Finding magnetic susceptibility of a quantum gas

[bhaubhau]

Homework Statement

H=H$$_{0}$$ -g$$\mu$$SB
H$$_{0}$$ = hamiltonian in absence of field
S=Spin operator in the direction of the fied (say along z-axis)

show that
1) M=1/$$\beta$$ (dLn Z/ dB)

2) $$\chi$$ = $$\beta$$(g $$\mu$$)$$^{2}$$ <(S-<S>)$$^{}2$$>

Dont know why it shows$$\mu$$ in superscript. It isn't meant to be!
That is the chemical potential.

Homework Equations

Canonical partition function for a grand ensemble is
Z=Tr{exp(-$$\beta$$(H-$$\mu$$N)}

The Attempt at a Solution

I know that M=g$$\mu$$<S>

I don't know how to go about differentiating Ln Z w.r.t B

For the second part, I used the given result and differentiated it w.r.t B again.
I get $$\beta$$times some junk! I can't get to the required result. I have a feeling its very poor math on my part. Any leads from here on would be well received.

 

[bhaubhau]

got it

Cant stop laighing at myself.