Update of Microwave Instability Study in SuperKEKB Damping Ring Using Vlasov Fokker-Planck Solver L. Wang, SLAC In collaboration with H. Ikeda, K. Ohmi, K. Oide, D. Zhou KEKB Physics Meeting 1

Motivation Systematic study the microwave instability using Vlasov solver to validate the results: We checked the numerical parameters: mesh size, domain, time step and the CSR impedance Our goal is to empirically find the appropriate parameters/way for the simulation of MWI. 2

Wakes Geometry wake CSR Impedance(rectangular geometry code (Japanese Journal of Applied Physics 51 (2012) ) The CSR Wake is given by the convolution of a 0.5 mm long Gaussian bunch with CSR impedance (frequency up to 477GHz) Geometry wake CSR wake 3

Microwave instability with different wakes (Yunhai’s code) With CSR wake only, the energy spread starts to increase near the nominal bunch current (N= 5  ) The microwave instability starts at bunch population 6  when both geometry wake and CSR wake are included Geometry wake only CSR wake only & both g-wake and CSR wake 4 Numerical parameters: qmax=8, nn=300, ndt=1024

Effect of domain, mesh size and time step The simulation is done in the normalized phase space, which is rectangular region with maximum domain qmax (and minimum –qmax). (16  X16  ) The whole domain contains (2*nn+1) mesh points: (1001X1001) The time step is given by ndt, the number of steps per synchrotron period. Requirement: nn: 500 qmax: 8 Time steps: 1024/syn. period 5

Effect of CSR impedance at high frequency Previous study using impedance with frequency up to 500GHz; CSR impedance with frequency up to 1.4 THz The wake is convoluted by a 0.2 mm Gaussian bunch More high frequency component 6 Although our bunch is long, but micro-wave instability occurs at micro-bunch level

With higher frequency CSR-1 With High frequency 1.5THz CSR impedance There is a similar threshold, but a much stronger instability above the threshold A clear Saw-tooth instability; Near threshold N=5.5E10 above threshold N=8.5E10 7 However, no saw-tooth instability found with the CSR impedance of f~477GHz

Higher frequency CSR effect above threshold N=8.5E10 High frequency CSR impedance is important: Saw-tooth type of instability is found with high frequency CSR impedance only 8

Higher frequency CSR effect above threshold N=8E10 High frequency CSR impedance is important: 9 f up to 500GHZ f up to 1500GHZ

With higher frequency CSR-2 instability near threshold N=5.5e10 10 Phase plot

With higher frequency CSR-3 MWI above threshold N=8.5e10 11  Clear Saw-tooth instability occurs  Phase plot shows high order modes Phase plot Saw-tooth instability driven by CSR Saw-tooth Period:110 syn. period 517Hz

Time step check above threshold N=8.5E10 Ndt (steps per synchrotron period) =1024 is good enough 12

Summary 13 The microwave instability in SuperKEKB Damping Ring is simulated using the Vlasov-Fokker-Planck code. It suggests a threshold slightly above the designed bunch current. PIC (Particle- In-Cell) code also confirms similar threshold The simulation of microwave instability with CSR impedance is nontrivial. We carefully checked the numerical parameters used in the simulation: the good numerical parameters are qmax=8 (domain), nn=500(mesh), ndt=1024 (time step) The most important finding: The high frequency part CSR impedance (f~1.4THz) plays an important role in the saw-tooth instability. High order modes observed

discussion 14 How to include high frequency impedance: (1)Calculate high frequency CSR impedance It is straightforward and only a matter of CPU, using parallel computation. (2) convolution with a short Gaussian bunch, <=0.1mm It would be very helpful (for both machine and beam dynamics) to observe the CSR & MWI in DR as proposed by Prof. Fukuma (3) Instead of convolution with a Gaussian bunch, directly using impedance (Demin IPAC12, Bob) green function (direct FFT of impedance)? other approaches?

Thank You! 15 Y. Cai and B. Warnock for fruitful discussions Thanks to Demin for his great help on the simulations