How does C affect molecular magnetism according to Olivier Kahn's book?

[assyrian_77]

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

$$\chi=C\sum_{M_S=-S}^{+S}\frac{{M_S}^2}{2S+1}$$

The book then states that this leads to

$$\chi=\frac{C}{3}S(S+1)$$

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

EDIT: Of course, $$M_S=-S,-S+1,...,S-1,S$$




PS. $$C=\frac{Ng^2\beta^2}{kT}$$ where $$\beta$$ is the Bohr magneton and $$k$$ is the Boltzmann constant.

 

[snooper007]

$$\sum_{k=1}^n k^2=\frac{1}{6}n(n+1)(2n+1)$$

 

[snooper007]

$$\chi=C\sum\limits_{M_s=-S}^S \frac{M_S^2}{2S+1} =C\times 2\times \frac{1}{6} \frac{S(S+1)(2S+1)}{(2S+1)}=\frac{C}{3}S(S+1)$$

 

[assyrian_77]

$$\sum_{k=1}^n k^2=\frac{1}{6}n(n+1)(2n+1)$$

Thanks a lot! Of course, I didn't remember that summation at all. It's been a while.