Rydberg constant
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The Rydberg constant, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to atomic spectra in the science of spectroscopy. Rydberg initially determined its value empirically from spectroscopy, but it was later found that its value could be calculated from more fundamental constants by using quantum mechanics.
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 lowestenergy 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.
Value of the Rydberg constantEdit
Making use of the simplifying assumption that the mass of the atomic nucleus is infinite compared to the mass of the electron, the constant is (according to 2002 CODATA results):

 where
 is the Planck's constant,
 is the rest mass of the electron,
 is the elementary charge,
 is the speed of light in vacuum, and
 is the permittivity of free space.
 where
This constant is often used in atomic physics in the form of an energy:
Two complications arise. One is that one may wish to discuss a hydrogenlike ion, i.e., an atom with atomic number Z that has only one electron. In this case, the wavenumbers and photon energies are scaled up by a factor of . The other is that the mass of the atomic nucleus is not actually infinite compared to the mass of the electron. The predicted spectrum must then be corrected by substituting the reduced mass for the mass of the electron, resulting in:
The Rydberg constant is one of the most welldetermined physical constants, with a relative experimental uncertainty of less than 7 parts per trillion. The ability to measure it directly to such a high precision constrains the proportions of the values of the other physical constants that define it.
Alternative expressionsEdit
The Rydberg constant can also be expressed as the following equations.
and
where
 is Planck's constant,
 is the speed of light in a vacuum,
 is the finestructure constant,
 is the Compton wavelength of the electron,
 is the Compton frequency of the electron,
 is the reduced Planck's constant, and
 is the Compton angular frequency of the electron.
The derivation of Rydberg constant from quantum mechanics Edit
Historically, the Rydberg equation was found empirically (experimentally), and it predated the development of quantum theory. (See Rydberg formula for a full discussion of its discovery.) To understand its significance in terms of the quantum theory, we can start from the equation
for the energy of the electron in the nth energy state, as can be derived either from the Bohr model or from a fully quantummechanical treatment of the hydrogen atom. Therefore a change in energy in an electron changing from one value of to another is
We simply change the units to wavelength and we get
where
 is Planck's constant,
 is the rest mass of the electron,
 is the elementary charge,
 is the speed of light in vacuum, and
 is the permittivity of free space.
 and being the electron shell number of the hydrogen atom
We have therefore found the Rydberg constant for hydrogen to be
See alsoEdit
 Rydberg formula, includes a discussion of Rydberg's original discovery.
ReferencesEdit
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