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e+/-

p

SPEAR

HERA

SLC

LEP

MIT/Bates

PETRA

Tristan

past and present polarized

beam facilities:

ZGS

AGS

IUCF

RHIC

+ many lower

energy

facilities

HERA-P

TeV-P (?)

possible future polarized

beam facilities:

NLC

JLC

TESLA

CLIC

outline:

- Thomas-BMT equation
- Spinor algebra
- Equation of motion for spin
- Periodic solution to the eom
- Depolarizing resonances
- Polarization preservation in storage rings
- Siberian snakes
- Partial Siberian snakes
- Resonance strength
- Summary

Thomas-Bargmann-Michel-Telegdi (Thomas-BMT) equation (1959)

Uhlenbeck and Goudsmit (1926):

gyromagnetic ratio

magnetic

moment

angular momentum, “spin”, |s|=hbar/2 (e+/-,p)

a=(g-2)/2 = anomolous part of the electron magnetic moment

G=(g-2)/2= anomolous part of the proton magnetic moment

a = 0.0011596 (e-)

G = 1.7928 (p)

x = 0.001166 (μ)

The “spin” (e.g. angular momentum) of the particle interacts with the external electromagnetic fields through influence on its magnetic moment. The equation

of motion in an external magnetic field B, in the rest frame of the particle is

orthogonal fields precess the spin

see e.g.

J.D. Jackson

“Classical

electro-

dynamics”

angular precession frequency:

It is convenient to normalize s and use S with the normalization |S|=1

Thomas-BMT equation (modern form)

In the laboratory frame, the spin precession of a relativistic particle is given

by the Thomas-BMT equation (derived from a Lorentz transformation of the

electro-magnetic fields including relativistic time dilation):

assuming that the particle velocity is along the direction of external electric fields and that there are no significant trans-verse electric fields; i.e. vE=0

the spin precession due to

B depends on the beam

energy (=E/m); the higher

the beam energy, the

more the spin precession

the spin precession due to

B|| is energy-independent

Beam polarization

the beam polarization is defined as the ensemble average over the spin vectors

S of the particles within the bunch:

N = number of particles per bunch

The polarization is the quantity that is measurable (e.g. by measuring a scattering

asymmetry of a fixed targets in a proton accelerator):

can transport

the (31)

spin vector

or, equivalently, the (21)

spin wave function

The transformation between the two representations is given by

with the Pauli matrices defined† by

y

†this is a cyclic permutation of the

“standard” Pauli matrix definitions

which conforms with the axes defi-

nitions prefered by the high-energy

physics community

v

s

x

Example: spinor representations for vertical polarization

given:

let

then

b=0

a=1

C. Spin equation of motion (reference: Courant and Ruth)

In 1980, Courant and Ruth expressed the magnetic fields of the Thomas-BMT equation

in terms of the particle coordinates and reexpressed the equation of motion for the spin in terms of a (complicated) Hamiltonian. In doing so a simple expression resulted:

where θ is the orbital angle:

θ

local bending radius

In the absence of depolarizing resonances, H has a simple form

where κ=G for protons and κ=a for electrons/positrons

Courant and Ruth introduced another (now conventional) form for the EOM.It is assumed that H is time-independent and that there are no perturbing fields. Then H can be reexpressed as a linear combination of the 3 components of the Pauli vector:

=|| is the amplitude of the precession frequency=[(g-2)/2]

Pauli matrices =<x, z, y>

unit vector aligned with

The solution to the eom is

with

After expanding the exponential, using the algebra of the -matrices, the solution for

the spinor is

D. Periodic solution to the spin equation of motion

M=M1M2…Mn

express the spin matrix M as the product of n precession matrices:

M0(+2)=M0()

the one-turn spin map M0 for the closed orbit is periodic:

for the single element precession the spin, we had

with giving the precession frequency and the

precession angle

for the one-turn-map, since M0 is unitary, it may also be expressed as

>

y

n0=“stable spin direction”

>

n0

P

s

0=/2=“spin tune”

with =(g-2)/2

x

>

n0 = stable stable spin direction (axis which returns to

same place in every turn around the ring)

0 = spin tune (number of times the spin precesses about

n0 in one turn around the ring)

>

Periodic solution to the spin equation of motion, cont’d

The one-turn-map is given, after algebra, by

(previous solution)

please remove subscript 2

in Eq. 10.24

stable spin

direction

spin tune

Expanding the Pauli matrices, the solution is given equivalently by

So, if the Hamiltonian is time-independent (e.g. the influence of spin resonances may be

neglected – as can be made the case with most low energy accelerators), the spin tune

and the stable spin direction may be easily evaluated.

