A Head-Tail Simulation Code for Electron Cloud January 30, 2004 Yunhai Cai Stanford Linear Accelerator Center

Outline Motivation and observation Physics and approximation Threshold of head-tail instability Spectrum of dipole oscillation Emittance growth along a bunch train Effects of chromaticity Conclusion

Bunch-by-Bunch Luminosity before installation of solenoid and bunches spaced with 4 RF buckets

Vacuum Pressure in Straights 4-RF bunch spacing 692 bunch train 30 Gauss solenoid Courtesy Artem Kulikov, 2001

Measurement of Beam Size

Longitudinally Sliced Bunch 1. Dynamical slices with static transverse size: 2. Initial longitudinal Gaussian distribution: where sz is the bunch length and sd is the energy spread of the positron bunch.

Positron Beam Center and Size 3. Centroid motion of the bunch: 4. Rms value of position for all the slices: 5. Effective size of the bunch: where sy is the size of the slices.

Presentation of Electron Cloud 1. Two-dimensional macro particles: 2. Initialized Gaussian distribution every turn: 3. Kick experienced by ith electron from nth slice: where is given by the Erskine-Bassetti formula.

Kick by a Gaussian Slice Erskine-Bassetti formula: where Relation to the complex error function W: where Nb is the bunch population.

Dynamics of the Positron Slices 1. Kick by the electron cloud: where 2. Lattice map for transverse coordinates: 3. Lattice map for longitudinal coordinates : where bz = sz/sd.

Beam and Machine Parameters E (Gev) energy 3.1 C(m) circumference 2200 bx/by(m) beta x/y 16.52/17.83 ex/ey(nm-rad) emittance x/y 24/1.5 tt(turn) damping time 9740 sz(cm) bunch length 1.3 sd energy spread 7.7E-4 nx/ny/ns tunes 0.65/0.56/0.025 Nb bunch population 1E11 Ns number of slices 1024

Parameters of Electron Cloud ne(m-3) density (1-13)E11 sxe(mm) x size 6.0 sye(mm) y size 3.0 Nm # of mp 10240

“Emittance Growth” at Different Intensities of Electron Cloud range of electron density: ne=(1-13)x1011m-3.

Threshold of Head-Tail Instability threshold of electron cloud density: ne=(4-5)x1011m-3.

Power Spectrum of Dipole Oscillation below the Threshold m = -1 mode betatron tune threshold of head-tail instability

Observation at PEP-II -1 mode? Courtesy Uli Wienands 2002 synchrotron sideband -1 mode? Courtesy Uli Wienands 2002

Build up of Electron Cloud Along a Bunch Train Densities of electron cloud were generated by Furman and Pivi with 4 RF spacing. t was about 50 ns and the saturation density was about 2x1011m-3.

Beam Blowup Along a Bunch Train Densities were below the threshold of head-tail instability.

Effects of Chromaticity on Beam Centroid Motion Density of electron cloud was above the threshold. ne = 8x1011m-3.

Effects of Chromaticity on Beam Size

Wake Field Due to Electron Cloud where and The wake field extracted from the code fitted well with the wake from LRC circuit.

Bunch Pattern and Luminosity L = 7.17x1033cm-2s-1, I- = 1310mA, I+ = 1950mA, Nb = 1319, 78 trains, December 23, 2003

Conclusion The low energy ring was operated (or limited) below the threshold of head-tail instability. The density of electron cloud was about 2x1011m-3. at the maximum beam current with a regular by-4 bunch pattern. Progress has been made to understand the dynamics between the electron cloud and positron beam. The simulation results can be qualitatively compared to the observations and experimental measurements. The main mechanism of beam blowup can be explained very well by the spread of centroid for all slices in a single bunch. Positive chromaticity can be used to damp down the centroid motion of a bunch but has little effects on the spread of slice centroid along the bunch.

Acknowledgement Thank F.J. Decker, M. Furman, S. Heifets, R. Holtzapple, A. Kulikov, K. Ohmi, M.Pivi, J. Seeman, and U. Wienands for many helpful discussion and sharing the idea and experimental data.