Lorentz–Lorenz equation

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The Lorentz–Lorenz equation, also known as the Clausius–Mossotti relation and Maxwell's formula[citation needed], relates the refractive index of a substance to its polarizability.

The most general form of the Lorentz–Lorenz equation is

 \frac{n^2 - 1}{n^2 + 2} = \frac{4 \pi}{3} N \alpha,

where n is the refractive index, N is the number of molecules per unit volume, and \alpha is the mean polarizability. This equation is only valid for certain crystal structures.[1][2]

A more specialized form of the Lorentz–Lorenz equation gives the refractive index n of a dilute gas as

  n \approx \sqrt{1 + \frac{3 A p}{R T}}

where A is the molar refractivity, p is the pressure of the gas, R is the universal gas constant, and T is the (absolute) temperature.


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.


  1. ^ Introduction to Solid State Physics/Charles Kittel. - 7th ed. (ISBN 0-471-11181-3) Chapter 13, or 8th ed. (ISBN 0-471-41526-X) p. 464
  2. ^ D. E. Aspnes, Am. J. Phys. 50, 704 (1982)
  • Born, Max, and Wolf, Emil, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (7th ed.), section 2.3.3, Cambridge University Press (1999) ISBN 0-521-64222-1