ionised and the lines weaken again, a competition the Boltzmann factor alone does not describe.

Rydberg's formula places every hydrogen line and says nothing else. It does not explain why the terms fall as 1/n² or why R_H = 1.09678 × 10⁷ m⁻¹; the next topic derives both, and any such derivation must also explain why deuterium's constant is larger by 2.7 parts in 10⁴, the 0.18 nm shift of Hα by which heavy hydrogen was discovered in 1932. It is silent on brightness. An emission line radiates power N Γ hf, where N counts the atoms in the upper level and Γ is the rate at which that level decays by that line, and Γ ranges from 6.3 × 10⁸ s⁻¹ for the fast states of n = 2 to almost zero for the metastable 2s state, which survives about 0.12 s. Those rates need wavefunctions. And the fit fails in two places: at a few parts in 10⁵, where fine structure splits each line, and at once for atoms with more than one electron, whose terms fit only as R/(n − δ)² with a quantum defect δ, about 1.35 for sodium's s levels.