The Cluster's Rydberg Formula
A page about the Rydberg formula — the empirical formula that predicted the hydrogen spectrum.
The Rydberg formula
The Rydberg formula gives the wavelengths of the hydrogen spectral lines: 1 / lambda = R_H * (1/n1^2 - 1/n2^2) where R_H = 1.097373 x 10^7 m^{-1} is the Rydberg constant, and n1 < n2 are integers. The series are:
- Lyman series: n1 = 1, n2 = 2, 3, 4... (UV, lambda > 91 nm)
- Balmer series: n1 = 2, n2 = 3, 4, 5... (visible, lambda > 365 nm)
- Paschen series: n1 = 3, n2 = 4, 5, 6... (IR, lambda > 820 nm)
- Brackett series: n1 = 4, n2 = 5, 6, 7...
- Pfund series: n1 = 5, n2 = 6, 7, 8...
In the cluster, the edit Rydberg formula gives the edit hydrogen spectral lines.
The Balmer series
The Balmer series is the most famous because its lines are in the visible spectrum:
- H-alpha: n2=3 -> n1=2, lambda = 656.3 nm (red)
- H-beta: n2=4 -> n1=2, lambda = 486.1 nm (blue-green)
- H-gamma: n2=5 -> n1=2, lambda = 434.0 nm (violet)
- H-delta: n2=6 -> n1=2, lambda = 410.2 nm (violet)
- H-epsilon: n2=7 -> n1=2, lambda = 397.0 nm (near-UV)
The series limit (n2 -> infinity): lambda = 364.6 nm. In the cluster, the edit Balmer series is the edit most famous.
The Bohr model prediction
Bohr's model gives the energy levels of hydrogen: E_n = - (m_e e^4) / (8 epsilon_0^2 h^2 n^2) = -13.6 eV / n^2 = -R_H h c / n^2 where R_inf = (m_e e^4) / (8 epsilon_0^2 h^3 c) = 1.097373 x 10^7 m^{-1} is the Rydberg constant for infinite nuclear mass. For finite nuclear mass, R_M = R_inf / (1 + m_e / M_nucleus). In the cluster, the edit Bohr model gives the edit hydrogen energy levels.
The fine structure
The fine structure correction (relativistic + spin-orbit) splits each level: Delta E_fs = - (E_n^2 / (2 m_e c^2)) [4n / (j + 1/2) - 3] where j = l +/- 1/2. For n = 2: the 2P_{1/2} and 2P_{3/2} levels split by about 4.5 x 10^{-5} eV (the Lamb shift adds a small additional correction). In the cluster, the edit fine structure correction splits each edit level.
The applications
The Rydberg formula is used in:
- Astronomical spectroscopy (identifying hydrogen lines in stars)
- Plasma diagnostics (measuring temperature and density)
- Laser physics (hydrogen maser, frequency standards)
- Fundamental physics tests (Lamb shift, 2S-2P splitting)
In the cluster, the edit Rydberg formula is used in:
- edit Astronomical spectroscopy
- edit Plasma diagnostics
- edit Laser physics
- edit Fundamental physics tests
This formula
This page is about the Rydberg formula. 1/lambda = R_H (1/n1^2 - 1/n2^2). R_H = 1.097373 x 10^7 m^{-1}. H-alpha: 656.3 nm. E_n = -13.6 eV / n^2. The formula is real.