Transverse mode-coupling instability and space charge

Abstract

Transverse mode-coupling instability (TMCI) is known to limit bunch intensity. Since space charge (SC) changes coherent spectra, it affects the TMCI threshold. Generally, there are only two types of TMCI with respect to SC: the vanishing type and the strong space charge (SSC) type. For the former, the threshold value of the wake tune shift is asymptotically proportional to the SC tune shift, as it was first observed twenty years ago by M. Blaskiewicz for exponential wakes. For the latter, the threshold value of the wake tune shift is asymptotically inversely proportional to the SC, as it was shown by one of the authors. In the presented studies of various wakes, potential wells, and bunch distributions, the second type of instability was always observed for cosine wakes; it was also seen for the sine wakes in the case of a bunch within a square potential well. The vanishing TMCI was observed for all other wakes and distributions we discuss in this paper: always for the negative wakes, and always, except the cosine wake, for parabolic potential wells. At the end of this paper, we consider high-frequency broadband wake, suggested as a model impedance for CERN SPS ring. As expected, TMCI is of the vanishing type in this case. Thus, SPS Q26 instability, observed at strong SC almost with the same bunch parameters as it would be observed without SC, cannot be TMCI.

Convective instabilities of bunched beams with space charge

“…According to Ref. [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 modecoupling instabilities vanishing, meaning that they vanish at high SC, when the SC tune shift sufficiently exceeds the wake-related coherent tune shifts.…”

“…By the same reason, the matrix elements U lm of Eq. (7) have to be computed with Left: Contour plot for natural logarithm of the maximal head-to-tail amplification log K versus wake phase advance kr, horizontally, and its amplitude parameter w, vertically, for the broadband case, Qr = 1, and no SC. Right: TMCI growth rate for the same parameters; the black dashed line is the no-SC TMCI threshold, Eq.…”

Comment on “Instability studies at the CERN Proton Synchrotron during transition crossing”

“…It is worth mentioning that there was a purely theoretical objection to the TMCI-explanation of the PS and SPS instabilities: with the SC tune shifts as high as they are in those machines, TMCI should not be possible. Vanishing of TMCI at high SC was first shown by M. Blaskiewicz in 1998 for some special cases [3], and later was confirmed by several authors [4][5][6] for practically all possible situations. In an apparent contradiction to this theoretical conclusion, the instabilities were observed at PS and SPS, approximately at the same parameters where TMCI at no SC should be.…”

Space Charge Effects for Collective Instabilities in Circular Machines

“…With the wake taken into account, the Blaskiewicz' result of linear growth of the TMCI wake threshold was confirmed in a series of publications, see Refs. [35,36] and references therein. A hidden obstacle with this problem, sometimes caused misleading results, was realized by V. Balbekov [37], who showed that convergence of the expansion of the sought-for eigenfunction over the zerowake basis degrades with SC, requiring more and more terms for higher SC parameter q.…”