Coupled-bunch instability for arbitrary multibunch configurations

The coupled-bunch instability for arbitrary multibunch configurations is investigated in its generality. The theoretical framework adopted is based on the general eigenvalue analysis based on the known formulas of the complex frequency shifts for the uniform multibunch configuration case. Closed formulas are derived for special cases. For the configuration consisting of a uniform filling pattern with a gap of missing bunches, the theoretical results are found to be in good agreement with measurements of the transverse coupled-bunch instability driven by the resistive wall impedance performed at the NSLS-II storage ring.

Figure 1

(a) One superperiod of the NSLS-II DBA lattice consisting of two cells with short and long straight sections. Vertical beta function (magenta trace) variation and its average (green trace) over the length of the impedance element (red rectangle): (b) aluminum vacuum chamber in the first half superperiod; (c) DWs in cell 8; (d) bending magnet chamber in cell 1.

Figure 2

Complex frequency shift for the uniform filling pattern with $M=h=1320$ at instability threshold $I_{\mathrm{th}}^U=12.7\mathrm{mA}$: growth rate (a); coherent tune shift (b). The fastest unstable mode $\mu =1303$ has growth rate ${\tau }_{\mu }^{-1}=1/{\tau }_y$, where ${\tau }_y$ is the vertical radiation damping time.

Figure 3

Largest imaginary part (largest growth rate) of the complex frequency shifts given by Eq. (22) for the equidistant multibunch configuration with $M=3$. $I_0=12.7\mathrm{mA}$.

Figure 4

Complex frequency shift with the largest imaginary part for all possible uniform filling patterns with $M\le h$, shown by the red dots, compared with the uniform filling pattern case with a gap $g=M-2$, shown by the blue squares: growth rate (a); coherent tune shift (b). $I_0=12.7\mathrm{mA}$.

Figure 5

Measurements at different average currents for the case $M_g=M-g=600$ (left frames) and $M_g=M-g=1200$ (right frames), where $g$ is the gap in the maximal ($M=1320$) uniform filling pattern. (a)–(h) time evolution of the average bunch centroid position over $50K$ turns, as detected from 6 of the 180 regular BPMs distributed around the NSLS-II storage ring. (a)–(d) and (e)–(h) showing unstable and stable betatron motion respectively. (i) and (j): average spectra of the time domain BPM measurements showing the transition from stable to unstable motion.

Figure 6

(a) Vertical coupled-bunch instability threshold $I_{\mathrm{th}}$ as a function of $M_g=M-g$, where $g$ is the gap in the maximal ($M=1320$) uniform filling pattern: comparison between theory and measurements. The blue trace shows the analytical result obtained by equating the fastest growth time (determined by the eigenvalue with the largest imaginary part) to the radiation damping time ${\tau }_y=22.5\mathrm{ms}$; the red dots represent the measured thresholds, obtained as the arithmetic mean of the currents slightly below and above threshold, as shown in Fig. 5. (b) Bunch trains close to instability threshold as measured from the NSLS-II filling pattern monitor: single bunch current $I_b=I_0/M_g$ as a function bunch number $m$.