Brillouin and Langevin functions
The Brillouin and Langevin functions are a pair of special functions that appear when studying an idealized paramagnetic material in statistical mechanics.
These functions are named after French physicists Paul Langevin and Léon Brillouin who contributed to the microscopic understanding of magnetic properties of matter.
The Langevin function is derived using statistical mechanics, and describes how magnetic dipoles are alignment by an applied field.
[1] The Brillouin function was developed later to give an explanation that considers quantum physics.
[3] The Brillouin function[4][5][2][6] arises when studying magnetization of an ideal paramagnet.
Considering the microscopic magnetic moments of the material.
(where g is the g-factor, μB is the Bohr magneton, and x is as defined in the text above).
The magnetization saturates with the magnetic moments completely aligned with the applied field: For low fields the curve appears almost linear, and could be replaced by a linear slope as in Curie's law of paramagnetism.
is small) the expression of the magnetization can be approximated by: and equivalent to Curie's law with the constant given by Using
[7] Ferromagnetic materials still has a spontaneous magnetization at low fields (below the Curie-temperature), and the susceptibility must then instead be explained by Curie–Weiss law.
[8][9][10] Then written as This could be linked to Ising's model, for a case with two possible spins: either up or down.
) was named after Paul Langevin who published two papers with this function in 1905[12][13] to describe paramagnetism by statistical mechanics.
[3] When Langevin published the theory paramagnetism in 1905[12][13] it was before the adoption of quantum physics.
Meaning that Langevin only used concepts of classical physics.
[17] Niels Bohr showed in his thesis that classical statistical mechanics can not be used to explain paramagnetism, and that quantum theory has to be used.
The magnetic moment would later be explained in quantum theory by the Bohr magneton (
This could be expressed as by J. H. Van Vleck: "When Langevin assumed that the magnetic moment of the atom or molecule had a fixed value
This makes the Langevin function to be in the borderland between classical statisitcal mechanics and quantum theory (as either semi-classical or semi-quantum).
(insead of the magntic equivalents), that is For small values of x, the Langevin function can be approximated by a truncation of its Taylor series: The first term of this series expansion is equivalent to Curie's law,[1] when writing it as An alternative, better behaved approximation can be derived from the Lambert's continued fraction expansion of tanh(x): For small enough x, both approximations are numerically better than a direct evaluation of the actual analytical expression, since the latter suffers from catastrophic cancellation for
The inverse Langevin function (L−1(x)) is without an explicit analytical form, but there exist several approximations.
[21] The inverse Langevin function L−1(x) is defined on the open interval (−1, 1).
One popular approximation, valid on the whole range (−1, 1), has been published by A. Cohen:[23] This has a maximum relative error of 4.9% at the vicinity of x = ±0.8.
Greater accuracy can be achieved by using the formula given by R. Jedynak:[24] valid for x ≥ 0.
The maximal relative error for this approximation is 1.5% at the vicinity of x = 0.85.
Even greater accuracy can be achieved by using the formula given by M. Kröger:[25] The maximal relative error for this approximation is less than 0.28%.
[27] Current state-of-the-art diagram of the approximants to the inverse Langevin function presents the figure below.
The table below reports the results with correct asymptotic behaviors,.
[25][27][28] Comparison of relative errors for the different optimal rational approximations, which were computed with constraints (Appendix 8 Table 1)[28] Also recently, an efficient near-machine precision approximant, based on spline interpolations, has been proposed by Benítez and Montáns,[29] where Matlab code is also given to generate the spline-based approximant and to compare many of the previously proposed approximants in all the function domain.
Approximations could also be used to express the inverse Brillouin function (
Takacs[30] proposed the following approximation to the inverse of the Brillouin function: where the constants
