Abstract
For a single hadron bunch in a circular accelerator at zero chromaticity, without multi-turn wakes and without electron clouds and other beams, only one transverse collective instability is possible, the mode-coupling instability, or TMCI. For sufficiently strong space charge (SC), the instability threshold of the wake-driven coherent tune shift normally increases linearly with the SC tune shift, as independently concluded by several authors using different methods. This stability condition has, however, a very strange feature: at strong SC, it is totally insensitive to the number of particles. Thus, were it correct, such a beam with sufficiently strong SC, being stable at some intensity, would remain stable at higher intensity, regardless of how much higher! This paper suggests a resolution of this conundrum: while SC suppresses TMCI, it introduces head-to-tail convective amplifications, which could make the beam even less stable than without SC, even if all the coherent tunes are real, i.e. all the modes are stable in the conventional {\it absolute} meaning of the word. This is done using an effective new method of analysis of the beam's transverse spectrum for arbitrary space charge and wake fields. Two new types of beam instabilities are introduced: the {\it saturating convective instability}, SCI, and the {\it absolute-convective instability}, ACI.
Highlights
Transverse mode coupling instabilities (TMCI) are believed to be one of the main limitations for the intensity of bunched beams
An important question is, how does TMCI depend on the space charge (SC) tune shift ΔQsc? Since the latter is typically high for low- and medium-energy machines, where the SC parameter q 1⁄4 ΔQsc=Qs ≫ 1 is even far from the transition energy, the question is really important
According to Refs. [2,3,4,5,6,7], TMCI intensity threshold increases, with rare exceptions, proportionally to the SC tune shift; SC makes the bunch more stable in this respect. We called this dominating class of the mode-coupling instabilities vanishing, meaning that they vanish at high SC, when the SC tune shift sufficiently exceeds the wakerelated coherent tune shifts
Summary
Transverse mode coupling instabilities (TMCI) are believed to be one of the main limitations for the intensity of bunched beams. The same mechanism of interaction may cause instability in the coasting beam and only head-to-tail signal amplification in the bunch, without making the entire bunch unstable To articulate this distinction, we may, following Ref. Amplification of the fast transverse microwave perturbation along the bunch was considered by Brandt and Gareyete [15,16] with respect to a long positron bunch in the CERN SPS in the LEP era; a BBU-type estimation for the amplification coefficient was found This no-SC formula was suggested later for the proton bunch in the same machine by Cappi, Metral and Metral [17]; based on that, the bunch lifetime with respect to the tail particle losses was estimated. In the last two sections, it is shown how the suggested understanding, supported by various computations, resolves the mentioned paradoxes and controversies between the theory and observations of the TMCI with SC
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