Longitudinal collective echoes in coasting particle beams - Ahmed Al-Khateeb; Oliver Boine-Frankenheim; Rainer W. Hasse; Ingo Hofmann (Physical Review Special Topics. Accelerators and Beams, 2003)
Longitudinal ballistic and collective beam echoes with diffusion effects are investigated theoretically. In the presence of the space-charge impedance, the collective echo amplitude is obtained as a closed form expression. In contrast to the ballistic case, the collective echo amplitude consists of one maximum at time t_{echo}. The echo amplitude grows up and damps down with a rate proportional to the Landau damping rate of space-charge waves. The effect of weak diffusion is found to modify the ballistic and the collective echo amplitudes in the same manner. This effect of diffusion was confirmed using a “noiseless,” grid-based simulation code. As a first application the amount of numerical diffusion in our simulation code was determined using the echo effect.