Abstract
Threshold of the transverse mode coupling instability is calculated in frameworks 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 without limit when the tune shift increases. A resemblance of the results to conventional case of the parabolic potential well is demonstrated and explained.
Highlights
Transverse mode coupling instability (TMCI) has been observed first in PETRA and explained by Kohaupt on the base of the two-particle model [1]
It was confirmed in both papers that the space charge heightens the TMCI threshold until the ratio of the SC tune shift to the synchrotron tune reaches the border in several tens or a hundred units
The dispersion equation is represented in the form of an infinite continued fraction as well as in the form of a recursive relation with an arbitrary number of basis functions involved
Summary
Transverse mode coupling instability of a single bunch with space charge (SC) and a wakefield is considered within the framework of the boxcar model. Eigenfunctions of the bunch without a wake are used as a basis for the solution of the equations with the wakefield included. A dispersion equation for a constant wake is presented in the form of an infinite continued fraction and as the recursive relation with an arbitrary number of basis functions. Realistic wakefields are considered as well including resistive wall, square, and oscillating wakes. It is shown that the transverse mode coupling instability threshold of the negative wake grows in absolute value when the SC tune shift increases. The threshold of the positive wake goes down at increasing the SC tune shift. The explanation is developed by an analysis of the bunch spectrum
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