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USPAS Lecture 17
LECTURE 17
Linear coupling (continued)
Coupling coefficients for distributions of skew quadrupoles
and solenoids
Pretzel Orbits
Motivation and applications
Implications
USPAS Lecture 17
Linear coupling (continued)
Coupling coefficients for distributions of skew quadrupoles
and solenoids
The previous discussion focused on a single skew quadrupole, for
simplicity. Actual machines typically have a distribution of skewquadrupoles, and also may include solenoids. The axial solenoid field couples to the slope of the trajectory; the end fields couple to
the trajectory itself: (c.f., Lecture 3, p 10:)
x
y
y
y
x
x
B
B
B
B
s
s
s
0
ρ
ρ
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Let’s see how to calculate the coupling coefficient for an arbitrary
distribution of skew quadrupole and solenoid strength around the
ring.
We’ll call the location at which we want to evaluate the trajectories
s
=0. At some other point in the ring,
s
, let the skew quadrupole
strength be
k s
, and the solenoid strength
s
. For the moment,
we assume that this is the only point of coupling in the ring. At theend of the discussion, we’ll integrate over the whole ring to get the
result for a distribution of strengths.
The incremental kick delivered to a trajectory at this point by these
fields, which extend a distance
s
is
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x s
Ty
x s
yk
Ty
y s
Tx
y s
xk
Tx
In Floquet coordinates, we have
ξ
α
ξ
β
ξ
β
ξ
α
ξ
β
ξ
ξ
β β
x
x
x
x
x
x
y
y
y
y
y
y
y
y
x
y
y x
Q
s Q
yk
y
k
Q
Q
s
This gives
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ξ
κ
ξ
κ
ξ
ξ
κ
ξ
κ
β β
α
β β
α
β β
κ
β β
κ
β β
x
x
x
y
x
x y
y
x
x
y
x
x
y
y
x y
x
y x
x
x y
x
y x
Q
Q Q
k
T
T
s
T
s
T
s
1
2
3
1 2
3 2
in which everything is evaluated at the point
s
There are similar equations for
y
, in which
y
and
x
are
interchanged, and
T
T
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Now consider a trajectory which starts at s=0, with phase space
coordinates
ξ
φ
ξ
φ
ξ
φ
ξ
φ
x
x
x
x
x x
x
y
y
y
y
y y
y
r
Q r
r
Q r
cos
sin
cos
sin
at that point. It travels to
s
, at which the betatron phase is
Φ
Φ
s
. The phase space coordinates there are
ξ
φ
ξ
φ
ξ
φ
ξ
φ
x
x
x
x
x
x x
x
x
y
y
y
y
y
y y
y
y
s
r
s
Q r
s
r
s
Q r
cos
sin
cos
sin
(
)
(
)
(
)
(
)
The changes in the Floquet coordinates at this point are then
USPAS Lecture 17
cos
sin
cos
cos
sin
cos
ξ
κ
φ
κ
φ
ξ
κ
φ
ξ
κ
φ
κ
φ
ξ
κ
φ
x
x
x y
y
y
x
x y
y
y
x
x y
y
y
y
y
y x
x
x
y
y x
x
x
y
y x
x
x
Q r
Q r
r Q r
Q r
r
(
) −
(
)
(
)
(
) −
(
)
(
)
1
2
3 1
2
3
We then continue from this point to
s
C
, where we started. The
Floquet coordinates at
s
C
are given by
ξ ξ
π
π
π
π
ξ
ξ
ξ
ξ
x x
C
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Q
Q Q
Q
Q
Q
s s
cos
sin
sin
cos
(
)
(
)
(
)
(
)
with a similar equation for
y
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We then calculate the changes in the phase-amplitude variables
over the turn, using
dr
dn
C
C
Q
x
x
x
x
x
x
2
2
2
2
2
2
ξ
ξ
ξ
ξ
d
dn
Q
C
C
Q
x
x
x
x
x
x
x
φ
ξ
ξ
ξ
ξ
−
−
tan
tan
1
1
In the following results, the parameters
κ
are assumed to be small,
so only the linear terms are retained. The trigonometric functions
have also been expanded, only terms driving the difference
resonance have been retained, and the change of variables to the
rotating coordinate system has been made. The resulting equations
are
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in which ε
πδ
Q
The minimum tune split, on the difference resonance, is
Q
Q
2
1
(
)
min
π
Correction of coupling.
For a difference resonance corresponding to
Q
Q
m
Q
x
y
δ
, we
can approximate
x
y
x
y
s
s
Q
Q
m
Q
(
) =
(
)
θ
θ
δ
θ
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in which
θ
π
s
C
is the azimuthal angle. Then, for small
δ
Q
the coupling coefficients become
ds
im
s
C
k
T
i
T
C
x
y
x
y x
y
x y
y x
x y
0
exp
π
β β
α
β β
α
β β
β β
β β
The coefficients which drive the
Q
Q
m
x
y
difference resonance
are the
m
th Fourier components of the coupling strength.
