A Nature research paper shows that the plasma in a plasma-wakefield beam-driven accelerator relaxes back to its initial state over nanosecond timescales, suggesting that wakefield accelerators could achieve MHz repetition rates
used a similar starting point to model the motion of ions within the initial plasma cavity. However, in the present study the motion of the ions is negligible on the timescale of the plasma-electron frequency. Instead, the total radial impulse,
$$\varDelta {p}_{r}\left=\int {F}_{r}\left{\rm{d}}t=\frac{e{Z}_{i}{Qr}}{4{\rm{\pi }}c{\sigma }_{r}^{2}{\varepsilon }_{0}},$$is the total bunch charge. Assuming that plasma electrons do not significantly alter the collective ion motion, the ions are all ‘focused’ onto the axis in a time $$\varDelta {t}_{{\rm{spike}}}=\frac{r}{{v}_{r}}=\frac{4{\rm{\pi }}c{\varepsilon }_{0}{m}_{{\rm{i}}}{\sigma }_{r}^{2}}{e{Z}_{{\rm{i}}}Q},$$
which represents an approximate upper bound to the timescale of the on-axis ion-peak generation. For this experiment , operating in singly ionized argon ) with an average leading-bunch charge of 590 pC and r.m.s. transverse beam size of 5 ± 1 µm, the resulting formation time for the density spike is estimated to be approximately 0.5 ± 0.2 ns.
In this regime, the focusing and decelerating fields at a given longitudinal slice are linked with the focusing-field strength
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