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

2. 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:
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.
U. Wienands, SLAC-ASD
SuperB WS, Elba, 1-Jun-08

3. Introduction (cont’d)
Polarization build-up time for SuperB:
HER: g = (7 GeV), r = 110 m, R = 263 m: 5…6 h
> inject polarized electrons into HER.
A polarized source of 15 nC/sec is needed to maintain beam current in the SuperB HER. Sources like this are available The SLC gun e.g. delivers 15 nC=1E11 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 SuperB proposal at present.
U. Wienands, SLAC-ASD
SuperB WS, Elba, 1-Jun-08

4. 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”, =gG for a flat ring, # spin precessions per turn.
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 ring (vertical orbit & correctors, detector solenoid, vertical betatron oscillations). d tends to large values at spin tune near integer values. ^  ^ ^  ^
G=(g-2)/2≈0.0012, gG(7 GeV) ≈ 16, for electrons   
U. Wienands, SLAC-ASD
SuperB WS, Elba, 1-Jun-08

5. 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 gG*BL/Br
solenoids introduce x–y coupling. 
U. Wienands, SLAC-ASD
SuperB WS, Elba, 1-Jun-08

6. Solenoid Rotator
spin=(1+G)*BL/(Br) => 36.6 Tm for 90° spin rotation
2.5 T field => m total length, 30E6 Amp turns
Dipole: spin=(gG)*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
SuperB WS, Elba, 1-Jun-08

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

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

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

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

11. 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
SuperB WS, Elba, 1-Jun-08

12. 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.
Like the symmetric solenoid rotators it adds rotation, but d stays 0.
May have somewhat more intrinsic depolarization from the dipoles than the solenoid rotator, but likely to be still small effect.
U. Wienands, SLAC-ASD
SuperB WS, Elba, 1-Jun-08

13. 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
SuperB WS, Elba, 1-Jun-08

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

15. Polarization Estimate for SuperB
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. ^
Preliminary!
U. Wienands, SLAC-ASD
SuperB WS, Elba, 1-Jun-08

16. Polarization Estimate for SuperB
Reasonable spin match (antisymmetric solenoid rotator, detector left uncompensated). 20 min. beam lifetime. Pinj = 0.9
Note:
Energy spread & synchr. oscillation not incl.
Preliminary!
U. Wienands, SLAC-ASD
SuperB WS, Elba, 1-Jun-08

17. Beyond these integer spin resonances, there are other effects limiting the energy choice:
“intrinsic” resonances, where k*nspin = ny
in case of SuperB 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, and GeV.
Corresponding LER energies (for Ecm=10.58 GeV) are 4.091, 3.969, GeV.
U. Wienands, SLAC-ASD
SuperB WS, Elba, 1-Jun-08

18. Alternate Schemes Bogomyagkov, Nikitin: Koop:
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
SuperB WS, Elba, 1-Jun-08

19. Spin Rotator Summary Comments
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
SuperB WS, Elba, 1-Jun-08

20. 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
SuperB WS, Elba, 1-Jun-08

21. Summary
Polarized beam in SuperB 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
SuperB WS, Elba, 1-Jun-08

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

23. Polarization scheme with odd number of Siberian Snakes in a ring
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:
I. Koop
U. Wienands, SLAC-ASD
SuperB WS, Elba, 1-Jun-08

24. 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 while other 6 arcs bends equals to So, the depolarization time at 7 GeV for the last case equals to:
I. Koop
U. Wienands, SLAC-ASD
SuperB WS, Elba, 1-Jun-08

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

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

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

28. Siberian Snake Parameters
Siberian Snake of the presented here example consists of two 10 m long solenoids with B= T and 7 usual quadrupole lenses in between, see the Fig.4. The length of one insertion is 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).
I. Koop
U. Wienands, SLAC-ASD
SuperB WS, Elba, 1-Jun-08

29. Siberian Snake Layout QF1 QD1 QF2 QD2 I. Koop Solenoid Solenoid
U. Wienands, SLAC-ASD
SuperB WS, Elba, 1-Jun-08

30. Conclusion (7-Snakes)
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 GeV seven Snakes are required to ensure tau >1000 s.
At 7 GeV 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 GeV.
I. Koop
U. Wienands, SLAC-ASD
SuperB WS, Elba, 1-Jun-08