To correct a general set of coupling errors, at least two correctors
are needed, to cancel the two Fourier harmonics (real and
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imaginary parts of
). If the coupling errors and the lattice
functions have superperiodicity
N
, this will suppress Fourier
harmonics which do not satisfy
m
jN
, for integral
j
Pretzel Orbits
Motivation and applications
The term “pretzel orbits” refers to the deliberate introduction of
closed orbit distortions, through the use of electric fields, in order to provide orbit separation at undesired collision points in multiple
bunch particle-antiparticle colliders.
Pretzel orbits were invented and first developed at CESR. They are
now in use here, and also in LEP at CERN, and in the Tevatron at Fermilab, to allow multiple bunch operation and higher luminosity.
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Why do more bunches give higher luminosity?
Recall, Lecture 1, p 38, luminosity formula:
L
f
N
c
b 2
2
πσ
Here
N
b
=number of particles per colliding bunch, and
f
c
=collision
frequency. If there are
B
bunches per species, then
f
fB
c
, where
f
is the revolution frequency, and so
L
f
BN
b 2
2
πσ
If there is some limit on
N
b
(e.g, the beam-beam limit, which is
proportional to
N
b
, then more bunches will give more luminosity.
USPAS Lecture 17
If, however, I can make
N
b
as big as I want, but have a fixed total
number of particles
N
BN
b
, then I can write
L
(
)
f
B
BN
f B
N
b
2
2
2
2
πσ
πσ
and I want to make
B
as small as I can (i.e., 1) to maximize
luminosity.
The typical situation in particle-antiparticle colliders is operation at
the beam-beam limit, and we want to have as many bunches as
possible. However,
B
bunches have 2
B
collision points, while
typically there are only one or two detectors. At each collision point, we suffer from the beam-beam interaction, so we want to
minimize the number of collision points. Thus, we want to separate
the bunches everywhere in the machine, so they do not collide,
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except at the collision points where we have detectors. This is the
purpose of “pretzel orbits”.
5
10
15
-0.
1
λ
C
p
λ
λ
λ Bunches
Collision points
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The figure above illustrates a possible ideal realization of the basic
idea, providing two collision points with 8 bunches. Two closed orbit distortions are generated, of wavelength
λ
and amplitude
p
The bunch spacing is equal to
λ
. The bunches are arranged as
shown, so that while two are at the collision points, the others are
at the pretzel antinodes. The orbit distortion is generated using
electric fields (typically electrostatic separators), so that the
oppositely charged, counter-rotating bunches follow an orbit with the opposite sign. The bunches passing at the pretzel antinodes are
separated by a separation 2
p
, while those at the collision points collide.
The scheme accommodates
B
C
λ
bunches, where
λ
is the
betatron wavelength. Since
Q
C
λ
, the value of the tune sets the
maximum number of bunches.
USPAS Lecture 17
This limitation has been overcome at CESR and LEP by using
trains
of bunches, with a spacing much smaller than
λ
. The trains
must be short enough to fit in the region of pretzel antinode. A
small
crossing angle
is introduced in the straight sections to
prevent undesired collisions for bunches in a single train.
The pretzel shown above is symmetric about each collision point.
An antisymmetric pretzel is also possible, and in fact desireable:
5
10
15
-0.
1
USPAS Lecture 17
closed orbit. It also causes quadrupole errors in both planes, which
in turn result in tune shifts, beta function distortion, and second
order resonance enhancement.
If the pretzel is vertical: The closed orbit deformation in the
sextupoles causes horizontal dipole errors, and skew quadrupole
errors in both planes, which increases the coupling.
Particle-antiparticle energy differences: If the pretzel is presentin the rf cavities, and the rf field varies with position, there maybe energy differences between the two beams.
Nonlinear resonances from field errors. The large amplitude
excursions of the beams may allow them to enter nonlinear field
regions, increasing the sensitivity to resonances.
USPAS Lecture 17
Injection. During the damped betatron oscillations which occur
after injection, the separation between the bunches may be
reduced, potentially leading to beam loss.
Electrostatic separators. The requirements on these devices are challenging. In addition to having to provide high electric fields
(typically > 100 kV/cm), for high current electron-positron
machines, they must have low impedance. For proton-antiproton colliders, they must be very reliable, as sparks often cause loss of
the stored beam.
Machines that operate with flat beams must strictly limit the
amount of vertical dispersion and coupling, in order to minimize
the vertical emittance. Vertical pretzel closure errors at the collision point are also very damaging, because of the small
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vertical beam size. Hence, electron colliders typically choose the
pretzel to be in the horizontal plane.
Let’s examine some of these effects quantitatively, for the case of
horizontally separated orbits.
Long-range beam-beam collisions.
