July 28th - 31st, 2015, IBS, Daejeon, Korea
THz image
Longitudinal Beam Dynamics (Phase Space,
Synchrotron Oscillation, Wakefields, and
Longitudinal Parameters)
Yujong Kim
Accelerator R&D Team
Radiation Equipment Research Division
Advanced Radiation Technology Laboratory
Korea Atomic Energy Research Institute (KAERI), Korea
yjkim@isu.edu / yjkim@kaeri.re.kr, http://www2.cose.isu.edu/~yjkim KAERI-2015-029
Outline
 Acknowledgements & Download Web Site
 Phase Space, Phase Slip Factor, and Transition Energy
 Below Transition, Above Transition, and Phase Stability
 Beam Acceleration, RF System, and RF Cavity
 Phase Oscillation Equation and Synchrotron Oscillation Frequency
 Equation of Small Phase Oscillations
 Wakefield, Beam Instabilities, and Large Phase Oscillation
 Bucket, Area of Bucket, Bucket Height, and Bunch Heights
 Maximum Momentum Acceptance of a Bucket
 Normalized Longitudinal Emittance
Textbook:
An Introduction to the Physics of Particle Accelerators by Mario Conte
Other Reference:
Accelerator Physics by S. Y. Lee
Particle Accelerator Physics by H. Wiedemann
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Acknowledgements
Y. Kim gives his sincere thanks to KoPAS2015 Organizing Committee and
KAERI supervisors for their allowance of this tutorial, and also to following
friends, references, and former & current supervisors:
PAL & POSTECH: Prof. W. Namkung, Prof. I. S. Ko, Prof. M. H. Cho,
Prof. M. H. Yoon, and Prof. J. Y. Huang
SPring-8: Prof. T. Shintake (now at OIST)
KEK: Prof. K. Yokoya and Prof. H. Matsumoto
PSI: Dr. S. Reiche, Dr. M. Pedrozzi, Dr. H. Braun, and Dr. T. Garvey,
DESY: Dr. K. Floettmann, Dr. S. Schreiber, Director R. Brinkmann,
Prof. J. Rossbach
APS: Dr. M. Borland, Dr. Y. Chae, and Prof. Kwang-Je Kim
LANL: Dr. B. Carlsten
Indiana University: Prof. S. Y. Lee
INFN: Dr. M. Ferrario
Jefferson Lab: Dr. A. Hutton, Dr. H. Areti, and Dr. S. Benson
Duke University: Prof. Y. Wu
Idaho State University & Idaho Accelerator Center: Prof. D. Wells
RTX: Mr. Pikad Buaphad, Dr. K. B. Song, Dr. H. D. Park, and Mr. S. Y. Ryu
IBS: Prof. Sunchan Jeong, Dr. D. Jeon, Mr. C. H. Park, and Miss. J. Cho
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Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Download Website
Yujong Kim's Lecture Notes on
 Basic Accelerator Physics
 Magnets and Transverse Motion in Accelerators
 RF System and Longitudinal Motion in Accelerators
 Advanced Accelerator Physics Tutorial for XFEL Projects
 Accelerator Beam Diagnostics
 Linux Basic for Physicists
can be downloaded from:
http://www2.cose.isu.edu/~yjkim/course/
Contact: yjkim@isu.edu / yjkim@kaeri.re.kr
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Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Beam Phase Space
The transverse phase space is described with the beam position and angle,
(x, x') and (y, y')
x'
y'
x
y
beam diameter = full width beam size ~ 6σx for Gaussian beam
Similarly, the longitudinal phase space is described with the beam longitudinal position
and energy (or time and energy spread, or phase and momentum)
(z, E), (dz, dE), (dz = -cdt, dE/E = dp/p), (dt, dE/E=dp/p), and (d = rfdt, dp/p)
dp/p
dE/E
tail
tail
dz
head
dt
head
full width bunch length ~ 6σz for Gaussian beam
http://www.slac.stanford.edu
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Phase Slip Factor and Transition Energy
Up to now, we ignored the effects of the RF electric field on the motion of
charged particles, which supplies an acceleration by Lorentz force
and gives a motion in longitudinal phase space (∆z - ∆p/p)
(∆t - ∆p/p), or (∆ϕ - ∆p/p) → synchrotron oscillation.
RF bucket : stable region where particles can be grouped or bunched.
continuous
acceleration
in an RF cavity
motion of positive
bunches in a TESLA
type SRF cavity
180 deg. later
Courtesy of Fermilab
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Phase Slip Factor and Transition Energy
The angular frequency ω of a particle circulating in a synchrotron with
a circumference L and a revolution time τ.
Differentiating ln(ω) →
Define phase slip factor tr
note that many other books define tr differently.
For tr= 0, there is a transition in longitudinal phase space →
∆p/p
phase jumping from ϕs to π - ϕs
below transition for positron
∆z ≈ -c
   tr  tr  0
∆p/p
above transition for positron
∆z ≈ -c
   tr  tr  0
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
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Phase Slip Factor and Transition Energy
Example, p of PLS storage ring = 0.00181
→ γtr = (0.00181)-1/2 =23.5
→ Etr ~ 12 MeV << EPLS = 2.0-2.5 GeV
→ PLS is always operating under the above transition condition.
above transition:    tr  U  mc 2  U tr   tr mc 2
note that p is large for weak focusing
below transition:    tr  U  mc 2  U tr   tr mc 2
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Phase Stability - Below Transition
Let's assume that circulating charge particles go through an RF cavity in the ring.
Synchronous particle can cross an RF cavity in perfect synchronism, while all other
particles will cross the RF cavity with different arrival times or phases with respect to
the RF oscillation (arriving earlier or later with respect to the synchronous particle).
For below transition,
   tr  tr  0
This means that a higher energy particle (dp > 0) arrives earlier (dτ < 0) at the RF
cavity in next turn.
→
acceleration for positive charge
To get stable oscillations under the below
transition, the RF slope of
should be positive and synchronous RF
phase should be between 0 < ϕs < π/2 for
a positive charged particle.
head tail
→
lower energy particle (head) will get
a higher energy later (stable oscillation).
acceleration for negative charge
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Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Phase Stability - Above Transition
Let's assume that circulating charges particles go through an RF cavity in the ring.
Synchronous particle can cross an RF cavity in perfect synchronism, while all other
particles will cross the RF cavity with different arrival times or phases with respect to
the RF oscillation (arriving earlier or later with respect to the synchronous particle).
For above transition,
   tr  tr  0
This means that a higher energy particle (dp > 0) arrives later (dτ > 0) at the RF
cavity in next turn.
→
acceleration for positive charge
To get stable oscillations under the above
