Coherent Synchrotron Radiation and Longitudinal Beam Dynamics in Rings

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Coherent Synchrotron Radiation and Longitudinal
Beam Dynamics in Rings
ICFA Workshop on High Brightness Beams
Sardinia, July 1-5, 2002

• M. Venturini and R. Warnock
• Stanford Linear Accelerator Center

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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.

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Observations of CSR - NSLS VUV RingCarr et al.
NIM-A 463 (2001) p. 387
Spectrum of CSR Signal (wavelength 7 mm)
Current Threshold for Detection of
Coherent Signal
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Observations of CSR - NSLS VUV RingCarr et al.
NIM-A 463 (2001) p. 387
Detector Signal vs. Time

• CSR is emitted in bursts.
• Duration of bursts is
• Separation of bursts is of the order
• of few ms but varies with current.

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NSLS VUV RingParameters
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
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X-Ring Parameters (R. Ruth et al.)
Can a CSR-driven instability limit performance?
Energy 25
MeV Circumference 6.3
m Local radius of curv. R25 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!
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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?

Radiated Power
incoherent
coherent
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Dynamical Effects of CSR
Presence of modulation
(microbunches) in bunch density

CSR may become significant
Collective forces associated with CSR induce
instability
Instability feeds back, enhances
microbunching
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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.

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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).

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Parallel Plate Model for CSR Geometry Outline
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Analytic Expression for CSR Impedance(Parallel
Plates)
By definition
Impedance
With
,
Argument of Bessel functions
,
beam height
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Collective Force due CSR
FT of (normalized) charge density of
bunch.
Assume charge distribution doesnt change much
over one turn (rigid bunch approx).
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Parallel Plate Model Two Examples
X-Ring
NSLS VUV Ring
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Properties of CSR Impedance

• Shielding cut off
• Peak value
• Low frequency limit of impedance

energy-dependent term
curvature term
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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 focusing
collective force
is distance from synchronous particle.
is relative momentum (or energy) deviation.
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Vlasov Equation

• Scaled variables
  
• Scale time
  1 sync. Period.

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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
  ).

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Haissinski Equilibrium for X-Ring

• If potential-well distortion is small, Haissinski
  can be approximated
• as Gauss with modified rms-length

Haissinski Equilibrium (close to Gaussian
with rms length ) gtBunch
Shortening. I0.844 pC/V corresponding to
2 cm
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Linearized Vlasov Equation

• Set and linearize about
  equilibrium
• Equilibrium distribution
• Equilibrium distribution for
• equivalent coasting beam
• (Boussard criterion)

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(Linear) Stability Analysis

• Ansatz
• Dispersion relation
• with ,
• and
• Look for for instability.

Error function of complex arg
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Keil-Schnell Stability Diagram for X-Ring
(stability boundary)
Most unstable harmonic
Threshold (linear theory)
Keil-Schnell criterion
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Numerical Solution of Vlasov Equation
coasting beam linear regime
Amplitude of perturbation vs time
(different currents)
Initial wave-like perturbation grows
exponentially. Wavelength of perturbation
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Validation of Code Against Linear Theory
(coasting beam)
Growth rate vs. current for 3 different mesh
sizes
Theory
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Coasting Beam Nonlinear Regime (I is 25 gt
threshold).
Density Contours in Phase space
2 mm
Energy Spread Distribution
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Coasting Beam Asymptotic Solution
Energy Spread vs. Time
Density Contours in Phase Space
Energy Spread Distribution
Large scale structures have disappeared.
Distribution approaches some kind of steady
state.
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Bunched Beam Numerical Solutions of Vlasov Eq. -
Linear Regime.
Amplitude of perturbation
vs. time
Wavelength of initial perturbation
RF focusing spoils exponential growth. Current
threshold (5 larger than
predicted by Boussard).
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Bunched Beam Nonlinear Regime (I is 25 gt
threshold).
Density Plots in Phase space
2 cm
Charge Distribution
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Bunched Beam Asymptotic Solutions
Bunch Length and Energy Spread vs. Time
Quadrupole-like mode oscillations continue
indefinitely.

Microbunching disappears within 1-2 synchr.
oscillations
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X-Ring
Evolution of Charge Density and Bunch Length
Charge Density
Bunch Length (rms)
(25 above instability threshold)
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Inclusion of Radiation Damping and Quantum
Excitations

• Add Fokker-Planck term to Vlasov equation

damping
quantum excit.
Case study NSLS VUV Ring
Synch. Oscill. frequency
Longitudinal damping time
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Keil-Schnell Diagram for NSLS VUV Ring
Current theshold
Most unstable harmonic
G. Carr et al., NIM-A 463 (2001) p. 387
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NSLS VUV Model
Bunch Length vs. Time
10 ms
Incoherent SR Power
CSR Power
vs. Time
current 338 mA, (I12.5 pC/V)
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Bunch Length vs. Time
vs. Time
1.5 ms
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Snapshots of Charge Density and CSR Power
Spectrum
5 cm
current 311 mA, (I11.5 pC/V)
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NSLS VUV Storage Ring
Charge Density
Bunch Length (rms)
Radiation Power
Radiation Spectrum
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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.