electrons/positrons · harmonic oscillator with periodic spring constant. periodic conditions:...

��

Protons:

• Transverse beam dynamics.

� Simple model of the proton.

� Spin dynamics.

� Depolarizing resonances.

� Siberian snakes.

� The real machines: RHIC and injectors.

Electrons/Positrons:

• Longitudinal beam dynamics.• Synchrotron oscillations and tune.• Electrons: Synchrotron radiation

� Radiative polarization.

� Quantum fluctuations ⇒ Spin Diffusion

� Polarization in some real e± machines.

� Measurements with polarized e± beams.

1 �

QCDSP: Spin Dynamics in AcceleratorsWaldo MacKay 7 June, 2004

� � � � � � ��

+ - ++ + +

+ ++++ - -

- - -

-- -

-

� � � ��

Linac: ~F = q ~E(t). Ring with rf cavity

• Must maintain synchronism of bunch with rf phase.

• Particles oscillate in energy about the stable synchronous phase.

2 �


��

� ��

Dipole magnets bend the beamaround the ring.

B

p

proton

q=+1

F

Bend magnet (dipole)

Field out of screen

Charged particles are deflected bymagnetic fields. Lorentz Force:

~F =q

γm~p× ~B

Quadrupole magnets focus thebeam for stability.

MagneticField

Protons moving into screen

S

N

N

S

Magnetic Lens (quadrupole)

Force

Vertically focusing

Horizontallydefocusing

3 �


� ��

H(X,PX , Y, PY , Z, PZ ; t) =

√

(~P − q ~A)2 +m2c4 + qφ

After a bunch of canonical transformations and φ = 0, ~A = (0, 0, As):

H(x, x′, y, y′, z, δp/p0; s) ' − q

p0

As −(

1 +x

ρ

)(

1 +δp

p0

− 1

2(x′2 + y′2) + · · ·

)

s= ρθρ

xz

y

Z

X

Y

Design trajectory coordinate systemLocal traveling

Fixed lab coordinatesystem

θ

ρ =p

qB⊥

x′ =dx

ds

y′ =dy

ds

Paraxial approx.: |x′|, |y′| � 1

4 �


��

x′′ + kx(s)x =δ

ρ(s),

y′′ + ky(s)y = 0,

with δ =δp

p0

.

For quadrupoles:

kx =q

p

∂By

∂x

ky = −qp

∂By

∂x

Harmonic oscillator with periodic spring constant.

Periodic conditions: kj(s+ L) = kj(s), ρ(s+ L) = ρ(s)

where L is length of periodic cell.

• Horizontal motion has inhomogeneous dispersion term.◦ Ignore it for now.

5 �


� � � � ��

Use Floquet’s (Block’s) Theorem ⇒Quasi-periodic solutions of form:

x(s) =√

Wβ(s) cos(ψ(s)), with

ψ′(s) =1

β(s).

Periodicity of β-function: β(s+ L) = β(s).

Note: In general ψ(s+ L) 6= ψ(s) + n2π. Resonances are bad!

x′(s) = −√

Wβ

(α cosψ + sinψ) ,

with α = − 12β′.

6 �


� ��

Alternate focusing and defocusing lenses for stability.

Horizontal Betatron Oscillationwith tune: Qh = 6.3,

i.e., 6.3 oscillations per turn.

Vertical Betatron Oscillationwith tune: Qv = 7.5,

i.e., 7.5 oscillations per turn.

7 �


� � � ��

For a particular trajectory with initial conditions:

• Solve for sinψ and cosψ from equations for x and x′.

• Use sin2 ψ + cos2 ψ = 1 to get an invariant:

W =1

β

[

y2 + (αy + βy′)2]

(1)

• Functions of s: y(s), y′(s), β(s), α(s). (β and α are periodic.)

• Eq. (1) is the equation for an ellipse.• Area of ellipse = πW.