transition, the RF slope of
should be negative and synchronous RF
phase should be between π/2 < ϕs < π for
a positive charged particle.
head tail
→
lower energy particle (tail) will get
a higher energy later (stable oscillation).
acceleration for negative charge
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Acceleration - DC Accelerator
Electrostatic (DC) Accelerator with a battery
+
Electrostatic (DC) Accelerator with multiple batteries to get a higher beam energy.
But there is also a limitation in this method due to the arc between two electrodes.
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Acceleration - Traveling Wave (TW) Accelerator
To avoid any arc between two electrodes, and to get a much higher beam energy gain,
we use an Alternating Current (AC) type accelerator → RF Accelerator.
To get the best acceleration, we need a good synchronization between charged beams
and RF wave (phase velocity of electromagnetic wave = velocity of electron beams).
→ Principle of Traveling Wave (TW) Accelerator, whose position of electromagnetic
wave is continuously moving.
> c without discs
~ c with discs
2π/4 mode TW structure
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Acceleration - Traveling Wave (TW) Accelerator
TW accelerator
SiC RF Dummy Load
for TW accelerator
a higher RF power loss &
a higher energy along
structure
larger iris
smaller iris for phase & gradient
controls
C-band TW accelerator
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Acceleration - Traveling Wave (TW) Accelerator
IAC S-band TW accelerator
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Energy Gain of TW Accelerator
150 m long 2.5 GeV S-band linac @ Pohang Accelerator Laboratory (PAL) in Korea
Forty four 3 m long S-band linac structures were installed at PAL in 1992.
2π/3 mode Traveling Wave (TW) structure
about 55 + 2×0.5 cells for 2 m
about 84 + 2  0.5 cells for 3 m
shunt impedance ~ 55 MΩ/m
Q0 = 11000-13000
attenuation parameter ~ 0.42 for 2 m = ω/2vgQL
filling time ~ 0.6 µs for 2 m = L/vg
gradient ~ 25 MV/m @ 50 MW, 3 m
~ 6.9 MV/m @ 3.0 MW, 2 m
~ 8.8 MV/m @ 5.0 MW, 2 m
Vgain(MV )  Pin Rsh L(1  exp(2 )
~ 35 mm
G  Vgain / L (MV/m )
Pin  RF input power (MW)
Rsh  average shunt impdeance per length (M/m)
L  lengthof structure (m)
velectron  v RF, phase   k
  attenuation parameter
v g  d dk
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
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Beam Acceleration - Standing Wave Accelerator
In case of Standing Wave (SW) accelerator, there are two electromagnetic waves
(forward and backward) as shown below. Therefore, one stationary standing wave
(black line at bottom-left figure) can be generated. Beams can be accelerated by this
standing wave at π phase advance as shown right figures.
two opposite waves
node
one stationary standing wave
http://www.wikipedia.org
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Coupling Cells in Standing Wave Accelerator
Varian Linac
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Beam Acceleration - Standing Wave Accelerator
SW accelerators have a much higher shunt impedance than TW accelerator.
Shunt impedance of S-band SW = 80 MΩ/m (effective acceleration → rings uses this)
Shunt impedance of S-band TW = 55 MΩ/m
Varian Medical Linac
SW cavity
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Beam Acceleration - Standing Wave Accelerator
In case of Standing Wave (SW) accelerator, we do not need any output coupler.
But we need a circulator to protect klystron from reflected RF power.
Energy gain of π mode Side-Coupled SW linac without attenuation: Vgain(MV )  Pin Rsh L
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RF Frequency, Microwave / Radar Bands
Radio Frequency (RF) is a rate of oscillation of electromagnetic waves in the range of
about 30 kHz to 300 GHz. Frequency Ranges of Microwaves = 300 MHz to 300 GHz.
Frequency Range
216 — 450 MHz
1 — 2 GHz
2 — 4 GHz
4 — 8 GHz
8 — 12 GHz
12 — 18 GHz
18 — 26.5 GHz
26.5 — 40 GHz
30 — 50 GHz
40 — 60 GHz
50 — 75 GHz
60 — 90 GHz
75 — 110 GHz
90 — 140 GHz
110 — 170 GHz
110 — 300 GHz
Microwave / Radar Bands
P-Band
L-Band
S-Band
C-Band
X-Band
Ku-Band
K-Band
Ka-Band
Q-Band
U-Band
V-Band
E-Band
W-Band
F-Band
D-Band
mm-Band
IEEE US Bands
30 - 300 kHz : LF-band
300 - 3000 kHz : MF-band
3 - 30 MHz : HF-band
30 - 300 MHz : VHF-band
300 - 1000 MHz : UHF-band
Bands for
RF Accelerators
Standard American Frequency
UHF or P-Band : 357 MHz
L-band : 1428 MHz
S-band : 2856 MHz
C-band : 5712 MHz
X-band : 11424 MHz
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RF Distribution of Linac - IU ALPHA Linac
RF input of klystrons comes from RF driver ~ 200 W
The pulsed HV for the klystron gun comes from a modulator.
Each RF source has its own RF amplitude & phase controller.
phase shifter & attenuator
5W
25 W
10 mW
2856 MHz
Oscillator
200 W
200 W
200 W
200 W
200 W
Trigger
+ Delay
3.0 MW
Thermionic RF Gun
5.5 MW
5.5 MW
5.5 MW
5.5 MW
2 m Standard TW linac structures
Layout of RF distribution of IU CEEM ALPHA Phase-II Linac Option-IV
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RF Driver for Varian Linac
front panel of RF driver
2856.0881 MHz
power meter
status LED
RF frequency, modes
AFC sensitivity adjustments
fRF ~ 2856 MHz
total output ~ 360 W
max output for klystron ~ 200 W
pulse length ~ 12 s
water temperature ~ 40 deg
back panel of RF driver
control & tuning
max 200 W RF output
Type N connector
ext servo
trigger in
water outlet
output moniter
remote prog
water inlet operation hour counter
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Modulator and Klystron for RF Linac
Modulator is a pulsed high voltage (HV) power supply for the klystron gun.
PFN charging DC HV power supply
Pulse Forming Network (PFN)
L: series
C: parallel
THYRATRON
trigger signal
Klystron & Modulator of APS Linac - Prof. M. H. Cho
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Modulator and Klystron for RF Linac
modulator is a pulsed high voltage power supply (U0)
for the klystron gun (applying cathode and anode).
klystron is an RF power amplifier.
(several hundred W RF input  several or tens MW)
Pulse Forming Network (PFN)
In a klystron, the velocity modulation of electron beam is
converted to the density modulation. The electron beam power
is converted to RF power at the output cavity. More cavities can
improve bunching and klystron efficiency. Solenoids are needed
to focusing beam along cavities.
  
