ICFA Workshop on High Brightness Beams
Sardinia, July 1-5, 2002
Coherent Synchrotron
Radiation and Longitudinal
Beam Dynamics in Rings
M. Venturini and R. Warnock
Stanford Linear Accelerator Center
July 2002
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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
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Observations of CSR - NSLS VUV Ring
Carr et al. NIM-A 463 (2001) p. 387
Current Threshold for Detection
of Coherent Signal
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Spectrum of CSR Signal
(wavelength ~
7 mm)
3
Observations of CSR - NSLS VUV Ring
Carr et al. NIM-A 463 (2001) p. 387
Detector Signal vs. Time
•CSR is emitted in bursts.
•Duration of bursts is
 100s
•Separation of bursts is of the order
of few ms but varies with current.
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NSLS VUV Ring
Parameters
Energy
Average machine radius
Local radius of curvature
Vacuum chamber aperture
737 MeV
8.1 m
1.9 m
4.2 cm
Nominal bunch length (rms)
5 cm

4


5

10
Nominal energy spread (rms) 
Synchrotron tune
Longitudinal damping time
July 2002
 s 0.018
10 ms
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X-Ring Parameters (R. Ruth et al.)
Can a CSR-driven instability limit performance?
Energy
25 MeV
Circumference
6.3 m
Local radius of curv.
R=25 cm
Pipe aperture
h~ 1 cm
Bunch length (rms)
 z 1 cm
Energy spread
  3 103
 s 0.018
Synchrotron tune
Long. damping time
~ 1 sec
Filling rate
100 Hz
# of particles/bunch N 6.25 109
July 2002
M. Venturini
RADIATION DAMPING
UNIMPORTANT!
6
When Can CSR Be Observed ?
• CSR emissions require overlap between (single
particle) radiation spectrum and charge density
spectrum:
Radiated
Power:
P N  Pn N
n
incoherent
2
P
n
n
z
in
e  ( )d
2
coherent
• What causes the required modulation on top of
the bunch charge density?
Dynamical Effects of CSR
Presence of modulation
(microbunches)
in bunch density
CSR
may become significant
Collective forces associated
with CSR induce instability
Instability feeds back,
enhances microbunching
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.
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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)].
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Parallel Plate Model for CSR:
Geometry Outline
R
h
July 2002

M. Venturini

E
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Analytic Expression for CSR Impedance
(Parallel Plates)
By definition:

2RE (n,  ) Z (n,  ) I (n,  )
L
M
M
N
Z (n,  ) 2 2 Z0 R
R  (1)  p
(1)


J
H

J
H
Impedance

n n
n
 h p 1,3,... p nc n n
 p2
2
O
P
P
Q
 2 ( / c) 2  2
p
p
p
Argument of Bessel functions R p
p sin( x ) / x , x ph / 2h
With

p / h ,
h  beam height
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Collective Force due CSR
FT of (normalized) charge
density of bunch.
z
z

e 0
in

it
i ( 
n 0 ) t '
E 
e
de Z (n,  ) dt ' e
 n (t ' )

2
(2 ) R n





t
Assume charge distribution doesn’t change much
over one turn (rigid bunch approx).
e 0
in ( 
 0t )
E 
e
Z (n, n 0 ) n (t )

2R n
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Parallel Plate Model : Two Examples
Z (n) Z (n, n 0 )
X-Ring
NSLS VUV Ring
R 25 cm, h 1 cm, E 25 MeV
R 1.9 m, h 4.2 cm, E 737 MeV
n 1000   1.6 mm
n 1000   12 mm
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Properties of CSR Impedance
• Shielding cut off
• Peak value
nc  ( R / h )
Z (n)
Re
n
h
130
R
3/ 2
( )
• Low frequency limit of impedance
limn 0
L
F
I
3 1O
M
P
G
J
M
N H K 8 P
Q
p 1
Z (n, n 0 )
 Z0
h
i


n
 p 1,3,... p  2
pR
energy-dependent term
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2
2
curvature term
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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

c
dt
FI
G
J
HK
2

d c s
ce

z  E ( , t )
dt  R
E
RF focusing
z ( 
 0t ) R
 ( P P0 ) / P0
July 2002
0
collective force
is distance from synchronous particle.
is relative momentum (or energy) deviation.
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Vlasov Equation
F
G
H

f

f

f
R
p

q I 0



q 
p
z
I e N / 2 s E0 E .
2
• Scaled variables

 Z (n)
n
n 

z
e
inq z / R
IJ0
K
z
1
n 
dqe inq z / R f (q, p)dp
2
q z /  z 0 , p 
E0 /  E 0 .
• Scale time   st,  2  1 sync. Period.
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Equilibrium Distribution in the Presence of
CSR Impedance Only (Low Energy)
H)
• Haissinski equilibria i.e. f 0 exp(
• Only low-frequency part of impedance affects
equilibrium distribution.
• For small n impedance is purely capacitive
Z limn 0 Z / n iZ ,
Z 0
• If energy is not too high imaginary part of Z
may be significant (space-charge term Z 1 /  2).
July 2002
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Haissinski Equilibrium for X-Ring
•If potential-well distortion is small, Haissinski can be approximated
as Gauss with modified rms-length:
Z
I 0 R 2
 q  1
in n 0 4  z 2
F
I
G
HJ
K
Haissinski Equilibrium
(close to Gaussian with
rms length  q 0.96 )
=>Bunch Shortening.
2 cm
I=0.844 pC/V corresponding
to N 4.6 1010 part ./bnch
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Linearized Vlasov Equation
• Set f f 0 f1 and linearize about equilibrium:


