snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l...

Georg.Hoffstaetter@DESY.de

snake snake

COSY Summer School 2002Accelerator Rings with Polarized Beams and Spin Manipulation

ZEUSHERAH1 (318 GeV)

HERA-B (42 GeV)

HERMES (7 GeV)

Spin Physics in HERA

Phobos und Deimos

l Alle Monde waren vor Dämpfung in die Spin-Bahn-Resonanz chaotisch

l In den chaotischen Bereichen hatten alle Monde instabile Rotationsachsen

tuneemittance

dampedemittance

Damping of theemittance

The Electron Beam

Generation of the Emittance

Path for design energy

Pat for smaller energy

Photon

Dispersion:closed path for anoff-energy particle

Stronger focusing Smaller dispersion Smaller emittance

Self Polarization of the Electron Beam

Each 1010-th photon flips the spin of the electron

In HERA every 38.5 minutes In HERA every 16.2 hours

Ideal ring: equilibrium polarization 92.38%HERA: routine operation with 60-65% polarization

First longitudinal lepton polarization

27.5 GeV70%1994HERA47 GeV57%1993LEP

16.5 GeV70%1982PETRA5.0 GeV80%1983DORIS5.0 GeV30%1883CESR5.0 GeV60%1982VEPP-43.7 GeV90%1975SPEAR2.0 GeV80%1976VEPP-3

0.65 GeV90%1974VEPP-2M0.53 GeV90%1070ACO0.65 GeV80%1970VEPP

(longitudinal)

Longitudinal Electron Polarization

80transverse polarization [%]

longitudinal polarization [%]

0.5 1 1.5 2 2.5time [h]

Polarized Proton Beamsl Resonance excitation by the Stern-Gerlach Effect

è requires extremely difficult phase space gymnasticsl Spin flip by scatterin of polarized electrons

è very long polarization timel Spin filter with polarized target (FILTEX at TSR)

è very long polarization times and for low energiesl Acceleration of polarized protons from rest

0.59 GeV/cPSI Cyclotron

0.7 GeV/cIUCF

3.6 GeV/cSATURN II

3.65 GeV/cCOSY

12 GeV/cZGS

25 GeV/cAGS

100 GeV/cRHIC

The Structure of the Proton

Proton:Ground state of a system oftwo u and one d quark

Then one should find:l Proton momentum = Sum of quark momental Proton spin = Sum of quark spins

The Momentum of the Proton

The momentum puzzle

∫ =⋅⋅1

5.0)( dxxqx

Gluons carry 50% of the proton’s momentum

q(x):Probability of scattering on a quarkor anti-quark which carries the fractionx of the proton’s total momentum

QCD Model of the Proton

The proton is a highlyrelativistic, bound state ofu and d quarks. Additionallygluons, as field quanta, createquark / anti-quark pairs.

The Spin PuzzleS : quark contribution to the proton spin

DG : gluon contribution to the proton spin

DLq,G : angular momentum contribution

Measurements at SLAC-CERN-DESY and integration:

∫ ≈⋅1

27.0... dx Quarks carry only 20-30%of the proton’s total spin

Goals• Determination of S, DG, DLq,G

• Spin contribution of the individual quark types

• Test of QCD

• Scattering between polarized photon and proton

• How relativistic is the proton ?

Experiment:Scattering of polarized proton beams on polarized electron beams

Polarized Protons at DESY

HERMES

HERA-B HERA

6336 m long

SY Polarized Electrons

Protons

Protons Electrons

20 keV Source Source 150 keV750 keV RFQ Linac II 450 MeV50 MeV Linac III Pia 450 MeV

8 GeV DESY III DESY II 7 GeV40 GeV PETRA PETRA 12 GeV

920 GeV HERA-p HERA-e 27.5 GeV

HERA and its Pre-Accelerator Chain

Challenges:l Polarized 20 mA H- sourcel Accelerationl Storage at 920 GeVl Polarimetry

Polarized H source-

6 mA plarized protons But there is potential forhigher polarization and forhigher current

