Parametric mapping of the efficiency$\unicode{x2013}$instability relation in plasma-wakefield accelerators

High efficiency is essential for plasma-wakefield accelerators to be a cost-effective alternative in high-power applications, such as a linear collider. However, in a plasma-wakefield accelerator the beam-breakup instability can be seeded by a transverse offset between the driver and trailing bunch. This instability, which rapidly increases the oscillation amplitude of the trailing bunch, grows with higher power-transfer efficiency from the driver to the trailing bunch [V. Lebedev et al., Phys. Rev. Accel. Beams 21, 059901 (2018)]. In this paper, we use particle-in-cell simulations to investigate the efficiency$\unicode{x2013}$instability relation that constrains the driver-to-trailing-bunch power-transfer efficiency in beam-driven plasma accelerators. We test the relation using a grid of simulations across all parameters that affect the beam-breakup instability, assuming a uniform accelerating field (optimal beam loading) and no ion motion. We find that the previously proposed efficiency$\unicode{x2013}$instability relation represents a lower limit on the strength of the instability for a given efficiency. For each normalized wake radius, only a certain accelerating field reaches this lowest value of the transverse instability; deviating from this point can increase the growth rate by several orders of magnitude. Lastly, we highlight how the oscillation-amplitude growth of the trailing bunch can be reduced or damped with an initial uncorrelated energy spread and the presence of ion motion.