pulsed HV
  
solenoids
RF driver ~ 180 W
PLS 80 MW klystron + 200 MW modulator
~ a few or tens MW
http://www.wikipedia.org
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Modulator Pulse - Duke University Modulator 3
PFN Tank of Modulator 3 - before upgrade
Non-uniform and short HV pulse in modulator 3 makes a long energy tail.
Before PFN upgrade
capacitance = 10 nF, inductance = 0.63 H
flat HV pulse width < 1.0 s
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Modulator Pulse - Duke University Modulator 3
Uniform and longer HV pulse in modulator 3 will reduce energy spread.
After PFN upgrade
capacitance = 50 nF, inductance = 2.15 H
flat HV pulse width < 3.5 s
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Modulator Pulse - Duke University Modulator 3
After the upgrade, a broken capacitor destoryed uniformity and length of HV pulse!
After PFN upgrade due to a broken capacitor - repaired later.
capacitance = 50 nF, inductance = 2.15 H
flat HV pulse width < 3.5 s
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Resonance Frequency of an RF Cavity
If a charged beam crosses an RF cavity at t, then an acceleration by the RF cavity is
Gdcos(2frft + ϕ0), where G is gradient of the cavity, d is length of the cavity, ϕ0 is the
initial RF phase, frf (~ f 010) is RF frequency. In this case, the longitudinal electric field
Ez and resonance angular frequency mnp of the TMmnp mode is given by
d
2R
Here, xmn is the nth solution of Jm, Jm(xmn) = 0, and p is 0, 1, 2, ... .
For the lowest TM010 mode, the resonance frequency f010 and
EM fields of the cavity is given by
f 010 
d ~ v
c 2.405  0


2
R
2
rf / 2
c
   rf / 2 for  mode SWRF cavity
Note that m,n,p are the number of nodes of the mode in the , r, z direction.
See Resonant Cavities in J.D. Jackson's Classical Electrodynamics.
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
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Resonant Acceleration
In a long term, there is a net acceleration only if the charged beam is synchronized
with an electromagnetic wave at a SW RF cavity in a storage ring.
All other un-synchronized waves will sometimes accelerate, and sometimes decelerate
the beam, which gives no net acceleration in time average.
Under following conditions, a synchronous particle crosses an RF gap or cavity at
t = 0, when the RF phase is ϕs, and the voltage across the gap is Vsinϕs:
1. there is only one accelerating gap of length g, located at s = 0.
2. the accelerating gap is much shorter than the distance traveled by the beam
during one RF period Trf.
g
rf
g   v  Trf  v 
   rf
c
3. the rf angular frequency ωrf is an integer multiple of the
2R
angular revolution frequency ωs. Here h is the harmonic number
or bucket number in a storage ring.
 rf  h s  f rf  hf s

