Spin Dynamics - agsrhichome.bnl.gov · µ θ Rotator axis y s x Beam Therotationaxisof...

Spin Dynamics in AGS and RHIC

Waldo MacKay

L. Ahrens, M. Bai, K. Brown, E. D. Courant, H. Huang,A. U. Luccio, V. Ptitsyn, T. Roser,J. W. Glenn,T. Satogata, N. Tsoupas, and A. N. Zelenski

BNL, Upton, NY 11973, USA. 1�

PAC2003: Spin Dynamics in AGS and RHICWaldo MacKay 14, May, 2003

ÆAccelerator Complex (Pol. Protons) �

AGSLINAC

BOOSTER

RHIC

ε = 20 π mm mrad

SOURCE

LINAC: Linear AcceleratorAGS: Alternating Gradient SynchrotronRHIC: Relativistic Heavy Ion Collider

2�


ÆThomas|Frenkel (BMT) Equation �

In the local rest frame of the proton, the spin precession of the proton obeysthe Thomas-Frenkel equation:

Torque :d�S�

dt=q

γm�S� ×

[(1 +Gγ) �B⊥ + (1 +G) �B‖

]TF

Force :d�p

dt=q

γm�p × �B⊥ Lorentz

(This is a mixed description: t, and �B in the lab frame, but spin �S� in local restframe of the proton.)

G =g − 22

= 1.7928, γ =Energymc2

.

3�


Æ Spin Precession in a Ring �

Example with 6 precessions of spin in oneturn:

Gγ + 1 = 6.

Spin tune: number of precessions per turnrelative to beam’s direction.

So we subtract one:

νspin = Gγ ∝ energy,

i.e., 5 in this example.

4�


ÆMisalignments or Imperfections �

misalignedquad

• A misaligned quadrupole creates a steering error which propagates throughthe lattice.

• For an accelerator ring, this shifts the closed orbit away from the designtrajectory.

• If the misalignment is vertical, then the design trajectory will have a periodicset of small vertical bends interspersed with the normal horizontal bends ofthe bending magnets.

• This leads to an integer resonance condition for the spin tune.

5�


Æ In general, rotations don't commute. �

x

y

zRz(90°) Rx(−90°)

1

2

3x

y

z

1

2

3

Rx(−90°) Rz(90°)

6�


ÆDepolarizing Resonances �

Simple Resonance Condition:

νspin = N + NvQv,

(imperfection) (intrinsic)

where N and Nv are integers.

MagneticField

Protons moving into screen

S

N S

Magnetic Lens (quadrupole)Vertically focusing

NForce

7�


ÆAGS Intrinsic Resonances �

0 + Qy

24 - Qy

12 + Qy

36 - Qy

24 + Qy 48 - Qy

36 + Qy

0 5 10 15 20 25 30 35 40 45 50

Gγ

0

0.01

0.02

0.03

Res

onan

ce S

tren

gth

AGS Intrinsic Resonances

8�


ÆDepolarizing Resonances �

0 20 40 60 80 100 120 140 160 180 200 220 240 260

γ = Energy / Rest Mass

0

0.1

0.2

0.3

0.4

0.5

Intrinsic Resonances in RHICQx=29.19 Qy=28.23 Emy=10 pi

Res

onan

ce S

tren

gth

Will depolarize beamduring acceleration.

Increase in strengthwith energy.

9�


ÆPartial Snakes �

Adding a partial snake opens up stopbands around the integer imperfectionresonances.

At the snake the stable spin directionpoints along the snake’s rotation axiswhen Gγ = integer.

Partial snake strength: µπ

cosπνs = cos(Gγπ) cos µ2

46 47 4846

47

48 SnakeStrength

0%25%50%100%

Gγ

ν s

10�


ÆCrossing an Isolated Spin Resonance �

Froissart—Stora Formula:

Pf

Pi= 2 exp

(−π|ε|

2

2α

)− 1.

Ramp rate: α = dGγdθ , (θ : 2π/turn.)

Resonance strength: ε =Fourier amplitude.

11�


Æ Simulation of Resonance Xing in AGS �

7.5 7.7 7.9 8.1 8.3 8.5 8.7 8.9 9.1 9.3 9.5Gγ

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Sy

0+ν, (Qx,Qy)=(8.7,8.8), (εx,εy)=(20,10)πwesting house

δ=0.0

Coupling res.νs=Qx

Intrinsic res.νs=Qy

Imperfection resonancesνs=N

7.5 7.7 7.9 8.1 8.3 8.5 8.7 8.9 9.1 9.3 9.5Gγ

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Sy

0+ν, (Qx,Qy)=(8.7,8.8), (εx,εy)=(20,10)πwesting house, w. 10mm coherence amplitude(Qm=0.215, ∆BL=18Gm)

δ=0.0, no coupling

AC dipole used to increase strengthof νs = Qy resonance.

