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 second pulse in the train could be adversely affected by dynamic thermal distortion induced by the preceding pulse.