Polarization in Super B U Wienands SLACASD Super

Polarization in Super. B U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 1

Introduction • How to Polarize Electrons: – Use radiative polarization: • Sometimes emission of a photon induces spin flip. The slight difference in probability between parallel and antiparallel to the B-field causes a net polarization built-up: Sokolov-Ternov effect. • (DKM formula) – Inject already polarized electrons: U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 • In a planar ring, the polarization vector (“spin”) precesses (mostly) about the vertical guide field => inject vertically polarized beam • The above mentioned radiative (de-)polarization effect still applies. 2

Introduction (cont’d) – Polarization build-up time for Super. B: • HER: g = 13700 (7 Ge. V), r = 110 m, R = 263 m: 5… 6 h – > inject polarized electrons into HER. • A polarized source of 15 n. C/sec is needed to maintain beam current in the Super. B HER. Sources like this are available The SLC gun e. g. delivers 15 n. C=1 E 11 e–/pulse at 120 Hz (≈2 µA). Polarization can be up to 90%. – Polarized positrons would require a polarized positron source. This is not a part of the Super. B proposal at present. U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 3

• Stable spin direction ^ ^ • n=n(s) is the closed solution for the spin motion around the ^ ring. A polarization vector P || n remains stationary turn after ^ turn. n is usually close to vertical due to the vertical guide field. • “spin tune”, =g. G for a flat ring, # spin precessions per turn. G=(g-2)/2≈0. 0012, g. G(7 Ge. V) ≈ 16, for electrons • To maintain polarization need to watch the quantity d in the DKM formula. It quantifies the variation of the ^n-axis with momentum: Large values of d cause radiative depolarization. • d becomes non-zero due to horizontal field components in the U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 ring (vertical orbit & correctors, detector solenoid, vertical betatron oscillations). d tends to large values at spin tune near integer values. 4

Spin Rotation – Polarization in the ring will normally be vertical. But needs to be longitudinal at the IP • => spin rotators needed before and after the IP to align P longitudinally & restore to vertical. • This is achieved with dipole fields (horizontal and vertical fields) and/or with solenoids – The net rotation wanted is by 90° about the transverse horizontal axis • Most straightforward way is to use a solenoid (90° about longitudinal axis => radial polarization) followed by a horizontal dipole (90° about vertical axis => longitudinal polarization). – solenoids need to be strong since spin ≈ (1+G)*BL/Br vs g. G*BL/Br – solenoids introduce x–y coupling. U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 5

Solenoid Rotator • spin=(1+G)*BL/(Br) => 36. 6 Tm for 90° spin rotation – 2. 5 T field => 14. 66 m total length, 30 E 6 Amp turns • Dipole: spin=(g. G)*BL/Br => 2. 3 Tm, 5. 7° orbit for 90° spin • Zholents & Litvinenko have shown how to compensate the plane rotation of the solenoid by optics in between two 45° solenoids. solenoid 45° Total length ≈ 55 m/side U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 6

Original IR Insertion Optics 7 U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08

IR with Solenoid Rotator, inner W. Wittmer Note: Not a fully developed solution yet! 8 U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08

IR with Solenoid Rotator, outer W. Wittmer Dipoles made antisymm. for spin match Note: U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 Not a fully developed solution yet! 9

IR with Solenoid Rotator, outer W. Wittmer Dipoles made antisymm. for spin match • Outer s. r. avoids interference with IR optics – at a cost of much larger geometry change. • Dipole “hinges” at either end to restore geometry – DBA cells => effect on emittance likely very small. Note: Not a fully developed solution yet! U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 I P 10

Dipole Spin Rotator – Dipole rotator: • 90°(v) — 90°(h) — – 90°(v) — 90°(h) • Add a vertical “dogleg” to restore vertical elevation. – use 6*2 HER dipoles, arranged in DBA cells (per side) – need to watch vertical dispersion to maintain vertical emittance. • need ≈ 150 m space on either side, only ≈ 100 m “new” I. Koop U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 11

Polarization with Rotators • Solenoid Rotators: – A pair of antisymmetric rotators will be spin matched • True for all beam energies, but only for compensated detector sole. • Small depolarizing effect from non field-aligned spins in IR – A pair of symmetric rotators will not be spin matched • Adds a net rotation, which has no effect on energy, but does have an effect off energy since is ≠ 0 (it is about 1. 5… 2). This may (will) reduce polarization achievable. • The dipole rotator shown earlier is spin matched. U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 • Like the symmetric solenoid rotators it adds rotation, but d stays 0. • May have somewhat more intrinsic depolarization from the 12 dipoles than the solenoid rotator, but likely to be still small effect.

Expected Polarization (HER) – With trickle injection, equilibrium between decay of stored beam and build-up due to injection • For HER, assume tstor is 1 h (low current, no collisions) – optimal spin match (tpol = 5… 6 h): P ≥ 0. 85*Pinj – symmetric solenoid rotator (d≈1. 6, tpol ≈ 2 h): P ≥ 0. 67*Pinj • For HER at full collision tstor is 5 m (<0. 1 h), P ≥ 0. 95… 0. 98*Pinj • These estimates assume Peq = 0. The symmetric solenoid rotator case benefits most from Peq > 0. U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 13

Resonant Energies ^ – At integer values of the spin tune, the n-axis rotates into the horizontal plane • longitudinal polarization @ IP –> 0. • d becomes large => depolarization. – This happens every 0. 441 Ge. V near g. G = integer. – The width of these depolarizing resonances depends on the degree of spin matching achieved. U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 14

