July 2002
M. Venturini
1
Coherent Synchrotron
Radiation and Longitudinal
Beam Dynamics in Rings
M. Venturini and R. Warnock
Stanford Linear Accelerator Center
ICFA Workshop on High Brightness Beams
Sardinia, July 1

5, 2002
July 2002
M. Venturini
2
Outline
•
Review of recent
observations of CSR
in electron
storage rings. Radiation bursts.
•
Two case studies:
–
Compact e

ring for a
X

rays Compton Source
.
–
Brookhaven
NSLS VUV Storage Ring
.
•
Model of
CSR impedance
.
•
Modelling of beam dynamics
with CSR
in terms of
1D Vlasov and Vlasov

Fokker

Planck equation
.
–
Linear theory CSR

driven instability.
–
Numerical solutions of VFP equation. Effect of nonlinearities.
–
Model reproduces main features of observed CSR
.
July 2002
M. Venturini
3
Observations of CSR

NSLS VUV Ring
Carr
et al
. NIM

A
463
(2001) p. 387
Current Threshold for Detection
of Coherent Signal
Spectrum of CSR Signal
(
wavelength ~ 7 mm
)
July 2002
M. Venturini
4
Observations of CSR

NSLS VUV Ring
Carr
et al
. NIM

A
463
(2001) p. 387
•
CSR is emitted in bursts.
•
Duration of bursts is
•
Separation of bursts is of the order
of few ms but varies with current.
s
100
Detector Signal vs. Time
July 2002
M. Venturini
5
NSLS VUV Ring
Parameters
Energy 737 MeV
Average machine radius 8.1 m
Local radius
of curvature 1.9 m
Vacuum chamber
aperture
4.2 cm
Nominal bunch length (rms) 5 cm
Nominal energy spread (rms)
Synchrotron tune
Longitudinal damping time 10 ms
5
1
0
4
s
0
018
.
July 2002
M. Venturini
6
X

Ring Parameters
(R. Ruth
et al
.)
Energy 25 MeV
Circumference 6.3 m
Local radius of curv. R=25 cm
Pipe aperture h~ 1 cm
Bunch length (rms) cm
Energy spread
Synchrotron tune
Long. damping time ~ 1 sec
Filling rate 100 Hz
# of particles/bunch
z
1
3
1
0
3
s
0
018
.
N
6
25
10
9
.
RADIATION DAMPING
UNIMPORTANT!
Can a CSR

driven instability limit performance?
When Can CSR Be Observed ?
•
CSR emissions require overlap between (single
particle) radiation spectrum and charge density
spectrum:
•
What causes the required modulation on top of
the bunch charge density?
P
N
P
N
P
e
d
n
n
n
in
n
z
2
2
(
)
incoherent
coherent
Radiated
Power:
Presence of
modulation
(microbunches)
in bunch density
CSR
may become significant
Collective forces
associated
with CSR induce
instability
Instability
feeds back,
enhances microbunching
Dynamical Effects of CSR
July 2002
M. Venturini
9
Content of Dynamical Model
•
CSR emission is sustained by a CSR driven
instability
[
first suggested by Heifets and Stupakov
]
•
Self

consistent
treatment of CSR and effects
of CSR fields on beam distribution.
•
No additional machine impedance.
•
Radiation
damping
and
excitations
.
July 2002
M. Venturini
10
Model of CSR Impedance
•
Instability driven by CSR is
similar to ordinary
microwave instability
. Use familiar formalism,
impedance, etc.
•
Closed
analytical expressions
for CSR
impedance in the presence of shielding exist only
for
simplified geometries
(parallel plates,
rectangular toroidal chamber, etc.)
•
Choose model of
parallel conducting plates
.
•
Assume e

bunch follows
circular trajectory
.
•
Relevant expressions are already available in the
literature
[Schwinger (1946), Nodvick & Saxon (1954), Warnock & Morton (1990)].
July 2002
M. Venturini
11
Parallel Plate Model for CSR
:
Geometry Outline
E
R
h
July 2002
M. Venturini
12
Analytic
Expression for CSR Impedance
(Parallel Plates)
By definition:
2
RE
n
Z
n
I
n
(
,
)
(
,
)
(
,
)
Z
n
n
Z
R
h
R
nc
J
H
J
H
p
p
n
n
p
p
n
n
(
,
)
,
,
.
.
.
(
)
(
)
L
N
M
M
O
Q
P
P
2
2
0
1
3
1
2
2
1
Argument of Bessel functions
R
p
With
p
p
h
/
p
c
p
2
2
2
(
/
)
,
Impedance
p
x
x
sin(
)
/
x
p
h
h
/
2
h
,
beam height
July 2002
M. Venturini
13
Collective Force due CSR
FT of (normalized) charge
density of bunch.
Assume charge distribution doesn’t change much
over one turn (rigid bunch approx).
E
e
R
e
d
e
Z
n
dt
e
t
in
i
t
i
n
t
t
n
n
z
z
0
2
2
0
(
)
(
,
)
'
(
'
)
(
)
'
E
e
R
e
Z
n
n
t
in
t
n
n
0
0
2
0
(
)
(
,
)
(
)
July 2002
M. Venturini
14
Parallel Plate Model : Two Examples
n
mm
1000
12
n
mm
1000
1
6
.
R
cm
h
cm
E
MeV
25
1
25
,
,
R
m
h
cm
E
MeV
1
9
4
2
737
.
,
.
,
X

