Ultrasound Velocity Measurements in the Geometrically Frustrated Spinel MgCr$_2$O$_4$

Magnesium chromite spinel MgCr$_2$O$_4$ is a geometrically frustrated magnet with the N$\acute{e}$el temperature $T_N\simeq$13 K, and the Weiss temperature $\theta_W = -390$ K. Recent inelastic neutron scattering experiments provided a compelling evidence for the spin molecular ground states in not only the paramagnetic phase but also the antiferromagnetic phase. We performed ultrasound velocity measurements of MgCr$_2$O$_4$ in all the symmetrically independent elastic moduli of $C_{11}$, $(C_{11}-C_{12})/2$, and $C_{44}$. Temperature dependence of all of these elastic moduli exhibits a remarkable softening in the paramagnetic phase. Taking into account the absence of orbital degrees of freedom in Cr$^{3+}$ (3$d^3$) in MgCr$_2$O$_4$, the spin degrees of freedom should play a significant role for the elastic softening. The most probable origin for the elastic softening in the paramagnetic phase is the strong coupling of the acoustic phonons to the molecular spin fluctuations.