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Kahn's Molecular Magnetism

  1. Mar 24, 2006 #1
    I am going through Olivier Kahn's book "Molecular Magnetism". I am stuck on something that seems so simple. On page 10, it is stated that

    [tex]\chi=C\sum_{M_S=-S}^{+S}\frac{{M_S}^2}{2S+1}[/tex]

    The book then states that this leads to

    [tex]\chi=\frac{C}{3}S(S+1)[/tex]

    I've tried to figure the steps between but I can't get anywhere. What am I missing here?

    EDIT: Of course, [tex]M_S=-S,-S+1,...,S-1,S[/tex]




    PS. [tex]C=\frac{Ng^2\beta^2}{kT}[/tex] where [tex]\beta[/tex] is the Bohr magneton and [tex]k[/tex] is the Boltzmann constant.
     
  2. jcsd
  3. Mar 24, 2006 #2
    [tex]\sum_{k=1}^n k^2=\frac{1}{6}n(n+1)(2n+1)[/tex]
     
  4. Mar 24, 2006 #3
    [tex]\chi=C\sum\limits_{M_s=-S}^S \frac{M_S^2}{2S+1}
    =C\times 2\times \frac{1}{6} \frac{S(S+1)(2S+1)}{(2S+1)}=\frac{C}{3}S(S+1)[/tex]
     
  5. Mar 24, 2006 #4
    Thanks a lot! Of course, I didn't remember that summation at all. It's been a while. :smile:
     
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