Polarization Estimate for Super. B – Equilibrium beetween injection & depolarization, Pinj = 0. 9 • Bad spin match (symmetric solenoid rotators), 20 min beam lifetime Estimate from variation ^ of d and n Energy spread & synchr. oscillation not incl. U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 Preliminary! 15

Polarization Estimate for Super. B • Reasonable spin match (antisymmetric solenoid rotator, detector left uncompensated). 20 min. beam lifetime. Pinj = 0. 9 Note: Energy spread & synchr. oscillation not incl. U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 Preliminary! 16

– Beyond these integer spin resonances, there are other effects limiting the energy choice: • “intrinsic” resonances, where k*nspin = ny – in case of Super. B ny ≈ 0. 5 => halfway between integer resonances in energy => more restrictive. • synchrotron satellites to all of these resonances, which effectively increases the width of each resonance. – Ignoring any shift from the rotators, the bad energies would be 6. 830, 7. 050 and 7. 271 Ge. V. • Corresponding LER energies (for Ecm=10. 58 Ge. V) are 4. 091, 3. 969, 3. 848 Ge. V. U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 17

Alternate Schemes • Bogomyagkov, Nikitin: – Match solenoid rotator into the local arc, without disturbing optics • Koop: – Use Siberian Snake (avoid s. r. in IR straight) – Add pairs of snakes to reduce thus maintain polarization U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 18

Spin Rotator Summary Note: Quads not enumerated No optics matching considered • Comments – Dipole rotator has v-bends => emittance? – Solenoid rotator has plane twister => tuning, emittance? – ? ? ? 7 Snakes require √(7*2) ≈ 4 times tuning effort ? ? ? U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 19

Trade-offs between Polarization & Luminosity – The solenoid rotator eschews vertical dipoles but has strong solenoids, thus introducing plane coupling. Mitigated by: • Compensation of the coupling effect by the plane twister in between the half-solenoids. • Lengthening the solenoids thus minimizing effect of the end fields. – The effect of the dipoles on emittance should be small. – The rotator adds optical elements in a critical location close to IP • Need to get phases right for crab waist & local sextupoles • Will add its own chromatic terms… how bad? ? • How sensitive will the plane-twister quad settings be? U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 20

Summary • Polarized beam in Super. B can be achieved by injection of polarized electrons. • Feasible spin-rotator designs have been proposed that will provide longitudinal polarization at the IP. – Length varies between 50 and 150 m per side of the IP – Some of these designs can provide part of the bending required to close the ring. • Good spin matching can be achieved • With trickle injection, polarization can exceed 95% of that of the injected beam. • Integration of rotator with IR critical task – Lattice space & geometry, geometric & chromatic aberrations – Compare the merits of each rotator design U. Wienands, SLAC-ASD 21 Super. B WS, Elba, 1 -Jun-08

End of Presentation END U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 22

Polarization scheme with odd number of Siberian Snakes in a ring I. Koop • The use of 1800 spin rotator, or of the so called Siberian Snake [1], has the very important advantage: being a solution, which provides perfectly aligned longitudinal spins at IP at any arbitrary energy. The only difficulty is that the depolarization time decreases with the beam energy very rapidly: U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 23

I. Koop • By inserting not a single but the odd number of Snakes one would expect increase of depolarization time quadratically with the number of Snakes. The optical structure of HER ring, being divided in seven arcs, naturally fits to the idea of use of seven Snakes, see the Fig. 1. This would result in dramatic increase of the depolarization time: by factor 72=49 in case of equal bending angles in all arcs and slightly less, approximately by factor 31, if the FF bend equals to 44. 3260 while other 6 arcs bends equals to 52. 6120. So, the depolarization time at 7 Ge. V for the last case equals to: U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 24

Layout with 7 Snakes I. Koop • Each snake: 180° spin rotator about z U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 25

Polarization vs E (7 Snakes) I. Koop The solid line corresponds to a case of equal arc sections 51. 430. The dashed line corresponds to the FF be 44. 3260 and six other arcs be 52. 6120. U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 26

Depolarization Time vs E I. Koop The solid line corresponds to a case of equal arc sections 51. 430. The dashed line corresponds to the FF be 44. 3260 and six other arcs be 52. 6120. U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 27

Siberian Snake Parameters I. Koop • Siberian Snake of the presented here example consists of two 10 m long solenoids with B=3. 6635 T and 7 usual quadrupole lenses in between, see the Fig. 4. The length of one insertion is 40. 72 m. This corresponds to 15% increase in the circumference. • The spin transparency condition requires the unity transformation matrix for the horizontal motion and the minus unity matrix for the vertical motion. This is fulfilled. Fields and gradients are listed in the Table 1. The solenoids could be switched off if polarization is not needed (last column in the Table 1). U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 28

Solenoid U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 QF 1 QD 1 QF 2 QD 2 QF 2 I. Koop QD 1 QF 1 Siberian Snake Layout Solenoid 29

Conclusion (7 -Snakes) I. Koop • The use of the odd number of Snakes greatly reduces (quadratically!) the depolarization effects from quantum fluctuations of the synchrotron radiation. • For HER-ring of Super-B at 7 Ge. V seven Snakes are required to ensure tau >1000 s. • At 7 Ge. V the needed field integral is 73. 3 T*m/snake. • Spin is directed longitudinally at IP independently of energy. • The geometry of a ring also is energy independent. • Spin tune equals exactly 0. 5 at any energy. Therefore all spin resonances are eliminated. • Same approach is applicable to the LER-ring if polarized positrons will be required at 4 Ge. V. U. Wienands, SLAC-ASD Super. B WS, Elba, 1 -Jun-08 30