Abstract
In this paper, we derive the longitudinal and transverse resistive wall impedance for a beam traveling in a round pipe with $vlc$. We argue that from the general formulas for the impedance obtained from the solution of the Maxwell equations one has to subtract the space charge component. After such subtraction, we find that the classical expressions for the impedance derived in the ultrarelativistic limit are also applicable for long bunches even when they are nonrelativistic---in contrast to the conclusion of Zimmermann and Oide [Phys. Rev. Accel. Beams 7, 044201 (2004)]. We also calculate the resistive wall Green-function wake for $vlc$ at small distances and show that the jump at the origin in the longitudinal wake and in the derivative of the transverse wake are smeared out and the wake propagates in front of the source charge.
Highlights
Resistive wall (RW) wakefield plays an important, often dominant, role in modern accelerators, especially in those with a small transverse size of the vacuum chamber
These corrections scale as jzj−7=2 for the longitudinal and jzj−5=2 for the transverse wake, where z is the distance between the source and the test particles
Solving Maxwell’s equations in the cylindrical coordinate system r, θ, z, we find the magnetic and the Published by the American Physical Society
Summary
Resistive wall (RW) wakefield plays an important, often dominant, role in modern accelerators, especially in those with a small transverse size of the vacuum chamber Calculations of this wake go back to the 1960s [1] and, in the relativistic limit, γ → ∞ (γ is the Lorentz factor), the results are well established and covered in textbooks [2,3]. The authors made an expansion of the impedance and derived corrections to the infinite-γ Green-function wake These corrections scale as jzj−7=2 for the longitudinal and jzj−5=2 for the transverse wake, where z is the distance between the source and the test particles. We define the space charge part as the impedance in the same pipe in the limit of perfect conductivity We subtract this impedance from the full expression to obtain the resistive wall part. To convert our expressions for the impedance and wake to the MKS system they should be multiplied by the factor Z0c=4π; in the case of the conductivity, the cgs value is divided by the factor Z0c=4π
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