Final energy gains (in keV) of the protons plotted vs. ξ_x at their initial positions (see Sect. 2.1.1 for more details). Panel a: for RCS with d = 40 m. Panel b: For RCS with different thickness: d = 1 m, 10 m, 40 m, 80 m, 100 m. The “x+, x+”/“x+, x−” and “x−, x+”/“x−, x−” are using the same markers to keep the picture visually clear. Equations (6) and (7) for different d are not plotted in panel b for the same reason. The other parameters are the same as described in Sect. 2.1.1: B₀ = 0.01T, E = 100 V m⁻¹, ξ_y = 0.1.