Strain susceptibilities and elastic-constant anomalies of TmVO$_4$ in magnetic field

Based on the real energy spectrum and wave function of the Tm-ion in the vanadate crystal field, all strain susceptibilities and the $\Delta E$ effect were calculated. A good description of the $C^{\delta}(T)$ and $C^{\gamma}(T)$ experimental elastic constants and their variations in magnetic field in the tetragonal and rhombic phases was obtained. The $B^{\delta}$ and $B^{\gamma}$ magnetoelastic and $K^{\delta}$ and $K^{\gamma}$ pair quadrupole coefficients were determined. The influence of the magnetic field oriented along various symmetrical directions in the crystal on the quadrupole ordering in TmVO$_{4}$ was studied.