wavenumber

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In the
physical science Physical science is a branch of natural science that studies non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the "physical sciences". Definition Phy ...
s, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. Waves can be Periodic function, periodic, in which case those quantities ...
, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temporal
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in Hertz (unit), hertz (H ...
, 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 In physics, a wave vector (or wavevector) is a vector (geometric), vector used in describing a wave, with a typical unit being cycle per metre. It has a Euclidean vector, magnitude and direction. Its magnitude is the wavenumber of the wave (invers ...
''. The space of wave vectors is called '' reciprocal space''. Wave numbers and wave vectors play an essential role in
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
and the physics of wave
scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
, such as
X-ray diffraction X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to Diffraction, diffract into many specific directions. By measurin ...
,
neutron diffraction 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 ...
,
electron diffraction Electron diffraction refers to the bending of electron beams around atom Every atom is composed of a atomic nucleus, nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutro ...
, and
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, antiqu ...
physics. For
quantum mechanical Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...
waves, the wavenumber multiplied by the reduced Planck's 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 The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special relativity, special theory of relativity, is ...
.

# Definition

Wavenumber, as used in
spectroscopy Spectroscopy is the field of study that measures and interprets the electromagnetic spectrum, electromagnetic spectra that result from the interaction between Electromagnetism, electromagnetic radiation and matter as a function of the wavelengt ...
and most chemistry fields, is defined as the number of
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase (waves), phase on the wave, such as two adjac ...
s per unit distance, typically centimeters (cm−1): :$\tilde \;=\; \frac,$ where ''λ'' is the wavelength. It is sometimes called the "spectroscopic wavenumber". It equals the spatial frequency. 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). 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: :$k \;=\; \frac$ When wavenumber is represented by the symbol , a
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in Hertz (unit), hertz (H ...
is still being represented, albeit indirectly. As described in the spectroscopy section, this is done through the relationship $\frac \;=\; \frac \;\equiv\; \tilde$, where s is a frequency in
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or Cycle per second, cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, me ...
. This is done for convenience as frequencies tend to be very large. Wavenumber has
dimensions In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of reciprocal length, so its
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
is the reciprocal of meters (m−1). In
spectroscopy Spectroscopy is the field of study that measures and interprets the electromagnetic spectrum, electromagnetic spectra that result from the interaction between Electromagnetism, electromagnetic radiation and matter as a function of the wavelengt ...
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 Heinrich Gustav Johannes Kayser Fellow of the Royal Society, ForMemRS (; 16 March 1853 – 14 October 1940) was a German physicist and spectroscopist. Biography Kayser was born at Bingen am Rhein. Kayser's early work was concerned with the char ...
(some older scientific papers used this unit, abbreviated as ''K'', where 1K = 1cm−1). The angular wavenumber may be expressed in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s per meter (rad⋅m−1), or as above, since the
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
is
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or ...
. For
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
in vacuum, wavenumber is directly proportional to frequency and to
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
energy. Because of this, wavenumbers are used as a convenient
unit of energy Energy is defined via Mechanical work, work, so the SI unit of energy is the same as the unit of work – the joule (J), named in honour of James Prescott Joule and his experiments on the mechanical equivalent of heat. In slightly more fundame ...
in spectroscopy.

## Complex

A complex-valued wavenumber can be defined for a medium with complex-valued relative
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (Epsilon, epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes ...
$\varepsilon_r$, relative permeability $\mu_r$ and
refraction index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
''n'' as: :$k = k_0 \sqrt = k_0 n$ where ''k''0 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.

## Plane waves in linear media

The propagation factor of a sinusoidal plane wave propagating in the x direction in a linear material is given by :$P = e^$ where *$k = k\text{'} - jk\text{'}\text{'} = \sqrt\;$ *$k\text{'} =$ phase constant in the units of
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s/meter *$k\text{'}\text{'} =$ attenuation constant in the units of
neper The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain (electronics), gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the invento ...
s/metre *$\omega =$ frequency in the units of
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s/metre *$x =$ distance traveled in the ''x'' direction *$\sigma =$ conductivity in
Siemens Siemens AG ( ) is a German Multinational corporation, multinational Conglomerate (company), conglomerate corporation and the largest industrial manufacturing company in Europe headquartered in Munich with branch offices abroad. The principal ...
/metre *$\varepsilon = \varepsilon\text{'} - j\varepsilon\text{'}\text{'} =$ complex permittivity *$\mu = \mu\text{'} - j\mu\text{'}\text{'} =$ complex permeability *$j=\sqrt$ 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 In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase (waves), phase on the wave, such as two adjac ...
,
phase velocity The phase velocity of a wave is the rate at which the wave Wave propagation, propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wa ...
, and
skin depth Skin effect is the tendency of an alternating current, alternating electric current (AC) to become distributed within a Conductor (material), conductor such that the current density is largest near the surface of the conductor and decreases exp ...
have simple relationships to the components of the wavenumber: :$\lambda = \frac \qquad v_p = \frac \qquad \delta = \frac 1$

