Thermoelastic analysis of a silicon surface under x-ray free-electron-laser irradiation

We present an analysis of the time evolution of a highly excited silicon substrate after partial absorption of a femtosecond soft x-ray pulse. The detailed time-dependent thermoelastic behavior of the substrate in terms of the displacements u(r,t) is derived for time delays for which the usual local thermodynamic variables, temperature T(r,t) and density n(r,t), become well-defined, namely, a few hundred femtoseconds after x-ray pulse absorption. For practical optical components under present conditions of operation with trains of pulses, we find that in a worst case scenario, already the 50th pulse in the train could be adversely affected by dynamic distortion induced by the preceding pulses [corrected]