Brillouin function

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=gJμ_B

M_s: staturation magnetization M_s=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 T_C. Next, we will find out the relation between T_C and γ and replace γ using T_C in the SCE.



To find T_C, 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 T_C 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/M_s and T/T_C 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/M_s=f(T/T_C), in stead of the function B_J(x).