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 techo. 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. © 2003 The American Physical Society.
CITATION STYLE
Al-Khateeb, A., Boine-Frankenheim, O., Hasse, R. W., & Hofmann, I. (2003). Longitudinal collective echoes in coasting particle beams. Physical Review Special Topics - Accelerators and Beams, 6(1), 101–112. https://doi.org/10.1103/PhysRevSTAB.6.014205
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