Linear Collider Damping Rings
Andy Wolski
Lawrence Berkeley National Laboratory
USPAS Santa Barbara, June 2003
2
300 m Main Damping Ring
3 Trains of 192 bunches
1.4 ns bunch spacing
231 m
Predamping Ring
2 Trains of 192 bunches
30 m Wiggler
30 m Wiggler
Injection and RF
Circumference
Correction and
Extraction
110 m
Injection
Line
110 m
Transfer
Line
90 m
Extraction
Line
Spin
Rotation
What do they look like?
NLC
Positron
Rings
3
Operating Cycle in NLC/JLC MDRs
•
Each bunch train is stored for three machine cycles
–
25 ms or 25,000 turns in NLC
•
Transverse damping time
4 ms
•
Horizontal emittance
×
1/50, vertical
×
1/7500
300 m Main Damping Ring
3 Trains of 192 bunches
1.4 ns bunch spacing
30 m Wiggler
30 m Wiggler
Injection and RF
Circumference
Correction and
Extraction
103 m
Injection
Line
160 m
Extraction
Line
Spin
Rotation
4
What do they look like?
TESLA Damping Rings
5
Performance Specifications
NLC MDR
TESLA e
+
Injected
γε
150 µm rad
10 000 µm rad
Extracted Horizontal
γε
3 µm rad
8 µm rad
Extracted Vertical
γε
0.02 µm rad
0.02 µm rad
Injected Energy Spread
1% full width
1% full width
Extracted Energy Spread
0.1% rms
0.13%
Extracted Bunch Length
4 mm
6 mm
Bunch Spacing
1.4 ns
20 ns
Bunches per Train
192
2820
Repetition Rate
120 Hz
5 Hz
6
Radiation Damping…
•
Longitudinal phase space
–
Particles perform synchrotron oscillations in RF focusing potential
–
Higher energy particles radiate energy more quickly in bends
–
At the equilibrium energy, the revolution period is an integer times the
RF period (the synchrotron principle…)
•
Transverse phase space
–
Particles perform betatron oscillations around the closed orbit
–
Radiation is emitted in a narrow cone centered on the
instantaneous
direction of motion
–
Energy is restored by the RF cavities
longitudinally
–
Combined effect of radiation and RF is a loss in transverse momentum
•
Damping time in all planes is given by:
0
0
0
2
T
U
E
J
7
…and Quantum Excitation
•
Radiation is emitted in discrete quanta
•
Number and energy distribution etc. of photons obey
statistical laws
•
Radiation process can be modeled as a series of “kicks” that
excite longitudinal and transverse oscillations
8
Synchrotron Oscillations
p
t
d
d
E
U
U
T
T
E
eV
t
RF
s
RF
d
d
1
sin
d
d
0
0
0
0
0
d
d
2
d
d
2
2
2
s
E
t
t
Equilibrium orbit
Dispersive orbit
9
Longitudinal Damping
s
p
RF
RF
s
T
E
eV
cos
0
0
2
s
s
t
p
s
s
s
t
t
t
E
E
sin
e
ˆ
cos
e
ˆ
2
4
0
0
0
2
2
I
I
T
U
E
E
E
E
J
J
s
k
I
s
I
d
2
1
d
1
1
2
4
2
2
Problem 1
Show that:
s
C
I
C
p
d
1
1
0
1
0
10
Quantum Excitation (Longitudinal)
δ
τ
0
1
1
1
1
1
1
sin
ˆ
sin
ˆ
cos
ˆ
cos
ˆ
E
u
s
p
s
p
sin
ˆ
2
ˆ
ˆ
0
2
0
2
2
2
1
E
u
E
u
E
s
u
N
C
E
t
2
2
0
2
0
2
ˆ
2
d
1
d
ˆ
d
Including damping:
11
Equilibrium Longitudinal Emittance
•
We have found that:
•
From synchrotron radiation theory:
s
I
I
I
E
C
s
u
N
C
E
E
q
d
1
4
d
1
3
3
2
3
2
0
2
2
0
J
Problem 2
Find an expression for the equilibrium energy spread,
and show that:
t
equ
t
e
1
e
,
0
E
s
u
N
C
E
t
2
2
0
2
0
2
ˆ
2
d
1
d
ˆ
d
12
•
It is often more convenient to describe betatron oscillations
using action

angle variables:
•
The old variables are related to the new ones by:
•
The equations of motion take the simple form:
Betatron Oscillations: Action

