\Delta E < 0.17\,\mathrm{meV}$. At MIRA, a vertically focusing monochromator was combined with a flat analyser. No collimation was installed in the incident and scattered beams. Higher order neutrons were removed by means of a cooled beryllium filter. At MIRA, we used the $hk0$ scattering plane and collected data in the $(220)$ Brillouin zone.

Furthermore, we employed the time-of-flight spectrometer (TOF) MERLIN [@MERLIN] at the Rutherford Appleton Laboratory in Harwell, UK. MERLIN uses a multi repetition-rate chopper, where we selected incident energies of $E_i = 24\,\mathrm{meV}$ and $E_i = 49\,\mathrm{meV}$ and used a Fermi chopper frequency of 350 Hz. Data was collected with the sample rotation covering the full range of $90^{\circ}$ around the $(440)$ Bragg reflection and the crystal oriented in the $hhl$ plane. Analysis was performed using the software HORACE [@Horace].

The primary goal of our experiments was to establish the dispersion of the TA1, TA2, and LA phonon modes. Note that in our notation, the elastic constant $C_{44}$ is related to the sound velocity of the twofold degenerate transverse mode TA1 propagating along a $[001]$-direction with a polarisation along $[100]$ or $[010]$. $\left(C_{11}-C_{12}\right)/2$ corresponds to the TA2-mode propagating along $[110]$ with a polarisation along $[1\overline{1}0]$. $\left(C_{11}+C_{12}+2C_{44}\right)/2$ is 