View source for Clausius–Mossotti relation

{{Short description|Equation for a material's  dielectric constant given its atomic polarizability}}

In [[electromagnetism]], the '''Clausius–Mossotti relation''', named for [[Ottaviano-Fabrizio Mossotti|O. F. Mossotti]] and [[Rudolf Clausius]], expresses the [[dielectric constant]] (relative [[permittivity]], {{math|''ε''<sub>r</sub>}}) of a material in terms of the atomic [[polarizability]], {{mvar|α}}, of the material's constituent atoms and/or molecules, or a [[homogeneous mixture]] thereof. It is equivalent to the '''Lorentz–Lorenz equation''', which relates the [[refractive index]] (rather than the dielectric constant) of a substance to its polarizability. It may be expressed as:<ref>{{cite journal |last=Rysselberghe|first=P. V.| title=Remarks concerning the Clausius–Mossotti Law|journal=J. Phys. Chem.| date=January 1932|volume=36|issue=4| pages=1152–1155|doi=10.1021/j150334a007}}</ref><ref name=Atkins>{{cite book| title=Atkins' Physical Chemistry|year=2010| publisher=Oxford University Press| isbn=978-0-19-954337-3|pages=622–629| last1=Atkins|first1=Peter| last2=de Paula|first2=Julio|chapter=Chapter 17}}</ref>

<math display="block">\frac{\varepsilon_\mathrm{r} - 1}{\varepsilon_\mathrm{r} + 2} = \frac{N \alpha}{3\varepsilon_0}</math>

where
*<math>\varepsilon_r = \tfrac{\varepsilon}{\varepsilon_0}</math> is the [[dielectric constant]] of the material, which for non-magnetic materials is equal to {{math|''n''{{sup|2}}}}, where {{mvar|n}} is the [[refractive index]];
*{{math|''&epsilon;''{{sub|0}}}} is the [[permittivity of free space]];
*{{mvar|N}} is the [[number density]] of the molecules (number per cubic meter);
*{{mvar|&alpha;}} is the [[Electric susceptibility#Molecular polarizability|molecular polarizability]] in SI-units [C·m<sup>2</sup>/V].

In the case that the material consists of a mixture of two or more species, the right hand side of the above equation would consist of the sum of the molecular polarizability contribution from each species, indexed by {{mvar|i}} in the following form:<ref>{{Cite book|title=Introduction to electromagnetic fields and waves | last1=Corson|first1=Dale R| last2=Lorrain|first2=Paul| date=1962|publisher=W.H. Freeman| location=San Francisco|language=en |oclc=398313 |page=116}}</ref>

<math display="block">\frac{\varepsilon_\mathrm{r} - 1}{\varepsilon_\mathrm{r} + 2}   = \sum_i \frac{N_i \alpha_i}{3\varepsilon_0}</math>

In the [[Centimetre–gram–second system of units|CGS system of units]] the Clausius–Mossotti relation is typically rewritten to show the [[Polarizability#Definition|molecular polarizability ''volume'']] <math>\alpha' = \tfrac{\alpha}{4\pi\varepsilon_0}</math> which has units of volume [m<sup>3</sup>].<ref name=Atkins/> Confusion may arise from the practice of using the shorter name "molecular polarizability" for both <math>\alpha</math> and <math>\alpha'</math> within literature intended for the respective unit system.

The Clausius–Mossotti relation assumes only an induced dipole relevant to its polarizability and is thus inapplicable for substances with a significant permanent [[dipole]]. It is applicable to gases such as {{chem2|[[Nitrogen|N2]], [[CO2]], [[CH4]]}} and {{chem2|[[Hydrogen|H2]]}} at sufficiently low densities and pressures.<ref>{{Cite journal|last=Uhlig|first=H. H.|last2=Keyes|first2=F. G.|date=1933-02-01|title=The Dependence of the Dielectric Constants of Gases on Temperature and Density|url=https://aip.scitation.org/doi/10.1063/1.3247827|journal=The Journal of Chemical Physics|volume=1|issue=2|pages=155–159|doi=10.1063/1.3247827|issn=0021-9606|url-access=subscription}}</ref> For example, the Clausius–Mossotti relation is accurate for N<sub>2</sub> gas up to 1000 atm between 25&nbsp;°C and 125&nbsp;°C.<ref>{{Cite journal|last=Michels|first=A.|last2=Jaspers|first2=A.|last3=Sanders|first3=P.|date=1934-05-01|title=Dielectric constant of nitrogen up to 1000 atms. Between 25&nbsp;°C and 150&nbsp;°C|url=https://www.sciencedirect.com/science/article/pii/S0031891434802509|journal=Physica|language=en|volume=1|issue=7|pages=627–633|doi=10.1016/S0031-8914(34)80250-9|issn=0031-8914|url-access=subscription}}</ref> Moreover, the Clausius–Mossotti relation may be applicable to substances if the applied electric field is at a sufficiently high frequencies such that any permanent dipole modes are inactive.<ref>{{Cite book|last=Böttcher|first=C.J.F.|url=https://linkinghub.elsevier.com/retrieve/pii/C20090155794|title=Theory of Electric Polarization|date=1973|publisher=Elsevier|isbn=978-0-444-41019-1|language=en|doi=10.1016/c2009-0-15579-4}}</ref>

