# Rydberg Formula Calculator 1/λ = R·Z²(1/n₁² − 1/n₂²) for hydrogen-like ions with reduced-mass correction: wavelength, photon energy, frequency, band, Pickering and Balmer notes. > Canonical page: https://elysiatools.com/en/tools/rydberg-formula-calculator - **Category:** Science & Education - **Keywords:** rydberg formula, hydrogen-like ion, spectral wavelength, reduced mass, balmer series, pickering series, atomic physics ## Overview The Rydberg Formula Calculator computes photon wavelengths, transition frequencies, photon energies, and spectral bands for electron transitions in hydrogen-like ions. It supports exact atomic models using infinite nuclear mass or reduced-mass corrections for Hydrogen-1, Deuterium-2, Helium-4, and custom isotopes. ## Inputs - **Nuclear charge Z** (number): e.g. 2 - **Nucleus model** (select) - **Custom nuclear mass (u)** (number): e.g. 7.016004 - **Lower level n₁** (number): e.g. 2 - **Upper level n₂** (number): e.g. 3 ## When to use - Predicting spectral line wavelengths and photon energies for one-electron atoms and ions. - Accounting for isotope shifts using reduced-mass corrections instead of the idealized infinite-mass constant. - Analyzing astrophysical and laboratory spectral series such as Balmer emission lines or the ionized helium Pickering series. ## How it works - Enter the nuclear charge Z along with the lower principal quantum number n₁ and upper quantum number n₂. - Select the nuclear mass model (infinite mass, Hydrogen-1, Deuterium-2, Helium-4, or a custom mass in unified atomic mass units). - The tool calculates the adjusted Rydberg constant R using the electron-to-nucleus mass ratio and computes the transition via 1/λ = R·Z²·(1/n₁² − 1/n₂²). - It outputs the emitted wavelength, photon energy in electronvolts, frequency in hertz, spectral region, and relevant spectroscopic series notes. ## Use cases - Calculating the Balmer-alpha (Hα) emission wavelength for optical spectroscopy and nebula emission studies. - Resolving spectral overlaps between ionized helium (He⁺) Pickering transitions and hydrogen Lyman lines. - Verifying quantum mechanics and atomic physics coursework calculations involving single-electron energy levels. ## Frequently asked questions ### What is the Rydberg formula used for? It calculates the wavelengths, frequencies, and energies of photons emitted or absorbed during electron transitions between energy levels in hydrogen-like ions. ### What constitutes a hydrogen-like ion? A hydrogen-like ion is any atomic species with a single bound electron orbiting a nucleus with charge Z, such as neutral H (Z=1), He⁺ (Z=2), or Li²⁺ (Z=3). ### Why is the reduced-mass correction important? Because the nucleus is not infinitely heavy, the electron and nucleus orbit their common center of mass. Reduced mass adjusts the Rydberg constant R, providing precise wavelengths that match experimental spectroscopy. ### How do I specify a custom isotope? Select 'Custom nuclear mass' in the nucleus model menu and enter the atomic mass of the nucleus in unified atomic mass units (u). ### What are the allowed ranges for the quantum numbers n₁ and n₂? 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