\right)$. Here, $A$ and $x$ serve as scaling factors for the energy and reduced momentum of the dispersion, respectively.

$S(q,E)$ as given by Eq. (\[eq:Sqw\]) was convoluted with the Eckold-Sobolev resolution function [@Eckold2014] of a TAS spectrometer using Monte-Carlo integration. This novel algorithm was used because it reproduces the resolution function of focusing TAS better than the algorithm of Ref. . Fitting the data was performed using Minuit’s simplex minimiser [@Root2011]. A full description of the software tool *Takin* that was developed by some of the authors and used for the convolution fits can be found in Refs. .

The $q$-dependence of $\Gamma_p$ of the TA2 branch is shown in Fig. \[fig:linewidths\_q\]. While $\Gamma_p$ is essentially $p$ independent at 10 K, it attains a maximum about half-way from the $\Gamma$-point to the K-point at 80 K. The maximum occurs at the $q$-position where we also observe the strongest softening of the TA2 phonon branch (Fig. \[fig:energies\]). No appreciable changes were observed for the other acoustic phonon branches or for the linewidth of the TA2 phonons in the tetragonal phase.

![\[fig:linewidths\_q\] Linewidth $\Gamma_p$ of the TA2 phonon branch. In the tetragonal phase at 10 K (blue triangles), $\Gamma_p$ is independent of 