Rydberg constant

Rydberg found his formula empirically in 1888 by fitting the wavelengths of alkali and hydrogen lines; Bohr derived it from first principles in 1913, and getting R∞ right out of a model built on quantised angular momentum is what convinced physicists to take quantum theory seriously. Every hydrogen line follows 1/λ = R∞(1/n₁² − 1/n₂²): the Balmer series lands in the visible, Lyman in the ultraviolet, Paschen in the infrared.

Two-photon laser spectroscopy of the 1S–2S transition, whose natural linewidth is a mere 1 Hz, now fixes R∞ to 1.1 parts in 10¹². The subscript ∞ means infinite nuclear mass; for real hydrogen you must apply the reduced-mass correction, giving R_H = R∞/(1 + m_e/m_p), smaller by 1 part in 1836 — a shift big enough to separate hydrogen from deuterium lines in a spectrograph, which is precisely how Urey found deuterium.