c
rf

hc
s
  s  hrf
fc 
c 2.405

2
R
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Harmonic Number, Buckets & Circumference
Meaning of the harmonic number or bucket number h in a storage ring.
ring circumference = λs
length of one RF bucket = λrf
ring circumference = multiple integer (= h) of the RF bucket
Example, PLS storage ring
frf = 500.087 MHz
λrf = 2.99792458×108 m/s / 500.087 MHz
= 0.5994806064 m in length (~ 2 ns in time)
harmonic number = 468
= number of RF cycling in circumference
circumference = 468× 0.5994806064 m
~ 280.556923 m
 rf  h s
f rf  hf s
 s  h rf
buckets = 468
bucket length
~ 0.6 m ~ 2.0 ns
revolution time
~ 468×2.0 ns ~ 936 ns
revolution frequency
~ 500 MHz / 468
~ 1.068 MHz
ring circumference
~ 468 × 0.6 m
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Energy Gain per Revolution
At the RF cavity, synchronous particle gets an energy gain per revolution:
The effective electric field along the ring can be given by
Here, L is circumference of the synchronous particle's orbit.
By using Fourier expansion of -function:
The effective electric field can be given re-written as
velocity of synchronous particle:
 rf  h s
sin(   )  sin cos   cos  sin 
cavity @ s =0, nL
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Energy Gain per Revolution
Energy gain of synchronous particle per revolution:
Passing time at s can be written for synchronous and generic particles:
Here note that generic particle has a time lag or deviation t w.r.t the synchronous
particle.
If we assume that the longitudinal oscillation (= synchrotron oscillation in ring)
frequency is much slower than the revolution frequency (typically s ~ s/ 100).
Then, let's perform time average of the electric field over one revolution to get
effective field seen by a generic particle whose a time lag is t (assume ts → 0 @ s = 0).
energy gain per turn : q<V>=qL<E(t)>
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Phase Oscillation Equation
Relationship between physical parameters of a generic particle and a synchronous
particle with subscript s are given as:
here
because a higher frequency has a shorter revolution period.
To get a net acceleration at an RF cavity, the synchronous particle must arrive at the
cavity with the same RF phase →
If the RF phase of the synchronous particle at the cavity is ϕs, and the RF phase of the
generic particle at the cavity is ϕ, then the relative RF phase difference between two
particle is given:
Energy gain of the generic particle per turn is given by
→
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Phase Oscillation Equation
Energy gain of the synchronous particle per turn is given by
Then, the energy deviation of the generic particle from that of the synchronous
particle at the beginning of the n-th turn is given by:
After the n-th turn, the deviation will be:
: deviation between turns
 : deviation between particles
for slowing varying oscillation about Us
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Phase Oscillation Equation
Defining a new variable W:
 W
Again after assuming that the oscillation is slow, after one revolution, change of phase
between two particles (
) after one turn :
 
d
 s     rf t   rf t
dt
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Phase Oscillation Equation
 
d
 s     rf (t )   rf t
dt
 A
Here t means the difference in the arrival time at the cavity of two particles.
Due to a slow oscillation, its change after one revolution (t →τ) is given by:
 t   t       s  

p
 tr
B

p
From A & B, we can find a time derivate of  for slow varying oscillation (τ ~ τs):
 rf
d  rf

(t ) 
dt
s
s

p  
  tr   rf
p  s

 1 p 



p 
   rf2 tr  1 U 
   s tr    rf2  tr 
 p 
   2U 
p

 rf

 rf

From chapter 1,
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Phase Oscillation Equation
If we use U ~ Us & definition of W at page 33 →
 1 U 
 1 U   rf2 tr
d
2
2
   rf  tr 

  rf  tr 
2
2



  2U
dt


U


U
rf
rf
s
s




 U   rf2 tr


W
    2U
rf 
s

If we apply time derivation →
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Phase Oscillation Equation
If we assume a small oscillation in  →
And if we define synchrotron oscillation frequency s :
-
U s  mc 2
Synchrotron Oscillation Equation
for a Small Oscillation
Synchrotron Tune is defined as:
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Synchrotron Oscillation in the Storage Ring
re
4
E4

3 ( mc 2 )3 
 Beam energy is lost due to the synchrotron radiation loss. UT , BM
 RF system (RF cavity) compensates the energy loss in the ring.
 Longitudinal focusing electric field from RF cavity generates a longitudinal
motion, synchrotron oscillation around the synchronous particle.
RF Cavity
Synchrotron Oscillation for E > ET
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Beam Instability & Wakefields
Up to now, we assumed that the amplitude of RF phase oscillation is small. But
generally, its amplitude becomes large if there is the longitudinal beam instability
in the storage ring.
Interaction of Charged Beams & Discontinuous Surroundings
Moving Charged Beams  Fields from Beams  Response of Surroundings
 Fields from Surroundings  Response of Beams  Change of following beam energy
Specially, beam instabilities are generated
when beams interact with higher order
modes (HOMs) of RF cavity.
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Wakefields - Type
 Filling & Decay Time of EM fields in RF Cavity is very long,   2QL r
 Time domain
– Wakefields
– High Q RF Cavity : Long range wakefields
 Coupled Bunch Mode Beam Instabilities
– Vacuum Components : Short range wakefields
 Single Bunch Beam Instabilities
 Frequency domain
– Impedance
– High Q RF Cavity : Narrow band Impedance
– Vacuum Components : Broad band Impedance
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Narrow band & Broad band Impedances
 Longitudinal narrowband impedance due to the RF cavity and broadband
impedance due to the vacuum chamber can be described by an equivalent
r
RLC resonance circuit.
Z (ωbeam ) 
||
l
Rsh
beam
1  iQ(beam  r   r beam )
lower
upper → unstable
where beam is circulating beam frequency,
Accelerator Physics - Prof. S. Y. Lee
Q and Rsh is the quality factor and
the shunt impedance at the resonance frequency r of HOMs, respectively.
 Impedance is inductive at the lower side beam  r (Im[Z||]  0)
 Impedance is capacitive at the upper side beam  r (Im[Z||]  0) at E > ET
Stronger Longitudinal focusing  Bunch length compression
 Higher peak current  Higher wakefields  Harmful longitudinal CBMI’s
42
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Coupled Bunch Mode Beam Instability (CBMI)
 Coupling of successive bunches through RF cavities or chamber
 Phase difference between evenly adjacent coupled bunches = 2πn M
Long range wakefields will be generated due to the interaction between the first
short charged bunch and HOMs of the RF Cavity  Following second short
bunch is received the first bunch induced wakefields  Energy deviation (RF
Cavity supplied normal E + wakefields supplied E) of the second bunch is
generated  Energy deviation will be increased after a few turns  Coupled
Bunch Mode Instabilities  Sidebands & CB Mode Oscillations.
V  V0e  t / ,   2QL / r
43
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
CBMIs - Time Domain Oscillations
 Period of Azimuthal Mode Oscillations = One Synchrotron Period
A)
B)
C)
Dipole Mode
m=1
Quadpole Mode
m=2
Sextupole Mode Octupole Mode
m=3
m=4
A) Mountain-range display for one synchrotron period (2 kHz  0.5 ms)
B) Superimposed
C) Phase Space (-E/E ) Oscillations, Time Step =  m   s 
44
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
CBMIs - Frequency Domain Sidebands
Under CBMIs, Circulating Beam induced Spectrum Frequency fbeam → fp,n,m
SideBand frequencies f p,n,m  p  M  n  m  s  f 0

