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E. Métral

FOLLOW-UP OF THE FAST VERTICAL SINGLE- BUNCH INSTABILITY AT SPS INJECTION. E. Métral. Animation of the BBU theory ( direction of propagation solved! ) Review of Laclare’s theory (azimuthal and radial modes  mq ) Animation of the TMC theory ( direction of propagation not solved! )

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E. Métral

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  1. FOLLOW-UP OF THE FAST VERTICAL SINGLE- BUNCH INSTABILITY AT SPS INJECTION E. Métral • Animation of the BBU theory (direction of propagation solved!) • Review of Laclare’s theory (azimuthal and radial modes  mq) • Animation of the TMC theory (direction of propagation not solved!) • Coupling between modes -33 and -44 (bad direction!) • Coupling between modes -24 and -35 (good direction!) • Animation of SPS measurements • With a relative (measured) chromaticity of 0.14 • With a relative (measured) chromaticity of 0.54 • With a relative (measured) chromaticity of 2.04 • Animation of the BBU theory for the PS compared to measurements

  2. BBU THEORY BB resonator impedance 1st trace = turn 1 Last trace = turn 115 Every turn shown

  3. LACLARE’S FORMALISM (1/6) • Follow-up of last RLC meeting when I said that the spectrum of mode mq (σmq) is peaked at and extend 2 modes with same q are peaked at the same frequency (same line density), and therefore have ~ the same sensitivity in the same frequency range. They however correspond to entirely different patterns in phase space…

  4. (2/6) LACLARE’S RESULTS – CERN 87-03, p. 315 (1/2) and not specified with

  5. (3/6) LACLARE’S RESULTS – CERN 87-03, p. 316 (2/2)

  6. (4/6) MY RESULTS (1/2)

  7. (5/6) MY RESULTS (2/2)

  8. LACLARE’S FORMALISM (6/6) • I have the same results as Laclare (I use his formalism…) except for σ02 , σ11 , σ33 , where I have an opposite sign •  I also have an opposite sign for X02 , X11 , X33 • However, it has no effect on the signal observed at a Pick-Up • And it has also no effect in the computation of the tune shift as it is σmq2 which enters into the formula

  9. TMC THEORY: coupling between modes -33 and -44 1st trace = turn 1 Last trace = turn 115 Every turn shown

  10. TMC THEORY: coupling between modes -24 and -35 (1/2) 1st trace = turn 1 Last trace = turn 115 Every turn shown

  11. TMC THEORY: coupling between modes -24 and -35 (2/2) • I asked to Elena Shaposhnikova if one can identify which (radial) modes couple: -2? with -3? (In her paper CERN/SL/93-43 (RFS) on MOSES’s results, it is said that if coupling of mode m is important, radial modes up to k ~ |m| / 2 have to be considered) • If k = 1 then it is the same case as the one of the previous page

  12. MEASUREMENTS WITH A CHROMATICITY OF 0.14 1st trace (in red) = turn 2 Last trace = turn 150 Every turn shown

  13. MEASUREMENTS WITH A CHROMATICITY OF 0.54 1st trace (in red) = turn 2 Last trace = turn 150 Every turn shown

  14. MEASUREMENTS WITH A CHROMATICITY OF 2.04 1st trace (in red) = turn 2 Last trace = turn 150 Every turn shown

  15. BBU THEORY FOR THE PS AT TRANSITION (1/2) BB resonator impedance 1st trace = turn 1 Last trace = turn 90 Every turn shown

  16. BBU THEORY FOR THE PS AT TRANSITION (2/2) MEASUREMENTS IN 2000 BBU THEORY AFTER 90 TURNS (~ 200 μs)  Seem very close except head and tail exchanged!!! Does it start only for the maximum peak intensity, i.e. in the middle???

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