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

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1 Analtical Estimation of Dnamic Aperture Limited b Wigglers in a Storage Ring J. GAO Institute of High Energ Phsics Chinese Academ of Sciences Snomass ILC orshop, August 4-7, 005

2 Contents Dnamic Apertures of Limited b Multipoles in a Storage Ring Dnamic Apertures Limited b Wigglers in a Storage Ring Application to TESLA damping ring Conclusions

3 Dnamic Aperturs of Multipoles H Hamiltonian of a single multipole m p K s = + + B z m m L ( s* L m! ( δ Bρ = Eq.. Where L is the circumference of the storage ring, and s* is the place here the multipole locates (m=3 corresponds to a setupole, for eample.

4 Important Steps to Treat the Perturbed Hamiltonian Using action-angle variables Hamiltonian differential equations should be replaced b difference equations dq = H dt p dp = H dt q Since under some conditions the Hamiltonian don t have even numerical solutions

5 Standard Map Near the nonlinear resonance, simplif the difference equations to the form of STANDARD MAP I = I + K 0 sin θ θ = θ + I

6 Some eplanations Definition of TWIST MAP = + Kf (θ θ = θ + g ( (mod here f ( θ + = f ( θ dg ( d 0,

7 Some eplanations Classification of various orbits in a Tist Map, Standard Map is a special case of a Tist Map.

8 Stochastic motions For Standard Map, K hen global stochastic motion starts. Statistical descriptions of the nonlinear chaotic motions of particles are subjects of research noadas. As a preliminar method, one can resort to Foer-Planc equation.

9 m=4 Octupole as an eample Step Let m=4 in Eq., and use canonical variables obtained from the unperturbed problem. Step Integrate the Hamiltonian differential equation over a natural periodicit of L, the circumference of the ring

10 m=4 Octupole as an eample Step 3 4 sin Φ + = A J J J B + Φ = Φ = = ρ β L b s J A m 3 4 ( = = ρ β L b s B m 3 4 ( K 4 AB 0 =

11 m=4 Octupole as an eample Step 4 ( < = AB K < = L b s J m ( 3 4 ρ β / / /,, ( ( ( ( ( = = = = L b s s s s J A m m oct dna ρ β β β β One gets finall

12 A General Formulae for the Dnamic Apertures of Multipoles dna,m = β ( s mβ m ( s(m ( m b ρ L m m A dnatotal, = A + + A A i j dnaseti,, dnaoct,, j dna, set, β ( s = β ( s A dnadeca,, ( A dna, set,... Eq. Eq. 3

13 Super-ACO Lattice Woring point

14 Single octupole limited dnamic aperture simulated b using BETA - plane -p phase plane

15 Comparisions beteen analtical and numerical results Setupole Octupole

16 D dnamic apertures of a setupole Simulation result Analtical result

17 Wiggler Ideal iggler magnetic fields B = B0 sinh( sinh( cos( s B = B0 cosh( cosh( cos( s B z = B0cosh( sinh( sin( s + = = λ π

18 Hamiltonian describing particle s motion H = ( p A sin( s ( p A sin( s z + + ( p here A = ρ cosh( cosh( A = sinh( sinh( ρ

19 Particle s transverse motion after averaging over one iggler period 3 ( 3 ds d + + = ρ 3 ( 3 ds d + + = ρ In the folloing e consider plane iggler ith K=0

20 H One cell iggler One cell iggler Hamiltonian ( s il ρ Eq. 4 4 = + +, H0 λ δ 4 ρ i= After comparing Eq. 4 ith Eq. one gets b ρ 3 L 3 ρ Using Eq. one gets one cell iggler limited dnamic aperture A = ( s ( s 3 λ, ( = s β β ρ λ /

21 A full iggler Using Eq. 3 one finds dnamic aperture for a full iggler N N λ = = ( β si, ( i i, i 3 ρβ ( N = = A s A s N, or approimatel β A here,m is the beta function in the middle of the iggler N, ( s = 3β ( s β, m ρ L

22 Multi-igglers Man igglers (M A total, ( s = M + j = j, A ( s A, ( s Dnamic aperture in horizontal plane A dna = β β ( A, m, igl, dna, igl,, m

23 Numerical eample: Super-ACO Super-ACO lattice ith iggler sitched off

24 Super-ACO (one iggler ρ ( m =.7 A, n( m = A, a( m = β ( m 3 ( m = , m = l L( m =

25 Super-ACO (one iggler ρ ( m = 3 A, n( m = A, a( m = β ( m 0.7 ( m = , m = l L( m =

26 Super-ACO (one iggler ρ ( m = 4 A, n( m = A, a( m = β ( m 9.5 ( m = , m = l L( m =

27 Super-ACO (one iggler ρ ( m = 4, m( m = 9.5 β L( m = l l l ( m = ( m = ( m = , ( m = 0.06 A, a( m = A n ( m = 0.033, A, a( m = A n ( m = 0.067, A, a( m = A n

28 Super-ACO (to igglers ρ ( m = 6 A, n( m = A, a( m = β ( m 3.75 ( m = , m = l L( m =

29 Application to TESLA Damping Ring E=5GeV Bo=.68T λ = 0.4 N = β = 9m, β = 5, m m (at the entrance of the iggler (at the eit of the iggler The total number of igglers in the damping ring is 45. The vertical dnamic aperture due to 45 iggler is = Atotal.cm,

30 Conclusions Analtical formulae for the dnamic apertures limited b multipoles in general in a storage ring are derived. Analtical formulae for the dnamic apertures limited b igglers in a storage ring are derived. 3 Both sets of formulae are checed ith numerical simulation results. 4 These analtical formulae are useful both for eperimentalists and theorists in an sense.

31 References R.Z. Sagdeev, D.A. Usiov, and G.M. Zaslavs, Nonlinear Phsics, from the pendulum to turbulence and chaos, Harood Academic Publishers, 988. R. Balescu, Statistical dnamics, matter our of equilibrium, Imperial College Press, J. Gao, Analtical estimation on the dnamic apertures of circular accelerators, NIM-A45 (000, p J. Gao, Analtical estimation of dnamic apertures limited b the igglers in storage rings, NIM-A56 (004, p. 43.

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