Thomas RoserDerbenev Symposium

August 2-3, 2010

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.

Spin Dynamics in Rings

Precession Equation in Laboratory Frame:(Thomas [1927], Bargmann, Michel, Telegdi [1959])

dS/dt = - (e/m) [(1+GB + (1+G) BII ] S

Lorentz Force equation:

dv/dt = - (e/m) [ B ] v

• For pure vertical field:Spin rotates Gtimes faster than motion, sp = G

• For spin manipulation:At low energy, use longitudinal fieldsAt high energy, use transverse fields

Spin Tune and Depolarizing Resonances

Depolarizing resonance condition:

Number of spin rotations per turn = Number of spin kicks per turnSpin resonance strength = spin rotation per turn / 2

Imperfection resonance (magnet errors and misalignments):

sp = n

Intrinsic resonance (Vertical focusing fields):

sp = Pn± Qy

P: Superperiodicity [AGS: 12]Qy : Betatron tune [AGS: 8.75]

Weak resonances: some depolarizationStrong resonances: partial or complete spin flip

Illustration by W.W. MacKay

Spin Resonance Crossing

Froissart-Stora: : crossing speed]

Non-adiabatic (2/1) Adiabatic (2/1)

Pf /Pi = 1 Pf /Pi = 1

122

2

- e P

P α

επ-

i

f

KG

GK

GK

KG

G=K

Spin Resonance Crossing (cont’d)

Imperfection Resonances:

Correction Dipoles (small) Partial Snake (large)

Intrinsic Resonances:

Pulsed Quadrupoles (large) RF Dipole (large)

Lattice modifications (small) Strong Partial Snake (large)

Siberian Snakes (Local Spin Rotators)

cos(180 sp) = cos(/2) · cos(180 G)

0 sp n

No imperfection resonancesPartial Siberian snake (AGS)

= 180 sp = ½

No imperfection resonances andNo Intrinsic resonancesFull Siberian Snake(Ya.S. Derbenev and A.M. Kondratenko)

Two Siberian Snakes (RHIC):sp = (/180

(: angles between snake axis and beam direction)

Orthogonal snake axis: sp = ½ and independent of beam emittance (SRM, S. Mane)

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

2.6 m 2.6 m

Polarized Protons in the AGS

G

Pol

arim

eter

asy

mm

etry

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.

Gn+1n

n-νx

n+νx

n-νy

n+νy

νsp

Spin Resonances in RHIC w/o Snakes

Intrinsic resonance strength for 10 mm mrad particle

Imperfection resonance strength for corrected orbit ( = 0.15 mm)

Imperfection resonance strength for uncorrected orbit ( = 28 mm)

Beam Polarization Near a Single Strong Intrinsic Resonance

1

1

n3 .32 Gg .3( )

n3 .32 Gg .6( )

n3ns .32 Gg .3( )

n3ns .32 Gg .6( )

55 Gg4 2 0 2 4

1

0.5

0

0.5

1

Without snakes: spin flip, width ~ ± 5With snakes: opening/closing of “spin cone”, nodes at ± 2

Resonance strength = 0.3, 0.6

G

With Snakes:Resonance crossing during acceleration is adiabatic with no polarization loss.

G

Snake Resonances

single snakeor two snakes with orbit errors

two snakes (m: odd)RHIC tune working point

1

1

n3 Q Gg0( )

0.50 Q0 0.1 0.2 0.3 0.4

1

0.5

0

0.5

1

1/6

3/14

3/10

1/10

0.60.70.80.9

Stable polarizationon resonance, = 0.3

• Higher order resonance condition sp + mQy = k (m, k = integer) driven by interaction of intrinsic resonance G + Qy = k with large spin rotations of dipoles and snakes.

• “Snake resonance strength” depends on intrinsic resonance strength and therefore energy• For sp=1/2+sp Qy = (2k-1)/2m-sp/m• First analytical solution of isolated resonance with snakes by S.R. Mane, NIM A 498

(2003) 1

1/4

3/8

1/61/10

1/8

1/12 3/10

0.60.70.80.9

Stable polarizationon resonance, = 0.3

Limits for Siberian Snakes

Spin rotation of Siberian snake () > Spin rotation of resonance driving fields ()“Spin rotation of Siberian snake drives strong imperfection resonance”

More realistically: tot ~ 2 max

Imperfection resonances Energy

Intrinsic resonances Energy

Emax/GeV Emax/GeV

Partial Siberian snakes (AGS, ~ 27° ) < 24 5

One full snake < 0.25

Two full snakes (RHIC) < 0.5 250 16

16 full snakes (LHC?) < 4 7000 84

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 sp = 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):

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) = 90

4 Snakes (HERA-p?) = 45

6 Snakes (Tevatron?) = 30, 90

8 Snakes (HERA-p?) = 22.5, 67.5

16 Snakes (LHC, replace 2 dipoles per arc with snakes, E ~ 1.4%) = 11.25, 33.75, 56.25, 78.75

Global imperfection resonances – ultimate energy limit?

BPM

QuadCorrector

Correct orbit to minimize kicks:

Orbit going through center of BPM’sOrbit without kicks

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 m residual orbit error (OK)At LHC (7 TeV) this corresponds to ~10 m residual orbit error !

(LHC orbit accuracy ~ 200 m)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:

Summary

Polarized beam acceleration has followed the path that Slava has laid out more than 30 years ago.

Happy Birthday!