• Beam envelope: ±√

β(s)ε• πε is the rms emittance

8 �


� � � � ��

• Most beams have a low enough density, so that we ignore hard collisionsbetween particles.• Thus we can use a 6d phase space rather than a 6N-d phase space.

• In the phase space of coordinates and their corresponding canonical mo-menta, the phase flow of the particle trajectories evolves so that the volumesof differential volume elements are preserved.• In other words, the Jacobian determinant is 1.

• Emittance is the area of the projection of the beam’s phase-space volumeonto a particular (xi, Pi) plane.

9 �


� ��

Horizontal Betatron Oscillationwith tune: Qx = 3.28,tracked through 10 turnswith 8 periodic cells.

x

x’

Poincare plot of proton on suc-cessive turns for one location inthe ring.

10 �


� � � ��

.Energy =−

B

µ B

µ B

r

µx=dF Bdq v

µ= πr2

dFi

x

dj=dq v

Torque =

� 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.)

11 �


� � � � � � � � ��

Bar magnet+ "proton"=

+ =+

+ Charge

N

SS

N

Gyroscope

MagneticDipoleSpin

Moment

++ +

Polarization: Average spin of the ensemble of protons.

~P =1

N

N∑

j=1

~S

|S|

12 �


� � ��

Energy-momentum tensor (a la Weinberg)

Tαβ(x) = T βα(x) =∑

n

pαnp

βn

Enδ3(x− xn(t))

For isolated system∂

∂xαTαβ = 0.

Define 4d analogue of ~r × ~p:

Mαβγ = xαT βγ − xβTαγ

Jαβ =

∫

M0αβd3x =

∫

xαT β0 − xβTα0d3x

Spin (intrinsic angular momentum):

Sα = 12cεαβγδJ

βγuδ, proper velocity: uδ = dxδ

dτ .

13 �


For a particle at rest with CM at rest at the origin:

J�µν :

0 0 0 00 0 S�

z −S�y

0 −S�z 0 S�

x

0 S�y −S�

x 0

, ( ~J� = ~S�)

Boost along z:

Jµν :

0 γβS�y −γβS�

x 0−γβS�

y 0 S�z −γS�

y

γβS�x −S�

z 0 γS�x

0 γS�y −γS�

x 0

, ⇒ ~J =

γS�x

γS�y

S�z

Sµ :

γβS�z

S�x

S�y

γS�z

, ⇒ ~S =

S�x

S�y

γS�z

, S0 = ~β · ~S

~J − ~S =

(γ − 1)S�x

(γ − 1)S�y

(1 − γ)S�z

14 �


� � � ��

~rCM × ~pCM = ( ~J − ~S)⊥

γβmc(−xCM y + yCM x) = (γ − 1)~S�⊥

γβmc(~xCM + ~yCM) = (γ − 1) z × ~S�⊥

~r⊥CM =γ

γ + 1

~β × ~S

mc

CM shifts to right.v=aω

CM at rest.S

FasterSlower

SBoost into screen

Center of charge wobbles: classical “Zitterbewegung”

15 �


� ��

1. Boost observer to left.2. Boost observer downward.3. Boost back to rest.

• Net rotation of rest frame.

Rest Sytem: SSytem: S'

Sytem: S'' Rest Sytem: S'''

Boost alonghorizontal (-y)

γ=2

Boo

st a

long

ver

tical

(-x

)

γ=2

63.43°

γ=4

Boo

st b

ack

36.87°

16 �


� ��

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

d~S�

dt=

q

γm~S� ×

[

(1 +Gγ) ~B⊥ + (1 +G) ~B‖ +

(

Gγ +γ

γ + 1

) ~E × ~v

c2

]

.

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

Proton: G =g − 2

2= 1.792847, 523.34 MeV/unit Gγ

Electron: a = G =g − 2

2= 0.001159652, 440.65 MeV/unit aγ

γ =Energy

mc2.

