Langevin function
The Langevin function (after the physicist Paul Langevin (1872-1946)) is a mathematical function used to calculate from orientation polarization , polarization , magnetization is used, and resistance.
definition
The Langevin function is defined by
- ,
wherein the cotangent hyperbolic designated.
An application
The best known application is the semi-classical description of a paramagnet in an external magnetic field. For this purpose, the Langevin parameter is introduced:
The individual symbols stand for the following quantities :
- : Magnetic moment of a particle
- : Amount of the applied external magnetic field
- : Boltzmann constant
- : Absolute temperature
For the magnetization of a paramagnet we get:
stands for the amount of substance and for the magnetic moment of the individual spins of the paramagnet. Another quantum mechanical description of paramagnetism is given by the Brillouin function .
Approximations
An approximation of the Langevin function for is
- .
The approximation applies to
- .
Inverse function
Since the Langevin function does not have an inverse function that can be represented in a closed manner, there are different approximations. A common approximation that holds in the interval was published by A. Cohen:
The largest relative error of this approximation is 4.9% um . There are other approximations that have much smaller relative errors.
See also
Individual evidence
- ↑ a b c Siegmund Brandt : Electrodynamics . Springer, Berlin 2005, ISBN 3-540-21458-5 , pp. 293 .
- ^ A. Cohen: A Padé approximant to the inverse Langevin function . In: Rheologica Acta . 30, No. 3, 1991, pp. 270-273. doi : 10.1007 / BF00366640 .
- ^ R. Jedynak: New facts concerning the approximation of the inverse Langevin function . In: Journal of Non-Newtonian Fluid Mechanics . 249, 2017, pp. 8-25. doi : 10.1016 / j.jnnfm.2017.09.003 .
- ↑ M. Kröger : Simple, admissible, and accurate approximants of the inverse Langevin and Brillouin functions, relevant for strong polymer deformations and flows . In: Journal of Non-Newtonian Fluid Mechanics . 223, 2015, pp. 77-87. doi : 10.1016 / j.jnnfm.2015.05.007 .