Polarized Beam Acceleration

1. Polarized Beam Acceleration In their seminal paper “Radiative Polarization: Obtaining, Control, Using” Part. Accel. 8 (1978) 115 Ya. S. Derbenev, A.M. Kondratenko, et al. laid out the path for high energy polarized beam acceleration followed for the next thirty years.

2. Spin Dynamics in Rings • Precession Equation in Laboratory Frame: • (Thomas [1927], Bargmann, Michel, Telegdi [1959]) • dS/dt = - (e/gm) [(1+Gg)B + (1+G) BII] S • Lorentz Force equation: • dv/dt = - (e/gm) [ B ] v • For pure vertical field:Spin rotates Gg times faster than motion, nsp = Gg • For spin manipulation:At low energy, use longitudinal fieldsAt high energy, use transverse fields

3. Spin Tune and Depolarizing Resonances Depolarizing resonance condition: Number of spin rotations per turn = Number of spin kicks per turn Spin resonance strength e = spin rotation per turn / 2p Imperfection resonance (magnet errors and misalignments): nsp = n Intrinsic resonance (Vertical focusing fields): nsp = Pn± Qy P: Superperiodicity [AGS: 12] Qy: Betatron tune [AGS: 8.75] Weak resonances: some depolarization Strong resonances: partial or complete spin flip Illustration by W.W. MacKay

4. Spin Resonance Crossing Froissart-Stora: [a: crossing speed] Non-adiabatic (e2/a << 1)  Adiabatic (e2/a >> 1) Pf /Pi = 1 Pf /Pi = - 1 e Gg-K Gg-K e e K-Gg K-Gg e Gg=K

5. Spin Resonance Crossing (cont’d) Imperfection Resonances: Correction Dipoles (e small) Partial Snake (e large) Intrinsic Resonances: Pulsed Quadrupoles (a large) RF Dipole (e large) Lattice modifications (e small) Strong Partial Snake (e large)

6. Siberian Snakes (Local Spin Rotators) cos(180 nsp) = cos(d/2) · cos(180 Gg) d 0 nsp  n No imperfection resonances Partial Siberian snake (AGS) d= 180 nsp = ½ No imperfection resonances and No Intrinsic resonances Full Siberian Snake (Ya.S. Derbenev and A.M. Kondratenko) Two Siberian Snakes (RHIC): nsp = (a2-a1)/180(a1,2: angles between snake axis and beam direction) Orthogonal snake axis: nsp = ½ and independent of beam emittance (SRM, S. Mane)

7. 2.6 m 2.6 m Siberian Snakes • AGS Siberian Snakes: variable twist helical dipoles, 1.5 T (RT) and 3 T (SC), 2.6 m long • RHIC Siberian Snakes: 4 SC helical dipoles, 4 T, each 2.4 m long and full 360 twist

8. Polarimeter asymmetry Gg Polarized Protons in the AGS • Two strong partial Siberian snakes • Vertical betatron tune at 8.98 • Pulsed quadrupoles to jump across the many weak horizontal spin resonances driven by the partial snakes. n+νy n+νx n-νx νsp n-νy n n+1 Gg

9. Spin Resonances in RHIC w/o Snakes Imperfection resonance strength for uncorrected orbit (s = 28 mm) Intrinsic resonance strength for 10 p mm mrad particle Imperfection resonance strength for corrected orbit (s = 0.15 mm)

10. Beam Polarization Near a Single Strong Intrinsic Resonance Without snakes: spin flip, width ~ ± 5e With snakes: opening/closing of “spin cone”, nodes at ± 2 With Snakes:Resonance crossing during acceleration is adiabatic with no polarization loss. Resonance strength e = 0.3, 0.6 Gg DGg

11. 1/12 3/10 1/8 3/8 1/6 1/10 1/4 Stable polarization on resonance, e = 0.3 0.9 0.8 0.7 0.6 Snake Resonances • Higher order resonance condition nsp + mQy = k (m, k = integer) driven by interaction of intrinsic resonance Gg + Qy = k with large spin rotations of dipoles and snakes. • “Snake resonance strength” depends on intrinsic resonance strength and therefore energy • For nsp=1/2+Dnsp Qy = (2k-1)/2m-Dnsp/m • First analytical solution of isolated resonance with snakes by S.R. Mane, NIM A 498 (2003) 1 single snakeor two snakes with orbit errors two snakes (m: odd) RHIC tune working point 3/14 3/10 1/10 1/6 Stable polarization on resonance, e = 0.3 0.9 0.8 0.7 0.6

12. Limits for Siberian Snakes • Spin rotation of Siberian snake (d) > Spin rotation of resonance driving fields (e) • “Spin rotation of Siberian snake drives strong imperfection resonance” • More realistically: dtot ~ 2 emax • Imperfection resonances e Energy • Intrinsic resonances eEnergy • Emax/GeV Emax/GeV • Partial Siberian snakes(AGS, d ~ 27°)e < 0.07 24 5 • One full snakee < 0.25 • Two full snakes(RHIC)e < 0.5 250 16 • 16 full snakes(LHC?)e < 4 7000 84

13. Multiple Siberian Snakes • For high energy rings with resonance strengths larger than ~ 0.5 multiple snake pairs need to be used. • Many choices of snake axis angles give nsp = ½ ! Which is best? • K. Steffen (1985) and G. Hoffstaetter (2004) proposed to choose snake axes angles to minimize spin-orbit integrals or effective intrinsic resonance strength. Possible snake axis angles for 8 snakes in ring with 4-fold symmetry (HERA-p):

14. Multiple Siberian Snakes (cont’d) • S.R. Mane showed that for a single strong intrinsic resonance the spin tune does not depend on the beam emittance if the snake axes angle increases in equal steps from one snake to the next. This may be a good starting point for a multiple snake design. 2 Snakes (RHIC)Da = 90 4 Snakes (HERA-p?)Da = 45 6 Snakes (Tevatron?)Da = 30, 90 16 Snakes (LHC, replace 2 dipoles per arc with snakes, DE ~ 1.4%)Da = 11.25, 33.75, 56.25, 78.75 8 Snakes (HERA-p?)Da = 22.5, 67.5

15. Orbit going through center of BPM’s Orbit without kicks Quad Corrector BPM Global imperfection resonances – ultimate energy limit? • Residual orbit distortion after orbit correction drives imperfection resonance with a strength that is not affected by (multiple) Siberian snakes • Resonance strength needs to be less than 0.05 ( S. Y. Lee and E. D. Courant, Phys. Rev. D 41, 292 (1990)) • At RHIC (250 GeV) this corresponds to ~250 mm residual orbit error (OK) • At LHC (7 TeV) this corresponds to ~10 mm residual orbit error ! (LHC orbit accuracy ~ 200 mm) • Need beam based quadrupole offset measurement, using trim-quadrupoles (?) • Flatten actual beam orbit using H,V - beam position monitors () and correctors (no) at each quadrupole: Correct orbit to minimize kicks:

16. Summary • Polarized beam acceleration has followed the path that Slava has laid out more than 30 years ago. • Happy Birthday!