Brillouin and Langevin functions

Brillouin Function

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

$B_{J}(x)={\frac {2J+1}{2J}}\coth \left({\frac {2J+1}{2J}}x\right)-{\frac {1}{2J}}\coth \left({\frac {1}{2J}}x\right)$

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 $x\to +\infty$ and -1 as $x\to -\infty$.

The function is best known for arising in the calculation of the magnetization of an ideal paramagnet. In particular, it describes the dependency of the magnetization $M$ on the applied magnetic field $B$ and the total angular momentum quantum number J of the microscopic magnetic moments of the material. The magnetization is given by:

$M=Ng\mu _{B}J\cdot B_{J}(x)$

where

• $N$ is the number of atoms per unit volume,
• $g$ the g-factor,
• $\mu _{B}$ the Bohr magneton,
• $x$ is the ratio of the Zeeman energy of the magnetic moment in the external field to the thermal energy $k_{B}T$:

$x={\frac {g\mu _{B}JB}{k_{B}T}}$

• $k_{B}$ is the Boltzmann constant and $T$ the temperature.

Click "show" to see a derivation of this law:

A derivation of this law describing the magnetization of an ideal paramagnet is as follows. 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 $E_{m}=-mg\mu _{B}B=-k_{B}Txm/J$ (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: $P(m)=e^{-E_{m}/(k_{B}T)}/Z=e^{xm/J}/Z$ where Z (the partition function) is a normalization constant such that the probabilities sum to unity. Calculating Z, the result is: $P(m)=e^{xm/J}/\left(\sum _{m'=-J}^{J}e^{xm'/J}\right)$. All told, the expectation value of the azimuthal quantum number m is $\langle m\rangle =(-J)\times P(-J)+\cdots +J\times P(J)=\left(\sum _{m=-J}^{J}me^{xm/J}\right)/\left(\sum _{m=-J}^{J}e^{xm/J}\right)$. The denominator is a geometric series and the numerator is a type of arithmetic-geometric series, so the series can be explicitly summed. After some algebra, the result turns out to be $\langle m\rangle =JB_{J}(x)$ With N magnetic moments per unit volume, the magnetization density is $M=Ng\mu _{B}\langle m\rangle =NgJ\mu _{B}B_{J}(x)$.

Langevin Function

Langevin function (red line), compared with $tanh(x/3)$ (blue line).

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

$L(x)=\coth(x)-{\frac {1}{x}}$

For small values of x, the Langevin function can be approximated by a truncation of its Taylor series:

$L(x)={\tfrac {1}{3}}x-{\tfrac {1}{45}}x^{3}+{\tfrac {2}{945}}x^{5}-{\tfrac {1}{4725}}x^{7}+\dots$

An alternative better behaved approximation can be derived from the Lambert's continued fraction expansion of tanh(x):

$L(x)={\frac {x}{3+{\tfrac {x^{2}}{5+{\tfrac {x^{2}}{7+{\tfrac {x^{2}}{9+\ldots }}}}}}}}$

For small enough x, both approximations are numerically better than a direct evaluation of the actual analytical expression, since the later suffers from Loss of significance.

The inverse Langevin function can also be approximated by a series of the form

$L^{-1}(x)=3x+{\tfrac {9}{5}}x^{3}+{\tfrac {297}{175}}x^{5}+{\tfrac {1539}{875}}x^{7}+\dots$

The inverse Langevin function can also be approximated by a Padé approximant

$L^{-1}(x)=x{\frac {3-x^{2}}{1-x^{2}}}+O(x^{6})$

High Temperature Limit

When $x\ll 1$ i.e. when $\mu _{B}B/k_{B}T$ is small, the expression of the magnetization can be approximated by the Curie's law:

$M=C\cdot {\frac {H}{T}}$

where $C={\frac {Ng^{2}J(J+1)\mu _{B}^{2}}{3k_{B}}}$ is a constant. One can note that $g{\sqrt {J(J+1)}}$ 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:

$M=Ng\mu _{B}J$