Wavenumber

Last updated April 25, 2024

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.

Definition

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

where λ is the wavelength. It is sometimes called the "spectroscopic wavenumber".^[1] It equals the spatial frequency.

For example, a wavenumber in inverse centimeters can be converted to a frequency expressed in the unit gigahertz by multiplying by 29.9792458 cm/ns (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\;=\;{\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 ${\textstyle {\frac {\nu _{\text{s}}}{c}}\;=\;{\frac {1}{\lambda }}\;\equiv \;{\tilde {\nu }},}$ where $ν$ _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 1 K = 1 cm⁻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.

Complex

A complex-valued wavenumber can be defined for a medium with complex-valued relative permittivity $\varepsilon _{r}$ , relative permeability $\mu _{r}$ and refraction index n as:^[9]

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

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^[10]^: 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 of radians/meter
$k''=$ attenuation constant in the units of nepers/meter
$\omega =$ angular frequency
$x=$ distance traveled in the x direction
$\sigma =$ conductivity in Siemens/meter
$\varepsilon =\varepsilon '-j\varepsilon ''=$ complex permittivity
$\mu =\mu '-j\mu ''=$ complex permeability
$j={\sqrt {-1}}$

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:

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

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 ν is the frequency of the wave, λ is the wavelength, ω = 2πν is the angular frequency of the wave, and v_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}}={\frac {\omega }{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 constant.

Wavenumber is also used to define the group velocity.

In spectroscopy

In spectroscopy, "wavenumber" ${\tilde {\nu }}$ (in reciprocal centimeters, cm⁻¹) refers to a temporal frequency (in hertz) which has been divided by the speed 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 an interferometer : 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 n_i and n_f are the principal quantum numbers 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:

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.

Often spatial frequencies are stated by some authors "in wavenumbers",^[11] incorrectly transferring the name of the quantity to the CGS unit cm⁻¹ itself.^[12]

Related Research Articles

<span class="mw-page-title-main">Frequency</span> Number of occurrences or cycles per unit time

Frequency, most often measured in hertz, 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 to distinguish it from spatial frequency. Ordinary frequency is related to angular frequency by a factor of 2 $π$ . The period is the interval of time between events, so the period is the reciprocal of the frequency: f = 1/T.

In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings. Wavelength is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. 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.

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the dimensionless change in magnitude or phase per unit length. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

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.

A sine wave, sinusoidal wave, or sinusoid is a periodic wave whose waveform (shape) is the trigonometric sine function. In mechanics, as a linear motion over time, this is simple harmonic motion; as rotation, it corresponds to uniform circular motion. Sine waves occur often in physics, including wind waves, sound waves, and light waves, such as monochromatic radiation. In engineering, signal processing, and mathematics, Fourier analysis decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes.

Rotational spectroscopy is concerned with the measurement of the energies of transitions between quantized rotational states of molecules in the gas phase. The rotational spectrum of polar molecules can be measured in absorption or emission by microwave spectroscopy or by far infrared spectroscopy. The rotational spectra of non-polar molecules cannot be observed by those methods, but can be observed and measured by Raman spectroscopy. Rotational spectroscopy is sometimes referred to as pure rotational spectroscopy to distinguish it from rotational-vibrational spectroscopy where changes in rotational energy occur together with changes in vibrational energy, and also from ro-vibronic spectroscopy where rotational, vibrational and electronic energy changes occur simultaneously.

In physics, a wave vector is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave, and its direction is perpendicular to the wavefront. In isotropic media, this is also the direction of wave propagation.

<span class="mw-page-title-main">Rotational frequency</span> Number of rotations per unit time

Rotational frequency, also known as rotational speed or rate of rotation, is the frequency of rotation of an object around an axis. Its SI unit is the reciprocal seconds (s⁻¹); other common units of measurement include the hertz (Hz), cycles per second (cps), and revolutions per minute (rpm).

<span class="mw-page-title-main">Dispersion relation</span> Relation of wavelength/wavenumber as a function of a waves frequency

In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the dispersion relation, one can calculate the frequency-dependent phase velocity and group velocity of each sinusoidal component of a wave in the medium, as a function of frequency. In addition to the geometry-dependent and material-dependent dispersion relations, the overarching Kramers–Kronig relations describe the frequency-dependence of wave propagation and attenuation.

In optics, an optical medium is material through which light and other electromagnetic waves propagate. It is a form of transmission medium. The permittivity and permeability of the medium define how electromagnetic waves propagate in it.