17 �


� ��

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 − 2

2= 1.7928, γ =

Energy

mc2.

18 �


��

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.

19 �


��

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.

20 �


��

� � � � ��

x

y

zRz(90°) Rx(−90°)

1

2

3x

y

z

1

2

3

Rx(−90°) Rz(90°)

21 �


� � ��

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

22 �


��

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

23 �


��

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

24 �


� � � � ��

Froissart—Stora Formula:

Pf

Pi

= 2 exp

(

−π|ε|2

2α

)

− 1.

Ramp rate: α = dGγdθ , (θ : 2π/turn.)

Resonance strength: ε =Fourier amplitude.

25 �


� ��

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.

(Simulations by Mei Bai)

26 �


� � � � � ��

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%

(From Jeff Woods)

AC dipole pulses at resonances:• 0 +Qy

• 12 +Qy

• 36 −Qy

• 36 +Qy

27 �


28 �


� � ��

0 100 200 300 400 500Gγ

0

0.2

0.4

0.6

0.8

1

1.2

intr

insi

c re

sona

nce

stre

ngth

pp rhicb.2001.twiss(10mβ*)

Qx = 28.236

Qy = 29.219

πεy = 10π µm

Will depolarize beamduring acceleration.

Solution: Snakes

29 �


� � ��

• 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

30 �


� ��

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

d~S

dt= ~W × ~S

H(~q, ~P , ~S; s) = Horb + Hspin

= Horb + ~W · ~S +O(h2)

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.

31 �


� � � � � ��

SU(2) with usual spinor notation:

Pauli matrices: σx =

(

0 11 0

)

, σy =

(

0 −ii 0

)

, σz =

(

1 00 −1

)

.

Rn(θ) = ei n·~σ θ/2 =

(

cos θ2

+ inz sin θ2

(ny + inx) sin θ2

(−ny + inx) sin θ2

cos θ2− inz sin θ

2

)

.

SO(3) :Rn(θ) =I cos θ +

0 nz −ny

−nz 0 nx

ny −nx 0

sin θ

+

n2x nxny nxnz

nxny n2y nynz

nxnz nynz n2z

(1 − cos θ).

32 �


��

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 ~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) 6= q(s) and P (s+ L) 6= P (s).

33 �


�

� ��

a: HERA-p / 8 snakes / 4 pi mm mrad / 800 GeV

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1-0.8

-0.6-0.4

-0.20

0.20.4

0.60.8

1

00.10.20.30.40.50.60.70.80.9

1

n-vector at 1σ and 800 GeV

b: HERA-p / 8 snakes / 4 pi mm mrad / 802 GeV

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1-0.8

-0.6-0.4

-0.20

0.20.4

0.60.8

1

00.10.20.30.40.50.60.70.80.9

1


• Simulation with only vertical betatron motion.

• 802 GeV is closer to a resonance spin resonance than 800 GeV.

Des Barber et al.

34 �


�

� ��

a: HERA-p / 8 snakes / 64 pi mm mrad / 800 GeV

-1-0.5

00.5

1 -1-0.8

-0.6-0.4

-0.20

0.20.4

0.60.8

1

-0.8-0.6-0.4-0.2

00.20.40.60.8

1


b: HERA-p / 8 snakes / 64 pi mm mrad / 802 GeV

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1-0.8

-0.6-0.4

-0.20

0.20.4

0.60.8

1

-1

-0.5

0

0.5

1


• Larger amplitude oscillations have a larger tune shift due to nonlinear ele-ments.

• 802 GeV is closer to a resonance spin resonance than 800 GeV.

Des Barber et al.

