Kahn's Molecular Magnetism

  • #1
115
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Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
33
1
[tex]\sum_{k=1}^n k^2=\frac{1}{6}n(n+1)(2n+1)[/tex]
 
  • #3
33
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[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]
 
  • #4
115
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snooper007 said:
[tex]\sum_{k=1}^n k^2=\frac{1}{6}n(n+1)(2n+1)[/tex]
Thanks a lot! Of course, I didn't remember that summation at all. It's been a while. :smile:
 

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