The Equation of Spin Motion

Restframe: '2'

sd rrr

Spin 4-Vector: ),(]),,0[(),( ||0 ⊥+⋅=== ssssfoLorentzTraSSSrrrrrrr

γβγβµ

,00 =⋅−= γγµµ vScSUS

ηνη

µννν

νµνµµµµ

UUFSUUdd

UFSUdd

)(g)(fe

⋅+⋅+⋅+

⋅+⋅+⋅+⋅=

Allow linear dependence on velocity, acceleration, spin, and fields:

_ ___ e___c 2 m

0)( =+= µµµ

µµµ

τττU

Search for 4-vector EOM like:ν

µνµ

⋅= )( EBvqpdtd rrrr

+×⋅=

The Thomas BMT-Equationsprs

rrrrr×Ω= ),(

[(),( 222 EpGmc

rrrrrrrrr

⋅−+−=Ω

γγγγγ

Protons G = 1.79Deuterons G = -0.143Electrons G = 0.00116

Spins in a transverse magnetic field

• Relative to the direction of the momentum, the spin rotation ofrelativistic particles does not depend on energyProtons: 5.48 Tm rotate by f=pDeuterons: 137.2 Tm rotate by f=pElectrons: 4.62 Tm rotate by f=p

• Devices can be built which rotate spinsindependent of energy.

qGd ⊥−=φ

d p γφ ⊥−==

Spin-tune in a flat ring

Accelerator ring

920 GeV/c Protons: Gg =1756 and 1 more each 523 MeV920 GeV/c Deuterons: Gg = -70 and 1 more each 13119 MeV27.5GeV/c Electrons: Gg = 62.5 and 1 more each 442 MeV

3.3 GeV/c Protons: Gg = 6.543.3 GeV/c Deuterons: Gg =-0.29

COSYSpin-tune Gg: Number of spin revolutions per turn

Spin-kick

COSY spin-kick orbit deflection3.3 GeV/c Protons: p/2 11.9 deg3.3 GeV/c Deuterons: p/2 126.6 deg

HERA spin-kick orbit deflection920 GeV/c Protons: p/2 0.89 mrad920 GeV/c Deuterons: p/2 -22.1 mrad27.5GeV/c Electrons: p/2 24.8 mrad

Spin rotation in solenoids

COSY spin-rotation solenoid field3.3 GeV/c Protons: p 12.39 Tm3.3 GeV/c Deuterons: p 40.35 Tm

HERA spin-rotation solenoid field920 GeV/c Protons: p 3456 Tm920 GeV/c Deuterons: p 11250 Tm27.5GeV/c Electrons: p 288 Tm

+−=γ

Orbit rotation in solenoidsBrqrmr&r&&r ×=γ zzz eBeBB

rrr+−= ρ

' 0=⋅∇ Brr

with so that

,zezerrrr

+= ρρ

,zezeerr

&r&&r ++= φρ ϕρρ )

)2()( 2

eBeBezeemq

r&&&&r

&&&&&r

+−×++=

+++−=

ρφρ

ρϕρρ

ϕρϕρϕρρ

F component: )'2

(2 zz BzBmq ρ

ϕρϕρ &&&&&& +−=+

dtd ρ

γϕρ −=&

21−=ϕ dl

Gd ||)1( +−=φ

Orbit Spin

Spin motion in the Accelerator FrameIndependent variable:

lt πθ 2=⇒⇒

lxlx eeedld rrr

lyly eeedld rrr

yyxxyxl eeeeedld rrrrr

κκϑϑρ

−−=+−= )sin(cos1

lyyxxlyylyxxlx eSSSdld

eSSdld

sdld rrrr

)()()( κκκκ +++−+−=

llyyxx eSeSeSsrrrr

Spin motion in the Accelerator FrameIndependent variable:

lt πθ 2=⇒⇒

eSSSdld

eSSdld

BMTlyyxxlyylyxxlx

rrrrr×Ω=+++−+− )()()( κκκκ

Seprdldt

rrrrrrr××−Ω= ]),([ κ

,),( SzSdd rrrr

×Ω= θθ

]),([2

),( lBMT eprdldtL

zrrrrrrr

×−Ω=Ω κπ

),( θθ

zvzdd rrr

with )()()( κκκrrrrrrrrr

⋅−⋅=×× SeeSeS lll

rrrr⋅⋅= /

The tumbling of Hyperion

l 250km X 115km X 110km (a=3(B-A)/2C=0.4): largest non-round structure in the solar system

l Unique due to large a and e=0.1

l Voyager2 observed a rotation axis which was close to the orbit plane: no spin-orbit-coppling

l Change in luminousness shows a chaotic rotationl The only observed chaos in solar-system-dynamics

Model: rotation around the vertical

l While changing from rotation to libration around spin-orbit-coupplunga large chaotic region has to be crossed.