35 �


��

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

(Alfredo Luccio)

36 �


� � � � � ��

RHIC

AGSLINAC

BOOSTERSOURCE

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

37 �


� � � � � ��

STARPHENIX

PHOBOS BRAHMS

PolarimetersHjet CNI

SnakesRotators

AC Dipole

RHIC

AGSLINAC

BOOSTERSOURCE

Solenoid Snake

Helical SnakeACDipole

Polarimeters

38 �


� ��

KEK OPPIS∗

upgraded at TRIUMF

70 → 80% Polarization

15 × 1011 protons/pulse

at source

6 × 1011 protons/pulse

at end of LINAC

∗Optically Pumped Polarized Ion Source

39 �


� � ��

a) TRIUMF OPPIS b) ECR PROTON SOURCE

Fig. 1. 1) ECR Proton Source, 2) Superconducting Solenoid, 3) Optically-Pumped Rb Cell, 4) Deflection Plates, 5) SonaTransition Region, 6) Ionizer Cell, 7) Ionizer Solenoid, 8) Quartz Tube, 9) ECR Cavity, 10) Three Grid Extraction System, 11) Boron-Nitride End Cups, 12) Indium Seals.

Capture polarized

electron from

alkali atomoptically pumped

Transfer ofelectron-spinpolarization tonuclear-spinpolarization

(Sona transition)

Ionization

H - (p )(p )H 0H 0 (e )H +

protonsGenerate

40 �


� ��

� � � ��

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

41 �


y

x

sφsnake axis

µ

(beam)The rotation axis ofthe snake is φ, and µ isthe rotation angle.

Note that the φ con-tours shift slightlyfrom injection (red)at 25 GeV to storage(pink) at 250 GeV.

Rotation Angles for a Helical Snake-5

-4

-3

-2

-1

00 1 2 3 4 5

Bout [T]

Bin [T]

10

20

30

40 50 60 70 80 90100110120130140

150

160

170

190

180

-10

-20

-30

-40

-50

-60

-70

0 10 20 30 40 50 60 70 80

φ µ

42 �


� � ��

43 �


� � � �� ν yjM ,j 0 P ij

M ,j 1 P fjM ,j 2

eint = 0.5, εimp = 0.05, 2 Snakes, spin tune = 0.5

29.15 29.16 29.17 29.18 29.19 29.2 29.21 29.22 29.23 29.24 29.251

0.5

0

0.5

1

3/16

3/14

Pf

1/41/6

Vertical Betatron Tune

44 �


� � � ��

-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

45 �


� � � � � ��

-1-0.5

00.5

1

Sz

-1 -0.5 0 0.5 1

Sx

-1

-0.5

0

0.5

1Sy

E = 25 GeV

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

-1-0.5

00.5

1

Sz

-1 -0.5 0 0.5 1

Sx

-1

-0.5

0

0.5

1Sy

E = 250 GeV

DX

Rotator D0

DX

D0 Rotator

46 �


µ

θRotat

or ax

is

y

s

x

Beam

The rotation axis ofthe spin rotator is inthe x-y plane at an an-gle θ from the vertical.The spin is rotated bythe angle µ around therotation axis. 0 0.5 1 1.5 2 2.5 3 3.5 4

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 Angles for a Helical Spin Rotator

Note: Purple contour for rotation into horizontal plane.Black dots show settings for RHIC energies inincrements of 25 GeV from 25 to 250 GeV.

47 �


� � � � ��

x

y

z

Rotator’s spin vector at injection energy

x

y

z

Rotator’s spin vector at 250 GeV

48 �


� � � � � ��

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

49 �


��

Vertical polarizationwith rotators off.

Spin is down.

Rotators on

Spin is radially inwards!

OOPS!

Reverse all rotatorpower supplies and tryagain.

YES!

50 �


� � � � � � � Gγ = 0 +Qy

Top: AC dipole pulse amplitude(current)

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

Top: Beam coherence

Bottom: Tune spectrum

51 �


� � � � � ��

• Radiated power:

Pγ =2

3remc

3 γ4β4

ρ2, re =

e2

4πε0mc2.

