? ? ?
snake snake
COSY Summer School 2002Accelerator Rings with Polarized Beams and Spin Manipulation
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
dtdϑ
dtdϑ
ϑ ϑ
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
20
40
60
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
uu
d
The Momentum of the Proton
The momentum puzzle
∫ =⋅⋅1
0
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
0
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
H1
ZEUS
HERMES
HERA-B HERA
PETRA
778 m
6336 m long
DE
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-
25 kG
10 kG
6 mA plarized protons But there is potential forhigher polarization and forhigher current
The Equation of Spin Motion
Restframe: '2'
Bsm
gqdt
sd rrr
×=
Spin 4-Vector: ),(]),,0[(),( ||0 ⊥+⋅=== ssssfoLorentzTraSSSrrrrrrr
γβγβµ
,00 =⋅−= γγµµ vScSUS
rr
µν
ηνη
µννν
µν
νµνµµµµ
τ
ττ
UUFSUUdd
SSF
UFSUdd
USdd
)(g)(fe
dcba
⋅+⋅+⋅+
⋅+⋅+⋅+⋅=
Allow linear dependence on velocity, acceleration, spin, and fields:
1c 2
_ ___ e___c 2 m
gq2
e =
0)( =+= µµµ
µµµ
τττU
dd
SSdd
UUSdd
Search for 4-vector EOM like:ν
µνµ
τUF
mq
Udd
⋅= )( EBvqpdtd rrrr
+×⋅=
The Thomas BMT-Equationsprs
dtd
BMT
rrrrr×Ω= ),(
])1
1(
1)1(
)1
[(),( 222 EpGmc
pcm
BpGBG
mq
prBMT
rrrrrrrrr
×+
+−+
⋅−+−=Ω
γγγγγ
=−
=2
2gG
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.
vdl
Bm
qGd ⊥−=φ
pdrp
r
vdl
mqB
pdp
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
HERA
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
dlp
qBG
vdl
Bmq
Gd
||
||
)1(
)1(
+−=
+−=γ
φ
Orbit rotation in solenoidsBrqrmr&r&&r ×=γ zzz eBeBB
rrr+−= ρ
ρ2
' 0=⋅∇ Brr
with so that
,zezerrrr
+= ρρ
,zezeerr
&r
&r&&r ++= φρ ϕρρ )
2'()(
)2()( 2
zzzz
z
eBeBezeemq
ezeer
rrr&
r&r&
r&
r&&&&r
&&&&&r
+−×++=
+++−=
ρφρ
φρ
ρϕρρ
γ
ϕρϕρϕρρ
F component: )'2
(2 zz BzBmq ρ
ργ
ϕρϕρ &&&&&& +−=+
)2
()(2
2zB
dtd
mq
dtd ρ
γϕρ −=&
dlp
qBd z
21−=ϕ dl
pqB
Gd ||)1( +−=φ
Orbit Spin
Spin motion in the Accelerator FrameIndependent variable:
Ll
lt πθ 2=⇒⇒
lxlx eeedld rrr
κρ
ϑ==
cos
lyly eeedld rrr
κρ
ϑ==
sin
yyxxyxl eeeeedld rrrrr
κκϑϑρ
−−=+−= )sin(cos1
lyyxxlyylyxxlx eSSSdld
eSSdld
eSSdld
sdld rrrr
)()()( κκκκ +++−+−=
llyyxx eSeSeSsrrrr
++=
Spin motion in the Accelerator FrameIndependent variable:
Ll
lt πθ 2=⇒⇒
sdldt
eSSSdld
eSSdld
eSSdld
BMTlyyxxlyylyxxlx
rrrrr×Ω=+++−+− )()()( κκκκ
Seprdldt
Sdld
lBMT
rrrrrrr××−Ω= ]),([ κ
,),( SzSdd rrrr
×Ω= θθ
]),([2
),( lBMT eprdldtL
zrrrrrrr
×−Ω=Ω κπ
θ
),( θθ
zvzdd rrr
=
with )()()( κκκrrrrrrrrr
⋅−⋅=×× SeeSeS lll
rdtd
erdld
edldt
ll
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
2
2
tra
dttd
−=+
dtdϑ
ϑ
ϕ
ϑ
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 ,,:
r
θQrr
=ΦAction variables:
Angle variables: εrr
21
=J
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
zr
),( θzfrr
Nsr
fzfzvffdd
z
rrrrrrrrr ×Ω=∂⋅+∂= ),(]),([ θθ
θ θ
The spin transport matrix EOM
ii SzRSrrr
);,()( 0 θθθ =
),();,(),( 00 θθθθ ii zfzRzfrrrrr
=
iSd
Sd rrr
×Ω=θ
);,(0
00
);,( 0
12
13
23
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
rr=
iii SaaSaaSaaSrrrrrrrr
×+⋅+−= 022
0 2)](2)()(θ
er
α
kijkjiijij aaaaaaR εδ 022
0 22)( −+−=r
1220 =+ aa
rwith
lmnlmn aaRaRTr 02
0 4,14)( −=−= ε
The spin transport quaternion
σ
σσ
rrrrrrr
rrrr
⋅×++−⋅−=
⋅−⋅−=
)(1)(
)1)(1(
00200
2020
abababiabab
aiabibC
Ad
iaiad
aaiad
dA
σθ
σσσθ
σθ
rrrrrrrr
rrrrrr
⋅Ω−=⋅Ω+⋅⋅Ω−=
⋅×Ω+Ω+⋅Ω−=
2]))([(
2
])(1[2
0
02
lmnlmnml i δσεσσ +=,120 σrr
⋅−= aiaA
Propagation:
Infinitesimal rotation: ,2
12 σθ rr
⋅Ω−=d
iB dAAC +=
AiddA
σθ
rr⋅Ω−=
21
Equation of motion for Spinors
since
,)1()( 20 iaia Ψ⋅−=Ψ σθrr
