The Langevin function $\mathcal{L}(x) = \mathrm{coth}(x) - \frac{1}{x}$ is named after the French physicist Paul Langevin (1872-1946). The Wikipedia article on the Brillouin and Langevin functions does not cite a publication by Langevin and other sources I consulted merely indicate that it originates in Langevin's work on paramagnetism.

In which publication by Langevin did this function make its first appearance?


1 Answer 1


Best I could determine after searching for a couple of hours, this function was first introduced in the following publication, a scan of which is made available by the Bibliothèque nationale de France:

P. Langevin, "Magnétisme et théorie des électrons," Annales de chimie et de physique, 8:5 (1905), 70-127.

On page 117 one finds:

Le moment magnétique total des $\mathrm{N}$ molécules est évidemment dirigé parallèlement au champ et est égal à la somme des projections sur cette direction des moments composants. Pour l'unité de volume supposée contenir $\mathrm{N}$ molécules, ce moment résultant représente l'intensité d'aimantation $\mathrm{I}$ :

$$\mathrm{I}=\int{\mathrm{M}\cos \alpha}{\ dn} = \int_{-1}^{+1}{2\pi\mathrm{MK\ }x\ e^{ax}\ {dx}}.$$ Or $$\int_{-1}^{+1}{x\ }{e^{ax}}{\ dx} = 2 \left(\frac{\mathrm{Ch\ } {a}}{a}-\frac{\mathrm{Sh\ }a}{a^{2}}\right),$$ d'où $$\mathrm{I}=\mathrm{MN}\left(\frac{\mathrm{Ch\ }a}{\mathrm{Sh\ }{a}}-\frac{1}{a}\right).$$

Here $\mathrm{Ch}$ represents the hyperbolic cosine and $\mathrm{Sh}$ the hyperbolic sine, thus $\mathrm{I}=\mathrm {MN}\left(\mathrm{coth}(a) - \frac{1}{a}\right) = \mathrm{MN}\mathcal{L}(a).$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.