# Wavenumber

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

In the physical sciences, the wavenumber (also wave number or repetency  ) is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance. In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components of the structure repeat per unit of distance. The SI unit of spatial frequency is cycles per meter. In image-processing applications, spatial frequency is often expressed in units of cycles per millimeter or equivalently line pairs per millimeter. In physics, mathematics, and related fields, a wave is a disturbance of a field in which a physical attribute oscillates repeatedly at each point or propagates from each point to neighboring points, or seems to move through space.

Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.

## Contents

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, and elementary particle physics. For quantum mechanical waves, the wavenumber multiplied by the reduced Planck's constant is the canonical momentum.

In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation. Neutron diffraction or elastic neutron scattering is the application of neutron scattering to the determination of the atomic and/or magnetic structure of a material. A sample to be examined is placed in a beam of thermal or cold neutrons to obtain a diffraction pattern that provides information of the structure of the material. The technique is similar to X-ray diffraction but due to their different scattering properties, neutrons and X-rays provide complementary information: X-Rays are suited for superficial analysis, strong x-rays from synchrotron radiation are suited for shallow depths or thin specimens, while neutrons having high penetration depth are suited for bulk samples. In particle physics, an elementary particle or fundamental particle is a subatomic particle with no sub structure, thus not composed of other particles. Particles currently thought to be elementary include the fundamental fermions, which generally are "matter particles" and "antimatter particles", as well as the fundamental bosons, which generally are "force particles" that mediate interactions among fermions. A particle containing two or more elementary particles is a composite particle.

Wavenumber can be used to specify quantities other than spatial frequency. In optical spectroscopy, it is often used as a unit of temporal frequency assuming a certain speed of light. The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its exact value is 299,792,458 metres per second. It is exact because by international agreement a metre is defined as the length of the path travelled by light in vacuum during a time interval of 1/299792458 second. According to special relativity, c is the maximum speed at which all conventional matter and hence all known forms of information in the universe can travel. Though this speed is most commonly associated with light, it is in fact the speed at which all massless particles and changes of the associated fields travel in vacuum. Such particles and waves travel at c regardless of the motion of the source or the inertial reference frame of the observer. In the special and general theories of relativity, c interrelates space and time, and also appears in the famous equation of mass–energy equivalence E = mc2.

## Definition

Wavenumber, as used in spectroscopy and most chemistry fields, is defined as the number of wavelengths per unit distance, typically centimeters (cm−1): Spectroscopy is the study of the interaction between matter and electromagnetic radiation. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, by a prism. Later the concept was expanded greatly to include any interaction with radiative energy as a function of its wavelength or frequency, predominantly in the electromagnetic spectrum, though matter waves and acoustic waves can also be considered forms of radiative energy; recently, with tremendous difficulty, even gravitational waves have been associated with a spectral signature in the context of LIGO and laser interferometry. Spectroscopic data are often represented by an emission spectrum, a plot of the response of interest as a function of wavelength or frequency. In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

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

where λ is the wavelength. It is sometimes called the "spectroscopic wavenumber".  It equals the spatial frequency.

In theoretical physics, a wave number defined as the number of radians per unit distance, sometimes called "angular wavenumber", is more often used: 

$k\;=\;{\frac {2\pi }{\lambda }}$ When wavenumber is represented by the symbol ν, a frequency is still being represented, albeit indirectly. As described in the spectroscopy section, this is done through the relationship ${\frac {\nu _{s}}{c}}\;=\;{\frac {1}{\lambda }}\;\equiv \;{\tilde {\nu }}$ , where νs is a frequency in hertz. This is done for convenience as frequencies tend to be very large. 

It has dimensions of reciprocal length, so its SI unit is the reciprocal of meters (m1). In spectroscopy it is usual to give wavenumbers in cgs unit (i.e., reciprocal centimeters; cm1); in this context, the wavenumber was formerly called the kayser, after Heinrich Kayser (some older scientific papers used this unit, abbreviated as K, where 1 K = 1 cm−1).  The angular wavenumber may be expressed in radians per meter (rad·m−1), or as above, since the radian is dimensionless.

