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Homework Help: Finding magnetic susceptibility of a quantum gas

  1. Nov 22, 2007 #1
    1. The problem statement, all variables and given/known data
    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.

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




    3. 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.
     
  2. jcsd
  3. Nov 22, 2007 #2
    got it

    Cant stop laighing at myself.
     
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