Get Brillouin Function essential facts below. View Videos or join the Brillouin Function discussion. Add Brillouin Function to your PopFlock.com topic list for future reference or share this resource on social media.
The Brillouin function is a special function defined by the following equation:
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 .
Takacs proposed the following approximation to the inverse of the Brillouin function:
where the constants and are defined to be
Langevin function (blue line), compared with (magenta line).
In the classical limit, the moments can be continuously aligned in the field and can assume all values (). The Brillouin function is then simplified into the Langevin function, named after Paul Langevin:
For small values of x, the Langevin function can be approximated by a truncation of its Taylor series:
Graphs of relative error for x ? [0, 1) for Cohen and Jedynak approximations
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 x = ±0.8. 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,
Current state-of-the-art diagram of the approximants to the inverse Langevin function,
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)
Maximum relative error [%]
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.
When 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.
When , the Brillouin function goes to 1. The magnetization saturates with the magnetic moments completely aligned with the applied field: