Vlasov solver for longitudinal dynamics in beam delivery systems for x-ray free electron lasers

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19. Throughout this paper p will always denote an energy, not a momentum.
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27. Here ρ_i and ρ_f refer to the beam density in the absence of the sinusoidal modulation.
28. After completion of this work, we came to the realization that for Gaussian beams this choice may lead to overestimating the space-charge impedance while the expression suggested in Appendix A of [4] r_b=1.7(σ_x+σ_y)/2 appears to be a better fit. We will address this point more at length in a forthcoming paper.
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