Finding magnetic susceptibility of a quantum gas

[bhaubhau]

Homework Statement

H=H_{0} -g\muSB
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-\muN)}

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 \betatimes 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.