Angle Variables
x
x
x
x
x
x
J
tan
2
2
2
2
cos
sin
2
cos
2
J
x
J
x
0
d
d
1
d
d
s
J
s
13
Damping of Vertical Oscillations
•
Radiation is emitted in a narrow cone (angle ~1/
γ) around
instantaneous direction of motion, so vertical co

ordinate and
momentum are not changed by photon emission
•
RF cavity changes longitudinal momentum, and hence the
vertical direction of motion:
•
Averaging over all betatron phase angles gives (per turn):
•
Hence the equation of motion is:
p
p
y
p
p
p
p
p
p
p
y
y
y
y
y
1
1
1
1
J
E
U
J
0
0
J
T
E
U
t
J
0
0
0
d
d
Problem 3
Show this!
14
Damping of Horizontal Oscillations
•
When a photon is emitted at a point where there is some
dispersion, the co

ordinates with respect to the closed orbit
change:
0
1
0
1
E
u
x
x
E
u
x
x
J
T
E
U
J
T
E
U
I
I
t
J
x
0
0
0
0
0
0
2
4
1
d
d
J
•
Taking the energy loss to first order and averaging around the
ring, we find after some work:
15
Quantum Excitation of Betatron Motion
•
Let us now consider the second order effects. It is easy to
show that the change in the action depends to second order on
the photon energy as follows:
•
Averaging over the photon spectrum and around the ring, and
including the radiation damping gives:
2
2
2
0
2
2
1
η
β
η
αη
γη
E
u
J
H
H
x
x
x
x
q
x
x
x
J
I
I
C
J
s
u
N
C
E
t
J
2
2
2
d
2
1
d
d
2
5
2
2
0
2
0
J
H
s
I
d
3
5
H
16
Summary of Dynamics with Radiation
2
5
2
0
0
0
0
2
0
1
2
3
2
2
2
4
0
0
2
4
2
4
0
0
0
sin
cos
2
2
1
1
2
I
I
C
eV
U
T
E
eV
C
I
I
I
C
I
E
C
U
I
I
I
I
T
U
E
x
q
RF
s
s
p
RF
RF
s
p
s
p
E
q
E
y
x
E
E
y
y
x
x
J
J
J
J
J
J
J
J
t
t
t
2
equ
2
inj
e
1
e
d
d
2
3
5
1
1
2
4
3
3
2
2
1
2
d
1
d
2
1
d
1
d
1
d
2
H
H
s
I
x
B
B
k
s
k
I
s
I
s
I
s
I
y
17
The NLC TME Cell
Low dispersion and horizontal
beta function in the dipole
High field in dipole
Sextupoles at high dispersion
points, with separated betas
Vertical focusing in the dipole
Cell length
≈ 5 m
18
H Function in the NLC TME Cell
19
The TESLA TME Cell
Larger dispersion and horizontal
beta function in the dipole
Low field in dipole
Sextupoles at high dispersion points
No vertical focusing in the dipole
Cell length
≈ 15 m
20
NLC and TESLA TME Cells Compared
•
NLC
–
Compact cell to keep circumference as short as possible
–
High dipole field for greater energy loss, reducing wiggler length
–
Short dipole requires very low values for dispersion and beta function
–
Gradient in dipole field to improve transverse dynamics
•
TESLA
–
Circumference fixed by bunch train and kicker rise/fall time
–
Long dipole for larger momentum compaction, longer bunch
•
Optimum lattice functions at center of dipole:
–
Obtained by minimizing
I
5
for a ring without a wiggler
–
It is not usually possible to control the dispersion and beta function
independently
15
12
24
15
2
3
2
min
0
0
x
q
C
L
L
J
21
Two Simple Scaling Relationships
Problem 4
Show that for an isomagnetic ring with the lattice functions tuned
for minimum emittance:
3
1
0
cell
6
2
2
0
3
2
0
cell
6
2
2
0
cell
15
12
8
2
15
12
8
q
e
q
e
C
L
c
em
C
B
C
L
c
em
C
B
N
22
Scaling Relationships Applied to the NLC
1 bunch train
6 bunch trains
γε
0
= 3 μm
L
cell
= 6 m
τ = N
train
1.6 ms
23
Scaling Relationships Applied to the NLC
2 bunch trains
6 bunch trains
γε
0
= 1 μm
L
cell
= 6 m
τ = N
train
1.6 ms
24
Damping Wiggler
•
A wiggler reduces the damping time by increasing the energy
loss per turn:
•
Wiggler must be located where nominal dispersion is zero,
otherwise there can be a large increase in the natural emittance
•
If horizontal beta function is reasonably small, wiggler can
significantly
reduce
the natural emittance (through reduced
damping time)
•
Drawbacks include possible detrimental effect on beam
dynamics
s
B
E
C
c
e
I
E
C
U
d
2
2
2
2
0
2
2
2
4
0
0
25
Types of Wiggler
•
A wiggler is simply a periodic array of magnets, such that the
field is approximately sinusoidal
•
Different technologies are possible:
–
Electromagnetic
–
Permanent magnet
–
Hybrid (permanent magnets driving flux through steel poles)
•
Choice of technology comes down to cost optimization for
given requirements on field strength and quality
•
Both TESLA and NLC damping rings have opted for hybrid
technology
26
Modeling the Dynamics in the Wiggler
•
Magnet design is produced using a standard modeling code
•
Field representation must be obtained in a form convenient for
fast symplectic tracking
2
2
2
2
2
,
,
,
,
,
,
sin
sinh
cos
cos
cosh
cos
cos
sinh
sin
z
x
mn
y
z
mn
y
x
mn
y
z
mn
z
z
mn
y
x
mn
y
z
mn
y
x
mn
y
x
mn
x
k
n
k
m
k
z
nk
y
k
x
mk
k
nk
c
B
z
nk
y
k
x
mk
c
B
z
nk
y
k
x
mk
k
mk
c
B
27
Fitting the Wiggler Field
28
Tracking Through the Field
•
Using an appropriate field representation (that satisfies
Maxwell’s equations), one can construct a
symplectic
integrator
:
•
M
is an explicit function of the phase