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.

### Complex

A complex-valued wavenumber can be defined for a medium with complex-valued relative permittivity ε, relative permeability μ and refraction index n as: 

$k=k_{0}{\sqrt {\varepsilon \mu }}=k_{0}n$ where k0 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 decaying evanescent fields.

## In wave equations

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

$k={\frac {2\pi }{\lambda }}={\frac {2\pi \nu }{v_{\mathrm {p} }}}={\frac {\omega }{v_{\mathrm {p} }}}$ where $\nu$ is the frequency of the wave, $\lambda$ is the wavelength, $\omega =2\pi \nu$ is the angular frequency of the wave, and $v_{\mathrm {p} }$ 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.

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:

$k={\frac {E}{\hbar c}}$ where E is the energy of the wave, ħ is the reduced Planck constant, and c is the speed of light in a vacuum.

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):

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

Wavenumber is also used to define the group velocity.

## In spectroscopy

In spectroscopy, "wavenumber" ${\tilde {\nu }}$ often refers to a frequency which has been divided by the speed of light in vacuum:

${\tilde {\nu }}={\frac {\nu }{c}}={\frac {\omega }{2\pi c}}.$ The historical reason for using this spectroscopic wavenumber rather than frequency is that it proved to be convenient in the measurement of atomic spectra: the spectroscopic wavenumber is the reciprocal of the wavelength of light in vacuum:

$\lambda _{\rm {vac}}={\frac {1}{\tilde {\nu }}},$ which remains essentially the same in air, and so the spectroscopic wavenumber is directly related to the angles of light scattered from diffraction gratings and the distance between fringes in interferometers, when those instruments are operated in air or vacuum. Such 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 spectroscopic wavenumber rather than frequency or energy.

For example, the spectroscopic wavenumbers of the emission spectrum of atomic hydrogen are given by the Rydberg formula:

${\tilde {\nu }}=R\left({\frac {1}{{n_{\text{f}}}^{2}}}-{\frac {1}{{n_{\text{i}}}^{2}}}\right),$ 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 spectroscopic wavenumber can be converted into energy per photon E by Planck's relation:

$E=hc{\tilde {\nu }}.$ It can also be converted into wavelength of light:

$\lambda ={\frac {1}{n{\tilde {\nu }}}},$ where n is the refractive index of the medium. Note that the wavelength of light changes as it passes through different media, however, the spectroscopic wavenumber (i.e., frequency) remains constant.

Conventionally, inverse centimeter (cm−1) units are used for ${\tilde {\nu }}$ , so often that such spatial frequencies are stated by some authors "in wavenumbers",  incorrectly transferring the name of the quantity to the CGS unit cm−1 itself. 

A wavenumber in inverse cm can be converted to a frequency in GHz by multiplying by 29.9792458 (the speed of light in centimeters per nanosecond). 

## Related Research Articles

Brightness temperature or radiance temperature is the temperature a black body in thermal equilibrium with its surroundings would have to be to duplicate the observed intensity of a grey body object at a frequency . This concept is used in radio astronomy, planetary science and materials science.

Fourier-transform spectroscopy is a measurement technique whereby spectra are collected based on measurements of the coherence of a radiative source, using time-domain or space-domain measurements of the electromagnetic radiation or other type of radiation. It can be applied to a variety of types of spectroscopy including optical spectroscopy, infrared spectroscopy, nuclear magnetic resonance (NMR) and magnetic resonance spectroscopic imaging (MRSI), mass spectrometry and electron spin resonance spectroscopy. There are several methods for measuring the temporal coherence of the light, including the continuous wave Michelson or Fourier-transform spectrometer and the pulsed Fourier-transform spectrograph.

A wavenumber–frequency diagram is a plot displaying the relationship between the wavenumber and the frequency of certain phenomena. Usually frequencies are placed on the vertical axis, while wavenumbers are placed on the horizontal axis. In optics, a Fabry–Pérot interferometer (FPI) or etalon is an optical cavity made from two parallel reflecting surfaces. Optical waves can pass through the optical cavity only when they are in resonance with it. It is named after Charles Fabry and Alfred Perot, who developed the instrument in 1899. Etalon is from the French étalon, meaning "measuring gauge" or "standard".

