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Abstract
In this paper, we present a new two particle model for studying the strong head-tail instabilities in the presence of the space-charge force. It is a simple expansion of the well-known two particle model for strong head-tail instability and is still analytically solvable. No chromaticity effect is included. It leads to a formula for the growth rate as a function of the two dimensionless parameters: the space-charge tune shift parameter (normalized by the synchrotron tune) and the wakefield strength, Upsilon. The three-dimensional contour plot of the growth rate as a function of those two dimensionless parameters reveals stopband structures. Many simulation results generally indicate that a strong head-tail instability can be damped by a weak space-charge force, but the beam becomes unstable again when the space-charge force is further increased. The new two particle model indicates a similar behavior. In weak space-charge regions, additional tune shifts by the space-charge force dissolve the mode coupling. As the space-charge force is increased, they conversely restore the mode coupling, but then a further increase of the space-charge force decouples the modes again. Lastly, this mode coupling/decoupling behavior creates the stopband structures.
Highlights
In low energy high-intensity hadron machines, the spacecharge tune shift is an important parameter in the design and operation of the machines
Many theoretical and simulation studies have been made for a better understanding of their interplay [1,2,3,4,5,6,7,8,9]. They generally indicate that beam instability can be damped when the space-charge force is weak, but the beam becomes unstable again when it becomes too strong
Ðωβ þ ΔωβÞ · Ts 1⁄4 ωβ · Ts Æ π: ð14Þ. It shows that the strong head-tail instability occurs by the mode coupling between the two solutions when the difference of their betatron phase advances over one synchrotron period becomes exactly 2π
Summary
In low energy high-intensity hadron machines, the spacecharge tune shift is an important parameter in the design and operation of the machines. Many theoretical and simulation studies have been made for a better understanding of their interplay [1,2,3,4,5,6,7,8,9] They generally indicate that beam instability can be damped when the space-charge force is weak, but the beam becomes unstable again when it becomes too strong The purpose of the present simplified model is not to explain every effect of space-charge force on beam instabilities with numerical precision, but to suggest a simple picture of some of the essence of the physics of this complicated subject.
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