Longitudinal Dynamics in High Intensity / Bunch

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Longitudinal Dynamics in High Intensity / Bunch
Cécile Limborg SSRL / SLAC
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Introduction

• Light sources
• ? Energy spread minimum (high Intensity spectral
  lines Und.)
• ? Short bunches subps desired
• Time resolved experiments
• High Î for SRFEL
• Coherent Synchrotron Radiation
• Damping Rings
• ? Large Energy oscillations undesirable _at_
  injection in linac

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Nature of instability

• Usually no beam loss
• transverse instabilities fix Ithr
• Instability Threshold of energy widening
• 2 regimes
• - potential well
• lengthening, no energy widening
• - microwave instability
• lengthening, energy widening
• Coherent signals (fs, 2 fs, 3 fs ) pop up
  (saturation or sawtooth)

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Strong Bunch lengthening

• Natural bunch length
• Quasi-isochronous tuning
• Demonstrated _at_ (SuperAco, ESRF, ALS, UVSOR)
• _at_ high current bunch length independent of ? and
  Energy
• Slope of assymptotic curve for each ring
  determined by Z/neffective of the ring
• At high currents,

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Measurements

• ESRF, Super-Aco, ALS, APS, Daphne, HER, ATF,
  NSLS VUV,Elettra ? strong lengthening
• Some signs of bunch shortening
• SPEAR I ,CESR , LEP (before SC cavities)
• Threshold of microwave instability
• Strong coherent signals on sync. Sidebands
• Ex SLC DR, ALS, SuperAco
• Microwave Instability Threshold in number of
  particles

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Measurements
Elettra To the courtesy of E.Karantzoulis
Daphne To the courtesy of A.Ghigo
SuperAco EPAC 98 Nadji et al.
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ALS To the courtesy of J.Byrd
Measurements
Energy Spread
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Measurements
Vrf0.84 MV
Vrf1.68 MV
Vrf3.36MV
SPEAR C.Limborg- J.Sebek 98
Signs of bunch shortening, but at low currents
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Models Methods

• ? Evolution of distribution of particles in
  phase space (?,?) with increasing current in the
  presence of short range wakefields
• Vlasov equation conservation of charges
  radiation Fokker-Planck
• Stationary solution Haissinski equation
• Linearized form Vlasov ? mode coupling theory
• Non-linearized ? numerical solvers (Warnock,
  Novokhatski
• See Warnock Al. submitted submitted Word
  Scientific Feb 26 -00
• Multiparticle Tracking codes

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Impedance models

• Impedance from codes
• Wakefield extracted from codes
• (ABCI-TBCI- MAFIA- GdFidl- Urmel...)
• ? Computing Limitations for the high frequencies
• Analytical Impedance models
• - SPEAR model 1st attempt to fit impedance
  (P.Wilson)
• - Broadband RLC
  (A.Hofmann)
• - Heifets-Bane
• Zotter review
• see http//www-project.slac.stanford.edu/lc/wkshp/
  talks

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Academic case of Z//jL?

• Haissinski equation with purely inductive
  Z//
• ?There exists a solution ? ?gt0
• ? No solution for ?lt -1.55
• Interest of Purely inductive impedance
• Fits bunch lengthening curves
• Good benchmark for test numerical noise (tracking
  code solvers)

for ? gt 0, stable NO ?? increase
for ? lt 0, Negative mass instability STRONG ??
increases
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Broadband impedance

• Handy model analytically (Rs, fr, Q1)
• Tracking code
• Bunch Spectrum vs Resonant frequency
• 16 000 particles in 200 cells over ? 7 ??o
• I 3mA ?? 150ps

Radiation Damping
Fluctuations
RF Voltage -losses
Variation Energy
Wakefield
Variation Path Length
fr 30 GHz, ?? 5 ?r fr 15 GHz, ?? 2.5 ?r
fr 7 GHz, ?? 0.9 ?r fr 3.5 GHz, ?? 0.5
?r
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Mode Coupling theory

• A.Mosnier proved good agreement of thresholds
  between tracking and mode coupling theory
• p.w distorsion from Haissinski for stationary
  distribution
• uses Oide-Yokoya radial step function expansion
• for determining the stability of modes
• compares threshold with tracking code results
  (good agreement)
• - spread in fs
• - eventual presence of 2 bunchlets

fr??gt1, azimuthal mode coupling before
radial fr??lt1, radial mode coupling, sub-bunches
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• K.Bane simulations exhibit quadrupole form of
  perturbation
• (but 3 of total intensity)