p  f RF  n  f 0  m  f s
(unstable @ above transition)
At above transition (   th , f RF  5  f 0 )
Kicker Central frequency f c   p  f RF  ( p  1 2 )  f RF  2
 f c   p  1 4  f RF
* exciting frequencies  1  f RF  2  f 0  m  f s
* damping frequencies   2  f RF  3  f 0  m  f s
45
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
PLS Status - HOMs of RF cavities
 HOMs of the PLS RF cavities which generate CBMIs.
fHOM [MHz] n
758.6
1301.1
1707.0
1870.1
826.4
831.2
1072.4
241
279
194
125
161
158
400
HOM
Qo
TM011 37,000
TM020 112,000
TM013 34,000
TM030 34,000
TM110V 56,000
TM110H 56,000
TM111V 40,000
Direction
Longitudinal
Longitudinal
Longitudinal
Longitudinal
Vertical
Horizontal
Vertical
 If fbeam = fp,n,m is close to fHOM, resonance or CBMIs are occurred.
 We can avoid CBMIs somewhat by optimizing cavity temperature
to satisfy f p,n,m  f HOM .
47
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
PLS Status - Sidebands of CBMIs
 Several CBMI sidebands of harmful HOMs
 The beam current is 203.0 mA, and the energy is 2.04 GeV.
 Temperature of four RF cavities are 40.8oC, 46.8oC, 45.6oC, 36.9oC, respectively.
 Due to these modes, the normal operation current is about 170 mA @ 2.04 (2.5) GeV.
48
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
The block diagram of the PLS LFS
49
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
The Racks for the PLS LFS
Master Oscillator
DSP Farm
50
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
DSP Farm - VXI & VME Crates
VXI Crate
VME Crate
51
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Racks for PLS LFS
52
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
PLS LFS Kicker – 3D Geometry
Single Ridge Wavequide Overloaded RF cavity for PLS LFS Kicker
Y. Kim et al, IEEE Trans. on NS, 47, No, 2, 2000
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
53
PLS LFS Kicker – Kicking E field Distribution
HFSS Simulation Result
Y. Kim et al, IEEE Trans. on NS, 47, No, 2, 2000
54
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
420 mm
Fabricated PLS LFS Kicker Cavity
100 mm
RF cavity for PLS LFS Kicker
Y. Kim et al, IEEE Trans. on NS, 47, No, 2, 2000
55
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Longitudinal Large Oscillation & Bucket
To get synchrotron oscillation equation, up to now, we assumed that the amplitude of
RF phase oscillation is small. But generally, its amplitude becomes large if there are
longitudinal beam instabilities in the storage ring.
Followings are the longitudinal phase oscillation when there are longitudinal beam
instabilities in the PLS storage ring.
LFS On
237.0 mA, 2.04 GeV
70 dB damped
Small Amplitude Oscillation (less than 0.03 deg@RF)
No harmful CBMI
LFS Off
180.1 mA, 2.04 GeV
Large Amplitude Oscillation (~ 6.0 deg@RF)
Strong CBMI due to 1707.0 MHz HOM (TM013)
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
56
Commissioning Results of the PLS LFS
 Pseudo-spectrum
 Beam spectrum without revolution harmonics
LFS turn on
237.0 mA, 2.04 GeV
70 dB damped
No harmful CBMI
LFS turn off
180.1 mA, 2.04 GeV
Large amplitude motion
Strong CBMI due to 1707.0 MHz HOM (TM013)
57
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Commissioning Results of the PLS LFS
 Grow/Damp Process
 It is possible to measure the feedback damping time by grow/damp process.
153.1 mA, 2.04 GeV
Two natural grow/damp modes, n = 125 (1870.1 MHz), 194 (1707.0 MHz)
CBMI growth time = 5.42 ms, 4.53 ms
feedback damping time ~ 2.0 ms
58
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Layout of Duke Longitudinal Feedback System
Y. Kim, W. Wu, et al., PAC2007
59
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Assembled Components of Duke LFS
October, 2007
January, 2008
60
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
HOM damped RF Cavity for Duke LFS Kicker
type : two waveguide overloaded RF Cavity
kicker bandwidth : fRF/2 ~ 90 MHz
central frequency ~ 938 MHz for (5+1/4)* fRF
maximum shunt impedance ~ 1600 
61
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
HOM damped RF Cavity for Duke LFS Kicker
Key Members of Duke LFS Project Team
Yujong Kim = Project Leader (Concepts and Component Selection, Kicker Design Advisor)
Wenzhong Wu (Prof. Ying Wu's Ph.D. Course Student) = HFSS Simulation of Kicker
Matthew Busch = Mechanical Design of Kicker Cavity
Prof. Ying Wu = General Accelerator Physics Related Advisor
Wenzhong Wu - Ph.D. Student, Physics Department, Duke University
Wenzhong, Y. Kim, et al., PAC2009
62
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Commissioning Results of Duke LFS
Damping of all CBMIs at 574 MeV, 50 mA, with 64 full bucket filling - done
feedback damping time ~ 2.8 ms
Wenzhong, Y. Kim, et al., PAC2009
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
63
Large Longitudinal Beam Oscillation & Bucket
More details on beam instabilities & beam feedback system will be studied
during 2011 Fall semester or 2012 Spring semester.
→ Y. Kim wants to find a good student who can work for the beam instability
& beam feedback systems for the Jefferson Lab MEIC project.
Let's study more analytical things on large longitudinal beam oscillation &
bucket from now on.
64
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Large Longitudinal Beam Oscillation & Bucket
From the general phase oscillation equation (see page No. 27), we can rewrite the phase oscillation for a large oscillation:
Even though we can not solve the equation above exactly, we can convert it to
a first order equation by using followings (see next page too):
65
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Large Longitudinal Beam Oscillation & Bucket
d (2 )  2 d
After an integration, the equation above becomes:
Here ϕ → ϕ0 at t = 0.
66
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Large Longitudinal Beam Oscillation & Bucket
We can find a similar large amplitude oscillation from a pendulum.
http://www.cds.caltech.edu/~shawn/LCS-tutorial/motivation.html
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
67
Large Longitudinal Beam Oscillation & Bucket
Stable Large Pendulum Oscillation within the Separatrix
http://www.cds.caltech.edu/~shawn/LCS-tutorial/motivation.html
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
68
Large Longitudinal Beam Oscillation & Bucket
Unstable Pendulum Oscillation at the outside of Separatrix until it loose its Energy.
http://www.cds.caltech.edu/~shawn/LCS-tutorial/motivation.html
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
69
Large Longitudinal Beam Oscillation & Bucket
  0  trajectory moving to right
  0  trajectory moving to left
By using a simple computer program, for a given ϕs, we can plot equation above
for a phase space ( , ), ( ,  ), or (ϕ, W) because following relation is satisfied:
2

