Abstract

Transverse mode coupling instability of a bunch with space charge and wake field is considered in frameworks of the boxcar model. Eigenfunctions of the bunch without wake are used as the basis for solution of the equations with the wake field included. Dispersion equation for the bunch eigentunes is obtained in the form of an infinite continued fraction. It is shown that influence of space charge on the instability essentially depends on the wake sign. In particular, threshold of the negative wake increases in absolute value until the space charge tune shift is rather small, and goes to zero at higher space charge. The explanation of this behavior is developed by analysis of the bunch spectrum. A comparison of the results with published articles is represented.

Highlights

  • The transverse mode coupling instability (TMCI) of a bunch with space charge (SC) was considered in papers [1,2,3,4,5,6]

  • The authors have represented similar results concerning the SC impact on the TMCI at a moderate ratio of the SC tune shift to the synchrotron tune ðΔQ=QsÞ. It follows from these papers that the threshold of the instability caused by a negative wake increases when the ratio goes up from zero to several tens, at least

  • More confusing pictures appear at larger values of this ratio like a hundred or over it. It has been suggested in Ref. [2] that the threshold growth ceases above this border becoming to 0 at ΔQ=Qs → ∞. It was asserted in Ref. [3] that negative wake cannot excite the TMCI in this limiting case

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Summary

Balbekov*

Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, Illinois 60510, USA (Received 3 August 2017; published 30 November 2017). The threshold of the transverse mode coupling instability is calculated in framework of the square well model at arbitrary value of space charge tune shift. A new method of calculation is developed beyond the traditional expansion technique. The square, resistive, and exponential wakes are investigated. It is shown that the instability threshold goes up indefinitely when the tune shift increases. A comparison with conventional case of the parabolic potential well is performed

INTRODUCTION
General relations
BALBEKOV
Square potential well
HOLLOW BUNCH WITH A SQUARE WAKE
Solution by an expansion
Solution without expansion
RESISTIVE WALL WAKE
EXPONENTIAL WAKE
CONCLUSIONS

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