space co

ordinates, and
satisfies the symplectic condition (so the dynamics obey
Hamilton’s Equations):
old
new
x
m
x
0
1
1
0
T
S
x
m
M
S
M
S
M
j
i
ij
29
Dynamics in the NLC Wiggler
Horizontal Kicks and Phase Space
Vertical Kicks and Phase Space
30
Chromaticity
•
Chromaticity is the tune variation with energy
•
Quadrupole focusing strength gets smaller as particle energy
increases
•
It can easily be shown that:
•
Since beta functions peak at the focusing quadrupoles in the
appropriate plane, the natural chromaticity is always negative
•
Chromaticity is connected to beam instabilities
–
particles with large energy deviation cross resonance lines
–
some collective effects (e.g. head

tail instability) are sensitive to the
chromaticity
s
k
s
k
y
y
y
x
x
x
d
4
1
d
4
1
1
1
31
sextupole
k
1
=
x k
2
Correcting Chromaticity with Sextupoles
32
Dynamics with Sextupoles
•
Sextupoles can be used to correct chromatic aberrations…
•
…but introduce geometric aberrations and coupling:
•
It is important to keep the required strengths to a minimum by
designing the linear lattice functions for effective sextupole
location
x
Y
l
k
y
x
l
k
x
2
2
2
2
1
s
k
k
s
k