Radiation in forward direction with opening angle ∝ γ−1

• Energy loss per turn:

Uγ =

∮

Pγ

cds

• Critical energy: half the power is radiated by photons less than the criti-cal energy, and the other half, above.

uc = hωc =3hc

2ργ3

52 �


• Number of photons per second:

Nγ =

∫ Umax

0

nγ(uγ) duγ =5

2√

3

αc

ργ

here: α = 1/137)

• Number of photons per radian:

Nr =5α

2√

3γ

• Average photon energy and 2nd moment:

〈uγ〉 =1

Nγ

∫ Umax

0

unγ(u) du =8

15√

3uc ' 0.32uc

〈u2γ〉 =

1

Nγ

∫ Umax

0

u2nγ(u) du =11

27√

3u2

c ' 0.41u2c

53 �


• Energy spread: σu =√

Cq

Jsργ2mc2

with Cq = 3.8 × 10−8 m and Js ∼ 2 + D.

Ring Energy σu

[GeV] [MeV]CESR 5.5 3HERAe 27.5 3LEP 45 30LEP 60 53LEP 100 150

Remember: Integer resonances separated by only 440 MeV.

The polarization in LEP dropped down to nothing just above 60 GeV.

54 �


��

ωrf = hωrev

W = −U − Us

ωrf

dW

dt=

qV

2πh(sinφs − sinφ)

dϕ

dt' ω2

rfηph

β2UsW

dωrev

ωrev

=dβ

β− dL

L= ηph

dp

p

ϕs ϕs+2π

τ+|dτ|τ

τ−|dτ|

Vrf

ϕ=ωrft

ηph < 0 above transition energy.

Add in synchrotron oscillations to resonance condition:

νspin = N +NvQv +NhQh +NsyQsy

55 �


��

Canonical cordinate: ϕ and conjugate momentum: W

a) ηtr > 0W

φπ−π φs

b) ηtr < 0W

φ2πφs

56 �


��

In a homogenious magnetic field the transition rates are

W↑↓ =5√

3

16

e2γ5h

4πε0mec2|ρ|3(

1 +8

5√

3

)

W↑↓ =5√

3

16

e2γ5h

4πε0mec2|ρ|3(

1 − 8

5√

3

)

.

Evaluating the equilibrium polarization have (Sokolov Ternov)

PST =W↑↓ −W↓↑W↑↓ +W↓↑

=8

5√

3= 0.9238.

An unpolarized beam polarizes:

P (t) = PST [1 − exp(−t/τST)] ,

where the polarization rate is given by

τ−1ST

=5√

3

8

e2γ5h

4πε0m2ec

2

1

L

∮

ds

|ρ|3 .

57 �


� � ��

Ring Particle Energy Nγ ∆U τST

W↑↓

frevNγ

[GeV] [/turn] [loss/turn]

CESR e± 5.5 700 −1 MeV 167 min 1 × 10−13

HERAe e± 27.5 3600 −83 MeV 23 min 1 × 10−12

LEP e± 45 5800 −120 MeV 300 min 2 × 10−13

LEP e± 60 7800 −380 MeV 81 min 8 × 10−13

RHIC p 100 7 −3 meV 3 × 1014 yr 6 × 10−29

RHIC p 250 18 −0.13 eV 3 × 1012 yr 2 × 10−27

HERAp p 920 65 −8.5 eV 1 × 1011 yr 3 × 10−26

Tevatron p 1000 70 −8.5 eV 2 × 1011 yr 2 × 10−26

SSC p 20000 1400 −0.12 MeV 7 × 107 yr 3 × 10−23

58 �


� ��

uγ

θx pu

W

ϕ

In phase space quantum fluctuations cause instantaneous hops of momentumfrom one ellipse to another. (Hops in the Action.)

59 �


��

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 ~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) 6= q(s) and P (s+ L) 6= P (s).

