The inversion condition for the X-ray Balmer-α transition in consideration of the modeled time-dependent Lyman-α reabsorption in a rapidly recombining laser-produced plasma

Abstract

The time-dependent condition for population inversion in the X-ray 3 → 2 transition in hydrogen-like ions of a recombining plasma produced by a short laser pulse is studied. The population densities of the energy levels are theoretically obtained in a collisional-radiative plasma model; the reabsorption of the 2 → 1 resonance line is taken into account via the escape probability. A new feature is first that the escape probability is treated as an explicitly time-dependent function, E21(r,t). On the basis of a four-level model of the H-like ions, the rate differential equations governing the time development of the population densities are considered, including time-dependent pumping terms and coefficients. Furthermore, specially modeling the time dependence of E21 (r,t), and assuming the other rate coefficients to be approximately constant with respect to time, via the explicit closed-form solution of the corresponding rate equations, the Balmer-α inversion condition ΔN32(r,t) > 0 is given in terms containing confluent hypergeometric functions and depends on rate coefficients, pumping terms, and explicitly on the time. For the free-electron density Ne of the recombining plasma, this relation means a condition depending on atomic and plasma state parameters and qualitatively changing in the course of time.