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| In the [[physical science]]s, the '''wavenumber''' (also '''wave number''') is the [[spatial frequency]] of a [[wave]], either in cycles per unit distance or radians per unit distance. It can be envisaged as the number of waves that exist over a specified distance (analogous to [[frequency]] being the number of cycles or radians per unit time).
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| Because of the use of this term in applied physics, including spectroscopy, often the reference distance should be assumed to be cm. For example, a particle's energy may be given as a wavenumber in cm<sup>−1</sup>, which strictly speaking is not a unit of energy. However if one assumes this corresponds to electromagnetic radiation, then it can be directly converted to any unit of energy, e.g. 1 cm<sup>−1</sup> implies 1.23984×10<sup>−4</sup> eV and 8065.54 cm<sup>−1</sup> implies 1 eV.<ref>[http://physics.nist.gov/cuu/Constants/energy.html NIST Reference on Constants, Units and Uncertainty (CODATA 2010)], specifically [http://physics.nist.gov/cgi-bin/cuu/Convert?exp=0&num=100&From=minv&To=ev&Action=Convert+value+and+show+factor 100/m] and [http://physics.nist.gov/cgi-bin/cuu/Convert?exp=0&num=1&From=ev&To=minv&Action=Convert+value+and+show+factor 1 eV]. Retrieved April 25, 2013.</ref>
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| In multidimensional systems, the wavenumber is also the magnitude of the [[wave vector]].
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| ==Definition==
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| It can be defined as either
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| * <math>\scriptstyle \tilde{\nu} \;=\; \frac{1}{\lambda}</math>, the number of wavelengths per [[unit interval|unit distance]], where ''λ'' is the [[wavelength]], sometimes termed the '''spectroscopic wavenumber''', or
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| * <math>\scriptstyle k \;=\; \frac{2\pi}{\lambda}</math>,the number of wavelengths per 2π units of distance, sometimes termed the '''angular''' or '''circular wavenumber''', but more often simply ''wavenumber''.
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| Its usual symbols are <math>\scriptstyle\nu</math>, <math>\scriptstyle\tilde{\nu}</math>, ''σ'' or ''k'', the first three used for one definition, the last for another. It has [[dimensional analysis|dimensions]] of [[reciprocal length]], so its [[SI unit]] is m<sup>−1</sup> and [[cgs unit]] cm<sup>−1</sup> (in this context formerly called the ''kayser'', after [[Heinrich Kayser]]).
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| For [[electromagnetic radiation]] in vacuum, wavenumber is proportional to [[frequency]] and to [[photon]] energy. Because of this, wavenumbers are used as a [[unit of energy]] in [[spectroscopy]]. In the [[SI units]], wavenumber is expressed in units of [[reciprocal meters]] (m<sup>−1</sup>), but in spectroscopy it is usual to give wavenumbers in [[reciprocal length|reciprocal centimeters]] (cm<sup>−1</sup>). The angular wavenumber is expressed in [[radian]]s per meter (rad·m<sup>−1</sup>).
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| ==In wave equations==
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| In general, the angular wavenumber <math>k</math> (i.e. the [[magnitude (mathematics)|magnitude]] of the [[wave vector]]) is given by
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| :<math>k = \frac{2\pi}{\lambda} = \frac{2\pi\nu}{v_\mathrm{p}}=\frac{\omega}{v_\mathrm{p}}</math> | |
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| where <math>\nu</math> is the frequency of the wave, <math>\lambda</math> is the wavelength, <math>\omega = 2\pi\nu</math> is the [[angular frequency]] of the wave, and ''v''<sub>p</sub> is the [[phase velocity]] of the wave. The dependence of the wavenumber on the frequency (or more commonly the frequency on the wavenumber) is known as a [[dispersion relation]].
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| For the special case of an [[electromagnetic wave]] in vacuum, where ''v''<sub>p</sub> = ''c'', ''k'' is given by
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| :<math>k = \frac{E}{\hbar c}</math>
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| where ''E'' is the [[energy]] of the wave, ''ħ'' is the [[reduced Planck constant]], and ''c'' is the [[speed of light]] in a vacuum.
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| For the special case of a [[matter wave]], for example an electron wave, in the non-relativistic approximation:
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| :<math>k \equiv \frac{2\pi}{\lambda} = \frac{p}{\hbar}= \frac{\sqrt{2 m E }}{\hbar} </math>
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| Here ''p'' is the [[momentum]] of the particle, ''m'' is the [[mass]] of the particle, ''E'' is the [[kinetic energy]] of the particle, and ''ħ'' is the [[reduced Planck's constant]].
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| Wavenumber is also used to define the [[group velocity]].
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| ==In spectroscopy==
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| In [[spectroscopy]], the wavenumber <math>\scriptstyle \tilde{\nu}</math> of [[electromagnetic radiation]] is defined as
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| :<math> \tilde{\nu} = \frac{1}{\lambda} </math>
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| where ''λ'' is the [[wavelength]] of the radiation.
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| 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 mechanics|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]].
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| For example, the wavenumbers of the emissions lines of [[hydrogen]] atoms are given by
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| :<math> \tilde{\nu} = R\left(\frac{1}{{n_f}^2} - \frac{1}{{n_i}^2}\right) </math>
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| where ''R'' is the [[Rydberg constant]] and ''n''<sub>i</sub> and ''n''<sub>f</sub> are the principal quantum numbers of the initial and final levels, respectively (''n''<sub>i</sub> is greater than ''n''<sub>f</sub> for emission).
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| A wavenumber can be converted into [[energy]] ''E'' via [[Planck's relation]]:
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| :<math>E = hc\tilde{\nu}</math>
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| It can also be converted into frequency via
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| :<math>\tilde{\nu} = \frac{\nu}{c_\mathrm{n}}</math>
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| where <math>\scriptstyle \nu</math> is the frequency, and ''c''<sub>n</sub> is the speed of light in the [[optical medium|medium]].
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| In colloquial usage, the unit cm<sup>−1</sup> is sometimes referred to as a "wavenumber",<ref>See for example,
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| *{{cite journal |last1=Fiechtner |first1=G.
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| |year=2001
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| |title=Absorption and the dimensionless overlap integral for two-photon excitation
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| |journal=[[Journal of Quantitative Spectroscopy and Radiative Transfer]]
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| |volume=68 |issue=5 |pages=543
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| |doi=10.1016/S0022-4073(00)00044-3
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| |bibcode = 2001JQSRT..68..543F }}
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| *{{cite patent
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| |invent1=Ray, James C.
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| |invent2=Asari, Logan R.
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| |pubdate=1991-09-10
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| |title=Method and apparatus for spectroscopic comparison of compositions
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| |country=US |number=5046846
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| |class=G01N21/35
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| }}
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| *{{cite journal
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| |year=2005
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| |title=Boson Peaks and Glass Formation
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| |journal=[[Science (journal)|Science]]
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| |volume=308 |issue=5726 |pages=1221
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| |bibcode=
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| |doi=10.1126/science.308.5726.1221a
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| }}</ref> 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<sup>−1</sup> 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<sup>−1</sup>." (Analogous statements hold true for the unit m<sup>−1</sup>.)
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| ==See also==
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| * [[spatial frequency]]
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| ==References==
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| {{reflist}}
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| [[Category:Wave mechanics]]
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| [[Category:Concepts in physics]]
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