Magnetic susceptibility (Ising model)

[garyman]

Hi,

I'm slightly confused about how to prove that:

$$\chi$$=$$\vartheta$$<M>/$$\vartheta$$H

is equal to...

$$\chi$$=(<M²>-<M>² )/ T

I've expressed <M> as $$\sum$$M_sexp(-E/k_BT) /
$$\sum$$exp(-E/k_BT)

and know that E=-J$$\sum$$S_iS_j-H$$\sum$$S_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>/$$\vartheta$$H

is equal to...

$$\chi$$=(<M²>-<M>² )/ T

I've expressed <M> as $$\sum$$M_sexp(-E/k_BT) /
$$\sum$$exp(-E/k_BT)

and know that E=-J$$\sum$$S_iS_j-H$$\sum$$S_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 $$\frac{\partial \langle M \rangle}{\partial h}$$.