Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 3

Hund's rules maximizes total spin within the partially filled shell: parallel spins require an antisymmetric spatial wavefunction, reducing Coulomb overlap through exchange. Among terms with that spin, the second rule selects the largest total orbital angular momentum. The third rule, from spin-orbit coupling within the selected term, gives $J=|L-S|$ below half filling and $J=L+S$ above half filling, with the half-filled case treated directly.

For seven $4f$ electrons, put one parallel-spin electron in each $m_l=-3,-2,-1,0,1,2,3$ orbital. Their orbital projections sum to zero and form the half-filled $L=0$ shell; all seven spins add. ThusThere is no ambiguity from applying the below- or above-half-filled formula because $L=0$. The absence of a first-order orbital moment is important for the weak magnetic anisotropy discussed in c. These are isolated-ion term rules; a crystal field or strong covalency can alter other ions' orbital behavior, but the half-filled Eu${}^{2+}$ shell is unusually robust.

Write $\mathbf{M}=M\mathbf{n}$ with $|\mathbf{n}|=1$, and let $P=n_x^2{n}_y^2+n_x^2{n}_z^2+n_y^2{n}_z^2$. Since the nonnegative numbers $n_i^2$ sum to one,The quartic coefficient is $b+cP$. If $c\ge 0$ it is at least $b>0$. If $c<0$, the given condition $2b+c>0$ implies $b+c/3>b/3>0$. For $T>T_c$ the quadratic coefficient $a(T-T_c)$ is also positive. Hence every nonzero $M$ raises the free energy:The stability argument must apply to all orientations, rather than checking just one cubic direction. The stated inequality is sufficient for that purpose, although the sharp quartic stability condition for negative $c$ is the weaker $b+c/3>0$.

For $c>0$, minimize the nonnegative anisotropy invariant $P$. Its minimum is zero, achieved only when one squared component of the unit vector is one and the others vanish. Thus the easy directions areThe six signs and axes are symmetry-related ferromagnetic domains. Allowing the magnetization amplitude to relax does not change the result: its minimum free-energy reduction is proportional to $-1/(b+cP)$, which is most negative for the smallest positive quartic coefficient. A $⟨110⟩$ direction has $P=1/4$ and is not an equally stable easy direction.

For $c<0$, maximize $P$ to lower the quartic coefficient. The identity $P=[1-(n_x^4+n_y^4+n_z^4)]/2$ shows that its maximum $1/3$ occurs for $n_x^2=n_y^2=n_z^2=1/3$. ThereforeThere are eight symmetry-related easy directions. This is a cubic easy-axis anisotropy even though the selected directions are body diagonals; it is not a continuous easy plane. At $c=0$ the quartic free energy instead has a sphere of degenerate orientations to this order.

Along any $⟨111⟩$ direction the quartic term is $(b+c/3)M^4$. Minimizing the amplitude below $T_c$ givesFor the ordered solution,Its free energy is $F_{\min }=F_0-a^2(T_c-T{)}^2/[4(b+c/3)]$. The denominator is positive by the supplied stability assumptions. The square-root onset is the Landau theory exponent; sufficiently close to a three-dimensional critical point, fluctuations can change the actual critical exponent without changing this saddle-point calculation.

Eu${}^{2+}$ has the half-filled $4f^7$ term with $L=0$. First-order orbital magnetism is therefore quenched already at the ion level, making the dominant moment approximately spin-only. Spin-orbit-induced magnetocrystalline anisotropy arises only through weaker admixtures and other small interactions; magnetic dipole interactions can also contribute. This explains why a cubic anisotropy term is allowed by symmetry but numerically weak.

An appropriate insulating local-moment model is a nearly isotropic Heisenberg model,with dominant effective ferromagnetic couplings in the sign convention $J_{ij}>0$. The Eu sublattice in rocksalt EuO is face-centered cubic; nearest- and further-neighbor effective exchanges can be retained as needed. Because the solid is not metallic, an itinerant-electron Stoner model or a generic free-carrier RKKY interaction mechanism is not the natural starting point. Exchange can still arise through virtual hopping and ligand-mediated processes in an insulator; it does not require mobile dc carriers.