Past exam of the condensed matter physics major option course of the University of Oxford / 2016 / 3 / b

The tetrahedral crystal field splitting places the two $e$ orbitals below the three $t_2$ orbitals. In the weak-field, high-spin limit the occupancy is $e^2{t}_2^1$, with $S=3/2$. Its lowest cubic term is ${}^4{T}_1$: an orbital triplet, before spin degeneracy is counted. In the hypothetical sufficiently strong-field limit the occupancy is $e^3$, with $S=1/2$ and an orbital doublet ${}^{2}E$. This latter limit requires overcoming pairing/Hund energy; ordinary tetrahedral oxide complexes generally remain high-spin.

Both orbital degeneracies permit a Jahn-Teller effect, so a regular tetrahedron is not a stable static equilibrium in the elementary adiabatic treatment. The sign of a particular distortion requires an electron–lattice coupling model, not just the degeneracy count. To make that dependence explicit, let $Q>0$ stretch $z$ while contracting $x,y$. In a leading point-charge quadrupole model write the shiftsIn the high-spin configuration the two $e$ shifts cancel. For $Q>0$ the remaining electron selects $d_{xy}$, gaining $-2gQ$; for $Q<0$ it selects the $d_{xz},d_{yz}$ doublet, gaining $gQ$. Adding $K{Q}^2/2$ gives a tetragonally elongated minimum at $Q=2g/K$, with energy gain $2g^2/K$, versus only $g^2/(2K)$ along the compressed branch. The elongated branch has an orbital singlet. This is the usual elementary elongation argument; quantitative coupling includes higher crystal-field multipoles and configuration mixing.

For the strong-field $e^3$ configuration the lower $e$ orbital is doubly occupied. The shift of the three electrons is $-2g|Q|$, so the same model has both an elongated and a compressed minimum, $Q=\pm 2g/K$, with equal stabilization. Elongation favours double occupation of $d_{x^2-y^2}$; compression favours double occupation of $d_{z^2}$. This tetrahedral doublet distortion-sign ambiguity prevents a unique choice from the supplied data. For example, a small symmetry-allowed cubic elastic invariant $A{Q}_{\theta }(Q_{\theta }^2-3Q_{ϵ}^2)$ selects opposite tetragonal signs when $A$ changes sign, while leaving the regular-site electronic configuration unchanged. Higher couplings or host constraints are needed to choose. Dynamic motion between equivalent minima may also restore an average regular shape without stabilizing the undistorted adiabatic configuration.

Once a static axial distortion removes the orbital degeneracy, second-order spin-orbit coupling produces a time-reversal-even quadratic spin Hamiltonian operator. The residual tetragonal symmetry requires equal $x,y$ coefficients. Dropping a multiple of $S(S+1)$ leaves $H_{\mathrm{an}}=D{S}_z^2$ for the weak-field $S=3/2$ state. The sign and magnitude of $D$ depend on excited-state splittings. For the strong-field $S=1/2$ Kramers doublet, $S_z^2=1/4$ is constant: this term cannot cause zero-field splitting, although the Zeeman g-factor can be anisotropic.