Threshold of transverse mode coupling instability with arbitrary space charge

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.

Figure 1

The TMCI threshold of the hollow bunch in the square potential well against the SC tune shift. Different curves are obtained with different $N_{\max }=1,\dots 19$. Each of them has a restricted region of applicability which expands with $N_{\max }$ growing. The drop-down parts of the curves merge forming the sole black line. The rising lines do not confirm each other marking ends of the applicability regions.

Figure 2

Threshold bunch spectrum at $N_{\max }=8$. The solid black line is the TMCI threshold in this approximation. Other lines present the bunch eigentunes just before the instability appears. Coalescence of different multipoles is responsible for the instability at different $\mathrm{\Delta }Q/Q_{s0}$. The essential lines are emphasized by colors.

Figure 3

The threshold bunch spectrum obtained with different $N_{\max }$. Only essential (capable to coalesce) spectral lines are shown.

Figure 4

Lowest real eigennumbers of Eq. (24) and (25). Thin straight lines are tangent to the curves of corresponding color.

Figure 5

The bunch head-tail modes against the wake strength at different SC tune shifts. The extremal points where $d{q}_0=0$ mark the beginning of the instability region. Only negative wakes are displayed at $\mathrm{\Delta }Q/Q_{s0}\ge 6$.

Figure 6

Instability threshold of the lowest TMCI modes. The short black line is copied from Fig. 1 representing result of the expansion technique at $N_{\max }=19$. Its deviation from the current lowest mode does not exceed 1% at $\mathrm{\Delta }Q/Q_{s0}<15$.

Figure 7

Lowest real eigennumbers of the resistive wall wake, Eq. (33).

Figure 8

TMCI threshold of the resistive wall wake. Its asymptotic value is: $q_0=-1.1\mathrm{\Delta }Q$.

Figure 9

The lowest real eigennumbers of Eq. (30) at different $\alpha$.

Figure 10

Higher real eigennumbers of Eq. (34) at different $\alpha$.

Figure 11

Lowest eigentunes of the bunch against the exponential wake strength at $\mathrm{\Delta }Q=0$.

Figure 12

Lowest eigentunes of the bunch against the exponential wake strength at $\mathrm{\Delta }Q=4$.

Figure 13

Instability threshold of exponential wake with different $\alpha$ against the SC tune shift.

Figure 14

Asymptotic ratio $q_0/\mathrm{\Delta }Q$ of exponential wake against the damp coefficient.