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The self-similar solution of one dimensional non-stationary system of hydrodynamic equations together with Poisson equation was obtained for fast electrons and ions of laser plasma. Final energy of accelerated ions was obtained for initial energy, temperature and density of fast electrons. Ion distribution function was found at the different regimes of acceleration.
Alexander A. Andreev andKonstantin Yu. Platonov
"Self-similar regime without quasi-neutral approximation of ion acceleration in expanding plasma", Proc. SPIE 5482, Laser Optics 2003: Superintense Light Fields and Ultrafast Processes, (10 June 2004); https://doi.org/10.1117/12.558779
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Alexander A. Andreev, Konstantin Yu. Platonov, "Self-similar regime without quasi-neutral approximation of ion acceleration in expanding plasma," Proc. SPIE 5482, Laser Optics 2003: Superintense Light Fields and Ultrafast Processes, (10 June 2004); https://doi.org/10.1117/12.558779