Average magnetic moment of atom in magnetic field ##B##

[ergospherical]

from the partition function - am trying to show that $\langle \mu \rangle = \beta^{-1} (\partial \log Z / \partial B)$ where $Z$ is the canonical partition function for one atom, i.e. $Z = \sum_{m=-j}^{j} \mathrm{exp}(\mu_0 \beta B m)$, and $\mu = \mu_0 m$. The average energy:\begin{align*}
\langle E \rangle = - \frac{\partial}{\partial \beta} \log Z
\end{align*}and $\langle E \rangle = -\langle \mu \rangle B$. How do I get the derivative $\partial B/ \partial \beta$ to link the two results? Or is there another way.

 

[vanhees71]

I guess you mean $H_{\text{mag}}=-\vec{\mu}_{\text{mag}} \cdot \vec{B}$ in the Hamiltonian; unfortunately $\mu$ is already reserved for the chemical potential of some charge or particle-number like conserved quantity (or quantities).

Then your formula follows from the Feynman-Hellman theorem as usual.