f1

f1 
f0
p
 I 0 Z (n) n einq z / R 0



q

p
n 

• Equilibrium distribution:
•Equilibrium distribution for
equivalent coasting beam
(Boussard criterion):
July 2002
e
f0 

p2 / 2
e
2
e
f0 
M. Venturini

q 2 / 2 q 2
2 q
p 2 / 2
2
1
2 q
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(Linear) Stability Analysis
• Ansatz
f1 f1 ( p) exp[
i ( 
nq z 0 / R)]
I Z ( n)
1

• Dispersion relation 
I0 n
iW ( )
with
and
1
 0 R2 1

I0
2  2z 0  q
,
 R /  z 0n
W ( z)  1 
iz  / 2 w( z / 2 )
• Look for Im 0
for instability.
July 2002
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Error function
of complex arg
21
Keil-Schnell Stability Diagram for X-Ring
1
for Im  0
iW ( )
(stability boundary)
Most unstable harmonic:
n 702   2.2 mm
Threshold (linear theory): I th 0.8183 pC / V 4 1010 part ./bnch
I Z ( n)
Keil-Schnell criterion:
1  I th 0.63 pC / V
I0 n
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Numerical Solution of Vlasov Equation
coasting beam –linear regime
Amplitude of perturbation vs time
(different currents)
Initial wave-like perturbation
grows exponentially.
Wavelength of perturbation:
 pert 2.2 mm
Validation of Code Against Linear Theory
(coasting beam)
Growth rate vs. current
for 3 different mesh sizes
#grid pts
I th  res (mm)
400 2
0.8241
0.6
8002
0.8202
0.3
12002
0.8189
0.2
--
0.8183
--
Theory
 pert 2.2 mm
Coasting Beam: Nonlinear Regime
(I is 25% > threshold).
Density Contours in Phase space
2 mm
Energy Spread Distribution
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Coasting Beam: Asymptotic Solution
Density Contours in Phase Space
Energy Spread vs. Time
Energy Spread Distribution
Large scale structures have
disappeared.
Distribution approaches some
kind of steady state.
Bunched Beam
Numerical Solutions of Vlasov Eq. - Linear Regime.
Amplitude of perturbation
vs. time
Wavelength of initial perturbation:
 pert 2.2 mm
RF focusing spoils exponential growth.
.
Current threshold I th 0836
(5% larger than predicted by Boussard).
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Bunched Beam: Nonlinear Regime
(I is 25% > threshold).
Density Plots in Phase space
2 cm
Charge Distribution
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Bunched Beam: Asymptotic Solutions
Bunch Length and Energy Spread vs. Time
Quadrupole-like mode
oscillations continue
indefinitely.
Microbunching disappears within 1-2 synchr. oscillations
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X-Ring:
Evolution of Charge Density and Bunch Length
Bunch Length (rms)
Charge Density
I  1.02 pC / V  5.8 1010 part / bnch
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(25% above instability threshold)
30
Inclusion of Radiation Damping and
Quantum Excitations
• Add Fokker-Planck term to Vlasov equation
F
G
H

f

f

f
R
p

q I 0



q 
q
z

 Z (n)
n
n 

e
inq z / R
IJ 2
K 
s
d
F
G
H

f

f
pf 

p

p
IJ
K
damping
quantum excit.
Case study NSLS VUV Ring
Synch. Oscill. frequency  s 2 / Ts 2 12 kHz
Longitudinal damping time
July 2002
 d 10 ms 106 synch. periods
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Keil-Schnell Diagram for NSLS VUV Ring
Current theshold:
I th 6.2 pC / V
18
. 1011 part ./bnch
168 mA
Most unstable harmonic:
n 1764   6.7 mm
Measurements *
Current threshold
100 mA
CSR wavelength
7 mm
* G. Carr et al., NIM-A 463 (2001) p. 387
Model
168 mA
6.7 mm
NSLS VUV Model
Bunch Length vs. Time
Incoherent SR Power:
P
incoh
b gRe Z (n)
N e 0
2
n
CSR Power:
P
coh
b g 
N e 0
2
2
2
n
Re Z (n)
n
P
coh
/P
incoh
vs. Time
current =338 mA, (I=12.5 pC/V)
10 ms
Bunch Length vs. Time
P
coh
/P
incoh
vs. Time
1.5 ms
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Snapshots of Charge Density and
CSR Power Spectrum
5 cm
current =311 mA,
(I=11.5 pC/V)
NSLS VUV Storage Ring
Charge Density
Bunch Length (rms)
Radiation Power
Radiation Spectrum
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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.
July 2002
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