ϑαϕϑ

2sin))(

())(( 3

The world ends on Feb 1 2019(possibly)(Filed: 25/07/2002, Telegraph)

Scientists have detected a giant asteroid headingtowards Earth. It could wipe out humanity, but itcould miss us altogether. Astronomers say a hugeasteroid is scheduled to crash into the Earth at11.47 am on Feb 1 2019.

Objects the size of NT7 only hit the Earth every oneor two million years.

The dangers of NT7 have yet to be reviewed by theInternational Astronomical Union,the main international body responsible forannouncing such risks.

The Importance of Asteroids

tuneemittance

Phase space motion

lyx QQQQ ,,:r

Tunes:Emittances: lyx εεεε ,,:

=ΦAction variables:

Angle variables: εrr

Action-angle variables of aclassical periodic system likea pendulum or planetary motion

Equation of motion for spin fields

Spin field: Spin direction for each phase space point),( θzfrr

),( θzfrr

fzfzvffdd

rrrrrrrrr ×Ω=∂⋅+∂= ),(]),([ θθ

The spin transport matrix EOM

ii SzRSrrr

);,()( 0 θθθ =

),();,(),( 00 θθθθ ii zfzRzfrrrrr

Sd rrr

×Ω=θ

);,( 0

0 θθθθθ ii zRzRrr

ΩΩ−Ω−Ω

ΩΩ−=∂

The spin transport matrix

)]([sin)]([cos)()( iiiii SeeSeSeeSSeeSrrrrrrrrrrrrr

⋅−×⋅+⋅−+⋅= ααθ

Rotation vector: , Rotation angle .

,cos 20α=a 2sin αea

iii SaaSaaSaaSrrrrrrrr

×+⋅+−= 022

0 2)](2)()(θ

kijkjiijij aaaaaaR εδ 022

0 22)( −+−=r

1220 =+ aa

lmnlmn aaRaRTr 02

0 4,14)( −=−= ε

The spin transport quaternion

rrrrrrr

⋅×++−⋅−=

⋅−⋅−=

)1)(1(

abababiabab

aiabibC

σσσθ

rrrrrrrr

rrrrrr

⋅Ω−=⋅Ω+⋅⋅Ω−=

⋅×Ω+Ω+⋅Ω−=

2]))([(

])(1[2

lmnlmnml i δσεσσ +=,120 σrr

⋅−= aiaA

Propagation:

Infinitesimal rotation: ,2

12 σθ rr

⋅Ω−=d

iB dAAC +=

rr⋅Ω−=

Equation of motion for Spinors

,)1()( 20 iaia Ψ⋅−=Ψ σθrr

Ψ⋅Ω−=Ψ

ΨΨ= +σθrr

rrrrrrr

×Ω=Ψ×ΩΨ=

Ψ⋅Ω−⋅ΩΨ=

)]()[(21

σσσσθ

sincos

ϑφϑφϑ

cossinsincossin

with 122

21 =+ ψψ

Spinor pase

Ψ−=Ψ−ΨΨ−=

−−

−−=Ψ⋅−=

sincossincossincos

cossinsincos

ϑϑϑϑα

ϑϑϑϕ

ϑϑϕϑ

Rotation around the spin direction: αSrr

)(2 0 θ

θθα

Ψ=Ψ−i

Equation of motion for spin fields

Spin field: Spin direction for each phase space point),( θzfrr

),( θzfrr

Ψ⋅Ω−=Ψ∂⋅+Ψ∂=Ψ ]),([2

]),([ σθθθ θ

rrrrrr z

Spin perturbation on the closed orbitInteger values of the closed orbit spin-tune n0 lead to coherentdisturbances of spin motion calledimperfection resonances.

