Analytical Estimation of Dynamic Aperture Limited by Wigglers in a Storage Ring

1
Analytical Estimation of Dynamic
Aperture Limited by Wigglers in a Storage Ring

• ?? J. Gao
• ??
• Laboratoire de LAccélérateur Linéaire
• CNRS-IN2P3, FRANCE
• KEK, Feb. 24 2004

2
Contents

• Dynamic Apertures of Limited by Multipoles in a
  Storage Ring
• Dynamic Apertures Limited by Wigglers
• in a Storage Ring
• Discussions
• Perspective
• Conclusions
• References
• Acknowledgement

3
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).

Eq. 1
4
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

5
Standard Mapping

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

6
Some explanations

• Definition of TWIST MAP

where
7
Some explanations

• Classification of various orbits in a Twist Map,
• Standard Map is a special case of a Twist
  Map.

8
Stochastic motions

• For Standard Mapping, when
  global 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 .

9
m4 Octupole as an example

• Step 1) Let m4 in , and use canonical
  variables obtained from the unperturbed problem.
• Step 2) Integrate the Hamiltonian differential
  equation over a natural periodicity of L, the
  circumference of the ring

Eq. 1
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m4 Octupole as an example

• Step 3)

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m4 Octupole as an example

• Step 4)

One gets finally
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General Formulae for the Dynamic Apertures of
Multipoles

Eq. 2
Eq. 3
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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
17
Wiggler

• Ideal wiggler magnetic fields

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Hamiltonian describing particles motion
where
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Particles transverse motion after averaging over
one wiggler period
In the following we consider plane wiggler with
Kx0
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One cell wiggler

• One cell wiggler Hamiltonian
• After comparing with one gets
• one cell wiggler limited
• dynamic aperture

Eq. 4
Eq. 4
Eq. 1
Using one gets
Eq. 2
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A full wiggler

• Using one finds dynamic aperture for a
  full wiggler
• or approximately
• where is the beta function in the
  middle of the wiggler

Eq. 3
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Multi-wigglers

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

23
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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Discussions

• The method used here is very
• general and the analytical results
• have found many applications in
• solving problems such as beam-beam
• effects, bunch lengthening, halo
• formation in proton linacs, etc

30
Maximum Beam-Beam Parameter in ee- 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

35
Dynamic apertures limited by beam-beam
interactions

• Three cases
• Beam-beam effect limited lifetime

(RB)
(FB)
(FB)
36
Recall of Beam-beam tune shift definitions
37
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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First limit of beam-beam tune shift (lepton
machine)

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

41
Second limit of beam-beam tune shift (lepton
machine)

• Flat beam V plane

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Some Examples

• DAFNE E0.51GeV,xymax,theory0.043,xymax,exp0.02
  
• BEPC E1.89GeV,xymax,theory0.039,xymax,exp0.029
  
• PEP-II Low energy ring E3.12GeV,xymax,theory0.0
  63,xymax,exp0.06
• KEK-B Low energy ring E3.5GeV,xymax,theory0.083
  2,xymax,exp0.069
• LEP-II E91.5GeV,xymax,theory0.071,xymax,exp0.0
  7

43
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

47
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

49
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
52
Perspective

• It is interesting and important to study the
  tail distribution analytically using the discrete
  time statistical dynamics, technically to say,
  using Perron-Frobenius operator.

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Conclusions

• 1) Analytical formulae for the dynamic
• apertures limited by multipoles in general
• in a storage ring are derived.
• 2) Analytical formulae for the dynamic apertures
  limited by wigglers in a storage ring are
  derived.
• 3) Both sets of formulae are checked with
• numerical simulation results.
• 4) These analytical formulae are useful both for
  experimentalists and theorists in any sense.

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References

• R.Z. Sagdeev, D.A. Usikov, and G.M. Zaslavsky,
  Nonlinear Physics, from the pendulum to
  turbulence and chaos, Harwood Academic
  Publishers, 1988.
• R. Balescu, Statistical dynamics, matter our of
  equilibrium, Imperial College Press, 1997.
• J. Gao, Analytical estimation on the dynamic
  apertures of circular accelerators, NIM-A451
  (2000), p. 545.
• J. Gao, Analytical estimation of dynamic
  apertures limited by the wigglers in storage
  rings, NIM-A516 (2004), p. 243.

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Acknowledgement

• Thanks go to Dr. Junji Urakawa for
• inviting the speaker to work on ATF
• at KEK, and to have this opportunity
• to make scientific exchange with you
• all, i.e.????.