In the physical sciences, the wavenumber (or wave number), also known as repetency,[1] is the spatial frequency of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber).[2] [3] [4] It is analogous to temporal frequency, which is defined as the number of wave cycles per unit time (ordinary frequency) or radians per unit time (angular frequency).
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.
Wavenumber, as used in spectroscopy and most chemistry fields, is defined as the number of wavelengths per unit distance, typically centimeters (cm−1):
\tilde{\nu} =
| 1 | |
| λ |
,
For example, a wavenumber in inverse centimeters can be converted to a frequency expressed in the unit gigahertz by multiplying by (the speed of light, in centimeters per nanosecond);[5] conversely, an electromagnetic wave at 29.9792458 GHz has a wavelength of 1 cm in free space.
In theoretical physics, a wave number, defined as the number of radians per unit distance, sometimes called "angular wavenumber", is more often used:[6]
k =
| 2\pi | |
| λ |
When wavenumber is represented by the symbol, a frequency is still being represented, albeit indirectly. As described in the spectroscopy section, this is done through the relationshipwhere s is a frequency expressed in the unit hertz. This is done for convenience as frequencies tend to be very large.[7]
Wavenumber has dimensions of reciprocal length, so its SI unit is the reciprocal of meters (m-1). In spectroscopy it is usual to give wavenumbers in cgs unit (i.e., reciprocal centimeters; cm-1); in this context, the wavenumber was formerly called the kayser, after Heinrich Kayser (some older scientific papers used this unit, abbreviated as K, where 1K = 1cm−1).[8] The angular wavenumber may be expressed in the unit radian per meter (rad⋅m−1), or as above, since the radian is dimensionless.
For electromagnetic radiation in vacuum, wavenumber is directly proportional to frequency and to photon energy. Because of this, wavenumbers are used as a convenient unit of energy in spectroscopy.
\varepsilonr
\mur
k=k0\sqrt{\varepsilonr\mur}=k0n
The propagation factor of a sinusoidal plane wave propagating in the positive x direction in a linear material is given by
P=e-jkx
k=k'-jk''=\sqrt{-\left(\omega\mu''+j\omega\mu'\right)\left(\sigma+\omega\varepsilon''+j\omega\varepsilon'\right)}
k'=
k''=
\omega=
x=
\sigma=
\varepsilon=\varepsilon'-j\varepsilon''=
\mu=\mu'-j\mu''=
j=\sqrt{-1}
The sign convention is chosen for consistency with propagation in lossy media. If the attenuation constant is positive, then the wave amplitude decreases as the wave propagates in the x-direction.
Wavelength, phase velocity, and skin depth have simple relationships to the components of the wavenumber:
λ=
| 2\pi | |
| k' |
vp=
| \omega | |
| k' |
\delta=
| 1 | |
| k'' |
Here we assume that the wave is regular in the sense that the different quantities describing the wave such as the wavelength, frequency and thus the wavenumber are constants. See wavepacket for discussion of the case when these quantities are not constant.
In general, the angular wavenumber k (i.e. the magnitude of the wave vector) is given by
k=
| 2\pi | |
| λ |
=
| 2\pi\nu | = | |
| vp |
| \omega | |
| vp |
For the special case of an electromagnetic wave in a vacuum, in which the wave propagates at the speed of light, k is given by:
k=
| E | |
| \hbarc |
=
| \omega | |
| c |
For the special case of a matter wave, for example an electron wave, in the non-relativistic approximation (in the case of a free particle, that is, the particle has no potential energy):
k\equiv
| 2\pi | |
| λ |
=
| p | |
| \hbar |
=
| \sqrt{2mE | |
Wavenumber is also used to define the group velocity.
In spectroscopy, "wavenumber"
\tilde{\nu}
\tilde{\nu}=
| \nu | |
| c |
=
| \omega | |
| 2\pic |
.
λ\rm=
| 1 | |
| \tilde\nu |
,
For example, the spectroscopic wavenumbers of the emission spectrum of atomic hydrogen are given by the Rydberg formula:
\tilde{\nu}=R\left(
| 1 | |
| {nf |
2}-
| 1 | |
| {ni |
2}\right),
A spectroscopic wavenumber can be converted into energy per photon E by Planck's relation:
E=hc\tilde{\nu}.
λ=
| 1 | |
| n\tilde\nu |
,
Often spatial frequencies are stated by some authors "in wavenumbers",[10] incorrectly transferring the name of the quantity to the CGS unit cm−1 itself.[11]