60 �


� ��

Derbenev–Kondratenko formula for equilibrium polarization:

PDK =8

5√

3

∮

⟨

1|ρ|3 b ·

(

n− ∂n∂δ

)

⟩

sds

∮

⟨

1|ρ|3

[

1 − 29(n · s)2 + 11

18

(

∂n∂δ

)2]⟩

sds

1

τDK

=5√

3

8

reγ5h

me

1

L

∮

⟨

1|ρ|3

[

1 − 29(n · s) + 11

18

(

∂n∂δ

)2]⟩

sds

averaged over phase space at azimuth s.

δ = ∆p/p is the fractional momentum deviation from design.n is the invariant spin field.

b = s× ˙s| ˙s| is the direction of magnetic field if ~E = 0.

ρ is the cyclotron radius of the trajectory.L is circumference of synchrotron.

61 �


��

62 �


��

1τdep

' 5√

38

reγ5hme

1L

∮

⟨

1|ρ|3

(

∂n∂δ

)2⟩

sds

1

τpol

' 1

τST

+1

τdep

63 �


� � ��

As an example from CESR (CUSB): MΥ = 9459.97 ± 0.11 ± 0.07 MeV

[Phys Rev D29, 2483 (1984)].

64 �


� ��

From Angelika Drees’ Thesis

65 �


� ��

(Des Barber in “eRHIC Zeroth-Order Design Report)

66 �


��

67 �


� � � � � � � � � ��

1. M. Conte and W. W. MacKay, An Introduction to the Physics of Particle

Accelerators, World Sci., (1991).

2. L. H. Thomas, Phil. Mag. S. 7, 3, 1 (1927).

3. M. H. L. Pryce, Proc. Royal Soc. of London, A195, 62 (1949).

4. S. Weinberg, Gravitation and Cosmology: Principles and Applications of

the General Theory of Relativity, John Wiley & Sons, (1972).

5. Misner, Thorne, and Wheeler, Gravitation, Freeman, San Francisco,(1973).

6. R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications,John Wiley & Sons, (1974).

7. Ya. S. Derbenev and A. M. Kondratenko, JETP 37, 968 (1973).

8. Brian W. Montague, Phys. Rep., 113 (1984).

9. S. Y. Lee, Spin Dynamics and Snakes in Synchrotrons, World Sci., (1997).

10. K. Yokoya, DESY 86-057 (1986), and KEK Report 92-6 (1992).

11. D. P. Barber et al., DESY 98-096 (1998).

68 �


12. D. P. Barber, “Electrons are not protons”, EICA Workshop, BNL, UptonNY, BNL-52663 (Feb. 2002).

13. Y. Mori, “Present Status and Future Prospects of Optically Pumped Po-larized Ion Source”, AIP Conf. Proc. 293, 151 (1994).

14. A. N. Zelenski et al., Proc. of the 1995 Part. Accel. Conf. and Int. Conf.

on High-Energy Accel., Dallas, 864 (1996).

15. A.A. Sokolov and I.M. Ternov, Synchrotron Radiation, Pergamon Press(1968).

16. V.B. Berestetskii, E.M. Landau and L.P. Lifshitz, Quantum Electrodynam-

ics, Pergamon Press (1982). (Landau & Lifshitz Course on Theor. Phys.,Vol. 4)

17. W. W. MacKay et al., Phys. Rev. D29, 2483 (1984).

18. J. Johnson, “SPEAR Polarization” in Polarized Electron Acceleration and

Storage Workshop, DESY-M-82/09 (1982); S. R. Mane, Phys. Rev. A36,120 (1987).

19. W. W. MacKay et al., Phys. Rev. D29, 2483 (1984).

20. K. A. Drees, Thesis, WUB-DIS 97-5, Wuppertal (1997).

21. http://www.rhichome.bnl.gov/RHIC/Spin/

69 �


22. Eds. Farkhondeh and Ptitsyn, “eRHIC Zeroth-Order Design Report”, C-A/AP142 (2004).

70 �


electrons/positrons · harmonic oscillator with periodic spring constant. periodic conditions:...

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