Analytical Estimation of Dynamic Aperture Limited by Wigglers in a Storage Ring 高傑 J. Gao 弘毅 Laboratoire de L’Accélérateur Linéaire CNRS-IN2P3, FRANCE KEK, Feb

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

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). Eq. 1

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

Some explanations Definition of TWIST MAP where

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

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.

m=4 Octupole as an example Step 1) Let m=4 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

m=4 Octupole as an example Step 3)

m=4 Octupole as an example Step 4) One gets finally

General Formulae for the Dynamic Apertures of Multipoles Eq. 2 Eq. 3

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

Hamiltonian describing particle’s motion where

Particle’s transverse motion after averaging over one wiggler period In the following we consider plane wiggler with K x =0

One cell wiggler One cell wiggler Hamiltonian After comparing with one gets one cell wiggler limited dynamic aperture Eq. 4 Eq. 1 Using one getsEq. 2

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

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)

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…

Maximum Beam-Beam Parameter in e+e- 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

First limit of beam-beam tune shift (lepton machine) or, for an isomagnetic machine where H o =2845 *These expressions are derived from emittance blow up mechanism

Second limit of beam-beam tune shift (lepton machine) Flat beam V plane

Some Examples DAFNE: E=0.51GeV,  ymax, theory =0.043,  ymax,exp =0.02 BEPC: E=1.89GeV,  ymax, theory =0.039,  ymax,exp =0.029 PEP-II Low energy ring: E=3.12GeV,  ymax, theory =0.063,  ymax,exp =0.06 KEK-B Low energy ring: E=3.5GeV,  ymax, theory =0.0832,  ymax,exp =0.069 LEP-II: E=91.5GeV,  ymax, theory =0.071,  ymax,exp =0.07

Some Examples (continued) PEP-II High energy ring: E=8.99GeV,  ymax, theory =0.048,  ymax,exp =0.048 KEK-B High energy ring: E=8GeV,  ymax, theory =0.0533,  ymax,exp =0.05

Beam-beam effects with crossing angle Horizontal motion Hamiltonian Dynamic aperture limited by synchro- betatron coupling

Crossing angle effect Dynamic aperture limited by synchro-betatron coupling Total beam-beam limited dynamic aperture Where is Piwinski angle

KEK-B with crossing angle KEK-B luminosity reduction vs Piwinski angle

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

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.

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.

References 1)R.Z. Sagdeev, D.A. Usikov, and G.M. Zaslavsky, “Nonlinear Physics, from the pendulum to turbulence and chaos”, Harwood Academic Publishers, )R. Balescu, “Statistical dynamics, matter our of equilibrium”, Imperial College Press, )J. Gao, “Analytical estimation on the dynamic apertures of circular accelerators”, NIM-A451 (2000), p )J. Gao, “Analytical estimation of dynamic apertures limited by the wigglers in storage rings, NIM-A516 (2004), p. 243.

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. 以文会友.