Exploring and understanding ultrafast processes at the atomic level is a scientific challenge. Femtosecond X-ray Absorption Spectroscopy (XAS) is an essential experimental probing technic, as it can simultaneously reveal both electronic and atomic structures, and thus unravel their non-equilibrium dynamic interplay which is at the origin of most of the ultrafast mechanisms. However, despite considerable efforts, there is still no femtosecond X-ray source suitable for routine experiments. Here we show that betatron radiation from relativistic laser-plasma interaction combines ideal features for femtosecond XAS. It has been used to investigate the non-equilibrium transition of a copper sample brought at extreme conditions of temperature and pressure by a femtosecond laser pulse. We measured a rise time of the electron temperature below 100 fs. This first experiment demonstrates the great potential of the betatron source and paves the way to a new class of ultrafast experiments.
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