Analytical Treatment of Some Nonlinear Beam Dynamics Problems in Storage Rings J. Gao Laboratoire de L’Accélérateur Linéaire CNRS-IN2P3, FRANCE Journées Accélérateurs Porquerolles, Oct. 5-7, 2003.

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

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 (m=3 corresponds to a sextupole, for example).

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 don’t have even numerical solutions

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

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.

General Formulae for the Dynamic Apertures of Multipoles

Super-ACO LatticeWorking point

Single octupole limited dynamic aperture simulated by using BETA x-y planex-xp phase plane

Comparisions between analytical and numerical results Sextupole Octupole

2D dynamic apertures of a sextupole Simulation resultAnalytical result

Wiggler Ideal wiggler magnetic fields

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

Full wiggler and multi-wigglers Dynamic aperture for a full wiggler or approximately where is the beta function in the middle of the wiggler

Full wiggler and multi-wigglers Many wigglers (M) Dynamic aperture in horizontal plane

Numerical example: Super-ACO Super-ACO lattice with wiggler switched off

Super-ACO (one wiggler)

Super-ACO (two wigglers)

Maximum Beam-Beam Tune Shift in Circular Colliders Luminosity of a circular collider where

Beam-beam interactions Kicks from beam-beam interaction at IP

Beam-beam effects on a beam We study three cases (RB) (FB)

Round colliding beam Hamiltonian

Flat colliding beams Hamiltonians

Dynamic apertures limited by beam-beam interactions Three cases Beam-beam effect limited lifetime (RB) (FB)

Recall of Beam-beam tune shift definitions

Beam-beam effects limited beam lifetimes Round beam Flat beam H plane Flat beam V plane

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.

Theoretical predictions for beam-beam tune shifts For example Relation between round and flat colliding beams

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

Space charge effect Relation between differential space charge and beam-beam forces

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

Space charge limited lifetime Space charge effect limited lifetime expressions Particle survival ratio

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

Nonlinear electron cloud effect Relation between differential electron cloud and beam-beam forces

Nonlinear electron cloud effect Normalized dynamic aperture due to electron cloud

Combined nonlinear beam- beam and electron cloud effect Normalized dynamic aperture due to combined beam-beam and electron cloud effects

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

PEP-II positron ring as an example Machine parameter

PEP-II positron ring as an example Machine parameter

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

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.