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Wavenumber

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Spatial frequency of a wave

Diagram illustrating the relationship between the wavenumber and the other properties of harmonic waves.

In thephysical sciences, thewavenumber (orwave number), also known asrepetency,^[1] is thespatial frequency of awave.Ordinary wavenumber is defined as the number ofwave cycles divided by length; it is aphysical quantity withdimension ofreciprocal length, expressed inSI units of cycles per metre orreciprocal metre (m⁻¹).Angular wavenumber, defined as thewave phase divided by time, is a quantity with dimension ofangle per length and SI units ofradians per metre.^[2]^[3]^[4] They are analogous to temporalfrequency, respectively theordinary frequency, defined as the number of wave cycles divided by time (in cycles per second orreciprocal seconds), and theangular frequency, defined as the phase angle divided by time (in radians per second).

Inmultidimensional systems, the wavenumber is the magnitude of thewave vector. The space of wave vectors is calledreciprocal space. Wave numbers and wave vectors play an essential role inoptics and the physics of wavescattering, such asX-ray diffraction,neutron diffraction,electron diffraction, andelementary particle physics. Forquantum mechanical waves, the wavenumber multiplied by thereduced Planck constant is thecanonical momentum.

Wavenumber can be used to specify quantities other than spatial frequency. For example, inoptical spectroscopy, it is often used as a unit of temporal frequency assuming a certainspeed of light.

Definition

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Wavenumber, as used inspectroscopy and most chemistry fields,^[5] is defined as the number ofwavelengths per unit distance:

{\tilde {\nu }}\;=\;{\frac {1}{\lambda }},

whereλ is the wavelength. It is sometimes called the "spectroscopic wavenumber".^[1] It equals thespatial frequency.^[6]

In theoretical physics, an angular wave number, defined as the number of radians per unit distance is more often used:^[7]

k\;=\;{\frac {2\pi }{\lambda }}=2\pi {\tilde {\nu }}

Units

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TheSI unit of spectroscopic wavenumber is the reciprocal m, written m⁻¹.However, it is more common, especially inspectroscopy, to give wavenumbers incgs units i.e., reciprocal centimeters or cm⁻¹, with

1~\mathrm {cm} ^{-1}=100~\mathrm {m} ^{-1}

Occasionally in older references, the unitkayser (afterHeinrich Kayser) is used;^[8] it is abbreviated asK orKy, where 1 K = 1 cm⁻¹.^[9]

Angular wavenumber may be expressed in the unitradian per meter (rad⋅m⁻¹), or as above, since theradian isdimensionless.

Unit conversions

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The frequency of light with wavenumber ${\tilde {\nu }}$ is

f={\frac {c}{\lambda }}=c{\tilde {\nu }}

where $c {\displaystyle c}$ is thespeed of light.The conversion from spectroscopic wavenumber to frequency is therefore^[10]

1~\mathrm {cm} ^{-1}\cdot c=29.9792458~\mathrm {GHz} .

Wavenumber can also be used asunit of energy, since a photon of frequency $f {\displaystyle f}$ has energy $h f {\displaystyle hf}$ , where $h {\displaystyle h}$ is thePlanck constant.The energy of a photon with wavenumber ${\tilde {\nu }}$ is

E=hf=hc{\tilde {\nu }}

The conversion from spectroscopic wavenumber to energy is therefore

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 inJ oreV.

Complex

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A complex-valued wavenumber can be defined for a medium with complex-valued relativepermittivity $\varepsilon _{r}$ , relativepermeability $\mu _{r}$ andrefraction indexn as:^[11]

k=k_{0}{\sqrt {\varepsilon _{r}\mu _{r}}}=k_{0}n

wherek₀ is the free-space wavenumber, as above. The imaginary part of the wavenumber expresses attenuation per unit distance and is useful in the study of exponentially decayingevanescent fields.

