Transverse coupled-bunch instability thresholds in the presence of a harmonic-cavity-flattened rf potential

A small vacuum chamber aperture is a present trend in the design of future synchrotron light sources. This leads to a large resistive-wall impedance that can drive coupled-bunch instabilities. Another trend is the use of passively driven cavities at a harmonic of the main radio frequency to lengthen the electron bunches in order to increase the Touschek lifetime and reduce emittance blowup due to intrabeam scattering. In some cases, the harmonic cavities may be tuned to fulfill the flat potential condition. With this condition met, it has been predicted in simulation that the threshold current for coupled-bunch resistive-wall instabilities is much higher than with no bunch lengthening at all. In this paper, the features of a bunch in the flat potential that would contribute toward this stabilization are identified and discussed. The threshold currents for these instabilities are estimated for the MAX IV 3 GeV storage ring at different values of chromaticity using macroparticle simulations in the time domain and, within the limits of the existing theory, frequency domain calculations. By comparing the results from these two methods and analyzing the spectra of the dominant head-tail modes, the impact of each of the distinguishing features of a bunch in the flat potential can be explained and quantified in terms of the change in threshold current. It is found that, above a certain chromaticity, the threshold current is determined by the radial structure of the zeroth-order head-tail mode. This happens at a lower chromaticity if the bunch length is longer.

Figure 1

Bunch profiles for the three non-Gaussian distributions listed in Table 2 compared to a Gaussian distribution and the distribution expected for the flat potential condition.

Figure 2

Threshold current for the coupled-bunch resistive-wall instability for the case with a short bunch in a single rf system. Results of macroparticle simulations in the time domain (dashed line) are compared with those obtained using a frequency domain computation (solid lines).

Figure 3

Comparison of the Laclare eigenvalue method with the Sacherer approximation when both are applied to the case of the short bunch in a single RF system.

Figure 4

Threshold current for the coupled-bunch resistive-wall instability for the case with a stationary, lengthened Gaussian bunch. Results of macroparticle simulations in the time domain (dashed line) are compared with those obtained using a frequency domain computation (solid lines).

Figure 5

Threshold current for the coupled-bunch resistive-wall instability for the case with a harmonic cavity flat potential. Results of macroparticle simulations (dashed line) in the time domain are compared with those obtained using a frequency domain computation of the $m=0$ head-tail mode for three non-Gaussian distributions (solid lines).

Figure 6

Fourier transform of the three non-Gaussian distributions listed in Table 2 compared to a Gaussian distribution and the distribution expected for the flat potential condition.

Figure 7

Fourier transform of the beam signal around the first betatron tune line from tracking of beam distributions produced in time domain simulations of coupled-bunch instabilities at a chromaticity of 2.4. The frequency axis is given in terms of the revolution frequency $f_0$.

Figure 8

Transverse bunch spectra for the short bunch case for the dominant mode at three different chromaticities. Time domain and frequency domain results are compared.

Figure 9

Transverse bunch spectra for the lengthened Gaussian bunch case for the dominant mode at three different chromaticities. Time domain and frequency domain results are compared.

Figure 10

Transverse bunch spectra for the lengthened Gaussian bunch case for the dominant mode at three different chromaticities.

Figure 11

Scatter plot of populated bins in synchrotron phase space for the lengthened Gaussian bunch at the end of a macroparticle simulation of a coupled-bunch instability at a chromaticity of 2.4. The color of each point represents the mean offset of particles within that bin.

Figure 12

Transverse bunch spectra in the flat potential condition for the $m=0$ mode at three different chromaticities. The frequency domain spectra are for the flattened bunch profile.

Figure 13

Scatter plot of populated bins in synchrotron phase space for the flat potential condition at the end of a macroparticle simulation of a coupled-bunch instability at a chromaticity of 2.4. The color of each point represents the mean offset of particles within that bin.

Figure 14

Comparison of the transverse bunch spectra ${\sigma }_m$ for three different beam types at a chromaticity of 2.4 with their corresponding shaker modes excited at zero frequency $f({\omega }_{mp},0)$.

Figure 15

Threshold current for the coupled-bunch resistive-wall instability for the case with a Gaussian bunch at an intermediate length of 100 ps. Results of macroparticle simulations in the time domain (dashed line) are compared with those obtained using a frequency domain computation (solid lines).

Figure 16

Threshold current for the coupled-bunch resistive-wall instability for the case with a potential flattened with a harmonic cavity at the 11th harmonic of the main rf. Results of macroparticle simulations in the time domain (dashed line) are compared with those obtained using a frequency domain computation (solid line).

Figure 17

Comparison of the transverse bunch spectra ${\sigma }_m$ for the flat potential condition, a chromaticity of 2.4 and two different bunch lengths with the corresponding shaker modes excited at zero frequency $f({\omega }_{mp},0)$.

Figure 18

Threshold current for the coupled-bunch resistive-wall instability in the 3 GeV ring with a BBR included for the case with a short bunch in a single rf system. Results of macroparticle simulations in the time domain (dashed line) are compared with those obtained using a frequency domain computation (solid lines).

Figure 19

Threshold current for the coupled-bunch resistive-wall instability in the 3 GeV ring with a BBR included for the case with a lengthened Gaussian bunch. Results of macroparticle simulations in the time domain (dashed line) are compared with those obtained using a frequency domain computation (solid lines).

Figure 20

Threshold current for the coupled-bunch resistive-wall instability in the 3 GeV ring with a BBR included for the case with a harmonic cavity flat potential. Results of macroparticle simulations in the time domain are compared with those obtained using a frequency domain computation of the $m=0$ head-tail mode for three non-Gaussian distributions.

Figure 21

Ratio of threshold currents predicted by time domain simulations of the flattened potential to those predicted by frequency domain calculations for a flattened bunch distribution with the same bunch length.