Finding magnetic susceptibility of a quantum gas

  • Thread starter bhaubhau
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  • #1
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Homework Statement


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

show that
1) M=1/[tex]\beta[/tex] (dLn Z/ dB)

2) [tex]\chi[/tex] = [tex]\beta[/tex](g [tex]\mu[/tex])[tex]^{2}[/tex] <(S-<S>)[tex]^{}2[/tex]>

Dont know why it shows[tex]\mu[/tex] in superscript. It isnt meant to be!
That is the chemical potential.

Homework Equations


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




The Attempt at a Solution



I know that M=g[tex]\mu[/tex]<S>

I dont 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 [tex]\beta[/tex]times some junk! I cant 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.
 

Answers and Replies

  • #2
6
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got it

Cant stop laighing at myself.
 

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