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.

Brillouin function

The Brillouin function is a special function defined by the following equation:

<blockquote style="border: 1px solid black; padding:10px;"> The function is usually applied (see below) in the context where x is a real variable and J is a positive integer or half-integer. In this case, the function varies from -1 to 1, approaching +1 as and -1 as. The function is best known for arising in the calculation of the [magnetization](https://bliptext.com/articles/magnetization) of an ideal [paramagnet](https://bliptext.com/articles/paramagnet). In particular, it describes the dependency of the magnetization M on the applied [magnetic field](https://bliptext.com/articles/magnetic-field) B and the [total angular momentum quantum number](https://bliptext.com/articles/total-angular-momentum-quantum-number) J of the microscopic [magnetic moments](https://bliptext.com/articles/magnetic-moment) of the material. The magnetization is given by: where Note that in the SI system of units B given in Tesla stands for the [magnetic field](https://bliptext.com/articles/magnetic-field), B=\mu_0 H, where H is the auxiliary [magnetic field](https://bliptext.com/articles/magnetic-field) given in A/m and \mu_0 is the [permeability of vacuum](https://bliptext.com/articles/permeability-of-vacuum). !Click "show" to see a derivation of this law: (where g is the [g-factor](https://bliptext.com/articles/g-factor-physics), μB is the [Bohr magneton](https://bliptext.com/articles/bohr-magneton), and x is as defined in the text above). The relative probability of each of these is given by the [Boltzmann factor](https://bliptext.com/articles/boltzmann-factor): where Z (the [partition function](https://bliptext.com/articles/partition-function-statistical-mechanics)) is a normalization constant such that the probabilities sum to unity. Calculating Z, the result is: All told, the [expectation value](https://bliptext.com/articles/expectation-value) of the azimuthal quantum number m is The denominator is a [geometric series](https://bliptext.com/articles/geometric-series) and the numerator is a type of arithmetico–[geometric series](https://bliptext.com/articles/geometric-series), so the series can be explicitly summed. After some algebra, the result turns out to be With N magnetic moments per unit volume, the magnetization density is Takacs proposed the following approximation to the inverse of the Brillouin function: where the constants a and b are defined to be

Langevin function

In the classical limit, the moments can be continuously aligned in the field and J can assume all values. The Brillouin function is then simplified into the Langevin function, named after Paul Langevin:

<blockquote style="border: 1px solid black; padding:10px; width:230px"> For small values of x , the Langevin function can be approximated by a truncation of its [Taylor series](https://bliptext.com/articles/taylor-series): An alternative, better behaved approximation can be derived from the [Lambert's continued fraction](https://bliptext.com/articles/gauss-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](https://bliptext.com/articles/catastrophic-cancellation) for x \approx 0 where. The inverse Langevin function L−1(x) is defined on the open interval (−1, 1). For small values of x, it can be approximated by a truncation of its [Taylor series](https://bliptext.com/articles/taylor-series) and by the [Padé approximant](https://bliptext.com/articles/pad-approximant) Cohen and Jedynak approximations.gif Since this function has no closed form, it is useful to have approximations valid for arbitrary values of x. One popular approximation, valid on the whole range (−1, 1), has been published by A. Cohen: This has a maximum relative error of 4.9% at the vicinity of . Greater accuracy can be achieved by using the formula given by R. Jedynak: 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: The maximal relative error for this approximation is less than 0.28%. More accurate approximation was reported by R. Petrosyan: valid for x ≥ 0 . The maximal relative error for the above formula is less than 0.18%. New approximation given by R. Jedynak, is the best reported approximant at complexity 11: valid for x ≥ 0 . Its maximum relative error is less than 0.076%. Current state-of-the-art diagram of the approximants to the inverse Langevin function presents the figure below. It is valid for the rational/Padé approximants, A recently published paper by R. Jedynak, provides a series of the optimal approximants to the inverse Langevin function. The table below reports the results with correct asymptotic behaviors,. **Comparison of relative errors for the different optimal rational approximations, which were computed with constraints (Appendix 8 Table 1) ** Also recently, an efficient near-machine precision approximant, based on spline interpolations, has been proposed by Benítez and Montáns, 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. # [Langevin function (blue line), compared with \tanh(x/3) (magenta line). | upload.wikimedia.org/wikipedia/commons/0/07/Mplwp///Langevin-function///tanhx3.svg] # [Current state-of-the-art diagram of the approximants to the inverse Langevin function, | upload.wikimedia.org/wikipedia/commons/4/41/Approximants///to///the///inverse///Langevin///function.png]

High-temperature limit

When x \ll 1 i.e. when is small, the expression of the magnetization can be approximated by the Curie's law: where is a constant. One can note that is the effective number of Bohr magnetons.

High-field limit

When x\to\infty, the Brillouin function goes to 1. The magnetization saturates with the magnetic moments completely aligned with the applied field: