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The Rydberg equation is one of the most important formulas in atomic spectroscopy, providing an exact prediction of the wavelengths of all spectral lines emitted by hydrogen-like atoms. First derived empirically by Johannes Rydberg in 1888 by generalizing Balmer's formula, it was later given a theoretical foundation by Bohr's quantum model in 1913. The Rydberg Equation Calculator computes the wavelength, wavenumber, frequency, and photon energy for any transition between quantum levels n₁ and n₂ in a hydrogen-like atom with atomic number Z. The Rydberg constant R∞ = 1.097 × 10⁷ m⁻¹ is one of the most precisely measured physical constants and connects atomic structure to fundamental constants. This equation accurately describes the spectra of H, He⁺, Li²⁺, and other one-electron systems, and its success provided compelling evidence for the quantized nature of atomic energy levels. It remains essential in astrophysics, plasma diagnostics, and laser physics.
The generalized Rydberg equation for hydrogen-like atoms is:
$$\frac{1}{\lambda} = R_\infty Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$$
where R∞ = 1.0974 × 10⁷ m⁻¹ is the Rydberg constant, Z is the atomic number, and n₂ > n₁.
Related quantities:
$$\tilde{\nu} = \frac{1}{\lambda} = R_\infty Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \text{ (wavenumber, m⁻¹)}$$
$$\nu = c \tilde{\nu} = \frac{c}{\lambda} \text{ (frequency, Hz)}$$
$$E = h\nu = \frac{hc}{\lambda} \text{ (photon energy)}$$
The Rydberg constant is related to fundamental constants by:
$$R_\infty = \frac{m_e e^4}{8\epsilon_0^2 h^3 c}$$
For finite nuclear mass M, a correction applies: R = R∞ × M/(M + m_e), but this correction is less than 0.055% even for hydrogen.
The wavelength output directly gives the position of the spectral line in the electromagnetic spectrum. Lines in the Lyman series (n₁=1) fall in the UV (91–122 nm), Balmer series (n₁=2) spans UV to visible (365–656 nm), and Paschen (n₁=3) and higher series are in the infrared. For hydrogen-like ions (Z > 1), all wavelengths are shorter by a factor of Z², shifting spectral lines into higher-energy regions. He⁺ (Z=2) Balmer lines appear in the UV where H Lyman lines would be. The wavenumber is commonly used in spectroscopy as it is directly proportional to energy. The series limit (n₂ → ∞) gives the ionization threshold from level n₁: 1/λ_limit = R∞Z²/n₁².
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The iconic Hα red line at 656.5 nm is the most prominent Balmer line, used extensively in solar physics and emission nebula imaging.
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He⁺ Lyman-alpha is at 30.4 nm—deep UV—four times the energy of hydrogen Lyman-alpha due to the Z² scaling. This line is important in solar extreme UV observations.
R∞ = 1.0973731568160 × 10⁷ m⁻¹ is a fundamental physical constant representing the highest wavenumber (lowest wavelength) of any photon that can be emitted by hydrogen. It equals m_e·e⁴/(8ε₀²h³c) and is known to 12 significant figures.
Only for hydrogen-like ions (one-electron systems): H, He⁺, Li²⁺, Be³⁺, etc. Multi-electron atoms have electron-electron repulsion that modifies energy levels. Empirical quantum defect corrections can extend the formula to alkali atoms.
The convention ensures a positive wavenumber/wavelength. If n₂ > n₁, the photon is emitted (transition from higher to lower energy). The absolute energy difference is the same regardless of direction; n₂ < n₁ would represent absorption.
As n₂ → ∞, the spectral line approaches the series limit: 1/λ = R∞Z²/n₁². Beyond this, the atom is ionized and the spectrum becomes a continuum. The Lyman limit is 91.2 nm, the Balmer limit is 364.6 nm.
Through precision spectroscopy of hydrogen transitions, especially the 1S–2S two-photon transition measured to 15 significant figures using frequency combs. This measurement contributes to determining the proton charge radius.
The nuclear mass M is finite, so both electron and nucleus orbit the center of mass. The correction replaces m_e with the reduced mass μ = m_e·M/(m_e+M). For hydrogen, this changes R by 0.054%. For positronium (e⁺e⁻), μ = m_e/2, halving all energies.
All energies and wavenumbers scale as Z². He⁺ lines have 4× the energy of corresponding H lines; Li²⁺ has 9×. Radii scale as 1/Z. This Z² scaling is a direct consequence of the stronger Coulomb attraction.
For alkali atoms (Li, Na, K...), the Rydberg formula is modified: E_n = −R∞/(n−δ_l)² where δ_l is the quantum defect depending on angular momentum l. It accounts for inner electron shielding of the nuclear charge.
Hydrogen is the most abundant element in the universe. The Rydberg equation predicts all hydrogen spectral lines used to: measure redshifts/distances of galaxies, determine temperatures of stars and nebulae, and study the intergalactic medium through absorption spectroscopy.
R∞ assumes infinite nuclear mass. R_H = R∞ × M_p/(M_p + m_e) = 1.09678 × 10⁷ m⁻¹ corrects for the proton's finite mass. The difference is 0.054% but matters for precision spectroscopy.
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