Beam observation and Introduction to Collective Beam Instabilities Observation of collective beam instability Collective modes Wake fields and coupling impedances Head-tail instability Microwave instability Beyond T. Toyama KEK

Observation of collective beam instability Example: KEK-PS 12 GeV Main Ring At 500 MeV injection plat bottom a beam loss occurs (red curve). Amount and timing of the loss => random. Proton number N B (Feedback CT) Magnetic field

Example: KEK-PS 12 GeV Main Ring At phase transition energy~5.4 GeV (in kinetic energy) a large beam loss occurs (red curve). Amount of the loss is at random. Proton number N B (Feedback CT) Magnetic field

Observation NBNB Multi-trace of horizontal betatron oscillation NBNB Amplitude of betatron oscillation Magnetic field

Observation horizontal betatron tune during acceleration t f f rev 2f rev f rev  - f  f rev  - f  2f rev  - f  2f rev  f  Without external kick, coherent oscillation emerged

Measurement by a wall current monitor Real signals may be attenuated by the loss in the cable > 100 m and limited band width of the WCM.

Beam loss: collective instabilities --- at random, a kind of positive feedback starting from a random seed direct space charge effects --- regular some mistake in parameterrs --- regular (B, f RF, tune, …)

Collective modes Coasting beam / longitudinal n=3 Beam

Coasting beam / transverse Collective modes n=3 Beam betatton oscillation x or y

Bunched beam / logitudinal Collective modes l=1l=2l=3 dipole quadrupolesextupole z zz charge density Phase space ….. no momopole mode

Bunched beam / transverse Collective modes dipole mode density  z  x zz z l=0l=1l=2 monopole dipolequadrupole ….. superimposed

Wake fields and coupling impedance Electromagnetic fields is produced by the beam passed by.

Wake fields and coupling impedance

Wake fields due to a Gaussian beam in a resistive pipe Longitudinal wake potentialTransverse wake potential Acceleration Decceleration Dampen deflection Further deflection

Wake fields and coupling impedance Impedance of a resistive pipe

Wake fields and coupling impedance Wake fields by cavities Q=1Q=10

Wake fields and coupling impedance Impedance of cavities Q=1Q=10

Head-Tail Instability Transverse bunched beam instability Time domain picture

Head-Tail Instability Chromaticity = 0 Red full line: (z)x(z) Red dushed line: (z)x’(z) Blue: kick due to resistive wall Growth Damp No effect ~Totally no effect (1) (2) (3) (4) head tail

Head-Tail Instability Head-tail phase z  p/p   0 phase of betatron oscillation phase space of synchrotron oscillation

Head-Tail Instability Damp  ~ 1 Red full line: (z)x(z) Red dushed line: (z)x’(z) Blue: kick due to resistive wall (1) (2) (1) (2) ~Totally damping head tail

Head-Tail Instability Growth ~Totally growing (1) (2)  ~  Red full line: (z)x(z) Red dushed line: (z)x’(z) Blue: kick due to resistive wall (1) (2) head tail

Head-Tail Instability Summary of Growth rate vs. Chromaticity Head-tail phase Growth rate Chao’s text book mode = 0 mode = 1 mode =2 mode =3 Stable Unstable

Head-Tail Instability KEK-PS 12 GeV Main Ring T. Toyama et al., PAC97, APAC98, PAC99 mode=0 mode=1 mode=2 NBNB amplitude of dipole oscillation

Head-Tail Instability CERN PS higher order head-tail mode R. Cappi, NIM

Head-Tail Instability KEK-PS 12GeV MR Frequency domain analysis growth rate ∝ Re[Z(  )] F(   ) Re[Z T ] Form factor F  (freq. spectrum of the beam)  m=0 m=1 m=2

Head-Tail Instability Observation Growth rate mode=0

Head-Tail Instability Cure Chromaticity control Landau damping by octupole magnets …

Beam response and Landau damping Coasting beam Transverse motion

Beam response and Landau damping Driving force Response

Driving force Response of the beam Absorbed power by the beam The beam: ensemble of the particles Frequency distribution:  The beam motion approaches steady oscillation. Velocity d /dt: in phase with the force Work is done on the beam Absorbed power by the beam: constant Stored energy in the beam: Macroscopic aspect: a beam driven by a force approaches steady oscillation. Microscopic aspect: Small amount of resonant particles grows infinitely large. Response of particles

Longitudinal instability Microwave Instability uniform distribution Wake: V=  Z (z) The seed of density modulation is produced V 1 =  Z (z), slippage, 

Landau damping by the spread of  rev =  p/p  phase slippage factor = 1/  t 2  1/  2  t  phase transition energy  p/p Density modulation reduced! Larger  p/p more stable

Microwave Instability Observation & simulation K. Takayama et al., Phys. Rev. Lett. 78 (1997) 871

Microwave Instability Sources: Narrow-band resonances  res ~ 1GHz

Cures Reducing Impedance Landau damping Reducing local beam chaege line density Artificial increasing momentum spread  p/p  >  rev Methods Higher harmonic rf cavity Voltage modulation of foundamental rf cavity …

Cures Reducing Impedance Exchange ~ 2/3 BPMs new ESM BPM ~2/3 Pump port new one with slits Growth rate reduction

Reducing local beam chaege line density Increasing momentum spread  >  rev Voltage modulation of foundamental rf cavity T. Toyama, NIM A447 (2000) 317

Beyond  Impedance calculation  Impedance measurements Beam transfer function  Vlasov equation Coupled bunch instability Mode-coupling instability  Electron-cloud instability  feedback system feedback in RF control system feedback damper = pick-up & kicker

“… every increase in machine performance has accompanied by the discovery of new types of instabilities.” - J. Gareyte (CERN)

References Schools: CAS, USPAS, and OHO (Japanese) Conferences proceedings: APAC, EPAC, and PAC Textbook etc.: A. W. Chao, PHYSICS OF COLLECTIVE BEAM INSTABILITIES IN HIGH ENERGY ACCELERATORS Editors: A. Chao and M. Tigner, Handbook OF ACCELERATOR PHISICS AND ENGINEERING

Good Luck!