In spectroscopy, the Rydberg constant, symbol R ∞ {\displaystyle R_{\infty }} for heavy atoms or R H {\displaystyle R_{\text{H}}} for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. The constant first arose as an empirical fitting parameter in the Rydberg formula for the hydrogen spectral series, but Niels Bohr later showed that its value could be calculated from more fundamental constants according to his model of the atom. Before the 2019 revision of the SI, R ∞ {\displaystyle R_{\infty }} and the electron spin g-factor were the most accurately measured physical constants. The constant is expressed for either hydrogen as R H {\displaystyle R_{\text{H}}} , or at the limit of infinite nuclear mass as R ∞ {\displaystyle R_{\infty }} . In either case, the constant is used to express the limiting value of the highest wavenumber (inverse wavelength) of any photon that can be emitted from a hydrogen atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing a hydrogen atom from its ground state. The hydrogen spectral series can be expressed simply in terms of the Rydberg constant for hydrogen R H {\displaystyle R_{\text{H}}} and the Rydberg formula. In atomic physics, Rydberg unit of energy, symbol Ry, corresponds to the energy of the photon whose wavenumber is the Rydberg constant, i.e. the ionization energy of the hydrogen atom in a simplified Bohr model.
Value
Rydberg constant The CODATA value is
R ∞ = m e e 4 8 ε 0 2 h 3 c = {\displaystyle R_{\infty }={\frac {m_{\text{e}}e^{4}}{8\varepsilon _{0}^{2}h^{3}c}}=} 10973731.568157(12) m−1, where
m e {\displaystyle m_{\text{e}}} is the rest mass of the electron (i.e. the electron mass),
e {\displaystyle e} is the elementary charge,
ε 0 {\displaystyle \varepsilon _{0}} is the permittivity of free space,
h {\displaystyle h} is the Planck constant, and
c {\displaystyle c} is the speed of light in vacuum. The symbol ∞ {\displaystyle \infty } means that the nucleus is assumed to be infinitely heavy; an improvement of the value can be made using the reduced mass of the atom:
μ = 1 1 m e + 1 M {\displaystyle \mu ={\frac {1}{{\frac {1}{m_{\text{e}}}}+{\frac {1}{M}}}}}
with M {\displaystyle M} the mass of the nucleus. The corrected Rydberg constant is:
R M = μ m e R ∞ {\displaystyle R_{\text{M}}={\frac {\mu }{m_{\text{e}}}}R_{\infty }}
that for hydrogen, where M {\displaystyle M} is the mass m p {\displaystyle m_{\text{p}}} of the proton, becomes:
R H = m p m e + m p R ∞ ≈ 1.09678 × 10 7 m − 1 , {\displaystyle R_{\text{H}}={\frac {m_{\text{p}}}{m_{\text{e}}+m_{\text{p}}}}R_{\infty }\approx 1.09678\times 10^{7}{\text{ m}}^{-1},}
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