Effect of beam emittance on self-modulation of long beams in plasma wakefield accelerators

The initial beam emittance determines the maximum wakefield amplitude that can be reached as a result of beam self-modulation in the plasma. The wakefield excited by the fully self-modulated beam decreases linearly with the increase of the beam emittance. There is a value of initial emittance beyond which the self-modulation does not develop even if the instability is initiated by a strong seed perturbation. The emittance scale at which the wakefield is twice suppressed with respect to the zero-emittance case (the so called critical emittance) is determined by inability of the excited wave to confine beam particles radially and is related to beam and plasma parameters by a simple formula. The effect of beam emittance can be observed in several discussed self-modulation experiments.