## 2 Materials and Methods
#### Radiative Transfer Simulations  
Simulations were run with the one-dimensional radiative transfer code Cretin [15] to study the uniformity of the plasma conditions. These simulations served a dual purpose: firstly to identify any possible gradients and secondly to produce reliable synthetic spectra that could be directly compared with the experimental data. The one dimensional capability of the code was necessary to study the change in the plasma conditions as a function of the depth along the targets. These were modelled as planar slabs, with the number of computational nodes across them chosen so that convergence in the kinetics calculations was obtained. The radiation field from the X-ray laser pulse is assumed to be Gaussian in time with the profile given above. The finite bandwidth of the incident radiation was also taken into account, and also assumed to be a Gaussian distribution in frequency.  
In order to replicate the experimental conditions, where the profile of the XFEL spot caused different regions of the plasma to be illuminated with a different laser intensity, simulations were run over six orders of magnitude of the laser intensity, and the time-integrated spectra were then adequately weighted according to the experimentally measured spatial distribution of the XFEL spot, following the procedure in [PERSON] et al. (2016) [10]. This allows for direct comparison of the simulated spectra with the spectra measured in the experiment.  
Figure 2 shows an example of the simulated time-and-space integrated spectrum for a photon energy of the XFEL of 1400 eV (green line). By separating the bound-bound and bound-free contributions from the spectrum (blue and red lines respectively), it can be seen that, in this case, the emission above 1400 eV is mostly produced by bound-free transitions. The intensity of the free-free emission in this region is several orders of magnitude lower, and therefore is not shown here. Given that the bound-free spectral emission has a characteristic energy dependence given by \(I\propto e^{-E/k_{B}T}\), it is possible to obtain an equivalent temperature by fitting the slope of the spectrum at these photon energies. Although this method can be used to characterise the temperature of isothermal plasmas [20], care must be taken in this case with the interpretation of the equivalent temperature, as this is an integrated measurement and the spectrum does not correspond to a single temperature.  
Figure 2: Bound-bound and bound-free components of the time-and-space integrated spectrum from the Cretin simulations corresponding to an XFEL photon energy of 1400 eV. For energies of the emitted photon above 1400 eV, the spectral emission is dominated by bound-free transitions. The gray area shows the slope of the bound-free emission at the peak temperature \(\pm\)10%, which agrees with that of the integrated spectrum.  
However, in this experiment, the spectral emission of the plasma is gated by the XFEL pulse, and therefore the time-integrated spectrum is heavily weighted by the conditions at the peak of the pulse (this was confirmed by the simulations). Additionally, owing to the large effective area of the laser spot on target, the contribution to the overall spectra from the wings of the spot is negligible compared to that from the high intensity regions. These two characteristics combined cause the time-and-space integrated spectra to be representative of the plasma conditions at the peak intensity of the laser, and therefore the temperature that can be obtained from the bound-free slope is comparable to the peak temperature reached by the plasma. The Cretin simulations show that the difference between the peak temperature and that obtained from fitting the integrated spectra is below 10%. This is illustrated with the gray band in Figure 2, which corresponds to the slope of the bound-free emission at the peak temperature (\(\pm\)10%). It can be seen that the slope of the space- and time-integrated spectrum is representative of this condition.