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A Few other mechanisms

• Dyachkov-Baartman model of sawtooth 1stable
  fixed point -
• 1 unstable fixed point
• diffusion from u. to s.
• followed by collapse of the 2
• A controlled instability
• Modulation of RF voltage
• Byrd-Zimmerman experiment-
• Huang- Li et al PhysRev

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Observe enhanced emission from NSLS VUV ring at 7
mm wavelength - To the courtesy of J.Murphy
Emission occurs after a current threshold Ith is
exceeded, grows as (I - Ith)2.
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• Emission is not continuous, but occurs in
  quasi-periodic bursts.period 1 to 10 ms
  rise/fall times faster than synchrotron damping
  time.

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Ith varies linearly (quadratically) with (fs0).
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For discussion in W.G

• Quadrupole feedback at Super-Aco
• Stabilization with FEL operation
• Effect of bunch lengthening cavities on Ith
  (J.Jacob, A.Mosnier)
• Do Harmonic cavities help for
• - Pushing the threshold of energy widening?
• - Improve the better than in I 2/3 ?
• Computing Limitations of e.m structures codes
• How many Broadband resonator for a realistic
  wakefield?
• Probing high frequencies on existing rings3mm ?
  100 GHz
• (limit of S.Analyzers and strong problem of
  attenuation along cables)

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References
Longitudinal Dynamics Hofmann Single-beam
collective phenomena- Longitudinal CERN 77-13
CAS lectures Besnier Longitudinal Stability
PhD thesis, Rennes 1978 Laclare Bunched beam
coherent Instabilities CERN 87-03 CAS
lectures Oide-Yokoya Longitudinal Single Bunch
Instability in e storage rings KEK Preprint
90-10 Mosnier Microwave Instability and
impedance model PAC 99 Bane Low and Negative
Momentum Compaction C.Pellegrini, D.Robin
Quasi-Isochronous storage Rings
Nucl.Inst.Methods A301,27-36,1991 Nadji- Level
Experiments with low and lt0 ? with Super-Aco
EPAC 96 Limborg A Review of Diffculties in
Achieving Short Bunches in Storage Rings EPAC
98 Limborg Ultimate Brilliance of Storage Ring
Based Synchrotron Radiation Facilities of the 3rd
Generation- Potential of Storage Ring Based
Sources in the production of Short and Intense
X-ray Pulses PhD ESRF, Grenoble, 1996 Sawtooth
Instability Dyachkov-Baartman simulaiton of
sawtooth Instability PAC 95 Bane Simulations of
the Longitudinal Instability in the SLC Damping
Rings PAC 93 Podobedov Longitudinal Dynamics in
The SLC Damping Rings PhD Dec 1999
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References
Non-Linear Dynamics Byrd Non-linear
Longitudinal studies at ALSPAC99 Huang et al.
Experimental determination of the Hamiltonian
for synchrotorn motion with RF phase modulation
Phys Rev.E Vol48, Num.6 Dec 93 Vlasov equation
Solvers Warnock-Ellison A general method for
propagation of the phase space distribution, with
application to the sawtooth instability
Submitted to World Scientific Feb 26 2000
Novokhatski SLC ring simulations Proceedings
Impedance Workshop SLAC Feb 2000 Impedances Hofm
ann Improved impedance models for High Enrgy
Accelerator CERN, LEP Note 1979 Zotter- Kheifets
Impedance and Wakes in High-Energy Particle
Accelerator World Scientific Publishing
1998 Palumbo-Vaccaro Wakefields, Impedances and
Greenfunction CERN 87-03 CAS lectures
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Acknowledgements
ESRF (Nagaoka- Farvacque- Revol- Ropert- Gunzel-
Besnier) CEA ( Mosnier- Laclare) Super-Aco
(Nadji, Level, Couprie, Flynn) Elettra
(Karantzoulis) APS (Harkay, Lumpkin, Emery
) NSLS (Murphy, Podobedov) ALS (Byrd) SLAC
(Heifets- Bane- Krejcik) CERN (Hofmann-
Zotter) SSRL (Sebek) Daphne (Ghigo)
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Mode Coupling ambiguities
Radial-Azimuthal modes not well suited for some
impedance models Too Small perturbations w.r.t
streak camera data Does not exhibit the
importance of synchrotron motion and damping in
mechanisms REMOVE THIS SLIDE, FOR W.G