9 November 00 AGS Polarization Workshop Matching of Siberian Snakes snake snake???
Driven spin perturbation on a traectory Integer values of spin-tune n tune n y lead to coherent disturbances of spin motion ± Remedy: Siberian Snakes avoid resonances by making the spin-tune n = ½ independent of energy.
Equation of motion for spin s r N fields r r f ( z, θ ) r r Spin field: Spin direction f ( z, θ ) for each phase space point d r S dθ d r f dθ r r r = Ω( z, θ ) S r r r r r r = f + [ v( z, θ ) ] f = Ω( z, θ ) θ r z r f z r
The Invariant Spin Field A) Maximum polarization: P lim = Phase space For a large divergence, the average polarization is small, even if the local polarization is 00%. B) S is an adiabatic invariance! C) Defines an amplitude dependent spin tune!
First Order TheoriesA) DESY III Isolated resonance model: Spectrum Linear spin-field theory: P lim Low energies: First order theories agree Momentum (GeV/c)
First Order TheoriesB) PETRA Isolated resonance model: Spectrum Linear spin-field theory: P lim 3 0 3 34 36 38 Momentum (GeV/c) Medium energies: resonances still isolated
Isolated resonance model: Spectrum First Order TheoriesC) HERA Linear spin-field theory: P lim High energies: Resonances are no longer isolated. Momentum (GeV/c) Isolated resonance model becomes invalid
Polarized Deuterons depolarizing resonance strength depolarizing resonance strength Resonances are 5 times weaker and 5 times rarer for D than for p Transverse polarization could be achieved without Siberian Snakes Transverse RF dipoles could be used to rotate and stabilize longitudinal polarization
Siberian Snakes Siberian Snakes rotate spins at each energy ½ times snake snake??? Freedom: direction of the rotation axis in the horizontal
CO spin motion with N Siberian Snake ] ) sin( ) [cos( ) sin cos )( sin cos ( ) sin cos ( 3 3 3 3 3 ψ ψ ψ ψ ψ ψ = = + = = + + = + = N i N N i N N i i e i e i ie A N L 3 ψ + = i i N e A π ψ ν 0 + = e y n r = r 0, to make n0 independent of energy, to make n0=0.5 = 0 ψ π =
Siberian Snakes and Resonances Some structure of the st order resonances remains after Siberian Snakes have been installed.
Spin Tune at Higher Order Resonance P lim 0.8 0.6 Spin tune n(j y ) ν =ν y ν = 38ν y ν ν = 55ν y = 6ν y 4 0.4 ν = ν y 0. (GeV/c) The spin tune n deviates from ½ for particles which (GeV/c ) oscillate around the design traectory with amplitude J y.
High Order Resonance Strength The higher order Froissart-Stora formula Resonances up to 9th order can be observed Resonance strength can be determined from tune ump. Tracked depolarization as expected P lim r r =< n( z) > Spin tune Computations performed in SPRINT, Hoffstaetter and Vogt, DESY/00
Snake matching 8 Snakes: 4 Snakes: st Order:4 harmonics of the spin perturbation in each section. With 4 snakes only can be compensated With 8 snakes all 4 can be compensated
P lim after Snake Matching 8 matched snakes 8 snakes standard scheme
Spin Tune after Snake Matching ν = Qy ν = Qy 4 snakes in standard scheme ν = Qy ν = Qy 8 matched snakes
Matching the betatron phases A proper choice of betatron phase advances allows snake matching with 4 snakes
Spin Tune after Snake Matching ν = Qy ν = Qy 4 snakes in standard scheme ν = Qy ν = Qy 4 matched snakes
Allowed Beam Sizes 8 filtered snakes 4 snakes in Standard scheme 4 filtered snakes 8 filtered snakes 4 matched snakes Snake matching allows to have significantly larger beams.
TESLA with Röntgen FEL Röntgen FEL Superconducting Electron Linac Detector and Experimental Area Wiggler for the Positron Source Cryogenic Halls Superconducting Positron Linac Tunnel Damping Ring