1. Recent Progress Haisheng Xu

2. Outline Check the calculation of wake potential (together with Paolo) – Analytical method – PyHEADTAIL – mbtrack – elegant Check the estimation by Boussard Criterion (together with Paolo) With resistive-wall only, the threshold of microwave instability by mbtrack Outlook: estimation of multi-bunch instabilities 2

3. Wake Potential Calculation in Time Domain Seite 3

4. Wake Potential Calculation in Time Domain Seite 4

5. Wake Potential Calculation in Time Domain Seite 5

6. Wake Potential Calculation in Time Domain Mbtrack (old version) 6

7. Wake Potential Calculation in Time Domain Mbtrack (new version) 7

8. Average energy loss due to longitudinal wake along the whole ring (circumference is 288 m) after 1 turn Analytical wake convoluted with Gaussian bunch (0.1219 fs time step) 3.71 keV Wake potential directly from mbtrack (old version, bin size = 1.7765 ps) 1.11 keV Wake potential directly from mbtrack (new version, bin size = 1.7765 ps) 3.82 keV 3.3356 fs time step in elegant 9.27 keV 0.1219 fs time step in elegant 3.82 keV Calculated energy loss from the shift of centroid position generated by PyHEADTAIL (500 slices/bunch ~ 70 fs initial time step) 66.2 keV The convolution using 0.12fs time step in elegant is very slow. The new version of mbtrack is chosen for the simulation of microwave instability. 8

9. Boussard Criterion 9

10. 10

11. Boussard Criterion 2,3,4 are identical; Threshold from 1 is 2.84 times lower if F’=1; The calculation of effective impedance 11

12. Seite 12 Boussard Criterion

13. Threshold of Microwave Instability - 1 --- Requirements of Longitudinal Effective Impedance by Boussard Criterion Zero-Current Equilibrium Parameters Maximum Effective Impedance 100 MHz RF (Assume threshold current = 5 mA) without harmonic cavity with ideal 3 rd harmonic cavity 500 MHz RF (Assume threshold current = 2 mA) without harmonic cavity with ideal 3 rd harmonic cavity 13

14. Threshold of Microwave Instability - 2 --- Threshold current using RW impedance by Boussard Criterion Conditions (round chambers; radius = 10 mm; no NEG) (1) Effective RW Impedance (1) Threshold Current 100 MHz RF without harmonic cavity Copper chamber 4.7230 mA with 3 rd harmonic cavity Copper chamber 11.0267 mA 500 MHz RF without harmonic cavity Copper chamber 2.7199 mA with 3 rd harmonic cavity Copper chamber 5.5433 mA 14

15. Threshold of Microwave Instability - 3 --- Threshold current using RW impedance by mbtrack Conditions (round chambers; radius = 10 mm; no NEG) (1) Threshold Current 100 MHz RF without harmonic cavity Aluminium13.32 mA Copper16.66 mA Stainless Steel 3.33 mA with 3 rd harmonic cavity Aluminium Copper Stainless Steel 500 MHz RF without harmonic cavity Aluminium11.66 mA Copper14.99 mA Stainless Steel 1.67 mA with 3 rd harmonic cavity Aluminium Copper Stainless Steel 15

16. Simulation results from mbtrack 16

17. Others In the single bunch simulation, harmonic cavity which provided ideally flat RF potential was used. The detuning of HC hasn’t been considered yet. Short range wake of other components are necessary for further estimation of threshold of microwave instability. 17