tr
rf


    s      2 W
 Us
Stable Fixed Point (SFP)
yellow region
= RF bucket
Separatrix
below transition
Stationary RF Bucket
(largest bucket size)
Unstable Fixed Point (UFP)
Here closed curve show the stable trajectories where charge particles oscillate about
an equilibrium synchronous RF phase, ϕs, and the open curves are unstable trajectories.
70
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Large Longitudinal Beam Oscillation & Bucket
  0  trajectory moving to right
  0  trajectory moving to left
By using a simple computer program, for a given ϕs, we can plot equation above
for a phase space ( , ), ( ,  ), or (ϕ, W) because following relation is satisfied:
2

tr
rf


    s      2 W
 Us
Stable Fixed Point (SFP)
yellow region
= RF bucket
Separatrix
below transition
Accelerating RF Bucket
(reduced bucket size)
Unstable Fixed Point (UFP)
ϕs
RF bucket = closed curves within separatrix showing the stable trajectories.
Charge particles oscillate about the synchronous RF phase, ϕs in the RF bucket.
71
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Large Longitudinal Beam Oscillation & Bucket
  0  trajectory moving to right
  0  trajectory moving to left
By using a simple computer program, for a given ϕs, we can plot equation above
for a phase space ( , ), ( ,  ), or (ϕ, W) because following relation is satisfied:
2

tr
rf


    s      2 W
Stable Fixed Point (SFP)
 Us
Separatrix
yellow region
= RF bucket
below transition
Accelerating RF Bucket
(more reduced bucket size)
Unstable Fixed Point (UFP)
ϕs
RF bucket = closed curves within separatrix showing the stable trajectories.
Charge particles oscillate about the synchronous RF phase, ϕs in the RF bucket.
72
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Large Longitudinal Beam Oscillation & Bucket
  0  trajectory moving to right
  0  trajectory moving to left
1    s  0   & 0  0  unstableoscillation
2  0  1    s & 0  0  stable oscillation
ϕ2 can be found by inserting 0    s , 0  0,   2 ,   0 →
RF buckets are stable phase regions, which repeat every 2π.
ϕ2
bucket No. i - 1
below transition
ϕ1
bucket No. i
ϕs
bucket No. i + 1
π - ϕs
73
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Stationary & Accelerating RF Buckets
To get a positive synchrotron frequency
For accelerating bucket in below transition ηtr > 0, 0 < ϕs < π/2
For accelerating bucket in above transition ηtr < 0, π/2 < ϕs < π
accelerating bucket with sinϕs≠0
ϕs jumping at
transition energy
stationary bucket with sinϕs=0
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
74
Beam Acceleration & Deceleration
To give an acceleration for a positive charged particle, the synchronous RF phase
should be located at π/2 < ϕs < π (above transition) or 0 < ϕs < π/2 (below transition).
To give a deceleration for a positive charged particle (= an acceleration for a negative
charged particle), the synchronous RF phase should be located at -π < ϕs < -π/2
(above transition) or -π/2 < ϕs < 0 (below transition).
→ a mirror symmetry around π to have the positive synchrotron frequency (s > 0).
qV > 0 & Us > 0 for acceleration
acceleration for positron
acceleration for electron
H. Widemann, Particle Accelerator Physics
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
75
Bucket Area, Bucket & Bunch Heights
To capture injected particle in a storage ring, we need a stationary bucket because it
has the largest bucket size.
Let's calculate the bucket area, bucket height, and bunch height of the stationary
bucket analytically.
stable area = bucket area
max energy spread → bucket height
Separatrix
below transition
Stationary RF Bucket
(largest bucket size)
max phase amplitude → bunch height
below transition
Accelerating RF Bucket
(more reduced bucket size)
76
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Bucket Area, Bucket & Bunch Heights
Lets' calculate the bucket area, bucket height, and bunch height of the stationary
bucket analytically.
There is no acceleration of the synchronous particle in the stationary bucket.
no acceleration (qVsinϕs = 0)
of the synchronous particle
if ϕs = 0 (below transition)
or π (above transition)
stable phase oscillation of particles in stationary bucket (ϕs = 0) in below transition (E < Etr)
77
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Bucket Area, Bucket & Bunch Heights
In the stationary bucket (ϕs = 0), the first order phase oscillation equation (see page
No. 55) can be simplified:
Here, ϕ0 → ϕm where particle trajectory crosses the axis on the phase space plot
at (
,
).
The equation of separatrix can be found by putting ϕm = π.
The synchronous frequency for the stationary bucket:
under below transition
bunch within a stationary bucket
78
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Bucket Area, Bucket & Bunch Heights
If we re-write it by using W and
s
W
s 
2
s
-1