In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium.

In atomic, molecular, and optical physics, the Einstein coefficients are quantities describing the probability of absorption or emission of a photon by an atom or molecule. The Einstein A coefficients are related to the rate of spontaneous emission of light, and the Einstein B coefficients are related to the absorption and stimulated emission of light. Throughout this article, "light" refers to any electromagnetic radiation, not necessarily in the visible spectrum.

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field $E$ or the magnetic field $B$ , takes the form:

<span class="mw-page-title-main">Spatial frequency</span> Characteristic of any structure that is periodic across a position in space

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.

In radiometry, radiant exitance or radiant emittance is the radiant flux emitted by a surface per unit area, whereas spectral exitance or spectral emittance is the radiant exitance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. This is the emitted component of radiosity. The SI unit of radiant exitance is the watt per square metre, while that of spectral exitance in frequency is the watt per square metre per hertz (W·m⁻²·Hz⁻¹) and that of spectral exitance in wavelength is the watt per square metre per metre (W·m⁻³)—commonly the watt per square metre per nanometre. The CGS unit erg per square centimeter per second is often used in astronomy. Radiant exitance is often called "intensity" in branches of physics other than radiometry, but in radiometry this usage leads to confusion with radiant intensity.

The Rydberg–Ritz combination principle is an empirical rule proposed by Walther Ritz in 1908 to describe the relationship of the spectral lines for all atoms, as a generalization of an earlier rule by Johannes Rydberg for the hydrogen atom and the alkali metals. 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.

Plasma parameters define various characteristics of a plasma, an electrically conductive collection of charged and neutral particles of various species that responds collectively to electromagnetic forces. Such particle systems can be studied statistically, i.e., their behaviour can be described based on a limited number of global parameters instead of tracking each particle separately.

The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient value that is large represents a beam becoming 'attenuated' as it passes through a given medium, while a small value represents that the medium had little effect on loss. The (derived) SI unit of attenuation coefficient is the reciprocal metre (m⁻¹). Extinction coefficient is another term for this quantity, often used in meteorology and climatology. Most commonly, the quantity measures the exponential decay of intensity, that is, the value of downward e-folding distance of the original intensity as the energy of the intensity passes through a unit thickness of material, so that an attenuation coefficient of 1 m⁻¹ means that after passing through 1 metre, the radiation will be reduced by a factor of e, and for material with a coefficient of 2 m⁻¹, it will be reduced twice by e, or e². Other measures may use a different factor than e, such as the decadic attenuation coefficient below. The broad-beam attenuation coefficient counts forward-scattered radiation as transmitted rather than attenuated, and is more applicable to radiation shielding. The mass attenuation coefficient is the attenuation coefficient normalized by the density of the material.

When an electromagnetic wave travels through a medium in which it gets attenuated, it undergoes exponential decay as described by the Beer–Lambert law. However, there are many possible ways to characterize the wave and how quickly it is attenuated. This article describes the mathematical relationships among:

The Planck relation is a fundamental equation in quantum mechanics which states that the energy of a photon, $E$ , known as photon energy, is proportional to its frequency, $ν$ :

References

1 2 ISO 80000-3:2019 Quantities and units – Part 3: Space and time.
↑ Rodrigues, A.; Sardinha, R.A.; Pita, G. (2021). Fundamental Principles of Environmental Physics. Springer International Publishing. p. 73. ISBN 978-3-030-69025-0 . Retrieved 2022-12-04.
↑ Solimini, D. (2016). Understanding Earth Observation: The Electromagnetic Foundation of Remote Sensing. Remote Sensing and Digital Image Processing. Springer International Publishing. p. 679. ISBN 978-3-319-25633-7 . Retrieved 2022-12-04.
↑ Robinson, E.A.; Treitel, S. (2008). Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing. Geophysical references. Society of Exploration Geophysicists. p. 9. ISBN 978-1-56080-148-1 . Retrieved 2022-12-04.
↑ "NIST: Wavenumber Calibration Tables - Description". physics.nist.gov. Retrieved 19 March 2018.
↑ W., Weisstein, Eric. "Wavenumber -- from Eric Weisstein's World of Physics". scienceworld.wolfram.com. Retrieved 19 March 2018.{{cite web}}: CS1 maint: multiple names: authors list (link)
↑ "Wave number". Encyclopædia Britannica . Retrieved 19 April 2015.
↑ Murthy, V. L. R.; Lakshman, S. V. J. (1981). "Electronic absorption spectrum of cobalt antipyrine complex". Solid State Communications. 38 (7): 651–652. Bibcode:1981SSCom..38..651M. doi:10.1016/0038-1098(81)90960-1.
↑ , eq.(2.13.3)
↑ Harrington, Roger F. (1961), Time-Harmonic Electromagnetic Fields (1st ed.), McGraw-Hill, ISBN 0-07-026745-6
↑
See for example,
- Fiechtner, G. (2001). "Absorption and the dimensionless overlap integral for two-photon excitation". Journal of Quantitative Spectroscopy and Radiative Transfer . 68 (5): 543–557. Bibcode:2001JQSRT..68..543F. doi:10.1016/S0022-4073(00)00044-3.
- US 5046846,Ray, James C.&Asari, Logan R.,"Method and apparatus for spectroscopic comparison of compositions",published 1991-09-10
- "Boson Peaks and Glass Formation". Science . 308 (5726): 1221. 2005. doi:10.1126/science.308.5726.1221a. S2CID 220096687.
↑ Hollas, J. Michael (2004). Modern spectroscopy. John Wiley & Sons. p. xxii. ISBN 978-0470844151.

