Energy gains (in keV) by electrons and protons at the end of simulation vs. their initial positions ξ_x (t = 0). The blue triangles represent the particles starting at x = −1 m and the red dots are the ones beginning at x = 1 m. For electrons: panel a: initial velocity v_x(t = 0) = 10⁴ m s⁻¹, panel b: v_x(t = 0) = 4 × 10⁶ m s⁻¹. For protons: panel c: v_x(t = 0) = 10⁴ m s⁻¹, panel d: v_x(t = 0) = 10⁵ m s⁻¹. Also, there are protons in the transit group coming back to the lower boundary, and protons in the bounced group that cross the RCS in the ξ_x > 0.2 region; these are marked by the green stars and purple plus symbols, respectively. The estimated maximum energy gains from Eqs. (6) and (7) are again plotted in solid blue and orange lines. The injecting ξ_x- coordinates of the selected electrons and protons in Fig. 2 are also marked in panels a and c. The electromagnetic fields are described in Sect. 2.1.1: B₀ = 0.01T, E = 100 Vm s⁻¹, ξ_y = 0.1, d = 1 m.