Past exam of the condensed matter physics major option course of the University of Oxford / 2019 / 4 / e / Solution

Above the transition, $ϵ=0$ and both site types have the same g-factor $g$. For the given spin-only Zeeman Hamiltonian, adjacent $S=1$ levels have the same energy spacing, so fixed-frequency electron paramagnetic resonance gives a single resonance field $B_0=h\nu /(g{\mu }_B)$, ignoring unresolved additional interactions.

Below the transition, half the elongation axes lie along $x$ and half along $y$. A field along $x$ is parallel to the long bond on one sublattice and perpendicular on the other. The line splits intoThe separation grows on cooling as the distortion grows. Equal site populations give comparable integrated weights when the microwave coupling to the two sites is equivalent; populations alone do not fix all polarization-dependent intensities.

A suitable staggered structural order parameter isFor an $x$-elongated A site and a $y$-elongated B site, $\Psi =ϵ$, with the opposite sign for the domain interchanging the pattern. The electron paramagnetic resonance data determine its magnitude fromFor small splitting this is equivalently proportional to $B_{\perp }-B_{\parallel }$, but the reciprocal-field formula follows directly from the stated g-factors. It does not fix the sign of the structural domain. In a continuous transition $|\Psi |$ tends continuously to zero at $T_{\mathrm{JT}}$ and grows below it, eventually saturating. A mean-field sketch has $|\Psi |\propto (T_{\mathrm{JT}}-T{)}^{1/2}$ close to the transition; continuity alone does not determine that exponent. The last panel of the original orbital figure shows this illustrative behavior.