2
( L/v )
under below transition
bunch within a stationary bucket
Here L is circumference of ring.
After combining with separatrix equation & ϕs = 0 :
79
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Bucket Area, Bucket & Bunch Heights
Then, area of stationary bucket (ϕs = 0), under below transition is
Abk
L

8
c
qVU s
2h3 tr
ϕm = π for saparatrix = maximum bucket size
under below transition
bunch within a stationary bucket
Then, the first order phase oscillation for the stationary bucket
(see pages No. 67 & 68)
2c 2h3tr
16
 2(cos   cos m ) 
W
W
L
U s qV
Abk
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
80
Bucket Area, Bucket & Bunch Heights
bucket height @ ϕ = 0 & ϕm = π
If we use a relation:
bunch height @ ϕ = 0 & ϕm = ϕm
2c 2h3 tr
16
 2(cos   cos m ) 
W
W
L
U s qV
Abk
W 
Abk
2(cos   cos m ) 
16
under below transition
bunch within a stationary bucket
From the longitudinal phase space (ϕ, W), we can find the height of the bucket Wbk by
putting ϕ = 0 & ϕm = π in

L
c
qVU s
2h3 tr
In addition, the height of the bunch Wb for general ϕm can be obtained by putting ϕ =
0 & ϕm = ϕm in
 Wb  Wbk
81
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Maximum Momentum Acceptance of a Bucket
For the synchrotron oscillation with
a small amplitude, the solution of the phase
oscillation can be given by:
bucket height @ ϕ = 0 & ϕm = π
bunch height @ ϕ = 0 & ϕm = ϕm
under below transition
bunch within a stationary bucket
Here ψ0 is the initial phase of the oscillation at t = 0, and
φm is the amplitude of the phase oscillation (= ϕm in previous pages).
By reminding relations of the energy oscillation,
the energy oscillation and its oscillation amplitude Wm can be written by:
82
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Maximum Momentum Acceptance of a Bucket
Therefore, we can find energy oscillation W and Wm if we know φm (= ϕm). For a small
phase oscillation in a strong ring with a circumference of L and harmonic number h,
φm, satisfies a following relation:
hlb
  m  m 
2 m : 2  hlb : L
bucket
L
ring
Here lb is the bunch length.
one bucket in a ring
83
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Maximum Momentum Acceptance of a Bucket
By reminding a relation between momentum deviation and energy deviation
(see page No. 25):
Then, for a bunch with a momentum spread of
W → Wb, U → Us
From the bucket height, bunch height, and φm,