Ring
NSLS VUV Ring
Z
n
Z
n
n
(
)
(
,
)
0
July 2002
M. Venturini
15
Properties of CSR Impedance
•
Shielding cut off
•
Peak value
•
Low frequency limit of impedance
n
R
h
c
(
/
)
/
3
2
Re
(
)
(
)
Z
n
n
h
R
130
lim
(
,
)
,
,
.
.
.
n
p
p
Z
n
n
n
i
Z
p
h
pR
F
H
G
I
K
J
L
N
M
M
O
Q
P
P
0
0
0
1
3
2
2
2
1
3
1
8
energy

dependent term
curvature term
July 2002
M. Venturini
16
Longitudinal Dynamics
•
Zero transverse emittance but finite y

size.
•
Assume circular orbit (radius of curvature R).
•
External RF focusing + collective force due to CSR.
•
Equations of motion
(
)
1
dz
dt
c
d
dt
c
s
R
z
ce
E
E
t
F
H
G
I
K
J
2
0
(
,
)
RF focusing
collective force
z
t
R
(
)
0
is distance from synchronous particle.
(
)
/
P
P
P
0
0
is relative momentum (or energy) deviation.
:
July 2002
M. Venturini
17
Vlasov
Equation
•
Scaled variables
•
Scale time 1 sync. Period
.
q
z
p
E
z
E
/
,
/
.
0
0
0
s
t
,
I
e
N
E
s
E
2
0
2
/
.
F
H
G
I
K
J
f
p
f
q
f
p
q
I
R
Z
n
e
z
n
n
inq
R
z
0
0
(
)
/
n
inq
R
dq
e
f
q
p
dp
z
z
z
1
2
/
(
,
)
2
July 2002
M. Venturini
18
Equilibrium Distribution in the Presence of
CSR Impedance Only (Low Energy)
•
Haissinski equilibria
i.e.
•
Only low

frequency part of impedance affects
equilibrium distribution.
•
For small n
impedance is purely capacitive
•
If energy is not too high imaginary part of Z
may be significant (space

charge term ).
Z
Z
n
i
Z
n
lim
/
,
0
/
Z
1
2
Z
0
f
H
0
exp(
)
July 2002
M. Venturini
19
•
If potential

well distortion is small, Haissinski can be approximated
as Gauss with modified rms

length:
q
n
z
Z
i
n
I
R
F
H
G
I
K
J
1
4
0
0
2
2
Haissinski Equilibrium
(close to Gaussian with
rms length )
=>Bunch Shortening.
I=0.844 pC/V
corresponding
to
N
part
bnch
4
6
10
10
.
./
q
0
96
.
Haissinski Equilibrium for X

Ring
2 cm
July 2002
M. Venturini
20
Linearized Vlasov Equation
f
p
f
q
f
p
I
Z
n
e
n
n
inq
R
z
1
1
0
0
0
(
)
/
•
Set and linearize about equilibrium:
•
Equilibrium distribution:
•
Equilibrium distribution for
equivalent coasting beam
(Boussard criterion):
f
e
e
p
q
q
q
0
2
2
2
2
2
2
2
/
/
f
e
p
q
0
2
2
2
1
2
/
f
f
f
0
1
July 2002
M. Venturini
21
(Linear) Stability Analysis
•
Ansatz
•
Dispersion relation
with ,
and
•
Look for for instability.
f
f
p
i
nq
R
z
1
1
0
(
)
exp[
(
/
)]
I
I
Z
n
n
iW
0
1
(
)
(
)
1
2
1
0
0
2
0
2
I
R
z
q
R
n
z
/
0
W
z
iz
w
z
(
)
/
(
/
)
1
2
2
Im
0
Error function
of complex arg
July 2002
M. Venturini
22
Keil