η
k
y
y
y
x
x
x
d
4
1
d
4
1
2
x
1
2
x
1
33
Dynamic Aperture
•
Geometric aberrations from sextupoles (and other sources)
distort the transverse phase space, and limit the amplitude
range of stable betatron oscillations
Horizontal phase space of NLC TME cell
Vertical phase space of NLC TME cell
34
Transverse and Longitudinal Aperture
•
Damping rings require a “large” dynamic aperture
–
Injected beam power ~ 50 kW average, and radiation load from any
significant injection losses will destroy the ring
–
Nonlinear distortion of the phase space may lead to transient emittance
growth from inability properly to match injected beam to the ring
–
For NLC Main Damping Rings, the target dynamic aperture is 15 times
the injected rms beam size
•
We also need a large momentum acceptance
–
Injected beam has a large energy spread
–
Particles may be lost from insufficient physical aperture in dispersive
regions, or through poor off

momentum dynamics
–
Particles within a bunch can scatter off each other, leading to a
significant change in energy deviation (Touschek Effect)
•
It is important to perform tracking studies with full dynamic
model and physical apertures
35
NLC Main Damping Ring Dynamic Aperture
Dynamic Aperture On

Momentum
δ= +0.005
δ=

0.005
15
×
Injected Beam Size
36
Longitudinal Acceptance
•
The longitudinal acceptance has three major limitations:
–
Poor off

momentum dynamics
–
Physical aperture in dispersive regions
–
RF bucket height
•
Off

momentum dynamics can be difficult to quantify
–
see previous slides
•
Physical aperture can be a significant limitation
–
1% momentum deviation in 1 m dispersion is a 1 cm orbit offset
•
RF bucket height comes from non

linearity of the longitudinal
focusing
–
Previous study of longitudinal dynamics assumed a linear slope of RF
voltage around the synchronous phase
–
Valid for small oscillations with synchronous phase close to zero

crossing
37
RF Bucket Height
•
The “proper” equations of longitudinal motion (without
damping) are:
•
These may be derived from the Hamiltonian:
s
RF
s
RF
p
T
E
eV
t
t
sin
sin
d
d
d
d
0
0
RF
s
RF
s
RF
RF
p
T
E
eV

H
sin
cos
0
0
2
2
1
H
t
H
t
d
d
d
d
38
Longitudinal Phase Space
•
The Hamiltonian is a constant of the motion, which allows us
to draw a phase

space portrait
S
Stable fixed point
Unstable fixed point
Separatrix
S
V
RF
s
s
s
p
RF
RF
RF
T
E
eV
sin
2
cos
4
0
0
2
RF
39
Alignment Issues
•
The final luminosity of the collider is critically dependent on
the vertical emittance extracted from the damping rings
•
In a perfectly flat lattice, the lower limit on the vertical
emittance comes from the opening angle of the radiation
–
Gives about 10% of the specified values for NLC and TESLA
•
Magnet misalignments give the dominant contribution to the
vertical emittance
–
Quadrupole vertical misalignments
•
Vertical dispersion
•
Vertical beam offset in sextupoles
–
Quadrupole rotations and sextupole vertical misalignments
•
Couple horizontal dispersion into the vertical plane
•
Couple horizontal betatron oscillations into the vertical plane
40
Betatron Coupling
•
In a damping ring, the dominant sources of betatron coupling
are skew quadrupole fields
–
Normal quadrupoles have some “roll” about the beam axis
–
Sextupoles have some vertical offset with respect to the closed orbit
•
Particles with a horizontal offset get a vertical kick
Particle on
closed orbit
Particle with
horizontal
amplitude
Vertical kick
depends on
horizontal
amplitude
41
Effects of Betatron Coupling
•
In action

angle variables, the “averaged Hamiltonian” for a
coupled storage ring can be written:
•
The equations of motion are:
y
x
y
x
n
y
y
x
x
J
J
J
J
H
C
cos
~
2
0
y
x
y
x
y
y
y
x
y
x
y
y
x
x
y
x
x
y
x
y
x
x
J
J
C
s
J
J
s
J
J
J
C
s
J
J
s
J
cos
2
~
2
d
d
sin
~
d
d
cos
2
~
2
d
d
sin
~
d
d
0
0
42
Solutions to the Coupled Hamiltonian
•
The sum of the horizontal and vertical actions is conserved:
•
There are fixed points at:
•
With radiation, the actions will damp to the fixed points
0
d
d
0
0
s
J
J
J
J
y
x
2
2
0
2
2
0
~
1
2
1
~
1
2
1
J
J
J
J
y
x
y
x
43
The Difference Coupling Resonance
•
The equilibrium emittance ratio is given by:
•
The measured tunes are given by:
2
2
2
4
~
~
x
y
J
J
2
2
2
1
2
1
0
~
d
d
2
s
C
y
x
44
What is the Coupling Strength?
•
We add up all the skew fields around the ring with an
appropriate phase factor:
•
k
s
is the skew quadrupole
k

