Functions dependent on a greater number of parameters

[tobiaszowo]

hello :)

let's look at the following functions:

Langevin function L(x)=ctgh(x)-1/x
Brillouin function B(x)=f(x J); when J→∞, then B(x)=L(x)

to which I have the following questions:

- how was the Brillouin function B(x)=f(x,J) developed?
- are there other functions with the properties of the function B - dependent on a greater number of parameters, that would allow to model more objects?

I would appreciate any help, so please do ;) greets!

 

[LagrangeEuler]

Brillouin function describes magnetization of paramagnet.
$M=Ng\mu_B J B_J(x)$
where $N$ is number of atoms per unit volume, $g$ is Lande factor, $\mu_B$ Bohr magneton, $J$ total angular quantum number, $x$ is the ratio of the Zeeman energy of the magnetic moment in the external field to the thermal energy $k_B T$. $x=\frac{g\mu_B\mu_0 H}{k_B T}$.
$$B_J(x)=\frac{2J+1}{2J}\coth (\frac{2J+1}{2J}x)-\frac{1}{2J}\coth (\frac{1}{2J}x)$$
If $J \rightarrow \infty$ then $B_J(x)$ becomes $L(x)=\coth x-\frac{1}{x}$. How when you have $J$ in definition of $x$?