The spin tune is given by determining M (multiplying all

rotation matrices) and taking the trace of the spin-OTM:

>

y

n0

>

y

For the stable spin direction n0, it is convenient to parametrize n0

using directional cosines

s

s

with normalization

x

x

Example: spin tune and stable spin direction for a planar ring with perfect alignment

the one-turn spin map for a ring with only vertical dipole fields is

with

expanding the exponential,

i.e.

the spin tune is derived from the trace of the OTM:

The orientation of the stable spin direction is found by

equating components of the OTM. Recall,

>

or, from above,

y

n0

y

s

s

x

x

depolarizing resonances occur whenever the spin tune is harmonically related (“beats”) with any of the natural oscillation frequencies of the particle motion:

q,r,s,t,and u

are integers

m=t+uP, where P is the

superperiodicity

betatron tunes

synchrotron tune

resonance order: |m|+|q|+|r|+|s|

Types of depolarizing resonances

0=t+uP imperfection resonances

due to magnet imperfections, dipole rotations,

and vertical quadruple misalignments

these are in practice usually

the most significant for

existing accelerators with

polarized beams

0=(t+uP)+rQyintrinsic resonances

due to gradient errors

0=(t+uP)+sQssynchrotron sideband resonances

due to coupling between longitudinal

and transverse motion

these resonances become

increasingly important at

higher beam energies

0=(t+uP)+qQx+rQy(higher-order) betatron

coupling resonances

1 mile total length, E=45.6 GeV

(a~103), 23 achromats

108° phase advance per cell

Simulated particle and spin motion in the SLC arc (courtesy P. Emma, 1999)

orbit with initial

offset error of

500 m

longitudinal

polarization

vertical

polarization

in practice, vertical “spin bumps” were used to properly orient the spin (longitudinally) at

the interaction point

equation of motion

(Eq. 10.13)

solution

(Eq. 10.17)

y

periodic solution

>

(Eq. 10.24)

n0

>

P

s

n0 = stable stable spin direction (axis which returns to

same place in every turn around the ring)

0 = spin tune (number of times the spin precesses about

n0 in one turn around the ring)

>

x

spacing of (strong) imperfections resonances:

electrons: 0=a=E/0.411 [GeV]

protons: 0=G=E/0.523 [GeV]

(as will be shown) resonance strength (i.e. the Fourier harmonic of the off-diagonal

elements of H which couple the up and down components of )

(Eq. 10.49)

linear in the

particle energy

depends on the vertical displacement

F. Polarization preservation in storage rings

1. Injection

>

align the beam polarization of the injected beam Pinj with the stable spin direction n0

injected

polarization

stable spin direction

component of polarization

surviving injection

polarization that one would

measure

>

>

using directional cosines (’s for Pinj and ’s for n0), project Pinj onto n0:

the measured polarization is given by projection onto the plane of interest:

2. Harmonic correction (Petra, Tristan, AGS, HERA, LEP,…)

concept: correct those orbital harmonics close to 0

n is the harmonic

of interest

orbital angle

Fourier harmonics

example: correction of imperfection resonances using pulsed dipoles at the AGS

during proton ramp to 16.5 GeV (courtesy A. Krisch, 1999)

pulsed dipole

currents

main dipole

current (~E)

Example: lepton beam polarization at 27.5 GeV measured at HERA after correction of the strength of the nearest imperfection resonance (courtesy the HERMES experiment, 2002)

HERA-I

spin rotators

at fixed-target

experiment

spin rotators

at all experiments

particular concerns for the colliding-beam experiments (H1 and ZEUS):

solenoidal fields not locally compensated

(beam trajectories not perfectly parallel

to solenoid axis)

increased lepton beam emittance coupling

(for matched IP beam sizes)

effect of beam-beam interaction on lepton beam polarization

complicated

spin-matched

optics

closed-orbit control

and harmonic spin

matching

no validation of theory

by experiment

(2003

data)

Froissart-Stora formula (for describing spin transport through a single, isolated resonance):

final polarization

resonance strength

initial polarization

“ramp rate”

limiting behavior

+1 if is small and/or if is large

-1 if is large and/or if is small

Pfinal/Pinitial =

Example: spin flipping of a vertically

polarized beam (courtesy A. Krisch, 1999)

t~1/

t=10 ms

t=30 ms

Vsol~

From the Froissart-Stora equation,

If the resonance is crossed quickly ( large), then the polarization will be preserved.

Intrinisic resonances may therefore be crossed by rapidly pulsing a quadrupole at the appropriate time.

0

energy ramp

integer + Qy

example: correction of intrinsic

resonances using pulsed quad-

rupoles at the AGS

integer

resonances

integer - Qy

G

pulsed dipole

currents

pulsed quad-

rupole currents

rapid traversal

of resonance

main dipole

current (~E)

method particle type energy application

fixed

ramped

ramped

minimize || of

nearby resonances

(same) – empirical correction

as function of beam energy, E

maximize ||/2 at nearby

resonance as function of E

harmonic orbit

correction

lepton

proton

proton

harmonic orbit

correction

tune jump

For high energy polarized protons, the above methods were anticipated to be of limited

applicability (empirically determined corrections are time consuming to develop and dependent on the closed orbit; adiabatic spin flip harder as ||increases). The solution,

proposed in 1976, was first tested over a decade later and proved effective.

