Numerical Study of Low energy Electron Beam Dynamics in Over-Dense Plasma

The self-modulation of a long proton bunch, which is used for Advanced WAKefield Experiment (AWAKE)\footnote{P. Muggli et al. (AWAKE Collaboration), Plasma Phys. Controlled Fusion 60, 014046 (2018).} at CERN, can be seeded by a sharply rising bunch front, a relativistic ionization front or a short preceding electron bunch. % The electron bunch seeding scheme decouples seed wakefield parameters from those of the proton bunch. % It is therefore essential to determine the evolution of the electron bunch and how it generates wakefields in the 5-10\,MV/m range over meter scale propagation distances necessary for effective seeding. % With a low energy bunch (20\,MeV), electrons lose a large fraction of their energy. % That leads to significant evolution of the bunch longitudinal distribution and thus of the wakefields' phase. % This evolution makes the investigation of transverse matching condition to the linear wakefields complicated. % In this context, we search for a condition for the stable generation of the seed wakefields using Particle-In-Cell code FBPIC\footnote{R. Lehe et al., Comput. Phys. Comm. 203, 66-82 (2016)}. % We will present detailed simulation results and their implication for wakefields' seeding. %