1. Elias Métral /1 POWER SPECTRA COMPARISON BETWEEN DIFFERENT TYPES OF LONGITUDINAL BUNCH PROFILES FOR THE LHC BUNCH AT 3.5 TEV Elias Métral Elias Métral u Examples of measured bunch spectra u Comparison of 4 very different longitudinal bunch profiles and corresponding power spectra u Comparison of several longitudinal bunch profiles close to Gaussian ones (BUT with finite tails), and corresponding power spectra u What is the distribution fitting best (one) measurement? u Analysis by ThemisM from BQM data and associated spectrum u Conclusion 16

2. 2 EXAMPLES OF MEASURED BUNCH SPECTRA Measurements on B1 by Themis and Philippe on fill # 2261 Reminder on the Average power deposited per unit length [W/m] Elias Métral / It was mentioned that it is in fact the power spectrum 16

3. 3 4 THEORETICAL LONGITUDINAL BUNCH PROFILES Elias Métral / 16

4. 4 CORRESPONDING POWER SPECTRA (1/2) Elias Métral / 16

5. 5 CORRESPONDING POWER SPECTRA (2/2) Elias Métral / 16

6. 6 COMPARING LONGITUDINAL BUNCH PROFILES CLOSE TO GAUSSIAN BUT WITH FINITE TAILS (1/6) Elias Métral / 7 Gaussian Family of distributions, depending on n and converging to a Gaussian when n goes to infinity n = 1 n = 5

7. 7Elias Métral / COMPARING LONGITUDINAL BUNCH PROFILES CLOSE TO GAUSSIAN BUT WITH FINITE TAILS (2/6) Gaussian n = 5 16 Power spectrum

8. 8Elias Métral / COMPARING LONGITUDINAL BUNCH PROFILES CLOSE TO GAUSSIAN BUT WITH FINITE TAILS (3/6) Gaussian n = 4 16 Power spectrum

9. 9Elias Métral / COMPARING LONGITUDINAL BUNCH PROFILES CLOSE TO GAUSSIAN BUT WITH FINITE TAILS (4/6) Gaussian n = 3 16 Power spectrum

10. 10Elias Métral / COMPARING LONGITUDINAL BUNCH PROFILES CLOSE TO GAUSSIAN BUT WITH FINITE TAILS (5/6) Gaussian n = 2 16 Power spectrum

11. 11Elias Métral / COMPARING LONGITUDINAL BUNCH PROFILES CLOSE TO GAUSSIAN BUT WITH FINITE TAILS (6/6) Gaussian n = 1 16 Power spectrum

12. 12 WHAT IS THE DISTRIBUTION FITTING BEST (ONE) MEASUREMENT? Elias Métral / 16 Theoretical power spectrum for the Water-bag bunch with Measurements on B1 by Themis and Philippe on fill # 2261

13. 13 ANALYSIS BY THEMIS FROM BQM DATA u The BQM reports the full-width at half maximum (FWHM) scaled by sqrt(2/ln(2)). The scaling is set to report a 4-sigma width assuming a Gaussian bunch u If we assume a Water-bag bunch distribution, the FWHM equals tau*sqrt(3)/2, where tau is the full width of the Water-bag model. As such, for a Water-bag bunch, the BQM will report tau*sqrt(3/2/ln(2)) u Therefore, a reported value of 1.1 ns (from the BQM) corresponds to a width of just 0.75 ns for the Water-bag model u The BQM reported bunch lengths at the end of the ramp for fill 2261 of 1.2 and 1.15 ns for B1, B2 respectively. These values correspond to a bunch width of 0.81 and 0.78 ns respectively for a Water-bag bunch u => Seems that the bunch spectra together with the BQM data are consistent with a bunch distribution close to a Water-bag with ~ 0.75 – 0.8 ns (full bunch length at the base) Elias Métral / 16

14. 14 WHAT IS THE PREDICTED POWER SPECTRUM FOR THE VALUE DEDUCED BY THEMIS FROM BQM => ~ 0.8 ns? Elias Métral / 14 Theoretical power spectrum for the Water-bag bunch with Measurements on B1 by Themis and Philippe on fill # 2261 Theoretical power spectrum for the Water-bag bunch with

15. 15 CONCLUSION u The best fit of (one of) the measured power spectra in stable beams (of the previous slide) is obtained for a Water-bag bunch with a total bunch length (at the base) of 0.75 ns u This distribution is very different from Gaussian-types with finite tails u According to the analysis done by ThemisM from BQM data, it seems that the bunch spectra together with the BQM data are consistent with a bunch distribution close to a Water-bag with a full bunch length at the base close to 0.75 ns (a reported value of 1.1 ns from the BQM corresponds to a width of just 0.75 ns for the Water-bag model) u Comment from AlexeyB: Theoretically, the distribution after the emittance blow-up must be water-bag if the width of the excitation line is much smaller than its detuning from 2*(maximal synchrotron frequency) => To be followed-up (and discussed also with J. Tuckmantel et at.) Elias Métral / 16

16. 16 ACKNOWLEDGEMENTS u Many thanks to Alexej Grudiev, Alexey Burov, Benoit Salvant, Carlo Zannini, Elena Shaposhnikova, Hannes Bartosik, Hugo Alistair Day, Nicolo Biancacci‎, Olav Ejner Berrig, Philippe Baudrenghien, Themistoklis Mastoridis, etc. for very interesting discussions => To be followed up Elias Métral / 16