Beam Polarization

1
Beam Polarization
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

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

Uhlenbeck and Goudsmit (1926)
gyromagnetic ratio
magnetic moment
angular momentum, spin, shbar/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 S1
3
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. v?E0
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)
4
B. Spinor algebra using SU(2)
can transport the (3?1) spin vector
or, equivalently, the (2?1) spin wave function
The transformation between the two
representations is given by with the Pauli
matrices defined by
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
y
v
s
x
Example spinor representations for vertical
polarization
given
let
then
b0
a1
5
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 ? is the orbital angle
?
local bending radius
In the absence of depolarizing resonances, H has
a simple form
where ?G? for protons and ?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 ?lt?x, ?z, ?ygt
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
6
D. Periodic solution to the spin equation of
motion
MM1M2Mn
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
gt
y
n0stable spin direction
gt
n0
s
P
?0??/2?spin tune with ?(g-2)/2
x
gt
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)
gt
7
Periodic solution to the spin equation of motion,
contd
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
gt
y
n0
gt
?y
For the stable spin direction n0, it is
convenient to parametrize n0 using directional
cosines
?s
s
with normalization
?x
x
8
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,
gt
or, from above,
y
n0
?y
?s
s
?x
x
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E. Depolarizing resonances
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
mtuP, where P is the superperiodicity
betatron tunes
synchrotron tune
resonance order mqrs
Types of depolarizing resonances
?0tuP 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(tuP)rQy intrinsic resonances
due to gradient errors
?0(tuP)sQs synchrotron sideband resonances
due to coupling between longitudinal and
transverse motion
these resonances become increasingly important
at higher beam energies
?0(tuP)qQxrQy (higher-order) betatron
coupling resonances
10
Example SLC collider arc
1 mile total length, E45.6 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
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Intermediate summary
equation of motion
(Eq. 10.13)
solution
(Eq. 10.17)
y
periodic solution
gt
(Eq. 10.24)
n0
gt
s
P
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)
gt
x
spacing of (strong) imperfections resonances
electrons ?0a?E/0.411 GeV protons
?0G?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
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F. Polarization preservation in storage rings
1. Injection
gt
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
gt
gt
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
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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)
14
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)
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HERA-II
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)
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3. Adiabatic spin flip
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)
?t1/?
?t10 ms
?t30 ms
Vsol?
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4. Tune jump
From the Froissart-Stora equation,
If the resonance is crossed quickly (? large),
then the polarization will be preserved. Intrinisi
c 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)
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recap
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.
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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) ? ?sG? as before ??
(full snake) ? ?sn/2 with n odd
independent of the beam energy
with a spin tune of ½, the depolarizing resonance
condition can never be satisfied
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Types of Siberian snakes
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
21
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
gt
gt
gt
gt
gt
gt
gt
original notation
( ,x)( ,-x)( ,y)( ,-x)(?,-y)( ,x)(
,y)
spin precession axis (in direction of field)
spin precession angle
22
Example polarization preservation near an
imperfection resonance using a Siberian snake
(spin precession)
snake on
snake on
snake off
snake off
depolarization
polarization maintained at all beam energies
Result all high-energy polarized proton
facilities plan to or do use Siberian snakes
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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 Qy0.2
(?0)
imperfection resonance
location of intrinsic resonance with Qy0.2
(??)
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)
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Polarized beams at RHIC
25
I. Resonance strength, ?
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,kk
for imperfections resonances, ?res,kkQy 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
26
J. Summary
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
gt
n0 is the stable spin direction
27
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 ?01/2 (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)
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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 (jx,y,s) is
given by exp(i??j/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(i??j) cos(?)i?jsin(?).
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