Past exam of the condensed matter physics course of the University of Oxford / 2022 / 4 / b

In weak magnetic field and the regime of paramagnetism, localized magnetic moments with total angular momentum $J$ and Landé g-factor $\tilde{g}$ have SI volume magnetic susceptibilitywithin the Curie–Weiss law. Here $n_m$ counts magnetic ions per volume, ${\mu }_0$ is the vacuum permeability, ${\mu }_B$ is the Bohr magneton, and $\Theta$ is the Weiss temperature. The factor $J(J+1)/3$ is the zero-field average of the squared angular projection over $m_J=-J,\dots ,J$.

For one magnetic ion per formula unit, a molar volume $V_m$ contains $N_A$ magnetic ions, where $N_A$ is the Avogadro constant. Consequently $n_m{V}_m=N_A$, and the molar magnetic susceptibility isDefining the dimensionless effective paramagnetic moment $m_{\mathrm{eff}}$ by ${\mu }_{\mathrm{eff}}=m_{\mathrm{eff}}{\mu }_B$, numerical substitution givesThe normalization is per mole of formula units, which here also means per mole of magnetic ions.

The Curie constant measures the squared magnetic moment averaged over all orientations of the $J$ multiplet, givingFor the usual positive Landé g-factor of the ion here, the absolute-value sign can be omitted. This effective paramagnetic moment differs from the maximum projected or saturation magnetic moment, ${\mu }_{\mathrm{sat}}=|\tilde{g}|J{\mu }_B$.

The Weiss temperature $\Theta$ is the extrapolated zero of $1/\chi$ in a Curie–Weiss law fit. It summarizes the leading high-temperature effect of interactions between magnetic moments. For the sign conventionwhere ${\mathbf{J}}_i$ are dimensionless angular momentum operators, mean-field theory givesThus positive net exchange interaction in this convention favours parallel magnetic moments, giving $\Theta >0$ and tendencies towards ferromagnetism; negative net exchange interaction favours antiparallel magnetic moments, giving $\Theta <0$ and antiferromagnetic tendencies. Competing interactions can partially cancel in this sum. The Weiss temperature need not equal the actual ordering temperature, and its sign alone is not a complete determination of the low-temperature ordered structure.