# 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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...
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 In physics, a wave vector (or wavevector) is a vector (geometric), vector used in describing a wave, with a typical unit being cycle per metre. It has a Euclidean vector, magnitude and direction. Its magnitude is the wavenumber of the wave (invers ...
) is given by :$k = \frac = \frac=\frac$ where ''ν'' is the frequency of the wave, ''λ'' is the wavelength, ''ω'' = 2''πν'' is the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of the wave, and ''v''p is the
phase velocity The phase velocity of a wave is the rate at which the wave Wave propagation, propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wa ...
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 In the physical sciences and electrical engineering, dispersion relations describe the effect of #Dispersion, dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency ...
. For the special case of an
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
in a vacuum, in which the wave propagates at the speed of light, ''k'' is given by: :$k = \frac$ where ''E'' is the
energy In physics, energy (from Ancient Greek: wikt:ἐνέργεια#Ancient_Greek, ἐνέργεια, ''enérgeia'', “activity”) is the physical quantity, quantitative physical property, property that is #Energy transfer, transferred to a phy ...
of the wave, ''ħ'' is the
reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalenc ...
, and ''c'' is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special relativity, special theory of relativity, is ...
in a vacuum. For the special case of a
matter wave Matter waves are a central part of the theory of quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatom ...
, 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): :$k \equiv \frac = \frac= \frac$ Here ''p'' is the
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the Multiplication, product of the mass and velocity of an object. It is a Euclidean vector, vector quantity, possessing a magnitude and a dire ...
of the particle, ''m'' is the
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a body. It was traditionally believed to be related to the physical quantity, quantity of matter in a Physical object, physical body, until the discovery of the atom and par ...
of the particle, ''E'' is the
kinetic energy In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical sci ...
of the particle, and ''ħ'' is the
reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalenc ...
. Wavenumber is also used to define the
group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope (waves), envelope'' of the wave—propagates through space. For example, if a stone is thr ...
.

# In spectroscopy

In
spectroscopy Spectroscopy is the field of study that measures and interprets the electromagnetic spectrum, electromagnetic spectra that result from the interaction between Electromagnetism, electromagnetic radiation and matter as a function of the wavelengt ...
, "wavenumber" $\tilde$ refers to a frequency which has been divided by the
speed of light in vacuum The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special relativity, special theory of relativity, is ...
usually in centimeters per second (cm.s−1): : :$\tilde = \frac = \frac.$ 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 an
interferometer Interferometry is a technique which uses the ''interference (wave propagation), interference'' of Superposition principle, superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important inves ...
: the spectroscopic wavenumber is the reciprocal of the wavelength of light in vacuum: :$\lambda_ = \frac,$ which remains essentially the same in air, and so the spectroscopic wavenumber is directly related to the angles of light scattered from
diffraction grating In optics, a diffraction grating is an optical component with a periodic structure that diffraction, diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form ...
s and the distance between fringes in
interferometer Interferometry is a technique which uses the ''interference (wave propagation), interference'' of Superposition principle, superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important inves ...
s, when those instruments are operated in air or vacuum. Such wavenumbers were first used in the calculations of
Johannes Rydberg Johannes (Janne) Robert Rydberg (; 8 November 1854 – 28 December 1919) was a Swedish people, Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to describe the wavelengths of photons (of visible light an ...
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: :$\tilde = R\left\left(\frac - \frac\right\right),$ where ''R'' is the
Rydberg constant In spectroscopy, the Rydberg constant, symbol R_\infty for heavy atoms or R_\text for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectrum, spectra of an atom. The constant ...
, and ''n''i and ''n''f are the
principal quantum number In quantum mechanics, the principal quantum number (symbolized ''n'') is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Its values are natural numbers (from one, 1) making it a discrete variable. ...
s of the initial and final levels respectively (''n''i is greater than ''n''f for emission). A spectroscopic wavenumber can be converted into energy per photon ''E'' by
Planck's relation The Planck relationFrench & Taylor (1978), pp. 24, 55.Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11. (referred to as Planck's energy–frequency relation,Schwinger (2001), p. 203. the Planck relation, Planck equation, and Planck formula, ...
: :$E = hc\tilde.$ It can also be converted into wavelength of light: :$\lambda = \frac,$ where ''n'' is the
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
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 $\tilde$, 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.