Magnetic susceptibility (Ising model)

[garyman]

Hi,

I'm slightly confused about how to prove that:

\chi=\vartheta<M>/\varthetaH

is equal to...

\chi=(<M²>-<M>² )/ T

I've expressed <M> as \sumM_sexp(-E/k_BT) /
\sumexp(-E/k_BT)

and know that E=-J\sumS_iS_j-H\sumS_i

But seem to get lost in the differentiation. I am going aobut this the correct way?

 

[olgranpappy]

Hi,

I'm slightly confused about how to prove that:

\chi=\vartheta<M>/\varthetaH

is equal to...

\chi=(<M²>-<M>² )/ T

I've expressed <M> as \sumM_sexp(-E/k_BT) /
\sumexp(-E/k_BT)

and know that E=-J\sumS_iS_j-H\sumS_i

But seem to get lost in the differentiation. I am going aobut this the correct way?

what is "M_s"?

Also, you can use "\partial", i.e., "\partial" for partial derivatives.

 

[Cyosis]

Firstly M=\sum_i S_i \neq M_s? I can't really envision what M_s would be at all. Secondly I personally find it easiest to solve this problem by first noticing that \langle M \rangle= k_b T \frac{\partial \log Z}{\partial h} then compute \frac{\partial \langle M \rangle}{\partial h}.