G. Siberian snakes (Derbenev and Kondratenko, 1976)

(P denotes a polarimeter)

P

concept: make the spin tune 0 independent of energy

(and equal to some non-resonant value)

B

snake,

example: a type-I Siberian snake (rotation of

spin around longitudinal axis per turn)

A

one-turn spin matrix

snake

B

A

expanding M and taking the trace gives

=0 (no snake) s=G as before

= (full snake) s=n/2 with n odd

independent of the

beam energy

with a spin tune of ½, the depolarizing resonance condition can never be satisfied

Type I = about longitudinal axis

Type II = about radial axis

Type III = about vertical axis

Design of Siberian snakes

longitudinal

snake

depends linearly on

best suited for low energy beams

transverse

snake

independent of so fixed-field magnets

may be used. However, a dipole produces

an orbital deflection angle of /G which

is large at low beam energies

therefore best suited for high energy beams

Example: type-I (= about longitudinal axis), transverse snake (courtesy A. Chao, 1999)

magnet orientation:

H = horizontal dipole

V = vertical dipole

vertical orbit excursion

optically

transparent

horizontal orbit excursion

orbital angle chosen for a total spin precession of /2

>

>

>

>

>

>

>

original

notation

( ,x)( ,-x)( ,y)( ,-x)(,-y)( ,x)( ,y)

spin precession axis (in direction of field)

spin precession angle

Example: polarization preservation near an imperfection resonance using a Siberian snake

(spin precession)

snake on

snake off

snake on

snake off

depolarization

polarization maintained at all beam energies

Result: all high-energy polarized proton facilities plan to or do use Siberian snakes

H. Partial Siberian snakes (T. Roser)

again

dependence of the spin tune on G for various strengths of partial snake:

location of intrinsic

resonance with Qy=0.2

(=0)

imperfection resonance

location of intrinsic

resonance with Qy=0.2

(=)

only a few % snake is needed to avoid strong imperfection resonances

larger partial snakes can be used to avoid intrinsic resonances (over narrow energy range)

The spin equation of motion was solved previously disregarding the influence of depolarizing resonances

EOM

with

Courant and Ruth gave the general form of H, where t and r

are complicated functions of the particle coordinates

While we defer here the extension of their work (see text), the definition of resonance

strength warrants mention. Due to the periodic nature of a circular accelerator, the

coupling term may be expanded in terms of the Fourier components; i.e.

where is the particle orbital angle, ±res,k=k for imperfections resonances, ±res,k=k±Qy for first order intrinsic resonances, etc., and k is the resonance strength given by the Fourier amplitude

for the case of an imperfection resonance, is given approximately by summing over the

radial error fields encountered by a particle in one turn:

optics programs (e.g. DEPOL) exist to calculate given the magnetic optics

electrons and protons possess a magnetic moment proportional

to the spin angular momentum, or polarization:

magnetic fields orthogonal to the polarization change the

orientation of the polarization

the Thomas-BMT equation shows this explicitly in the rest frame of the particle

polarization transport can be equivalently described in terms of the

spin wave function, or spinors, given in terms of the Pauli matrices

in terms of spinors, the equation of motion

(Courant and Ruth) has a simple form

with solution

the periodic solution is

0 is the spin tune

>

n0 is the stable spin direction

depolarizing resonances result when the spin frequency is harmonically related

to any natural oscillation frequency of the beam

the resonance strengths can be evaluated (for widely spaced resonances):

which shows that the resonance strength increases with increasing energy

polarization preservation includes matching the polarization onto the stable

spin direction at injection

other preservation methods include:

adiabatic spin flip (ala Froissart and Stora)

betatron tune jump (AGS)

harmonic correction (AGS, LEP, HERA,…)

Siberian snakes (Derbenev and Kondratenko)

Siberian snakes force 0=1/2 (full snake) so the resonance condition is never

satisfied at any energy

snake designs generally are optically transparent. The choice of solenoidal or

dipole snakes depends on the beam energy

partial Siberian snakes (Roser) are useful for curing selected resonances (AGS)

Exercises (for Wednesday, June 30):

1. Derive the spinor representations for a) horizontal, b) vertical, and c) longitudinal polarization.

Recall the transformation:

y

s

particle

trajectory

(along s)

where the Pauli matrices are given by

x

2. The spinor rotation matrix for precession by an angle

about the j-axis (j=x,y,s) is given by exp(ij/2). Assuming an initial polarization that is purely

a) horizontal, b) vertical, c) longitudinal, determine the final polarization after precession about

the longitudinal (s) axis by an angle and sketch the final orientation in each of these cases.

Note: exp(ij) = cos()+ijsin().

3. Consider a hexagonally-shaped accelerator with a type-I Siberian snake.

a) with a proton kinetic energy of 108.4 MeV, what is G?

b) show that at the location of the snake the stable spin direction is in

the longitudinal direction (i.e. parallel to the nominal particle velocity)

c) draw the orientation of the stable spin direction in each of the 6 seg-

ments of the ring

d) assuming that the beam is fully polarized, what are the amplitudes of the

components of the transverse beam polarization measured at the location P?

snake

P

beam direction

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