L
c
Abk
L

8
c
qVU s
2h3 tr
m
2
m 
hlb
L
Wb
qVU s
Wb
 2U s p
 hlb 



sin


m
m

p
2
L
2h3 tr


rf
sin
sin
2
2
 hlb / L  1 
sin m  0 ,Wb  0
84
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Maximum Momentum Acceptance of a Bucket
Then, by using rf  h s  h
2
s
h
2
L/ c
and
the gap voltage of an RF cavity, V can be written by:
Therefore, for a given energy deviation (p/p), the gap voltage should be higher to get
a shorter bunch length lb. In this case, a simple scaling relation can be given by:
V → 0 gives a infinite bunch length → bunch diffusion.
When φm becomes π, bunch size becomes its maximum (= bucket). Then, the
maximum acceptable momentum of a bucket can be given by:
higher V gives a higher momentum acceptance.
85
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Longitudinal Normalized Emittance
Similar to the conserved normalized transverse emittance, there is a conserved beam
parameter in the longitudinal phase space: longitudinal normalized emittance
(see Conte's book page No. 107-108 & 155-156).
For at least one cycle of the synchrotron oscillation, the Poincaré-Cartan Integral
Invariant can be given by:
For a small oscillation,
IL = invariant, which is related with the longitudinal normalized emittance.
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
86
Longitudinal Normalized Emittance
IL = invariant, which is related with the longitudinal normalized emittance.
E,u × z = normalized longitudinal emittance = invariant
E,u : rms uncorrelated energy spread (= slice energy spread)
z : rms bunch length
Generally, the longitudinal normalized emittance is conserved well even during
bunch length compression.
(E,u × z)before BC ≈ (E,u × z)after BC
exceptional case : when CSR & ISR, and
wakefields are strong.
SCSS Bunch Compressor shows the conservation
of longitudinal normalized emittance.
87
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Longitudinal Normalized Emittance
 Uncorrelated energy deviation dEu,i can be obtained by polynomial fitting.
dEu ,i  dEc ,i  FIT[dE ] dzdz
i
FIT[dE ]  a1  a2dz  a3dz 2  a4dz 3  a5dz 4  .....
[After BC : dEc,i]
[After BC : FIT[dE]]
88
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
Longitudinal Normalized Emittance
 Uncorrelated energy spread E,u is increased in BC (about 5 ~ 10 times).
 Increase of SCSS Phase-I Stage : about 4.6 times (from 5.42E-5 to 2.50E-4).
 Increase of SCSS Phase-II Stage : about 10.2 times (from 5.75E-5 to 5.84E-4).
 What is source of this increase ?
→ mainly due to reduced bunch length (E,u × z)before BC ≈ (E,u × z)after BC
[After X-band Linac : dEu,i]
[After BC : dEu,i]
[BC for the SCSS Phase-I Stage]
89
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
4th Alternative ILC BC - Long. Emittance
Q=3.2 nC
e-beam
Initial parameters
E = 5.0 GeV
 = 0.13% (small !)
1/10.0
z = 6.0 mm
z = 6.00 mm
600 m
nx= 8.0 m, ny= 0.020 m
1/4.0
150 m
4 FODO Cells
Damping Ring
SACC12
SACC34
650 MHz
650 MHz
14.0 MV/m
-66.5 deg
ACC1
BC1
SACC5-13
SBC2
ACC2 ACC3
1300
650 MHz
MHz
26.3 MV/m
163.0 deg
E = 5.244 GeV
6.7 (14) MV/m
 ~ 1.81%
-27.7 deg
R56 ~ 302 mm
 = 5.3 deg
E = 6.4 (7.6) GeV
 ~ 1.55%
R56 ~ 30.3 mm
 = 1.41 deg
Up to main Linac with ELEGANT under CSR, ISR, and geometric short-range
wakefields. but without space charge
Y. Kim @ Snowmass2005 ILC workshop
Final parameters
E = 6.4 (7.6) GeV
Here SACC5-13 are eight 650 MHz subharmonic modules with
 = 1.55%
12 cavities, and four 90 deg FODO cells is used in SACC5-13
z = 150 m
modules. SACC5-13 can be replaceable with normal 1300 MHz
TESLA modules.
nx= 8.28 m, ny= 0.020 m
90
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
4th Alternative ILC BC – Long. Emittance
01AUG05 Version
Y. Kim @ Snowmass2005 ILC workshop
Nonlinearity was compensated by 1300 MHz TESLA module (ACC1)
z = 6.00 mm
z = 150 m
At BC2, we can compress further because we have good linearity in dz-dE
91
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
4th Alternative ILC BC – Long. Emittance
01AUG05 Version
A
B , E,u ~ 2.55 MeV (RMS)
zoom in
Y. Kim @ Snowmass2005 ILC workshop
RMS value through region A gives total (or projected) rms energy spread, E (or its relative RMS
energy spread  ). In our case, initial total rms relative energy spread  = 0.13%.
RMS value through region B gives uncorrelated (or slice) rms energy spread E,u. In our case,
peak-to-peak momentum change in region B is about 30 (=9800-9770). If we change its unit to
MeV, it is 30*0.511 ~ 15.3 MeV. Since RMS value is about six times smaller than peak-to-peak one,
its RMS value E,u is about 2.55 MeV. Please note that E,u × z should be constant during bunch
compression to conserve the longitudinal emittance. But unfortunately, many persons
misunderstand that δ × z (or E × z ) should be constant (wrong concept).
Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
92
4th Alternative ILC BC – Long. Emittance
01AUG05 Version
A
B , E,u ~ 98.4 MeV (RMS)
zoom in
Y. Kim @ Snowmass2005 ILC workshop
After BC2, longitudinal phase space is rotated about 90 deg such as shown in left plot. In this
special case, total rms energy spread, E (or  ) is very close to uncorrelated (or slice) rms
energy spread E,u (or ,u). after BC2, peak-to-peak momentum change in region B is about 1155
(=13005-11850). If we change its unit to MeV, it is 1155*0.511 ~ 590.2 MeV. Since RMS value is
about six times smaller than peak-to-peak one, its RMS value E,u is about 98.4 MeV. Therefore is
relative RMS uncorrelated energy spread ,u ~ 98.4 /6400 ~ 0.0154 or 1.54%, which is slightly
smaller than our total relative RMS energy spread  = 1.55% after BC2.
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Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI
4th Alternative ILC BC – Long. Emittance
At SACC1 (=before BC1), E,u × z = 6.0 mm*2.55 MeV ~ 15.3 mm.MeV
After BC2, E,u × z = 0.150 mm*98.4 MeV ~ 14.8 mm.MeV
Since (E,u × z)SACC1 ~ (E,u × z)BC2, the longitudinal emittance is well conserved
during bunch compression. Small difference is due to the short-range wakefield in
modules, CSR and ISR effects in BC.
Therefore there is no special problem in the longitudinal emittance conservation in the
4th alternative ILC bunch compressor.
Y. Kim @ Snowmass2005 ILC workshop
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Yujong Kim @ Advanced Radiation Technology Laboratory - KAERI