FCC electron cloud study plan K. Ohmi (KEK) Mar FCC electron cloud study meeting CERN

Simulation using cylindrical chamber in PEI code Space charge off, stop when line density is over 2x beam line density. 25 ns 5ns  2,max = , E max =300 eV

Eelctron cloud simulation using exact boundary Uniform mesh in the transverse plane. Solve difference (discrete Poisson) equation using Band Matrix Method.

Potential solver

Necessary revision on PEI code Electron production at the boundary Beam force satisfying the boundary condition

Coupled bunch instability Corrective motion between beam and electron cloud. based on buildup code, PEI. Instability due to electron cloud in bending magnet in DAFNE (rectangular chamber). White : beam center Violet: electron cloud

Single bunch instability Strong head-tail instability caused by electron cloud Threshold of electron density Electron frequency in the beam field Tune shift of the beam

Electron motion (3.3TeV-50TeV)  =200 m,  =3517,  z =8cm, s =0.002 – N p =10 11 (25ns),  n =0.44x10 -6 m.  e /2  =3.56 GHz (3.3TeV) 13.9 GHz (50TeV)  e  z /c=5.97 (3.3TeV) 23.3 (50TeV)  e,th = 4.4x10 10 (3.3) 5.7x10 11 m -3 (50)  (  e,th )= (3.3) (50) – N p =2x10 10 (5ns),  n =2.2x10 -6 m.  e /2  =3.58 GHz (3.3) 13.9 GHz (50TeV)  e  z /c=6.00 (3.3) 23.3 (50TeV)  e,th = 4.4x10 10 (3.3) 5.7x10 11 m -3 (50)  (  e,th )= (3.3) (50)

Electron density during interaction with beam Transverse electron profile along z. (drift space) Electron initial energy v 0 =10 6 m/s (3eV) is assumed. Variation of electron density and size along the bunch interaction, z. Density increase to 25x of initial. z=-3  z z=-2.4  z z=-0.4  z

In strong Bending magnet Tune shift increases 5x at interaction with bunch center, z~0. (drift). Variation of electron density and size along the bunch interaction, z. Density increase to 5x of initial. z=-2.4  z z=-1.8  z z=+0.6  z

Emittance growth caused by electron cloud Electron cloud potential induces tune spread and resonances. The strength of the resonances are characterized by the width in amplitude space,  J. Modulation of the resonances due to synchrotron motion etc. results in emittance growth.

Electron cloud induced tune spread and resonance term U(x,y): integrated effective potential due to electron cloud Resonance driving term is Fourier component of U. H 0 : lattice transformation of s’ to s

Tune shift and slope due to Gaussian electron cloud Tune shift Tune slope

Tune slope and resonance width Tune Example: J-PARC space charge

Tune shift is very small,  =0.0003, even at the threshold of coherent instability. The tune shift can be x5 higher due to pinching,  = Not very large. How electron cloud with the tune shift causes emittance growth.

Future plan Study Electron cloud Build up using accurate boundary. Evaluate growth rate and unstable mode of the coupled bunch instability. Evaluate Resonance width and simulation using resonance model.