Plane waves in linear media

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The propagation factor of asinusoidal plane wave propagating in the positive x direction in a linear material is given by^[12]^: 51

P=e^{-jkx}

where

$k=k'-jk''={\sqrt {-\left(\omega \mu ''+j\omega \mu '\right)\left(\sigma +\omega \varepsilon ''+j\omega \varepsilon '\right)}}\;$
$k'=$ phase constant in the units ofradians/meter
$k''=$ attenuation constant in the units ofnepers/meter
$\omega =$ angular frequency
$x=$ distance traveled in thex direction
$\sigma =$ conductivity inSiemens/meter
$\varepsilon =\varepsilon '-j\varepsilon ''=$ complex permittivity
$\mu =\mu '-j\mu ''=$ complex permeability
$j={\sqrt {-1}}$

Thesign 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, andskin depth have simple relationships to the components of the wavenumber:

\lambda ={\frac {2\pi }{k'}}\qquad v_{p}={\frac {\omega }{k'}}\qquad \delta ={\frac {1}{k''}}

In wave equations

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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. Seewavepacket for discussion of the case when these quantities are not constant.

In general, the angular wavenumberk (i.e. themagnitude of thewave vector) is given by

k={\frac {2\pi }{\lambda }}={\frac {2\pi \nu }{v_{\mathrm {p} }}}={\frac {\omega }{v_{\mathrm {p} }}}

whereν is the frequency of the wave,λ is the wavelength,ω = 2πν is theangular frequency of the wave, andv_p is thephase velocity of the wave. The dependence of the wavenumber on the frequency (or more commonly the frequency on the wavenumber) is known as adispersion relation.

For the special case of anelectromagnetic wave in a vacuum, in which the wave propagates at the speed of light,k is given by:

k={\frac {E}{\hbar c}}={\frac {\omega }{c}}

whereE is theenergy of the wave,ħ is thereduced Planck constant, andc is thespeed of light in a vacuum.

For the special case of amatter wave, for example an electron wave, in the non-relativistic approximation (in the case of afree particle, that is, the particle has no potential energy):

k\equiv {\frac {2\pi }{\lambda }}={\frac {p}{\hbar }}={\frac {\sqrt {2mE}}{\hbar }}

Herep is themomentum of the particle,m is themass of the particle,E is thekinetic energy of the particle, andħ is thereduced Planck constant.

Wavenumber is also used to define thegroup velocity.

In spectroscopy

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Inspectroscopy, "wavenumber" ${\tilde {\nu }}$ (inreciprocal centimeters, cm⁻¹) refers to a temporal frequency (in hertz) divided by thespeed of light in vacuum (usually in centimeters per second, cm⋅s⁻¹):

{\tilde {\nu }}={\frac {\nu }{c}}={\frac {\omega }{2\pi c}}.

The historical reason for using this spectroscopic wavenumber rather than frequency is that it is a convenient unit when studying atomic spectra by counting fringes per cm with aninterferometer : the spectroscopic wavenumber is the reciprocal of the wavelength of light in vacuum:

{\tilde {\nu }}={\frac {1}{\lambda _{\rm {vac}}}},

which remains essentially the same in air, and so the spectroscopic wavenumber is directly related to the angles of light scattered fromdiffraction gratings and the distance between fringes ininterferometers, when those instruments are operated in air or vacuum. Such wavenumbers were first used in the calculations ofJohannes Rydberg in the 1880s. TheRydberg–Ritz combination principle of 1908 was also formulated in terms of wavenumbers. A few years later spectral lines could be understood inquantum theory as differences between energy levels, energy being proportional to wavenumber, or frequency. However, spectroscopic data kept being tabulated in terms of spectroscopic wavenumber rather than frequency or energy.

For example, the spectroscopic wavenumbers of theemission spectrum of atomic hydrogen are given by theRydberg formula:

{\tilde {\nu }}=R\left({\frac {1}{{n_{\text{f}}}^{2}}}-{\frac {1}{{n_{\text{i}}}^{2}}}\right),

whereR is theRydberg constant, andn_i andn_f are theprincipal quantum numbers of the initial and final levels respectively (n_i is greater thann_f for emission).

A spectroscopic wavenumber can be converted intoenergy per photonE byPlanck's relation:

E=hc{\tilde {\nu }}.

It can also be converted into wavelength of light:

\lambda ={\frac {1}{n{\tilde {\nu }}}},

wheren is therefractive index of themedium. Note that the wavelength of light changes as it passes through different media, however, the spectroscopic wavenumber (i.e., frequency) remains constant.

Often spatial frequencies are stated by some authors "in wavenumbers",^[13] incorrectly transferring the name of the quantity to the CGS unit cm⁻¹ itself.^[14]