Mathematical function, used to describe magnetization
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[1][2] is a special function defined by the following equation:
The function is usually applied (see below) in the context where is a real variable and is a positive integer or half-integer. In this case, the function varies from -1 to 1, approaching +1 as and -1 as .
Note that in the SI system of units given in Tesla stands for the
magnetic field, , where is the auxiliary magnetic field given in A/m and is the
permeability of vacuum.
Click "show" to see a derivation of this law:
A derivation of this law describing the magnetization of an ideal paramagnet is as follows.[1] Let z be the direction of the magnetic field. The z-component of the angular momentum of each magnetic moment (a.k.a. the
azimuthal quantum number) can take on one of the 2J+1 possible values -J,-J+1,...,+J. Each of these has a different energy, due to the external field B: The energy associated with quantum number m is
(where g is the
g-factor, μB is the
Bohr magneton, and x is as defined in the text above). The relative probability of each of these is given by the
Boltzmann factor:
where Z (the
partition function) is a normalization constant such that the probabilities sum to unity. Calculating Z, the result is:
.
All told, the
expectation value of the azimuthal quantum number m is
.
The denominator is a
geometric series and the numerator is a type of
arithmetico–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[3] proposed the following approximation to the inverse of the Brillouin function:
where the constants and are defined to be
Langevin function
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:
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 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[4]
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:[5]
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:[6]
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:[7]
The maximal relative error for this approximation is less than 0.28%. More accurate approximation was reported by R. Petrosyan:[8]
valid for x ≥ 0. The maximal relative error for the above formula is less than 0.18%.[8]
New approximation given by R. Jedynak,[9] is the best reported approximant at complexity 11:
valid for x ≥ 0. Its maximum relative error is less than 0.076%.[9]
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,[7][9]
A recently published paper by R. Jedynak,[10] provides a series of the optimal approximants to the inverse Langevin function. The table below reports the results with correct asymptotic behaviors,.[7][9][10]
Comparison of relative errors for the different optimal rational approximations, which were computed with constraints (Appendix 8 Table 1)[10]
Complexity
Optimal approximation
Maximum relative error [%]
3
13
4
0.95
5
0.56
6
0.16
7
0.082
Also recently, an efficient near-machine precision approximant, based on spline interpolations, has been proposed by Benítez and Montáns,[11] 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.
High-temperature limit
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.
High-field limit
When , the Brillouin function goes to 1. The magnetization saturates with the magnetic moments completely aligned with the applied field:
^Takacs, Jeno (2016). "Approximations for Brillouin and its reverse function". COMPEL - the International Journal for Computation and Mathematics in Electrical and Electronic Engineering. 35 (6): 2095.
doi:
10.1108/COMPEL-06-2016-0278.