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X-ray lasing from diatomic molecules core-excited by the XFEL short intense pulses is studied theoretically. With the help of the solution of the Maxwell-Bloch equations in an open quantum system we show a strong dependence of the emitted light spectra on the propagation length, due to the interplay of various spectral components from different vibrational states of the molecule. In the saturated regime, the amplified x-ray emission spectrum transforms into a narrow line. The spectral width and the duration of the emitted x-ray pulse are determined by the lifetime of the core-ionized state. The amplified x-ray emission frequency can be tuned within the x-ray fluorescence band by manipulating the vibrational wave packet in the ground electronic state with the help of additional strong infra-red radiation. We discuss the effect of molecular alignment on the amplification process. The induced nuclear dynamics in the intermediate and final electronic states are strongly dependent on the intensity of the x-ray laser field.
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