The Rydberg constant, symbol R∞ for heavy atoms or RH for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to atomic spectra, in the science of spectroscopy. 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, explaining the relationship via his "Bohr model". As of 2012, R∞ and electron spin g-factor are the most accurately measured fundamental physical constants.
Contents
- Value of the Rydberg constant and Rydberg unit of energy
- Occurrence in Bohr model
- Precision measurement
- Alternative expressions
- References
The Rydberg constant represents the limiting value of the highest wavenumber (the inverse wavelength) of any photon that can be emitted from the hydrogen atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing the hydrogen atom from its ground state. The spectrum of hydrogen can be expressed simply in terms of the Rydberg constant, using the Rydberg formula.
The Rydberg unit of energy, symbol Ry, is closely related to the Rydberg constant. It corresponds to the energy of the photon whose wavenumber is the Rydberg constant, i.e. the ionization energy of the hydrogen atom.
Value of the Rydberg constant and Rydberg unit of energy
According to the 2014 CODATA, the constant is:
where
This constant is often used in atomic physics in the form of the Rydberg unit of energy:
Occurrence in Bohr model
The Bohr model explains the atomic spectrum of hydrogen (see hydrogen spectral series) as well as various other atoms and ions. It is not perfectly accurate, but is a remarkably good approximation in many cases, and historically played an important role in the development of quantum mechanics. The Bohr model posits that electrons revolve around the atomic nucleus in a manner analogous to planets revolving around the sun.
In the simplest version of the Bohr model, the mass of the atomic nucleus is considered to be infinite compared to the mass of the electron, so that the center of mass of the system lies at the barycenter of the nucleus. This infinite mass approximation is what is alluded to with the
where n1 and n2 are any two different positive integers (1, 2, 3, ...), and
A refinement of the Bohr model takes into account the fact that the mass of the atomic nucleus is not actually infinite compared to the mass of the electron. Then the formula is:
where
A generalization of the Bohr model describes a hydrogen-like ion; that is, an atom with atomic number Z that has only one electron, such as C5+. In this case, the wavenumbers and photon energies are scaled up by a factor of Z2 in the model.
Precision measurement
The Rydberg constant is one of the most well-determined physical constants, with a relative experimental uncertainty of fewer than 7 parts in 1012. The ability to measure it to such a high precision constrains the proportions of the values of the other physical constants that define it. See precision tests of QED.
Since the Bohr model is not perfectly accurate, due to fine structure, hyperfine splitting, and other such effects, the Rydberg constant
Alternative expressions
The Rydberg constant can also be expressed as in the following equations.
and
where
The last expression in the first equation shows that the wavelength of light needed to ionize a hydrogen atom is 4π/α times the Bohr radius of the atom.
The second equation is relevant because its value is the coefficient for the energy of the atomic orbitals of a hydrogen atom: