July 2002M. Venturini1 Coherent Synchrotron Radiation and Longitudinal Beam Dynamics in Rings M. Venturini and R. Warnock Stanford Linear Accelerator Center ICFA Workshop on High Brightness Beams Sardinia, July 1-5, 2002

July 2002M. Venturini2 Outline Review of recent observations of CSR in electron storage rings. Radiation bursts. Two case studies: –Compact e-ring for a X-rays Compton Source. –Brookhaven NSLS VUV Storage Ring. Model of CSR impedance. Modelling of beam dynamics with CSR in terms of 1D Vlasov and Vlasov-Fokker-Planck equation. –Linear theory CSR-driven instability. –Numerical solutions of VFP equation. Effect of nonlinearities. –Model reproduces main features of observed CSR.

July 2002M. Venturini3 Observations of CSR - NSLS VUV Ring Carr et al. NIM-A 463 (2001) p. 387 Current Threshold for Detection of Coherent Signal Spectrum of CSR Signal ( wavelength ~ 7 mm )

July 2002M. Venturini4 Observations of CSR - NSLS VUV Ring Carr et al. NIM-A 463 (2001) p. 387 CSR is emitted in bursts. Duration of bursts is Separation of bursts is of the order of few ms but varies with current. Detector Signal vs. Time

July 2002M. Venturini5 NSLS VUV Ring Parameters Energy 737 MeV Average machine radius 8.1 m Local radius of curvature 1.9 m Vacuum chamber aperture 4.2 cm Nominal bunch length (rms) 5 cm Nominal energy spread (rms) Synchrotron tune Longitudinal damping time 10 ms

July 2002M. Venturini6 X-Ring Parameters (R. Ruth et al.) Energy 25 MeV Circumference 6.3 m Local radius of curv. R=25 cm Pipe aperture h~ 1 cm Bunch length (rms) cm Energy spread Synchrotron tune Long. damping time ~ 1 sec Filling rate 100 Hz # of particles/bunch RADIATION DAMPING UNIMPORTANT! Can a CSR-driven instability limit performance?

When Can CSR Be Observed ? CSR emissions require overlap between (single particle) radiation spectrum and charge density spectrum: What causes the required modulation on top of the bunch charge density? incoherentcoherent Radiated Power:

Presence of modulation (microbunches) in bunch density CSR may become significant Collective forces associated with CSR induce instability Instability feeds back, enhances microbunching Dynamical Effects of CSR

July 2002M. Venturini9 Content of Dynamical Model CSR emission is sustained by a CSR driven instability [ first suggested by Heifets and Stupakov ] Self-consistent treatment of CSR and effects of CSR fields on beam distribution. No additional machine impedance. Radiation damping and excitations.

July 2002M. Venturini10 Model of CSR Impedance Instability driven by CSR is similar to ordinary microwave instability. Use familiar formalism, impedance, etc. Closed analytical expressions for CSR impedance in the presence of shielding exist only for simplified geometries (parallel plates, rectangular toroidal chamber, etc.) Choose model of parallel conducting plates. Assume e-bunch follows circular trajectory. Relevant expressions are already available in the literature [Schwinger (1946), Nodvick & Saxon (1954), Warnock & Morton (1990)].

July 2002M. Venturini11 Parallel Plate Model for CSR: Geometry Outline

July 2002M. Venturini12 Analytic Expression for CSR Impedance (Parallel Plates) By definition: Argument of Bessel functions With, Impedance, beam height

July 2002M. Venturini13 Collective Force due CSR FT of (normalized) charge density of bunch. Assume charge distribution doesn’t change much over one turn (rigid bunch approx).

July 2002M. Venturini14 Parallel Plate Model : Two Examples X-RingNSLS VUV Ring

July 2002M. Venturini15 Properties of CSR Impedance Shielding cut off Peak value Low frequency limit of impedance energy-dependent termcurvature term

July 2002M. Venturini16 Longitudinal Dynamics Zero transverse emittance but finite y-size. Assume circular orbit (radius of curvature R). External RF focusing + collective force due to CSR. Equations of motion RF focusingcollective force is distance from synchronous particle. is relative momentum (or energy) deviation. :

July 2002M. Venturini17 Vlasov Equation Scaled variables Scale time 1 sync. Period.