value.
•
For a rotated quadrupole or vertically misaligned sextupole,
the equivalent skew fields are given by:
s
k
C
s
s
y
x
n
n
d
e
2
1
~
0
0
i
0
2
C
s
n
s
y
x
y
x
n
2
sin
1
k
k
s
y
k
k
s
2
45
Vertical Dispersion
•
In an electron storage ring, the vertical dispersion is typically
dominated by betatron coupling
–
Emittance ratios of 1% are typical
•
For very low values of the vertical emittance, vertical
dispersion starts to make a significant contribution
•
Vertical dispersion is generated by:
–
Vertical steering
•
vertically misaligned quadrupoles
–
Coupling of horizontal dispersion into the vertical plane
•
quadrupole rotations
•
vertical sextupole misalignments
46
Vertical Steering: Closed Orbit Distortion
•
A quadrupole misalignment can be represented by a kick that
leads to a “cusp” in the closed orbit
•
We can write a condition for the closed orbit in the presence of
the kick:
•
We can solve to find the distortion resulting from many kicks:
x
x
x
x
x
x
x
x
x
x
sin
cos
sin
sin
sin
cos
M
0
0
0
0
y
y
y
y
M
s
s
s
s
s
s
s
y
y
y
y
y
y
y
d
cos
sin
2
1
1
1
47
•
The vertical dispersion obeys the same equation of motion as
the vertical orbit, but with a modified driving term:
•
We can immediately write down the vertical dispersion arising
from a set of steering errors:
•
Including the effect of dispersion coupling:
Vertical Steering: Vertical Dispersion
1
1
1
y
k
y
1
1
1
y
k
k
y
y
differentiate wrt
s
s
s
y
k
s
s
s
y
y
y
y
y
y
y
d
cos
1
sin
2
1
1
1
1
s
s
s
k
y
k
k
s
s
s
y
y
y
x
s
x
y
y
y
y
d
cos
1
sin
2
1
2
1
1
1
48
Effects of Uncorrelated Alignment Errors
•
Closed orbit distortion from quadrupole misalignments:
•
Vertical dispersion from quadrupole rotation and sextupole
misalignment:
•
Vertical emittance generated by vertical dispersion:
2
1
2
2
2
sin
8
l
k
Y
y
y
y
q
y
2
1
2
2
2
sin
2
x
y
y
q
y
y
l
k
2
2
2
2
2
sin
8
x
y
y
s
y
y
l
k
Y
2
2
y
2
y
J
J
y
E
y
49
Examples of Alignment Sensitivities
•
Note:
Sensitivity values give the random misalignments that will
generate a specified vertical emittance. In practice, coupling
correction schemes mean that significantly larger
misalignments can be tolerated.
APS
SLS
KEK

ATF
ALS
NLC MDR
TESLA DR
Energy [GeV]
7
2.4
1.3
1.9
1.98
5
Circumference [m]
1000
288
140
200
300
17,000
γε
x
[µm]
34
23
2.8
24
3
8
γε
y
[nm]
140
70
28
20
19
14
Sextupole vertical [µm]
74
71
87
30
53
11
Quadrupole roll [µrad]
240
374
1475
200
511
38
Quadrupole jitter [nm]
280
230
320
230
264
76
50
Collective Effects
•
Issues of damping, acceptance, coupling are all
single particle
effects

they are independent of the beam current
•
Particles in a storage ring interact with each other (directly or
via some intermediary e.g. the vacuum chamber)
•
A wide variety of collective effects limit the achievable beam
quality, depending on the bunch charge or total current
•
The consequences of collective effects are
–
Phase space distortion and/or emittance growth
–
Particle loss
•
Damping rings have high bunch charges, moderate energies
and small emittance
–
Vulnerable to a wide range of collective effects
•
Too wide a subject to enter into here!
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