To estimate the effect of these collisions, we need to know the fields produced by a bunch. Imagine the bunch to have a length
L
along the direction of motion. We will be seeking the “long-
range” fields, at a distance from the bunch large compared to its
transverse size. So, we imagine the bunch to have a very small
transverse size.
USPAS Lecture 17
Q
L
E
v
x
B
The bunch is taken to be composed of ultra-relativistic point
charges, which have “flattened” fields that are directed
perpendicular to the direction of motion (see figure above).To find the electric field at a point a distance
r
from the bunch,
we surround the bunch with a Gaussian surface as shown:
USPAS Lecture 17
v
x
y
Gaussian surface
Q
r
Applying Gauss’ Law to find the field gives
r
r
E
da
E
rL
Q
E
Q
rL
∫
0
0
π
ε
π
ε
To find the magnetic field at
r
, use Ampere’s Law
USPAS Lecture 17
x
y
B
Amperian loop
r
Q
r
r
B
dl
B
r
I
dQ
dt
dQ
dt
Q
t
Q
s
v
Qv
L
B
Qv
rL
∫
0
0
0
π
μ
μ
μ
π
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Now consider a point charge -
e
, moving opposite to the bunch, at
the point
r
. The effect of the long-range fields of the bunch on
the trajectory of this particle is given by (see Lect 2, p. 35):
x
eB
p
eE
vp
y
eB
p
eE
vp
y
x
x
y
r
y
x
E
B
θ
For small
θ,
we have
USPAS Lecture 17
E
E
E
E
y r
B
y r
B
B
B
x
y
x
y
So the total change in slopes of the trajectory produced by the
fields of the bunch is
∆
∆
∆
∆
∆
∆
x
eB
p
eE vp
s
eQ
m
c L
s
r
y
y r
eB
p
eE vp
s
eQ
m
c L
y
s
r
0
0
2
0
0
2
2
πε γ
πε γ
USPAS Lecture 17
In practice, it is not this tune shift itself which causes problems, but
rather smaller, higher order nonlinear effects which are difficult to
correct. Nevertheless, this simple estimate correctly sets the scale
of the required pretzel separation.
Sextupole effects of horizontal pretzel orbits
The vertical field of a sextupole is
B
B
x
y
y
2
2
. Let the
closed orbit deformation produced by the pretzel be
p
s
). Then, on
the pretzel, the sextupole field is
USPAS Lecture 17
B
B
x
p s
y
B
p s
xp s
x
y
y
(
)
2
2
2
2
2
in which (
x,y
) now refer to betatron oscillations about the pretzel
orbit. We see that the effect of the sextupoles is to produce a dipolefield error
B p
2
, which is the same for both species. This error can
be corrected with standard correction dipoles. There is also a tune
shift due to the quadrupole error
∆
k
B pB
mp
0
ρ
, in which
m
is
the sextupole strength. The total tune shift, integrated around the
ring, is
Q
dsm s
s p s
x
x
C
∫
0
π
β
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The tune shift per unit pretzel amplitude is called the
tonality
This tune shift will have opposite signs for particles and
antiparticles. If the ring has superperiodicity two, and the pretzel is
antisymmetric about the symmetry point, (
p s
C
p s
)then
∆
Q
dsm s
s p s
dsm s
s p s
x
x
C
x
C
C
∫
∫
0
2
2
π
β
β
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Q
dsm s
s p s
dsm s
C
s
C
p s
C
dsm s
s p s
dsm s
s p s
x
x
C
x
C
x
C
x
C
∫
∫
∫
∫
0
2
0
2
0
2
0
2
π
β
β
π
β
β
The tonality is zero to lowest order. The quadrupole errors produce
a lattice function distortion (from Lect 8, p 21)
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∆
Φ
Φ
β
β
π
β
π
x
x
x
C
x
x
x
x
s s
Q
ds m s
p s
s
s
s
Q
sin
)cos
0
0 0
0
0
0
0
(
)
[
]
∫
For tunes near a half-integer, this perturbation is maximally antisymmetric about C/2. The tonality, calculated using the
perturbed lattice functions, will thus be non-zero in next to lowest
order in pretzel amplitude.
USPAS Lecture 17
Path length changes.
In one of the homework problems, it was shown that a dipole error
θ
at a location where the dispersion is
η
produces a path length
change
C
ηθ
. If the separators that produce the pretzel are
located at dispersive points, then the path length change on the
pretzel will be
C
s
s
i
i
i
(
) (
)
∑
η
θ
where the sum is over all the pretzel kicks. This change is oppositefor the two species of particles. Since the circumference is fixed bythe rf wavelength and harmonic number, the path length change onthe pretzel results in an energy change given by
δ
α
C
C
C
. The two
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species will then have different energies, which can be a problem if
there is residual vertical dispersion at the interaction point. To lowest order in the pretzel amplitude (i.e., neglecting the
changes in
η
due to the pretzel itself)
C
is zero for an
antisymmetric pretzel in a superperiod 2 lattice
