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

For a fixed open-shell electron configuration in the LS coupling regime, Hund's rules select the free-ion ground term in the following order:

The first two rules come from the Coulomb interaction between electrons within the partially filled subshell, including its exchange interaction. Parallel spins require an antisymmetric spatial wavefunction, reducing close approach and often lowering the repulsive Coulomb interaction energy. Among the allowed high-spin terms, the angular dependence of the Coulomb interaction generally favours the largest $L$; a simple magnetic alignment argument alone does not derive this second ordering rule.

The weaker spin-orbit coupling then splits a selected $L,S$ term. In LS coupling, its term-level energy is proportional toThe effective sign appropriate to less-than-half versus more-than-half filling gives the third rule. These rules assume the electrostatic term separation dominates spin-orbit coupling; strong crystal field splitting or other coupling regimes require further analysis.

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.

Write the fitted inverse molar magnetic susceptibility asThe vertical scale is in $10^6\mathrm{mol}{\mathrm{m}}^{-3}$, not unscaled inverse SI magnetic susceptibility. The source line is approximately $8.8\times 10^6\mathrm{mol}{\mathrm{m}}^{-3}$ at $100\mathrm{K}$ and $19\times 10^6\mathrm{mol}{\mathrm{m}}^{-3}$ at $300\mathrm{K}$. These readings giveTherefore the effective paramagnetic moment and Weiss temperature areEquivalently, ${\mu }_{\mathrm{eff}}\simeq (3.5\text{–}3.6){\mu }_B$. An uncertainty of order $10\mathrm{K}$ in the graphical Weiss temperature is reasonable; the plotted fit does not justify exact integer parameters. Its positive intercept at $T=0$ requires a negative extrapolated temperature-axis intercept. Thus the dominant interactions have the antiferromagnetic sign in the interpretation of part (b).

The given electron configuration has an inert closed $[\mathrm{Xe}]$ core and two $4f$ electrons. The Aufbau principle identifies the occupied subshells; the free-ion term within the open $4f^2$ configuration is then selected by Hund's rules. A closed core contributes $L=S=0$.

Each $f$ electron has orbital angular momentum quantum number $l=3$. Place the two spins parallel, so $S=1$. The Pauli exclusion principle then requires distinct orbital projections. The largest allowed total projection is $M_L=3+2=5$, obtained from $m_l=3$ and $m_l=2$, giving the maximum allowed $L=5$. A larger $M_L=6$ would place two parallel-spin electrons in the same atomic orbital and is forbidden.

The $f$ subshell holds fourteen electrons; $f^2$ is less than half filled. The third of Hund's rules therefore selects $J=L-S=4$. HenceThe term symbol uses $2S+1=3$ and the spectroscopic letter $H$ for $L=5$. The praseodymium ion retains an orbital contribution, so a spin-only estimate would be inappropriate.

With $J=4$, the measured effective paramagnetic moment gives the Landé g-factorFor the free-ion Hund's rules values $S=1$, $L=5$, $J=4$, the stated formula yieldsThusThe source fit is consistent with the unquenched $4f$ orbital magnetic moment of the free praseodymium ion, within graphical accuracy. Small deviations need not indicate a change of electron configuration: crystal field splitting, interactions and the fit range can affect a measured magnetic susceptibility.