Remedy:Partial snakes avoidresonances by avoidingan integer spin-tune n

closed orbit

θπθθφφ 0

0)2mod( 00

kpSeay ∑=−∝∝ rr

Periodic spin direction for the closed orbit

Accelerator ring

Spin-tune n0: Number of spin revolutions per turn

A particle traveling on the closed orbit has the closed orbit spin direction if its spin is periodic after every turn.

00000 )2;,( nzRnrrr

πθθ +=

Closed orbit spin motionl When the design orbit spin tune n0 becomes integer, no net rotation has

occurred after one turn, so that perturbations from the design orbit dominate the motion

l If the perturbations are very small, the resonance region can be crossed quickly and the spins hardly react

l If the perturbation is large, the closed orbit spin direction changes slowly and spins can follow adiabatic changes of the periodic spin direction on the closed orbit.

l When the perturbations have intermediate strength, the polarization will be reduced

Remedies:l Correction of the closed orbit to reduce the perturbationl Increase of spin perturbation (for example by a solenoid partial snake) to

increase the perturbation

Closed orbit precession vector:

The periodic coordinate system)2()( 00 πθθ +Ω=Ω

)2()(,)( 00000 πθθθ

θ+=×Ω= nnn

dnd rrrrr

Non-periodicPerpendicular vectors:

000000 ,)( mnlm

dmd rrrrrr

×=×Ω= θθ

)2]([)]([ 002

000 πθθ πν ++=+ limelim i

Non-periodicPerpendicular vectors: )]([)]([ 00

0 θθ θν limelim irrrr

)(])([)(

000 limnd

limd rrrrrr

+×−Ω=+

νθθ

Closed orbit spin vector:

EOM in the periodic system

Srrrrrrrrr

×−Ω+++==×Ω ][)( 0003

0 νθθθθ

0,ˆˆ,ˆ 3021 ==+= ssisisss dd

θθ ν

)()()( 0321 θθθ nslsmsSrrrr

Smooth rotation around the periodic spin direction !

Adiabatic invariant on the closed orbit

Spin flip during crossing of a resonance with a partial snake

Spin flip at imperfection resonances in the AGS

Courtesy T.Roser

Slowly varying periodic system

rrrrrrrr

×+×−Ω+++=

ηθτ

νθθθ

),,(),(),,( 00 nlmnlmdd rrrrrrr

×= τθητ

),2(),( 00 τπθτθ +Ω=Ωrr

Slowly varying parameter :εθτ =

Slowly rotating coordinate system:

cossin1

sin1,cos1

θϕϕ

+−=−

−=−=

Slowly varying periodic system

−−

1)cossin[()(

1)cossin(

ηϕηϕηετν

ϕηϕηε

Averaging of two frequency systems:The system which is averaged over 2p of the 2 fast variables describes thetrue motion of the slow variables up to for .

εc< εθ 1<

εθθ css += )()( 033 is an adiabatic invariant

Driven spin perturbation on a trajectoryInteger values of spin-tune n tune ny lead to coherentdisturbances of spin motion

Remedy:Siberian Snakes avoidresonances by makingthe spin-tune n = ½independent of energy.

Crossing resonancesRemedy for DESY III:Tune jump, energy jump,and RF dipole excitation

Remedy for PETRA and HERA:Fixing the spin tune to ½ by Siberian Snakes

Problems with Resonances

At integer spin-tune n thespin returns without achange after one turn,and error fields add up.

Remedy: insert controlledperturbations of spin motion.

AGS Experiment E880

Controlled perturbation on

simulation

yνν = yνν −= 24 yνν += 12

Siberian Snakes

Freedom: direction of the rotation axis in the horizontal

snake snake

Siberian Snakes rotate spins at each energy ½ times

CO spin motion with 1 Siberian Snake

)]sin()cos([

)sin)(cossincos(

γπασγπασγπσγπασασ

−+−−=−+−=

Spin direction after the snake:

=Ψ⇔−+−= − )(220 2

)0()sin()cos( γαγγ αα Gi

eenrrr

−=Ψ −−

− )]([)(322

)sin(cos)(2

θπγαγπαγθγθ

σθ Gi

Snake angle a tunes spin direction anywhere in the ring, especially atit is independent of energy !

