A method for estimating the temperature in high energy density free electron laser experiments

Present and forthcoming free electron laser (FEL) large scale facilities deliver high fluence ultrafast soft and hard x-ray pulses able to create and probe warm dense matter (WDM). Proper diagnostic for basic physical quantities, like temperature and density, is necessary, but the short lifetime of the WDM state (few ps) makes their measurements a challenging task. In this work we propose a method to estimate the WDM temperature using the experimental information from a slow temperature pyrometric probe exploiting the properties of the heat diffusion equation. Numerical simulations show that for typical thin foil samples, a temperature measurement with 1-10 $\mu$s temporal resolution at the distance of about 300-500 $\mu m$ from the beam center contains sufficient information to retrieve the initial spatial temperature distribution with sufficient accuracy providing information on the temperature reached in the WDM regime. The inversion of the experimental information is obtained by means of a Bayesian approach exploiting a Metropolis Monte Carlo numerical procedure. The model and calculations presented in this work provide the theoretical background for the development of a device for temperature diagnostics of the TIMEX end-station at the Fermi@Elettra FEL facility.