==Lorentz–Lorenz equation==
The '''Lorentz–Lorenz equation''' is similar to the Clausius–Mossotti relation, except that it relates the refractive index (rather than the dielectric constant) of a substance to its [[polarizability]]. The Lorentz–Lorenz equation is named after the Danish mathematician and scientist [[Ludvig Lorenz]], who published it in 1869, and the Dutch physicist [[Hendrik Lorentz]], who discovered it independently in 1878.

The most general form of the Lorentz–Lorenz equation is (in [[Gaussian units|Gaussian-CGS]] units)
:<math> \frac{n^2 - 1}{n^2 + 2} = \frac{4 \pi}{3} N \alpha_\mathrm{m} </math>
where {{mvar|n}} is the refractive index, {{mvar|N}} is the number of molecules per unit volume, and <math>\alpha_\mathrm{m}</math> is the mean polarizability. 
This equation is approximately valid for homogeneous solids as well as liquids and gases.

When the square of the refractive index is <math> n^2 \approx 1 </math>, as it is for many gases, the equation reduces to:
:<math> n^2 - 1 \approx 4 \pi N \alpha_\mathrm{m}</math>
or simply
:<math> n - 1 \approx 2 \pi N \alpha_\mathrm{m}</math>

This applies to gases at ordinary pressures. The refractive index {{mvar|n}} of the gas can then be expressed in terms of the [[molar refractivity]]  {{mvar|A}} as:
:<math>  n \approx \sqrt{1 + \frac{3 A p}{R T}}</math>
where {{mvar|p}} is the pressure of the gas, {{mvar|R}} is the [[universal gas constant]], and {{mvar|T}} is the (absolute) temperature, which together determine the number density {{mvar|N}}.

==References==
{{reflist}}

==Bibliography==
*{{cite book | last=Lakhtakia | first=A | title=Selected papers on linear optical composite materials | publisher=SPIE Optical Engineering Press | location=Bellingham, Wash., USA | year=1996 | isbn=978-0-8194-2152-4 | oclc=34046175 |url=https://spie.org/Publications/Book/228828?SSO=1}}
*{{cite book | title=Theory of Electric Polarization | publisher=Elsevier | year=1973 | isbn=978-0-444-41019-1 | doi=10.1016/c2009-0-15579-4 | edition=2nd|first=C.J.F.|last=Böttcher}}
*{{cite book | last=Clausius | first=R. | title=Die Mechanische Behandlung der Electricität | publisher=Vieweg+Teubner Verlag | location=Wiesbaden | year=1879 | isbn=978-3-663-19891-8 | doi=10.1007/978-3-663-20232-5 }}
* {{cite book|author-link=Max Born|last1=Born|first1=Max|author-link2=Emil Wolf|last2=Wolf|first2=Emil|title=[[Principles of Optics|Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light]]|edition=7th |chapter=section 2.3.3|publisher=Cambridge University Press|location=Cambridge New York|year=1999|isbn=0-521-64222-1|oclc=40200160}}
* Lorenz, Ludvig, "Experimentale og theoretiske Undersogelser over Legemernes Brydningsforhold", Vidensk Slsk. Sckrifter 8,205 (1870) https://www.biodiversitylibrary.org/item/48423#page/5/mode/1up
* {{cite journal | last=Lorenz | first=L. | title=Ueber die Refractionsconstante | journal=Annalen der Physik und Chemie | publisher=Wiley | volume=247 | issue=9 | year=1880 | issn=0003-3804 | doi=10.1002/andp.18802470905 | pages=70–103 | bibcode=1880AnP...247...70L | language=de| url=https://zenodo.org/record/1423774 }}
* {{cite journal | last=Lorentz | first=H. A. | title=Ueber die Anwendung des Satzes vom Virial in der kinetischen Theorie der Gase | journal=Annalen der Physik | publisher=Wiley | volume=248 | issue=1 | year=1881 | issn=0003-3804 | doi=10.1002/andp.18812480110 | pages=127–136 | bibcode=1881AnP...248..127L | url=https://zenodo.org/record/1423778 | language=de}}
* O. F. Mossotti, Discussione analitica sull'influenza che l'azione di un mezzo dielettrico ha sulla distribuzione dell'elettricità alla superficie di più corpi electrici disseminati in esso, Memorie di Mathematica e di Fisica della Società Italiana della Scienza Residente in Modena, vol. 24, p.&nbsp;49-74 (1850).

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[[Category:Electromagnetism]]
[[Category:Electric and magnetic fields in matter]]
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