Ψ⋅Ω−=Ψ
)(21
σθ
rri
dd
ΨΨ= +σθrr
)(S
Si
id
Sd
rrrr
rrrrrrr
×Ω=Ψ×ΩΨ=
Ψ⋅Ω−⋅ΩΨ=
+
+
)(21
)]()[(21
σ
σσσσθ
=
=Ψ
2
2
2
1
sincos
ϑφ
ϑξ
ψψ
ii
ee
=
ϑφϑφϑ
cossinsincossin
Sr
with 122
21 =+ ψψ
Spinor pase
Ψ−=Ψ−ΨΨ−=
Ψ
−−
−=
Ψ
−−=Ψ⋅−=
Ψ
+
−
−
2)1(
sincossincossincos
cossinsincos
22
221
21
22
22
2221
22
αα
α
ϑϑϑϑα
σα
θ
ϑϑϑϕ
ϑϑϕϑ
ϕ
ϕ
ii
ee
i
ee
iSidd
i
i
i
irr
Rotation around the spin direction: αSrr
=Ω
)( 0
)(2 0 θ
θθα
Ψ=Ψ−i
e
Equation of motion for spin fields
Spin field: Spin direction for each phase space point),( θzfrr
zr
),( θzfrr
Nsr
Ψ⋅Ω−=Ψ∂⋅+Ψ∂=Ψ ]),([2
]),([ σθθθ θ
rrrrrr z
izv
dd
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
0)2mod( 00
ik
k
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.
0nr
0nr
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
rr
)2()(,)( 00000 πθθθ
θ+=×Ω= nnn
dnd rrrrr
Non-periodicPerpendicular vectors:
000000 ,)( mnlm
dmd rrrrrr
×=×Ω= θθ
)2]([)]([ 002
000 πθθ πν ++=+ limelim i
rrrr
Non-periodicPerpendicular vectors: )]([)]([ 00
0 θθ θν limelim irrrr
+=+
)(])([)(
000 limnd
limd rrrrrr
+×−Ω=+
νθθ
Closed orbit spin vector:
EOM in the periodic system
Sndds
ndds
ldds
mSdd
Srrrrrrrrr
×−Ω+++==×Ω ][)( 0003
021
0 νθθθθ
θ
0,ˆˆ,ˆ 3021 ==+= ssisisss dd
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
Slowly varying periodic system
Sdd
Sndds
ndds
ldds
m
SSdd
rrrrrrrr
rrr
×+×−Ω+++=
×Ω=
ηθτ
νθθθ
θθ
)(][
)(
0003
021
0
),,(),(),,( 00 nlmnlmdd rrrrrrr
×= τθητ
),2(),( 00 τπθτθ +Ω=Ωrr
Slowly varying parameter :εθτ =
Slowly rotating coordinate system:
212
3
232
231
cossin1
sin1,cos1
sdd
sdd
dd
s
ssss
θϕ
θϕϕ
θ
ϕϕ
+−=−
−=−=
Slowly varying periodic system
−−
++
−+=
]
1)cossin[()(
1)cossin(
323
3120
2312
3
ηϕηϕηετν
ϕηϕηε
ϕθs
sss
dd
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 .
=
1
~ε
θτ
θdd
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
vert
ical
pol
ariz
atio
n in
%
Controlled perturbation on
off
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(
21
321
γπασγπασγπσγπασασ
GGi
GiGiA
−+−−=−+−=
Spin direction after the snake:
=Ψ⇔−+−= − )(220 2
1
21
)0()sin()cos( γαγγ αα Gi
Gl
Gx e
eenrrr
=
−=Ψ −−
−
− )]([)(322
1
2
1
21
)sin(cos)(2
θπγαγπαγθγθ
γθ
σθ Gi
i
GiGG
ee
ei
G
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
1
2
122121
1
22212
21
2
1
2
3
31232
3
σαααα
ασασασασ
ασασ
σψ
σψψψψ
σψ
−−
=
∆−
−−
=
−+−−
=
−
−−−=
++=
+=
∏
∏
∏
jjjj
N
j
iN
jj
N
j
jj
iN
jj
N
j
i
iei
ei
ieA
N
j
L
322 σαψ ∆+∆−
=iNeiA π
αψν
22
0∆+∆
=
yenrr
=0, to make n0 independent of energy
, to make n0=0.5
0=∆ψ
2π
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)
30mm
45mm
12m
Particle Trajectories
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.
|Spe
ctru
m
First Order Theories A) DESY III
Low energies: First order theories agree
Isolatedresonancemodel:
Linearspin-fieldtheory:
Momentum (GeV/c)
Plim
First Order Theories B) PETRA
Medium energies: resonances still isolated
Linearspin-field theory:
|Spe
ctru
m
Plim
Isolatedresonancemodel:
36 38343230
Momentum (GeV/c)
First Order Theories C) HERA
High energies:Resonances are no longer isolated.
Isolatedresonancemodel:
Linearspin-fieldtheory:
The isolated resonance model becomes invalid
|Spe
ctru
m
Plim
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
0.2
0.4
0.6
0.8
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:
Spin Tune after Snake MatchingQy2=ν
Qy2=ν
Qy21−=ν
Qy21−=ν
4 snakes in standard scheme
4 matched snakes
Spin Tune after Snake Matching
Qy21−=ν
Qy2=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
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