July 2002M. Venturini18 Equilibrium Distribution in the Presence of CSR Impedance Only (Low Energy) Haissinski equilibria i.e. Only low-frequency part of impedance affects equilibrium distribution. For small n impedance is purely capacitive If energy is not too high imaginary part of Z may be significant (space-charge term ).

July 2002M. Venturini19 If potential-well distortion is small, Haissinski can be approximated as Gauss with modified rms-length: Haissinski Equilibrium (close to Gaussian with rms length ) =>Bunch Shortening. I=0.844 pC/V corresponding to Haissinski Equilibrium for X-Ring 2 cm

July 2002M. Venturini20 Linearized Vlasov Equation Set and linearize about equilibrium: Equilibrium distribution: Equilibrium distribution for equivalent coasting beam (Boussard criterion):

July 2002M. Venturini21 (Linear) Stability Analysis Ansatz Dispersion relation with, and Look for for instability. Error function of complex arg

July 2002M. Venturini22 Keil-Schnell Stability Diagram for X-Ring (stability boundary) Threshold (linear theory): Keil-Schnell criterion: Most unstable harmonic:

Amplitude of perturbation vs time (different currents) Numerical Solution of Vlasov Equation coasting beam –linear regime Initial wave-like perturbation grows exponentially. Wavelength of perturbation:

#grid pts Growth rate vs. current for 3 different mesh sizes Validation of Code Against Linear Theory (coasting beam ) Theory

July 2002M. Venturini25 Coasting Beam: Nonlinear Regime (I is 25% > threshold). Density Contours in Phase space Energy Spread Distribution 2 mm

Coasting Beam: Asymptotic Solution Large scale structures have disappeared. Distribution approaches some kind of steady state. Energy Spread vs. TimeDensity Contours in Phase Space Energy Spread Distribution

July 2002M. Venturini27 Bunched Beam Numerical Solutions of Vlasov Eq. - Linear Regime. RF focusing spoils exponential growth. Current threshold (5% larger than predicted by Boussard). Amplitude of perturbation vs. time Wavelength of initial perturbation:

July 2002M. Venturini28 Bunched Beam: Nonlinear Regime (I is 25% > threshold). Density Plots in Phase space Charge Distribution 2 cm

July 2002M. Venturini29 Bunched Beam: Asymptotic Solutions Bunch Length and Energy Spread vs. Time Quadrupole-like mode oscillations continue indefinitely. Microbunching disappears within 1-2 synchr. oscillations

July 2002M. Venturini30 Charge DensityBunch Length (rms ) (25% above instability threshold) X-Ring: Evolution of Charge Density and Bunch Length

July 2002M. Venturini31 Inclusion of Radiation Damping and Quantum Excitations Add Fokker-Planck term to Vlasov equation Case study NSLS VUV Ring Synch. Oscill. frequency Longitudinal damping time damping quantum excit.

G. Carr et al., NIM-A 463 (2001) p. 387 Measurements Model Current threshold 100 mA 168 mA CSR wavelength 7 mm 6.7 mm Keil-Schnell Diagram for NSLS VUV Ring Most unstable harmonic: Current theshold:

vs. Time CSR Power: Incoherent SR Power: Bunch Length vs. Time NSLS VUV Model current =338 mA, ( I=12.5 pC/V) 10 ms

July 2002M. Venturini34 Bunch Length vs. Time vs. Time 1.5 ms

Snapshots of Charge Density and CSR Power Spectrum 5 cm current =311 mA, (I=11.5 pC/V)

July 2002M. Venturini36 Charge Density Bunch Length (rms) Radiation Spectrum Radiation Power NSLS VUV Storage Ring

July 2002M. Venturini37 Conclusions Numerical model gives results consistent with linear theory, when this applies. CSR instability saturates quickly Saturation removes microbunching, enlarges bunch distribution in phase space. Relaxation due to radiation damping gradually restores conditions for CSR instability. In combination with CSR instability, radiation damping gives rise to a sawtooth-like behavior and a CSR bursting pattern that seems consistent with observations.