Lorentz-Lorenz relation: Difference between revisions
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In [[physics]], the '''Lorentz-Lorenz relation''' is an equation between the [[index of refraction]] ''n'' and the [[density]] ρ of a [[dielectricum]] ( | In [[physics]], the '''Lorentz-Lorenz relation''' is an equation between the [[index of refraction]] ''n'' and the [[density]] ρ of a [[dielectricum]] (non-conducting matter), | ||
:<math> | :<math> | ||
\frac{n^2-1}{n^2+2} = K\, \rho, | \frac{n^2-1}{n^2+2} = K\, \rho, | ||
</math> | </math> | ||
where ''K'' | where the proportionality constant ''K'' depends on the molar polarizability of the dielectricum. | ||
The relation is named after the Dutch physicist [[Hendrik Antoon Lorentz]] and the Danish physicist [[Ludvig Valentin Lorenz]]. | The relation is named after the Dutch physicist [[Hendrik Antoon Lorentz]] and the Danish physicist [[Ludvig Valentin Lorenz]]. | ||
For molecular | For a molecular dielectricum consisting of a single kind of non-polar molecules, the proportionality factor ''K'' (m<sup>3</sup>/kg) is to a good approximation, | ||
:<math> | :<math> | ||
K = \frac{P_M}{M} \times 10^3, | K = \frac{P_M}{M} \times 10^3, | ||
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where ''M'' (g/mol) is the the [[molar mass]] and ''P''<sub>''M''</sub> is the ''molar polarizability'' (in [[SI]] units): | where ''M'' (g/mol) is the the [[molar mass]] and ''P''<sub>''M''</sub> is the ''molar polarizability'' (in [[SI]] units): | ||
:<math> | :<math> | ||
P_M = \frac{1}{3} N_\mathrm{A} \alpha | P_M = \frac{1}{3} N_\mathrm{A} \alpha. | ||
</math> | </math> | ||
Here ''N''<sub>A</sub> is [[Avogadro's constant]] and α is the [[polarizability]] (with dimension volume) of one molecule. In this expression for ''P''<sub>''M''</sub> it is assumed that the molecular polarizabilities are additive; if this is not the case, the expression can still be used when α is replaced by an effective polarizability. The factor 1/3 arises from the assumption that a single molecule inside the dielectricum feels a nearly spherical field from the surrounding molecules. | |||
In [[Gaussian units]] (a non-rationalized centimer-gram-second system): | In [[Gaussian units]] (a non-rationalized centimer-gram-second system): | ||
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n \approx \sqrt{\varepsilon_r}. | n \approx \sqrt{\varepsilon_r}. | ||
</math> | </math> | ||
In this relation it is presupposed that the relative [[magnetic constant]] μ<sub>r</sub> equals unity, which is a reasonable assumption for [[diamagnetic]] and [[paramagnetic]] matter, but not for [[ferromagnetic]] materials. | |||
==References== | ==References== | ||
* H. A. Lorentz, ''Über die Beziehung zwischen der Fortpflanzungsgeschwindigkeit des Lichtes und der Körperdichte'' [On the relation between the propagation speed of light and density of a body], Ann. Phys. vol. '''9''', pp. 641-665 (1880). [http://gallica.bnf.fr/ark:/12148/bpt6k15253h/CadresFenetre?O=NUMM-15253&M=chemindefer Online] | * H. A. Lorentz, ''Über die Beziehung zwischen der Fortpflanzungsgeschwindigkeit des Lichtes und der Körperdichte'' [On the relation between the propagation speed of light and density of a body], Ann. Phys. vol. '''9''', pp. 641-665 (1880). [http://gallica.bnf.fr/ark:/12148/bpt6k15253h/CadresFenetre?O=NUMM-15253&M=chemindefer Online] | ||
Revision as of 03:13, 25 November 2008
In physics, the Lorentz-Lorenz relation is an equation between the index of refraction n and the density ρ of a dielectricum (non-conducting matter),
where the proportionality constant K depends on the molar polarizability of the dielectricum.
The relation is named after the Dutch physicist Hendrik Antoon Lorentz and the Danish physicist Ludvig Valentin Lorenz.
For a molecular dielectricum consisting of a single kind of non-polar molecules, the proportionality factor K (m3/kg) is to a good approximation,
where M (g/mol) is the the molar mass and PM is the molar polarizability (in SI units):
Here NA is Avogadro's constant and α is the polarizability (with dimension volume) of one molecule. In this expression for PM it is assumed that the molecular polarizabilities are additive; if this is not the case, the expression can still be used when α is replaced by an effective polarizability. The factor 1/3 arises from the assumption that a single molecule inside the dielectricum feels a nearly spherical field from the surrounding molecules.
In Gaussian units (a non-rationalized centimer-gram-second system):
and the factor 103 is absent from K.
For polar molecules a temperature dependent contribution due to the alignment of dipoles must be added to K.
The Lorentz-Lorenz law follows from the Clausius-Mossotti relation when we use that the index of refraction n is to a very good approximation (for non-conducting materials and long wavelengths) equal to the square root of the static relative permittivity (formerly known as relative dielectric constant) εr,
In this relation it is presupposed that the relative magnetic constant μr equals unity, which is a reasonable assumption for diamagnetic and paramagnetic matter, but not for ferromagnetic materials.
References
- H. A. Lorentz, Über die Beziehung zwischen der Fortpflanzungsgeschwindigkeit des Lichtes und der Körperdichte [On the relation between the propagation speed of light and density of a body], Ann. Phys. vol. 9, pp. 641-665 (1880). Online
- L. Lorenz, Über die Refractionsconstante [About the constant of refraction], Ann. Phys. vol. 11, pp. 70-103 (1880). Online