this experiment. The results are suggestive of a slight decrease in energy of the acoustic phonon at high magnetic fields at intermediate wavevectors. Further measurements to higher fields and also in other orbitally degenerate spinels would be helpful for establishing this effect.

![\[fig:field\_001\] Panel $(a)$ shows the field-dependence of the TA2 phonon at the $q$-position $\left(\overline{0.4}\ 0.4\ 0\right)$. Panels $(b)$–$(d)$ depict selected phonon data. A small softening of $\approx 0.2$ meV is observed at high fields in the tetragonal phase for $T = 40$ K as shown by blue triangles in panel $(a)$ and in panel $(d)$.](fieldscans){width="1\columnwidth"}

 \[sec:disc\] Discussion 
=========================

By measuring the phonon dispersion along the high-symmetry directions $[0\xi0]$ and $[\overline\xi\xi0]$ using neutron scattering, we have demonstrated an anomaly in the TA2 branch along the $\langle\xi\xi0\rangle$-directions associated with the structural and orbital phase transition in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} at $T_{OO}$ (Fig. \[fig:energies\]). An energy softening at intermediate wavevectors mid-way between the $\Gamma$- and the K-point is observed with increasing temperature. This softening leads to a spoon-like dispersion of the TA2 branch.

We first consider the possibility of thermal expansion [@Maradudin62] of the lattice as an explanation for the acoustic phonon anomalies in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{}. A change in the unit cell volume would 