In spectroscopy, the Rydberg constant, symbol R for heavy atoms or RH for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. The constant first arose as an empirical fitting parameter in the Rydberg formula for the hydrogen spectral series, but Niels Bohr later showed that its value could be calculated from more fundamental constants via his Bohr model. As of 2018, R and electron spin g-factor are the most accurately measured physical constants. In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. The formula was primarily presented as a generalization of the Balmer series for all atomic electron transitions of hydrogen. It was first empirically stated in 1888 by the Swedish physicist Johannes Rydberg, then theoretically by Niels Bohr in 1913, who used a primitive form of quantum mechanics. The formula directly generalizes the equations used to calculate the wavelengths of the hydrogen spectral series. In physics, the Rayleigh–Jeans Law is an approximation to the spectral radiance of electromagnetic radiation as a function of wavelength from a black body at a given temperature through classical arguments. For wavelength , it is: In thermodynamics and solid state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (heat capacity) in a solid. It treats the vibrations of the atomic lattice (heat) as phonons in a box, in contrast to the Einstein model, which treats the solid as many individual, non-interacting quantum harmonic oscillators. The Debye model correctly predicts the low temperature dependence of the heat capacity, which is proportional to – the Debye T3 law. Just like the Einstein model, it also recovers the Dulong–Petit law at high temperatures. But due to simplifying assumptions, its accuracy suffers at intermediate temperatures.

In radiometry, irradiance is the radiant flux (power) received by a surface per unit area. The SI unit of irradiance is the watt per square metre. The CGS unit erg per square centimetre per second is often used in astronomy. Irradiance is often called intensity because it has the same physical dimensions, but this term is avoided in radiometry where such usage leads to confusion with radiant intensity. Thomson scattering is the elastic scattering of electromagnetic radiation by a free charged particle, as described by classical electromagnetism. It is just the low-energy limit of Compton scattering: the particle's kinetic energy and photon frequency do not change as a result of the scattering. This limit is valid as long as the photon energy is much smaller than the mass energy of the particle: , or equivalently, if the wavelength of the light is much greater than the Compton wavelength of the particle. In physical sciences and electrical engineering, dispersion relations describe the effect of dispersion in a medium on the properties of a wave traveling within that medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. From this relation the phase velocity and group velocity of the wave have convenient expressions which then determine the refractive index of the medium. More general than the geometry-dependent and material-dependent dispersion relations, there are the overarching Kramers–Kronig relations that describe the frequency dependence of wave propagation and attenuation.

In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other except for instantaneous thermalizing collisions. This simple model can be used to describe the classical ideal gas as well as the various quantum ideal gases such as the ideal massive Fermi gas, the ideal massive Bose gas as well as black body radiation which may be treated as a massless Bose gas, in which thermalization is usually assumed to be facilitated by the interaction of the photons with an equilibrated mass.

The four-frequency of a massless particle, such as a photon, is a four-vector defined by

The Rydberg–Ritz combination principle is an empirical generalization proposed by Walther Ritz in 1908 to describe the relationship of the spectral lines for all atoms. The principle states that the spectral lines of any element include frequencies that are either the sum or the difference of the frequencies of two other lines. Lines of the spectra of elements could be predicted from existing lines. Since the frequency of light is proportional to the wavenumber or reciprocal wavelength, the principle can also be expressed in terms of wavenumbers which are the sum or difference of wavenumbers of two other lines.

Free spectral range (FSR) is the spacing in optical frequency or wavelength between two successive reflected or transmitted optical intensity maxima or minima of an interferometer or diffractive optical element.

The Planck–Einstein relation is also referred to as the Einstein relation, Planck's energy–frequency relation, the Planck relation, and the Planck equation. Also the eponym Planck formula belongs on this list, but also often refers to Planck's law instead. These various eponyms are far from standard; they are used only sporadically, neither regularly nor very widely. They refer to a formula integral to quantum mechanics, which states that the energy of a photon, E, known as photon energy, is proportional to its frequency, ν:

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