  We use particle-in-cell (PIC) simulations to study the effects of variations of the incoming 400 GeV proton bunch parameters on the amplitude and phase of the wakefields resulting from a seeded self-modulation (SSM) process. We find that these effects are largest during the growth of the SSM, i.e. over the first five to six meters of plasma with an electron density of $7 \times10^{14}$ cm$^{-3}$. However, for variations of any single parameter by $\pm$5%, effects after the SSM saturation point are small. In particular, the phase variations correspond to much less than a quarter wakefield period, making deterministic injection of electrons (or positrons) into the accelerating and focusing phase of the wakefields in principle possible. We use the wakefields from the simulations and a simple test electron model to estimate the same effects on the maximum final energies of electrons injected along the plasma, which are found to be below the initial variations of $\pm$5%. This analysis includes the dephasing of the electrons with respect to the wakefields that is expected during the growth of the SSM. Based on a PIC simulation, we also determine the injection position along the bunch and along the plasma leading to the largest energy gain. For the parameters taken here (ratio of peak beam density to plasma density $n_{b0}/n_0 \approx 0.003$), we find that the optimum position along the proton bunch is at $\xi \approx -1.5 \; \sigma_{zb}$, and that the optimal range for injection along the plasma (for a highest final energy of $\sim$1.6 GeV after 10 m) is 5-6 m. 
Influence of proton bunch parameters on a proton-driven plasma wakefield   acceleration experiment