In the physical sciences, the wavenumber (or wave number), also known as repetency, is the spatial frequency of a wave. Ordinary wavenumber is defined as the number of wave cycles divided by length; it is a physical quantity with dimension of reciprocal length, expressed in SI units of cycles per metre or reciprocal metre (m−1). Angular wavenumber, defined as the wave phase divided by length, is a quantity with dimension of angle per length and SI units of radians per metre. They are analogous to temporal frequency, respectively the ordinary frequency, defined as the number of wave cycles divided by time (in cycles per second or reciprocal seconds), and the angular frequency, defined as the phase angle divided by time (in radians per second). In multidimensional systems, the wavenumber is the magnitude of the wave vector. The space of wave vectors is called reciprocal space. Wave numbers and wave vectors play an essential role in optics and the physics of wave scattering, such as X-ray diffraction, neutron diffraction, electron diffraction, and elementary particle physics. For quantum mechanical waves, the wavenumber multiplied by the reduced Planck constant is the canonical momentum. Wavenumber can be used to specify quantities other than spatial frequency. For example, in optical spectroscopy, it is often used as a unit of temporal frequency assuming a certain speed of light.
Definition Wavenumber, as used in spectroscopy and most chemistry fields, is defined as the number of wavelengths per unit distance:
ν ~ = 1 λ , {\displaystyle {\tilde {\nu }}\;=\;{\frac {1}{\lambda }},}
where λ is the wavelength. It is sometimes called the "spectroscopic wavenumber". It equals the spatial frequency. In theoretical physics, an angular wave number, defined as the number of radians per unit distance is more often used:
k = 2 π λ = 2 π ν ~ {\displaystyle k\;=\;{\frac {2\pi }{\lambda }}=2\pi {\tilde {\nu }}} .
Units The SI unit of spectroscopic wavenumber is the reciprocal m, written m−1. However, it is more common, especially in spectroscopy, to give wavenumbers in cgs units i.e., reciprocal centimeters or cm−1, with
1 c m − 1 = 100 m − 1 {\displaystyle 1~\mathrm {cm} ^{-1}=100~\mathrm {m} ^{-1}} . Occasionally in older references, the unit kayser (after Heinrich Kayser) is used; it is abbreviated as K or Ky, where 1 K = 1 cm−1. Angular wavenumber may be expressed in the unit radian per meter (rad⋅m−1), or as above, since the radian is dimensionless.
Unit conversions The frequency of light with wavenumber ν ~ {\displaystyle {\tilde {\nu }}} is
f = c λ = c ν ~ {\displaystyle f={\frac {c}{\lambda }}=c{\tilde {\nu }}} , where c {\displaystyle c} is the speed of light. The conversion from spectroscopic wavenumber to frequency is therefore
1 c m − 1 ⋅ c = 29.9792458 G H z . {\displaystyle 1~\mathrm {cm} ^{-1}\cdot c=29.9792458~\mathrm {GHz} .}
Wavenumber can also be used as unit of energy, since a photon of frequency f {\displaystyle f} has energy h f {\displaystyle hf} , where h {\displaystyle h} is the Planck constant. The energy of a photon with wavenumber ν ~ {\displaystyle {\tilde {\nu }}} is
E = h f = h c ν ~ {\displaystyle E=hf=hc{\tilde {\nu }}} . The conversion from spectroscopic wavenumber to energy is therefore
1 c m − 1 ⋅ h c = 1.986446 × 10 − 23 J = 1.239842 × 10 − 4 e V {\displaystyle 1~\mathrm {cm} ^{-1}\cdot hc=1.986446\times 10^{-23}~\mathrm {J} =1.239842\times 10^{-4}~\mathrm {eV} }
where energy is expressed either in J or eV.
Complex A complex-valued wavenumber can be defined for a medium with complex-valued relative permittivity ε r {\displaystyle \varepsilon _{r}} , relative permeability μ r {\displaystyle \mu _{r}} and refraction index n as:
… excerpt ends here. Continue reading the full article.
