Analytical Treatment of Some Nonlinear Beam Dynamics Problems in Storage Rings

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Analytical Treatment of Some Nonlinear
Beam Dynamics Problems in Storage Rings

• J. Gao
• Laboratoire de LAccélérateur Linéaire
• CNRS-IN2P3, FRANCE
• 30th Advanced ICFA Beam Dynamics Workshop on High
  Luminosity ee- Colliders
• Stanford, California, Oct. 13-16, 2003.

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Contents

• Dynamic Apertures of Multipoles in a Storage Ring
• Dynamic Apertures limited by Wigglers
• Limitations on Luminosities in Lepton Circular
  Colliders from Beam-Beam Effects
• Nonlinear Space Charge Effect
• Nonlinear electron cloud effect

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Dynamic Aperturs of Multipoles

• Hamiltonian of a single multipole
• Where L is the circumference of the storage
  ring, and s is the place where the multipole
  locates (m3 corresponds to a sextupole, for
  example).

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Important Steps to Treat the Perturbed
Hamiltonian

• Using action-angle variables
• Hamiltonian differential equations should be
  replaced by difference equations
• Since under some conditions the Hamiltonian
  dont have even numerical solutions

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Standard Mapping

• Near the nonlinear resonance, simplify the
  difference equations to the form of STANDARD
  MAPPING

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Stochastic motions

• When stochastic motion
  starts. Statistical descriptions of the nonlinear
  chaotic motions of particles are subjects of
  research nowadays. As a preliminary method, one
  can resort to Fokker-Planck equation .

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General Formulae for the Dynamic Apertures of
Multipoles

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Super-ACO
Lattice
Working point
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Single octupole limited dynamic aperture
simulated by using BETA
x-y plane
x-xp phase plane
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Comparisions between analytical and numerical
results
Sextupole
Octupole
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2D dynamic apertures of a sextupole
Simulation result
Analytical result
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Wiggler

• Ideal wiggler magnetic fields

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One cell wiggler

• One cell wiggler Hamiltonian
• One cell wiggler limited dynamic aperture

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Full wiggler and multi-wigglers

• Dynamic aperture for a full wiggler
• or approximately
• where is the beta function in the
  middle of the wiggler

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Full wiggler and multi-wigglers

• Many wigglers (M)
• Dynamic aperture in horizontal plane

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Numerical example Super-ACO

• Super-ACO lattice with wiggler switched off

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Super-ACO (one wiggler)
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Super-ACO (one wiggler)
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Super-ACO (one wiggler)
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Super-ACO (one wiggler)
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Super-ACO (two wigglers)
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Maximum Beam-Beam Tune Shift in
Circular Colliders

• Luminosity of a circular collider

where
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Beam-beam interactions

• Kicks from beam-beam interaction at IP

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Beam-beam effects on a beam

• We study three cases

(RB)
(FB)
(FB)
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Round colliding beam

• Hamiltonian

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Flat colliding beams

• Hamiltonians

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Dynamic apertures limited by beam-beam
interactions

• Three cases
• Beam-beam effect limited lifetime

(RB)
(FB)
(FB)
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Recall of Beam-beam tune shift definitions
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Beam-beam effects limited beam lifetimes

• Round beam
• Flat beam H plane
• Flat beam V plane

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Important finding

• Defining normalized beam-beam effect limited
  beam lifetime as
• An important fact has been discovered that
  the beam-beam effect limited normalized beam
  lifetime depends on only one parameter linear
  beam-beam tune shift.

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Theoretical predictions for beam-beam tune shifts
Relation between round and flat colliding beams
For example
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The roles for higher order poles

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First limit of beam-beam tune shift (lepton
machine)

• or, for an isomagnetic machine
• where
• Ho2845
• These expersions are derived from emittance blow
  up mechanism

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Second limit of beam-beam tune shift (lepton
machine)

• Flat beam V plane
• where xy should be replaced by
• 0.0447 xy / xy,max,1

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Some Examples

• DAFNE E0.51GeV,xymax,theory0.043,xymax,exp0.02
  
• BEPC E1.89GeV,xymax,theory0.04,xymax,exp0.04
• PEP-II Low energy ring E3.12GeV,xymax,theory0.0
  63,xymax,exp0.06
• KEK-B Low energy ring (with crossing angle!)
  E3.5GeV,xymax,theory0.0832,xymax,exp0.069
• CESR E5.3GeV,xymax,theory0.048,xymax,exp0.025
• LEP-II E91.5GeV,xymax,theory0.071,xymax,exp0.0
  7

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Some Examples (continued)

• PEP-II High energy ring E8.99GeV,xymax,theory0.
  048,xymax,exp0.048
• KEK-B High energy ring E8GeV,xymax,theory0.0533
  ,xymax,exp0.05

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Beam-beam effects with crossing angle

• Horizontal motion Hamiltonian
• Dynamic aperture limited by synchro-betatron
  coupling

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Crossing angle effect

• Dynamic aperture limited by synchro-betatron
  coupling
• Total beam-beam limited dynamic aperture

Where
is Piwinski angle
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KEK-B with crossing angle

• KEK-B luminosity reduction vs Piwinski angle

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The Limitation from Space Charge Forces to TESLA
Dog-Borne Damping Ring

• Total space charge tune shift
• Differential space charge tune shift
• Beam-beam tune shift

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Space charge effect

• Relation between differential space charge and
  beam-beam forces

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Space charge effect limited dynamic apertures
Dynamic aperture limited by differential space
charge effect
Dynamic aperture limited by the total space
charge effect
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Space charge limited lifetime

• Space charge effect limited lifetime expressions
• Particle survival ratio

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TESLA Dog-Borne damping ring as an example

• Particle survival ratio vs linear space charge
  tune shift when the particles are ejected from
  the damping ring.

TESLA parameters
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Nonlinear electron cloud effect

• Relation between differential electron cloud and
  beam-beam forces

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Nonlinear electron cloud effect

• Normalized dynamic aperture due to electron cloud

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Combined nonlinear beam-beam and electron cloud
effect

• Normalized dynamic aperture due to combined
  beam-beam and electron cloud effects

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Combined nonlinear beam-beam and electron cloud
effect

• Beam lifetime due to the combined effect
• where is the damping time of positron
  in the vertical plane

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PEP-II positron ring as an example

• Machine parameter

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PEP-II positron ring as an example

• Machine parameter

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PEP-II positron ring as an example

• If the beam-beam alone limited maximum beam-beam
  tune shift is
• with
• the maximum beam-beam tune shift will be
  reduced to

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Conclusion

• Various nonlinear effects are the main limiting
  factors to the performance of storage rings.
• In addition to numerical simulations, analytical
  treatments are very helpful in understanding the
  physics behind the phenomena, are very economic.