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The initial beam emittance determines the maximum wakefield amplitude thatcan be reached as a result of beam self-modulation in the plasma. The wakefieldexcited by the fully self-modulated beam decreases linearly with the increaseof the beam emittance. There is a value of initial emittance beyond which theself-modulation does not develop even if the instability is initiated by astrong seed perturbation. The emittance scale at which the wakefield is twicesuppressed with respect to the zero-emittance case (the so called criticalemittance) is determined by inability of the excited wave to confine beamparticles radially and is related to beam and plasma parameters by a simpleformula. The effect of beam emittance can be observed in several discussedself-modulation experiments.

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