12�


ÆAGS Raw Asymmetry during Ramp �

12+Qy 36-Qy 24+Qy 48-Qy 36+Qy Extract

20 25 30 Gγ 35 40 45150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450

Scaled gauss clock counts

-0.015

-0.01

-0.005

0

0.005

0.01

AG

S R

aw A

sym

met

ry

AGS has 12 superperiods.Vertical betatron tune: 8.7Snake strength: 5%

AC dipole pulses at resonances:• 0 +Qy

• 12 +Qy

• 36−Qy

• 36 +Qy

13�


Æ Invariant Spin Field �

Turn: j Turn: j+1

Closed orbit

Unclosed trajectory

• For the closed orbit: �n0(s) = �n0(s+ L),with �q0(s) = �q0(s+ L) and polarization �P0(s) = �P0(s+ L).

• For other locations in phase space: �n(�q, �p, s) = �n(�q, �p, s+ L),even though in general q(s+ L) �= q(s) and �P (s+ L) �= �P (s).

14�


Æ Spin Tracking in RHIC �

Particles with largeramplitude betatronoscillations may expe-rience more precessionaway from the stablespin direction of thecenter of the beam

15�


Æ Snake Charming �

• 2 snakes: spin is up in one half of the ring,and down in the other half.

• Spin tune: νspin = 12

(It’s energy independent.)

• “The unwanted precession which happensto the spin in one half of the ring is un-wound in the other half.” Unwinds

Winds

16�


ÆTrajectory and Spin through Snakes �

0 5 10Longitudinal position (m)

−5

−3

−1

1

3

5

B fi

eld

(T)

Fields through first Snake100 GeV proton

BxByBz

0 5 10Longitudinal Position (m)

−0.005

0

0.005

0.01

Tra

nsve

rse

Dis

plac

emen

t (m

)

Orbit Trajectories through Snake100 GeV proton

XY

0 5 10Longitudinal position (m)

−1

−0.5

0

0.5

1

Spi

n P

olar

izat

ion

Spin motion through one Snake100 GeV proton

SxSySz

02

46

s [m] 810

12-4

-2

0

2

4

-10

-5

0

5

10

17�


ÆRHIC Beam Polarization �

18�


Æ Snake Resonances �

19�


ÆRotators for Longitudinal Polarization �

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 2 4 6 8 10

B [

T]

s [m]

BxByBz

Magnetic Field24.3 GeV proton

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0 2 4 6 8 10T

raje

ctor

y [m

]s [m]

xy

Beam Trajectory24.3 GeV proton

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

Spin

s [m]

SxSySz

Spin Components24.3 GeV proton

02

46

810

12 -10

-5

0

5

10

-4

-2

0

2

4

20�


ÆCompensation for D0-DX Bends �

x

y

z

E = 25 GeVD0DX: 10◦ precession

1.8 2.2 2.6 3.0 3.4 3.8B1 [T]

2.2

2.6

3.0

3.4

3.8

B2

[T]

γ

SPIN ROTATOR

25

75

3040

50

275

300

100 125

150

175

200

225

250

OH Fmap

y

z x

E = 250 GeVD0DX: 100◦ precession

DX

Rotator D0

DX

D0 Rotator

21�


ÆOrientation of PHENIX Polarimeters �

x

φ Spin −

Spin +

"Left−Right" Asymmetry(Tilted at 45 )

x

x

Schematic layout of PHENIX polarimetersYellow from left. Blue from right.

The PHENIX Local Polarimeter measures an asymmetry in small anglescattered neutrons which is proportional to transverse polarization.

ALR =

√L+R− −√

L−R+

√L+R− +

√L−R+

∝ Py

22�


ÆTale of the Blue Ring �

Vertical polarizationwith rotators off.

Spin is down.

Rotators on

Spin is radially inwards!

OOPS!

Reverse all rotatorpower supplies and tryagain.

YES!

23 �


Æ SimpleModel of Proton �

Bar magnet+ "proton"=

+ =+

+ Charge

N

SS

N

Gyroscope

MagneticDipoleSpin

Moment

++ +

Polarization: Average spin of the ensemble of protons.

