• Images
• Videos
• Documents
• Books
• Articles

Definition

Wavenumber, as used in spectroscopy and most chemistry fields,⁵ is defined as the number of wavelengths per unit distance:

ν ~ = 1 λ , {\displaystyle {\tilde {\nu }}\;=\;{\frac {1}{\lambda }},}

where λ is the wavelength. It is sometimes called the "spectroscopic wavenumber".⁶ It equals the spatial frequency.⁷

In theoretical physics, an angular wave number, defined as the number of radians per unit distance is more often used:⁸

k = 2 π λ = 2 π ν ~ {\displaystyle k\;=\;{\frac {2\pi }{\lambda }}=2\pi {\tilde {\nu }}} .

Units

The SI unit of spectroscopic wavenumber is the reciprocal m, written m−1. However, it is more common, especially in spectroscopy, to give wavenumbers in cgs units i.e., reciprocal centimeters or cm−1, with

1   c m − 1 = 100   m − 1 {\displaystyle 1~\mathrm {cm} ^{-1}=100~\mathrm {m} ^{-1}} .

Occasionally in older references, the unit kayser (after Heinrich Kayser) is used;⁹ it is abbreviated as K or Ky, where 1 K = 1 cm−1.¹⁰

Angular wavenumber may be expressed in the unit radian per meter (rad⋅m−1), or as above, since the radian is dimensionless.

Unit conversions

The frequency of light with wavenumber ν ~ {\displaystyle {\tilde {\nu }}} is

f = c λ = c ν ~ {\displaystyle f={\frac {c}{\lambda }}=c{\tilde {\nu }}} ,

where c {\displaystyle c} is the speed of light. The conversion from spectroscopic wavenumber to frequency is therefore¹¹

1   c m − 1 := 29.979245   G H z . {\displaystyle 1~\mathrm {cm} ^{-1}:=29.979245~\mathrm {GHz} .}

Wavenumber can also be used as unit of energy, since a photon of frequency f {\displaystyle f} has energy h f {\displaystyle hf} , where h {\displaystyle h} is the Planck constant. The energy of a photon with wavenumber ν ~ {\displaystyle {\tilde {\nu }}} is

E = h f = h c ν ~ {\displaystyle E=hf=hc{\tilde {\nu }}} .

The conversion from spectroscopic wavenumber to energy is therefore

1   c m − 1 := 1.986446 × 10 − 23   J = 1.239842 × 10 − 4   e V {\displaystyle 1~\mathrm {cm} ^{-1}:=1.986446\times 10^{-23}~\mathrm {J} =1.239842\times 10^{-4}~\mathrm {eV} }

where energy is expressed either in J or eV.

Complex

A complex-valued wavenumber can be defined for a medium with complex-valued relative permittivity ε r {\displaystyle \varepsilon _{r}} , relative permeability μ r {\displaystyle \mu _{r}} and refraction index n as:¹²

k = k 0 ε r μ r = k 0 n {\displaystyle 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 positive x direction in a linear material is given by¹³: 51 

P = e − j k x {\displaystyle P=e^{-jkx}}

where

• k = k ′ − j k ″ = − ( ω μ ″ + j ω μ ′ ) ( σ + ω ε ″ + j ω ε ′ ) {\displaystyle k=k'-jk''={\sqrt {-\left(\omega \mu ''+j\omega \mu '\right)\left(\sigma +\omega \varepsilon ''+j\omega \varepsilon '\right)}}\;}
• k ′ = {\displaystyle k'=} phase constant in the units of radians/meter
• k ″ = {\displaystyle k''=} attenuation constant in the units of nepers/meter
• ω = {\displaystyle \omega =} angular frequency
• x = {\displaystyle x=} distance traveled in the x direction
• σ = {\displaystyle \sigma =} conductivity in Siemens/meter
• ε = ε ′ − j ε ″ = {\displaystyle \varepsilon =\varepsilon '-j\varepsilon ''=} complex permittivity
• μ = μ ′ − j μ ″ = {\displaystyle \mu =\mu '-j\mu ''=} complex permeability
• j = − 1 {\displaystyle 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 π k ′ v p = ω k ′ δ = 1 k ″ {\displaystyle \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 = 2 π λ = 2 π ν v p = ω v p {\displaystyle 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 vp 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 = E ℏ c = ω c {\displaystyle 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 ≡ 2 π λ = p ℏ = 2 m E ℏ {\displaystyle 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" ν ~ {\displaystyle {\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):

ν ~ = ν c = ω 2 π c . {\displaystyle {\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:

λ v a c = 1 ν ~ , {\displaystyle \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:

ν ~ = R ( 1 n f 2 − 1 n i 2 ) , {\displaystyle {\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 = h c ν ~ . {\displaystyle E=hc{\tilde {\nu }}.}

It can also be converted into wavelength of light:

λ = 1 n ν ~ , {\displaystyle \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",¹⁴ incorrectly transferring the name of the quantity to the CGS unit cm−1 itself.¹⁵

See also

• Angular wavelength
• Spatial frequency
• Refractive index
• Zonal wavenumber

External links

• Media related to Wavenumber at Wikimedia Commons

References

1. ISO 80000-3:2019 Quantities and units – Part 3: Space and time. https://www.iso.org/standard/64974.html ↩

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. 978-3-030-69025-0 ↩

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. 978-3-319-25633-7 ↩

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. 978-1-56080-148-1 ↩

5. Gold, Victor, ed. (2019). The IUPAC Compendium of Chemical Terminology: The Gold Book (4 ed.). Research Triangle Park, NC: International Union of Pure and Applied Chemistry (IUPAC). doi:10.1351/goldbook.w06664. https://goldbook.iupac.org/ ↩

6. ISO 80000-3:2019 Quantities and units – Part 3: Space and time. https://www.iso.org/standard/64974.html ↩

7. Hecht, Eugene (2017). "2.2 Harmonic Waves". Optics (5 ed.). Boston: Pearson Education, Inc. p. 16. ISBN 978-0-13-397722-6. 978-0-13-397722-6 ↩

8. Weisstein, Eric W. "Wavenumber -- from Eric Weisstein's World of Physics". scienceworld.wolfram.com. Archived from the original on 27 June 2019. Retrieved 19 March 2018. https://web.archive.org/web/20190627132558/https://scienceworld.wolfram.com/physics/Wavenumber.html ↩

9. François Cardarelli (1997). Scientific Unit Conversion - A Practical Guide to Metrication. p. 209. ↩

10. 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. /wiki/Bibcode_(identifier) ↩

11. "Wave number". Encyclopædia Britannica. Retrieved 19 April 2015. http://www.britannica.com/EBchecked/topic/637882/wave-number ↩

12. [1], eq.(2.13.3) http://www.ece.rutgers.edu/~orfanidi/ewa/ch02.pdf ↩

13. Harrington, Roger F. (1961), Time-Harmonic Electromagnetic Fields (1st ed.), McGraw-Hill, ISBN 0-07-026745-6 {{citation}}: ISBN / Date incompatibility (help) 0-07-026745-6 ↩

14. 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. https://zenodo.org/record/1259655 ↩

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