TMCI and Space Charge

TL;DR

This work analyzes Transverse Mode Coupling Instability (TMCI) in the presence of strong space charge using two analytical frameworks: Blaskiewicz's AirBag Square (ABS) model and strong space charge (SSC) theory. It examines negative wakes and oscillating wakes across square-well and Hofmann-Pedersen HP$_0$ distributions, showing that $ΔQ_{\mathrm{sc}}$ drives TMCI behavior differently for each wake type; negative wakes lead to thresholds that grow roughly linearly with $ΔQ_{\mathrm{sc}}$ and vanish in the SSC limit, while oscillating wakes yield non-monotonic thresholds and possible non-vanishing TMCI at high SC due to positive-mode couplings. For exponential/constant and resistive-wall wakes, TMCI thresholds generally do not persist in the SSC limit when many SSC harmonics are included, indicating a vanishing TMCI in SSC. The results reconcile previous debates, reveal non-trivial SC dependence for oscillating wakes (including a possible $\sim 1/ΔQ_{sc}$ scaling), and demonstrate convergence between ABS and SSC predictions in the SSC regime, with the HP$_0$ distribution showing unique coupling behavior for certain wakes.

Abstract

Transverse mode-coupling instability (TMCI) is known to limit bunch intensity. Since space charge (SC) changes the spectra of the collective modes, it affects the TMCI threshold as well. In agreement with results of M. Blaskiewicz and V. Balbekov, we found that, when the wake is negative or it essentially negative, the instability threshold increases as fast as the space charge tune shift when the latter is large enough. In contrast, for oscillating wakes, when the oscillations are sufficiently pronounced, the threshold dependence on SC is non-monotonic; at sufficiently high SC tune shift the threshold goes inversely proportional to that.

Figures (15)

• Figure 1: First eight SSC harmonics $Y_k(\tau)$ for a bunch in a square potential well (red) and $\text{HP}_0$ distribution (blue). The harmonics are presented in dimensionless units. Note that for both distributions $\rho(\tau) = 1/2$.
• Figure 2: Eigenvalues (left) and normalized eigenvalues (right) for the ABS model as functions of the SC parameter for no-wake case, $k=-9,\ldots,10$.
• Figure 3: Normalized tune shifts for the ABS model and delta wake for different values of the SC parameter (first three plots). The last figure shows same values for the SSCSW (solid lines) and SSCHP (dashed lines) models.
• Figure 4: Spectra for the ABS model and constant (top row) or exponential (bottom row) wakes. Different columns correspond to different values of the space charge parameter $\Delta Q_\mathrm{sc}/ Q_{\text{s}}=0,2,20,200$. The TMCI thresholds are shown by black points.
• Figure 5: TMCI threshold as a function of space charge for the ABS model with the exponential wakes (reproduction of Ref. blaskiewicz1998fast).