Abstract
The usual formulas for the resistive-wall wake field are derived considering ultrarelativistic beams, traveling at the speed of light. This simplifies the calculation, and it leads to a cancellation between electric and magnetic fields. However, for proton beams below 10 GeV and for many heavy-ion beams, the velocities may significantly differ from the speed of light. In this paper, we compute the longitudinal and transverse wake fields for velocities smaller than $c$ and examine under which conditions nonrelativistic effects become important. We illustrate our results by a few examples.
Highlights
Several accelerators are under construction which aim to produce intense proton or ion beams at energies around 1 GeV, for example, the Spallation
A rare and early example is Ref. [6], but in the ultrarelativistic limit its wake field does not reduce to the conventional form
Some authors reserve the terms wake field and impedance exclusively for ultrarelativistic beams, where the longitudinal monopole and the transverse dipole wake fields have special features, e.g., they are equal to zero ahead of the source by virtue of causality and they do not depend on the transverse position of the test particle nor on the beam energy
Summary
Several accelerators are under construction which aim to produce intense proton or ion beams at energies around 1 GeV, for example, the Spallation. The conventional treatment of the resistive-wall wake field considers an ultrarelativistic beam; see, e.g., Refs. An expression for the longitudinal impedance due to space charge and the resistive wall for a beam of finite transverse size can be found in Ref. Our results differ from the earlier papers in that we (i) derive an explicit nonrelativistic correction of the resistive-wall impedance, (ii) start from the exact solution of Maxwell’s equations inside the wall for a circular geometry, (iii) derive the associated Green-function wake fields,.
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