75
[email protected] ? ? ? snake snake COSY Summer School 2002 Accelerator Rings with Polarized Beams and Spin Manipulation

snake snakehoff/talks/02-08-30/[email protected] Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

  • Upload
    others

  • View
    11

  • Download
    0

Embed Size (px)

Citation preview

Page 1: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

? ? ?

snake snake

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

Page 2: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

ZEUSHERAH1 (318 GeV)

HERA-B (42 GeV)

HERMES (7 GeV)

PETRA

Spin Physics in HERA

Page 3: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 4: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

tuneemittance

dampedemittance

Damping of theemittance

The Electron Beam

Page 5: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 6: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 7: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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)

Page 8: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

Longitudinal Electron Polarization

20

40

60

80transverse polarization [%]

longitudinal polarization [%]

0.5 1 1.5 2 2.5time [h]

Page 9: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 10: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 11: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 12: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

QCD Model of the Proton

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

Page 13: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 14: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 15: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 16: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

Polarized H source-

25 kG

10 kG

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

Page 17: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

+×⋅=

Page 18: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 19: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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 ⊥−==

Page 20: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 21: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 22: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 23: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 24: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

++=

Page 25: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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⋅⋅= /

Page 26: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

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

Page 27: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

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

Page 28: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 29: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 30: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 31: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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−=∂

Page 32: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 33: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 34: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

=

2

2

2

1

sincos

ϑφ

ϑξ

ψψ

ii

ee

=

ϑφϑφϑ

cossinsincossin

Sr

with 122

21 =+ ψψ

Page 35: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

)( 0

)(2 0 θ

θθα

Ψ=Ψ−i

e

Page 36: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 37: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 38: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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 +=

Page 39: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 40: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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:

Page 41: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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 !

Page 42: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

Adiabatic invariant on the closed orbit

Spin flip during crossing of a resonance with a partial snake

Page 43: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

Spin flip at imperfection resonances in the AGS

Courtesy T.Roser

Page 44: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

+−=−

−=−=

Page 45: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

θτ

θdd

εc< εθ 1<

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

Page 46: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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.

±

Page 47: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 48: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 49: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

Siberian Snakes

Freedom: direction of the rotation axis in the horizontal

? ? ?

snake snake

Siberian Snakes rotate spins at each energy ½ times

Page 50: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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 =

Page 51: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

α =∆

Page 52: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

RHIC SiberianSnakes (A)

l 2p helical dipoles

l 10 cm diameterl Superconducting

4Tesla magnets

Spin Motion

Page 53: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

Are the implemented devicesand the applied theorystill applicable at:

Trajectory at 25GeVRHIC SiberianSnakes (B)

30mm

45mm

12m

Particle Trajectories

Page 54: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

RHIC Siberian Snake (C)

Production

Page 55: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

Required installations in HERA

Cost: about 30M Euro

l Polarimetersl Flattening Snakesl Spin rotatorsl At least 4 Siberian Snakes

Page 56: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

Space for PETRA’s Siberian Snakes

Page 57: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

Space for Siberian Snakes in HERA

Page 58: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 59: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

The Isolated Resonance Model

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

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

Page 60: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 61: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

|Spe

ctru

m

First Order Theories A) DESY III

Low energies: First order theories agree

Isolatedresonancemodel:

Linearspin-fieldtheory:

Momentum (GeV/c)

Plim

Page 62: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

First Order Theories B) PETRA

Medium energies: resonances still isolated

Linearspin-field theory:

|Spe

ctru

m

Plim

Isolatedresonancemodel:

36 38343230

Momentum (GeV/c)

Page 63: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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)

Page 64: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

Siberian Snakes and Resonances

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

Page 65: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

Amplitude dependentspin-orbit resonances

Page 66: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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 .

Page 67: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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:

Page 68: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

Plim after Snake Matching

8 snakes standard scheme

8 matched snakes

Page 69: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

Spin Tune after Snake MatchingQy2=ν

Qy2=ν

Qy21−=ν

Qy21−=ν

4 snakes in standard scheme

4 matched snakes

Page 70: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

Spin Tune after Snake Matching

Qy21−=ν

Qy2=ν

Qy2=ν

Qy21−=ν

4 snakes in standard scheme

8 matched snakes

Page 71: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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

Page 72: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

Allowed Beam Sizes

Snake matching allows to have significantly larger beams.

4 snakes inStandard scheme

4 filteredsnakes

4 matchedsnakes

8 filteredsnakes

8 filteredsnakes

Page 73: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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:

Page 74: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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.

Page 75: snake snakehoff/talks/02-08-30/02cosy_ss.pdfGeorg.Hoffstaetter@DESY.de Polarized Proton Beams l Resonance excitation by the Stern-Gerlach Effect Łrequires extremely difficult phase

[email protected]

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