In Spectroscopy
In spectroscopy, the wavenumber of electromagnetic radiation is defined as
where λ is the wavelength of the radiation.
The historical reason for using this quantity is that it proved to be convenient in the analysis of atomic spectra. Wavenumbers were first used in the calculations of Johannes Rydberg in the 1880s. The Rydberg–Ritz combination principle of 1908 was also formulated in terms of wavenumbers. A few years later spectral lines could be understood in quantum theory as differences between energy levels, energy being proportional to wavenumber, or frequency. However, spectroscopic data kept being tabulated in terms of wavenumber rather than frequency or energy, since spectroscopic instruments are typically calibrated in terms of wavelength, independent of the value for the speed of light or Planck's constant.
For example, the wavenumbers of the emissions lines of hydrogen atoms are given by
where R is the Rydberg constant and ni and nf are the principal quantum numbers of the initial and final levels, respectively (ni is greater than nf for emission).
A wavenumber can be converted into energy E via Planck's relation:
It can also be converted into frequency via
where is the frequency, and cn is the speed of light in the medium.
In colloquial usage, the unit cm−1 is sometimes referred to as a "wavenumber", which confuses the name of a quantity with that of a unit. Furthermore, spectroscopists often express a quantity proportional to the wavenumber, such as frequency or energy, in cm−1 and leave the appropriate conversion factor as implied. Consequently, a phrase such as "the energy is 300 wavenumbers" should be interpreted or restated as "the energy corresponds to a wavenumber of 300 cm−1." (Analogous statements hold true for the unit m−1.)
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