Development of the Self-Modulation Instability of a Relativistic Proton Bunch in Plasma

Self-modulation is a beam-plasma instability that is useful to drive large-amplitude wakefields with bunches much longer than the plasma skin depth. We present experimental results showing that, when increasing the ratio between the initial transverse size of the bunch and the plasma skin depth, the instability occurs later along the bunch, or not at all, over a fixed plasma length, because the amplitude of the initial wakefields decreases. We show cases for which self-modulation does not develop and we introduce a simple model discussing the conditions for which it would not occur after any plasma length. Changing bunch size and plasma electron density also changes the growth rate of the instability. We discuss the impact of these results on the design of a particle accelerator based on the self-modulation instability seeded by a relativistic ionization front, such as the future upgrade of the AWAKE experiment.