## 3 Data Analysis
In order to compare dynamics of the collected data as a function of the time delay \(\tau\) for different samples and IR laser energies, the following dimensionless parameters were used:  
\[\frac{\Delta q(\tau)}{q}=\frac{\left\langle q_{on}(\tau)\right\rangle-\left\langle q _{off}\right\rangle}{\left\langle q_{off}\right\rangle} \tag{1}\]  
\[\frac{\Delta w(\tau)}{w}=\frac{\left\langle w_{on}(\tau)\right\rangle-\left\langle w _{off}\right\rangle}{\left\langle w_{off}\right\rangle} \tag{2}\]  
where \(q(\tau)\) is the distance between the opposite Bragg peaks, and \(w(\tau)\) is the radial or azimuthal FWHM of the Bragg peak. Subscripts \"on\" and \"off\" define measurements with and without the IR laser, respectively. Brackets <... > denote ensemble averaging of the chosen Bragg peak parameter for each time delay \(\tau\) over all XFEL pulses. Time dependencies of the momentum transfer vector \(q(\tau)\), as well as radial \(w_{q}(\tau)\) and azimuthal \(w_{\phi}(\tau)\) broadening of the Bragg peaks for three measured samples, are shown in Figure 7. Some points in this figure are excluded due to the unexpected drops of intensity of the XFEL (such drops of intensity were observed when operation of some clystrons in the accelerator complex was failing). The statistical error of these parameters was determined using the standard approach for the error determination in the multi-parameter equation [28]. As clearly seen in Figure 7, an error of the \(\Delta q(\tau)/q\) values is one order of magnitude lower than for the Bragg peaks broadening parameters. This could be explained by the influence of the XFEL pulse shape on the shape of the Bragg peaks with a relatively stable position of each pulse. Due to this, the Gaussian fit of the Bragg peaks shape was not accurate, while the influence of the XFEL pulse shape on the variation of the center of mass was insignificant. The error values for the silica sample are higher than for the PS due to lower statistics (for the PS crystals 600 shots and for the silica crystals 120 shots were collected with the IR laser). Figure 7: Time dependence of the relative change of the distance between the two opposite Bragg peaks \(\Delta q(\tau)/q\) (**a–c**) and Bragg peaks widths in radial \(\Delta w_{q}(\tau)/w_{q}\) (**d–f**) and azimuthal \(\Delta w_{\phi}(\tau)/w_{\phi}\) (**g–i**) directions for three selected samples. Analysis of the data was performed for the IR pump laser intensities shown in Figure 3 and 11.0 Bragg peaks indicated by a square in Figure 4.  
Figure 7 also clearly shows that variations of all parameters are around zero and in all cases, the signal is lower than the statistical error. To reveal characteristic frequencies excited by the IR laser, we performed the Fourier analysis of the time dependencies of the Bragg peak parameters described above. The corresponding Fourier spectra are shown in Figure 8. Due to the more precise measurements of the momentum transfer vector \(q(\tau)\), the average value of the Fourier components of \(q(\tau)\) is one order of magnitude lower than for the Bragg peaks broadening \(w(\tau)\). There was no significant enhancement of any particular Fourier component, for all Fourier spectra. We determined an average value of the Fourier components and the standard deviation \(\sigma\) from the distribution of these Fourier components. For all Fourier spectra shown in Figure 8, there is no Fourier component higher than \(3\sigma\) above the average Fourier component value, which indicates that the periodic signal could not be reliably detected in these conditions.