πθ =

CO spin motion with 2N Siberian Snake

])sin()([cos

)sincos()sincos(

)sincos(

3122122

122121

σαααα

ασασασασ

ασασ

σψψψψ

−−

∆−

−−

−+−−

−−−=

322 σαψ ∆+∆−

=iNeiA π

αψν

0∆+∆

=0, to make n0 independent of energy

, to make n0=0.5

0=∆ψ

α =∆

RHIC SiberianSnakes (A)

l 2p helical dipoles

l 10 cm diameterl Superconducting

4Tesla magnets

Spin Motion

Are the implemented devicesand the applied theorystill applicable at:

Trajectory at 25GeVRHIC SiberianSnakes (B)

Particle Trajectories

RHIC Siberian Snake (C)

Production

Required installations in HERA

Cost: about 30M Euro

l Polarimetersl Flattening Snakesl Spin rotatorsl At least 4 Siberian Snakes

Space for PETRA’s Siberian Snakes

Space for Siberian Snakes in HERA

For a large divergence, the average polarization is small, even if the local polarization is 100%.

Linearized can be analytically computed

Phase spaceA) Maximum polarization: Plim =

B) is an adiabatic invariance !S⋅

The InvariantSpin Field

The Isolated Resonance Model

Is this theory still applicable for HERA with 920 GeV ? ? ?

All Fourier components of ,except the dominant one, are neglected.

The Isolated Resonance ModelAll Fourier components of ,except the dominant one, are neglected.

First Order Theories A) DESY III

Low energies: First order theories agree

Isolatedresonancemodel:

Linearspin-fieldtheory:

Momentum (GeV/c)

First Order Theories B) PETRA

Medium energies: resonances still isolated

Linearspin-field theory:

36 38343230

Momentum (GeV/c)

First Order Theories C) HERA

High energies:Resonances are no longer isolated.

Linearspin-fieldtheory:

The isolated resonance model becomes invalid

Momentum (GeV/c)

Siberian Snakes and Resonances

Some structure of the 1st order resonances remains after Siberian Snakes have been installed.

Amplitude dependentspin-orbit resonances

Spin Tune at Higher Order ResonancePlim

Spin tune n(Jy)

yνν −=1yνν 83−=

yνν 155 −=

416 −= yνν

yνν 2=

(GeV/c) (GeV/c)

The spin tune n deviates from ½ for particles which oscillatearound the design trajectory with amplitude Jy .

1st Order: 4 harmonics of the spin perturbation in each section.With 4 snakes only 2 can be compensatedWith 8 snakes all 4 can be compensated

Snake matching

4 Snakes:

8 Snakes:

Plim after Snake Matching

8 snakes standard scheme

8 matched snakes

Spin Tune after Snake MatchingQy2=ν

Qy2=ν

Qy21−=ν

4 snakes in standard scheme

4 matched snakes

Spin Tune after Snake Matching

Qy21−=ν

Qy2=ν

Qy21−=ν

4 snakes in standard scheme

8 matched snakes

High Order Resonance Strength

Tracked depolarization as expected

Spin tune>=< )(lim znPrr

Computations performed in SPRINT, Hoffstaetter and Vogt, DESY/00

l Resonances up to 19th order can be observed

l Resonance strength can be determined from tune jump.

The higher order Froissart-Stora formula

Allowed Beam Sizes

Snake matching allows to have significantly larger beams.

4 snakes inStandard scheme

4 filteredsnakes

4 matchedsnakes

8 filteredsnakes

The Invariant Spin Field

l Fourier analysisl Stroboscopic averagingl Anti-dampingl Differential Algebra

Computations performed in SPRINT, Hoffstaetter and Vogt, DESY/02

l Computation of thel invariant spin field by

analyzing tracking data:

Higher Order Effectsl Overlapping resonancesl Deformation of the invariant spin fieldl Resonances of very high orderl Nonlinear spin transfer matrix

Siberian Snakes avoid 1st order resonances but higher orders become important for HERA.

Polarized Deuteronsdepolarizing resonance strength

depolarizing resonance strength

l Resonances are 25 times weaker and 25 times rarer for D than for p

l Transverse polarization could be achieved without Siberian Snakes

l Transverse RF dipoles could be used to rotate and stabilize longitudinal polarization

snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l...

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