Efficiency of feedbacks for suppression of transverse instabilities of bunched beams

Which gain and phase have to be set for a bunch-by-bunch transverse damper, and at which chromaticity is it better to stay? These questions are considered for three models: the two-particle model with possible quadrupole wake, the author’s nested head-tail (NHT) model with the broadband impedance, and the NHT with the LHC impedance model. Details of 2D areas of stability in the chromaticity-intensity and chromaticity-gain planes and possibilities to use them are discussed. It is shown that resistive feedbacks may generate asymmetry of the tune shift distribution, which requires positively-shifted stability diagrams.

Figure 1

Two-particle growth rate versus gain $g$ and constant wake value $w$ for reactive damper and zero chromaticity. All the values are in the units of the inverse synchrotron period $1/T$.

Figure 2

The same as Fig. 1, but for the resistive damper.

Figure 3

Low gain, low wake area for the resistive damper.

Figure 4

Growth rates in units of $1/\mathrm{T}$ for the round chamber and zero gain.

Figure 5

Same for the vertical case. Note that stability at zero chromaticity is reestablished at $w\ge 22$ (mode decoupling).

Figure 6

Same for the horizontal.

Figure 7

Round chamber, reactive damper, $gT=1$.

Figure 8

Same as Fig. 7, but $gT=10$.

Figure 9

Growth rates for the round chamber and resistive damper with the gain $gT=1$.

Figure 10

Growth rate for the round chamber and resistive damper for $w=8$ (twice above the TMCI threshold). Seas of stability are huge.

Figure 11

Same for the reactive damper. There are no seas or lakes of stability, only rivers.

Figure 12

NHT result for the growth rate $\mathrm{Im}{q}_{*}={({\omega }_s\tau )}^{-1}$ versus gain $g$ and intensity parameter $\mathit{ImpF}=N/N_0$ for reactive damper and zero chromaticity. Gain is in units of ${\omega }_s$; its positive sign corresponds to focusing.

Figure 13

Same, for a larger range of the variables.

Figure 14

The growth rate versus chromaticity and reactive gain for intensity twice exceeding zero-gain zero-chromaticity TMCI threshold, i.e. for $\mathit{ImpF}=3.2$.

Figure 15

Same, for three selected gains.

Figure 16

Growth rate versus intensity and chromaticity; the damper is off.

Figure 17

Same for resistive $g=1$.

Figure 18

Same for resistive $g=10$.

Figure 19

Same as Fig. 15, but for the resistive damper. Note the fjord of stability.

Figure 20

The growth rate for $\xi =-1$ vs intensity for selected resistive gains. At high gains, the threshold saturates approximately at four times higher value than its damper-off value ($6.4:1.6$).

Figure 21

Growth rate for zero chromaticity and almost reactive damper, which phase declines towards resistive by 18°.

Figure 22

Same as the previous figure, for selected intensities.

Figure 23

Growth rate for the resistive feedback and chromaticity $\xi =-1$.

Figure 24

Same for the feedback phase $\pi /4$, i.e. 50% resistive and 50% reactive. Comparison with the previous figure shows how robust the resistive damper is against the phase variations at the proper chromaticity.

Figure 25

Growth rate for a single bunch LHC beam and no feedback. The TMCI threshold is at $\mathit{ImpF}=2.4$.

Figure 26

Growth rate for 25 ns LHC beam with the resistive feedback and $\mathit{ImpF}=1.5$. Coupled bunch interaction is included. Note the lake of stability; for the multibunch regime, the lake is limited by $\mathrm{ImpF}=1.7$.

Figure 27

Single bunch growth rate for the resistive damper and TMCI threshold intensity $\mathit{ImpF}=2.4$. Note that the lake of stability (shown upside down) almost vanished.

Figure 28

The same damper phase and bunch intensity for the full 25 ns beam.

Figure 29

Threshold current of the Landau octupoles, in Amperes, for the same case as Fig. 28, computed according to Ref. [5].

Figure 30

Growth rate for the single LHC bunch, reactive damper and zero chromaticity. Compare with Fig. 12 for the broadband impedance and Fig. 1 for the two-particle model.

Figure 31

Same as above, but for the full 25 ns beam. The average linear slope toward higher intensity reflects contribution of the coupled-bunch motion. It can be approximated as $\mathrm{Im}{q}_{*}\approx 0.075\mathit{ImpF}$.

Figure 32

Tune shifts of unstable modes for the full 25 ns LHC beam at chromaticity $\xi =18$, $\mathit{ImpF}=2$, damper off (blue, 1), and with resistive gain $g=1.4$ (red, 2). Both with and without damper, there are no unstable modes with positive tune shifts. Seventeen representative coupled-bunch mode numbers are depicted.

Figure 33

Tune shifts of unstable modes for the APS single bunch broadband impedance model at chromaticity $\xi =10$, $\mathit{ImpF}=2$, damper off (blue, 1), and with resistive gain $g=1.4$ (red, 2).