24 �


ÆTorque onClassicalMagneticMoment �

.Energy =−

B

µ B

µ B

r

µx=dF Bdq v

µ= πr2

dFi

x

dj=dq v

Torque =

Ǳ Semiclassical model:• The spin �S has a constant magnitude in the rest frame.• The magnetic moment �µ ∝ �S.

• �µ has a constant magnitude in the rest frame.(Sort of like a loop of infinite inductance.)

25 �


ÆThomas|Frenkel (BMT)Equation �

In the local rest frame of the proton, the spin precession of the protonobeys the Thomas-Frenkel equation:

d�S�

dt=q

γm�S� ×

[(1 +Gγ) �B⊥ + (1 +G) �B‖ +

(Gγ +

γ

γ + 1

) �E × �vc2

].

This is a mixed description: t, �B, and �E in the lab frame, but spin �S� in localrest frame of the proton.

G =g − 22

= 1.7928, γ =Energymc2

.

26 �


ÆHamiltonianwith Spin �

(Here I drop the “�” superscript on �S.)

d�S

dt= �W × �S

H(�q, �P , �S; s) = Horb +Hspin

= Horb + �W · �S +O(h̄2)

Group symmetries:

• Orbital motion without spin: Sp(6,r).• Spin by itself: SU(2, c) ∼= SO(3, r) (homomorphic).

• Full blown symmetry: Sp(6, r)⊕ SU(2, c).• Spin dependence on orbit (Thomas-Frenkel).• Orbit dependence on spin (Stern-Gerlach Force)—Usually ignored.

27 �


ÆRepresentation of Rotations �

SU(2) with usual spinor notation:

Pauli matrices: σx =(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

).

Rn̂(θ) = ei n̂·�σ θ/2 =(cos θ

2 + inz sin θ2 (ny + inx) sin θ

2

(−ny + inx) sin θ2 cos θ

2 − inz sin θ2

).

SO(3) :Rn̂(θ) =I cos θ +

0 nz −ny

−nz 0 nx

ny −nx 0

sin θ

+

n2

x nxny nxnz

nxny n2y nynz

nxnz nynz n2z

(1− cos θ).

28 �


ÆAccelerator Complex for Protons �

PHOBOSBRAHMS & PP2PP (p)

STAR (p)

PHENIX (p)

AGS

LinacBooster

Pol. Proton Source

Spin Rotators

Partial Siberian Snake

Siberian Snakes

200 MeV Polarimeter AGS Internal PolarimeterRf Dipoles

RHIC pC Polarimeters

Absolute Polarimeter (H jet)

AC Dipole

29 �


ÆAGSCycle �

• Snake at 5% up the ramp.

• 4 ac dipole pulses.Notice beam loss at first pulse.

• Transition occurs at 1.426msbetween the first two ac dipolepulses.

Note: This cycle was not extracted.

30 �


ÆACDipole pulse at Gγ = 0 +Qy �

Top: AC dipole pulse amplitude(current)

Bottom: Beam current.(Just scrapes the beam pipe.)

Top: Beam coherence

Bottom: Tune spectrum

31 �


y

x

sφ

snake axis

µ

(beam)

Therotation

axisof

thesnake

isφ,and

µis

therotation

angle.

Theshaded

regionshow

sapproxim

atere-

strictionsfor

fieldlimit

(right)and

injectionaperture

limits(top

andlowerright).

-5-4-3-2-100

12

34

5 Bout [T

]

Bin [T

]

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

190

180

-10

-20

-30

-40

-50

-60

-70

010

20

30

40

50607080

φµ

32

�

PA

C2003:

Spin

Dynam

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inA

GS

and

RH

ICW

ald

oM

acK

ay

14,

May,

2003

µθ

Rotator axis

y

s

x

Beam

Therotation

axisof

thespin

rotatorisin

thex-y

planeatanan-

gleθfrom

thevertical.

Thespin

isrotated

bythe

angleµaround

therotation

axis.0

0.51

1.52

2.53

3.54

B1 [T

]

0 0.5

1 1.5

2 2.5

3 3.5

4

B2 [T]

µθ−10°

−30°

−60°

−90°

−120°

−150°−180°

0°

−10°

−20°

−30°

−40°

30°

25

100

200

250

Rotation A

ngles for a Helical Spin R

otator

Note: Purple contour for rotation into horizontal plane.

Black dots show

settings for RH

IC energies in

increments of 25 G

eV from

25 to 250 GeV

.

33

�

PA

C2003:

Spin

Dynam

ics

inA

GS

and

RH

ICW

ald

oM

acK

ay

14,

May,

2003

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