Brillouin function
People often talk about Brillouin function in the literature and use that to describe the magnetic property as a function of temperature with a parameter J, the total atomic angular momentum. I got confused at first when I looked up the definition of the Brillouin function in my text book, which says:
B_J=\frac{2J+1}{2J}coth(\frac{2J+1}{2J}x)-\frac{1}{2J}coth(\frac{x}{2J})
where x=\frac{\mu B}{k_BT}
Then the question is where is the field B when people show their Brillouin function.
The answer has to do with the context of ferromagnetism. In Weiss mean field theory, the ferromagnetism is explained in terms of the molecular field B m that is proportional to the magnetization of the material M:
B_m=\gamma M
Then the ferromagnetism can be found from the following self consitent eqations (SCE):
\frac{M}{M_s}=B_r(x)
\frac{M}{M_s}=\frac{k_BT}{\gamma M_s\mu}x
The first equation is the thermal dynamic average of the moment, second one describes the relation between x and the magetization. Here is the list of the meaning of the symbols:
M: magnetization
μ: magnetic moments of the sites μ=gJμB
Ms: staturation magnetization Ms=Nμ where N is the density of the magnetic sites.
The solution of the temperatures gives the magnetization at different temperature. The only parameter here is γ, unfortunately a microscopic interaction parameter we don't know. However, we have the knowledge of another important parameter, which is the phase transition temperature TC. Next, we will find out the relation between TC and γ and replace γ using TC in the SCE.
To find TC, we just have to remember when T→Tc, the magnetization M→0, therefore x→0. Hence, we can use the limit of Brillouin function at x→0, which is:
B_r(x)=\frac{2J+1}{3J}x
Therefore, the relation between γ and TC is found as:
\frac{k_BT_C}{\gamma M_s\mu}=\frac{J+1}{3J}
With this relation, one can easily change the SCE.
\frac{M}{M_s}=B_r(x)
\frac{M}{M_s}=\frac{T(J+1)}{2T_CJ}x
Now all the discussion can be based on the relation between M/Ms and T/TC with the only parameter J.
However, the so-called Brillouin function is a transcendental function which can only be found from solving the SCE.
In the end, sometime Brillouin function refers to the solution of the SCE: M/Ms=f(T/TC), in stead of the function BJ(x).