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[iso80000-3-1] 1 2 ISO 80000-3:2019 Quantities and units – Part 3: Space and time.

[Rodrigues_Sardinha_Pita_2021_p._73-2] Rodrigues, A.; Sardinha, R.A.; Pita, G. (2021). Fundamental Principles of Environmental Physics. Springer International Publishing. p. 73. ISBN 978-3-030-69025-0 . Retrieved 2022-12-04.

[Solimini_2016_p._679-3] Solimini, D. (2016). Understanding Earth Observation: The Electromagnetic Foundation of Remote Sensing. Remote Sensing and Digital Image Processing. Springer International Publishing. p. 679. ISBN 978-3-319-25633-7 . Retrieved 2022-12-04.

[Robinson_Treitel_2008_p._9-4] Robinson, E.A.; Treitel, S. (2008). Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing. Geophysical references. Society of Exploration Geophysicists. p. 9. ISBN 978-1-56080-148-1 . Retrieved 2022-12-04.

[5] "NIST: Wavenumber Calibration Tables - Description". physics.nist.gov. Retrieved 19 March 2018.

[6] W., Weisstein, Eric. "Wavenumber -- from Eric Weisstein's World of Physics". scienceworld.wolfram.com. Retrieved 19 March 2018.{{cite web}}: CS1 maint: multiple names: authors list (link)

[7] "Wave number". Encyclopædia Britannica . Retrieved 19 April 2015.

[8] Murthy, V. L. R.; Lakshman, S. V. J. (1981). "Electronic absorption spectrum of cobalt antipyrine complex". Solid State Communications. 38 (7): 651–652. Bibcode:1981SSCom..38..651M. doi:10.1016/0038-1098(81)90960-1.

[9] , eq.(2.13.3)

[Harrington_1961-10] Harrington, Roger F. (1961), Time-Harmonic Electromagnetic Fields (1st ed.), McGraw-Hill, ISBN 0-07-026745-6

[11] See for example,
Fiechtner, G. (2001). "Absorption and the dimensionless overlap integral for two-photon excitation". Journal of Quantitative Spectroscopy and Radiative Transfer . 68 (5): 543–557. Bibcode:2001JQSRT..68..543F. doi:10.1016/S0022-4073(00)00044-3.
US 5046846,Ray, James C.&Asari, Logan R.,"Method and apparatus for spectroscopic comparison of compositions",published 1991-09-10
"Boson Peaks and Glass Formation". Science . 308 (5726): 1221. 2005. doi:10.1126/science.308.5726.1221a. S2CID 220096687.

[mwAZg] Fiechtner, G. (2001). "Absorption and the dimensionless overlap integral for two-photon excitation". Journal of Quantitative Spectroscopy and Radiative Transfer . 68 (5): 543–557. Bibcode:2001JQSRT..68..543F. doi:10.1016/S0022-4073(00)00044-3.

[mwAaM] US 5046846,Ray, James C.&Asari, Logan R.,"Method and apparatus for spectroscopic comparison of compositions",published 1991-09-10

[mwAa8] "Boson Peaks and Glass Formation". Science . 308 (5726): 1221. 2005. doi:10.1126/science.308.5726.1221a. S2CID 220096687.

[12] Hollas, J. Michael (2004). Modern spectroscopy. John Wiley & Sons. p. xxii. ISBN 978-0470844151.

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