Schnell Stability Diagram
for X

Ring
1
0
iW
for
(
)
Im
(stability boundary)
Threshold (linear theory):
I
pC
V
part
bnch
th
0
8183
4
10
10
.
/
./
Keil

Schnell criterion:
I
I
Z
n
n
0
1
(
)
I
pC
V
th
0
63
.
/
n
mm
702
2
2
.
Most unstable harmonic:
Amplitude of perturbation vs time
(different currents)
pert
mm
2
2
.
Numerical Solution of Vlasov Equation
coasting beam
–
linear regime
Initial wave

like perturbation
grows exponentially.
Wavelength of perturbation:
pert
mm
2
2
.
#grid pts
0.8241
0.6
0.8202
0.3
0.8189
0.2

0.8183

res
mm
(
)
2
400
2
800
2
1200
Growth rate vs. current
for 3 different mesh sizes
I
th
Validation of Code Against Linear Theory
(
coasting beam
)
Theory
July 2002
M. Venturini
25
Coasting Beam: Nonlinear Regime
(I is 25% > threshold).
Density Contours in Phase space
Energy Spread Distribution
2 mm
Coasting Beam: Asymptotic Solution
Large scale structures have
disappeared.
Distribution approaches some
kind of steady state.
Energy Spread vs. Time
Density Contours in Phase Space
Energy Spread Distribution
July 2002
M. Venturini
27
Bunched Beam
Numerical Solutions of Vlasov Eq
.

Linear Regime.
RF focusing spoils exponential growth.
Current threshold (5% larger than predicted by Boussard).
pert
mm
2
2
.
Amplitude of perturbation
vs. time
I
th
0
836
.
Wavelength of initial perturbation:
July 2002
M. Venturini
28
Bunched Beam: Nonlinear Regime
(I is 25% > threshold).
Density Plots in Phase space
Charge Distribution
2 cm
July 2002
M. Venturini
29
Bunched Beam: Asymptotic Solutions
Bunch Length and Energy Spread vs. Time
Quadrupole

like mode
oscillations continue
indefinitely.
Microbunching disappears within 1

2 synchr. oscillations
July 2002
M. Venturini
30
Charge Density
Bunch Length (rms
)
bnch
part
V
pC
I
/
10
8
.
5
/
02
.
1
10
(25% above instability threshold)
X

Ring:
Evolution of Charge Density and Bunch Length
July 2002
M. Venturini
31
Inclusion of Radiation Damping and
Quantum Excitations
•
Add Fokker

Planck term to Vlasov equation
F
H
G
I
K
J
f
p
f
q
f
q
q
I
R
Z
n
e
z
n
n
inq
R
z
0
(
)
/
2
s
d
f
p
pf
f
p
F
H
G
I
K
J
Case study NSLS VUV Ring
s
s
T
kHz
2
2
12
/
d
ms
synch
periods
10
106
.
Synch. Oscill. frequency
Longitudinal damping time
damping
quantum excit
.
G. Carr
et al
., NIM

A 463 (2001) p. 387
Measurements
Model
Current threshold
100 mA
168 mA
CSR wavelength
7 mm
6.7 mm
Keil

Schnell Diagram for NSLS VUV Ring
Most unstable harmonic:
n
mm
1764
6
7
.
I
pC
V
part
bnch
mA
th
6
2
1
8
10
168
11
.
/
.
./
Current theshold:
*
*
P
N
e
Z
n
coh
n
n
2
0
2
2
b
g
Re
(
)
P
N
e
Z
n
incoh
n
0
2
b
g
Re
(
)
P
P
coh
incoh
/
vs. Time
CSR Power:
Incoherent SR Power:
Bunch Length vs. Time
NSLS VUV Model
current =338 mA, (
I=12.5 pC/V)
10 ms
July 2002
M. Venturini
34
Bunch Length vs. Time
P
P
coh
incoh
/
vs. Time
1.5 ms
Snapshots of Charge Density and
CSR Power Spectrum
5 cm
current =311 mA,
(I=11.5 pC/V)
July 2002
M. Venturini
36
Charge Density
Bunch Length (rms)
Radiation Spectrum
Radiation Power
NSLS VUV Storage Ring
July 2002
M. Venturini
37
Conclusions
•
Numerical model gives results
consistent with
linear theory,
when this applies.
•
CSR
instability saturates quickly
•
Saturation removes microbunching,
enlarges
bunch distribution in phase space.
•
Relaxation due to
radiation damping gradually
restores conditions for CSR instability
.
•
In combination with CSR instability, radiation
damping gives rise to a
sawtooth

like behavior
and
a
CSR bursting
pattern that seems consistent with
observations.
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