The recombination X-ray laser scheme is considered in the case of recombining plasmas produced by laser pulses which are short to such a degree that the condition for inversion in X-ray transitions of the ions is essentially time-dependent. Using an analytic four-level model, applicable for H-like and Li-like ions, we calculate the inversion conditions for the 4→3, 3→2, and 2→l transitions in H-like ions as functions of time. As we have already shown, a drastic change occurs, especially for the 3→2 transition. To achieve inversion, for short times the free-electron density as a function of the temperature Te, Neinv(Te), must be greater than a critical density Nesh (Te), Neinv(Te) > Nesh(Te), whereas for longer times in the quasi-steady state, Neinv(Te) must be smaller than a critical density Neqv(Te), Neinv(Te) < Nequ(Te). We also consider the evolution of the inversion condition in the intermediate time domain. The time duration of the existence of the short-time 3→2 inversion condition–in the time interval from the initial state with unoccupied lower levels to the quasi-steady state—is in the main determined by t α 1/A21; for example, one has t ≈ 730 fs for Na.
In a plasma produced in particular by optical-field ionization, during the initial cascade populating the excited levels, an inversion may be possible in the 2→1 resonance line transition in the transient regime. The inversion in this transition is of interest especially because of the arising possibility to achieve X-ray gain at shorter wavelengths. In the transient case, an inversion occurs for a small time depending on the temperature Te and the density N.
In addition, we discuss generally the time behavior of the population densities (especially of N3) and show that an optimal population density occurs, on the one hand, for very short times (fs region, depending on Z). On the other hand, however, the optimal (ion-)density-temperature relation is determined via a time of the optimal population density required with regard to the experimental conditions.