Transcript Longitudinal Dynamics

LONGITUDINAL DYNAMICS
Frank Tecker
with lots of material
from the course by
Joël Le Duff
Many Thanks!
Introductory Level Accelerator Physics Course
Granada, 28 October - 9 November 2012
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
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Overview
• Methods of Acceleration
• Accelerating Structures
• Synchronism Condition and Phase Stability (Linac)
• Bunching and bunch compression
• Circular accelerators: Cyclotron / Synchrotron
• Dispersion Effects in Synchrotron
• Synchrotron Oscillations
• Energy-Phase Equations
• Longitudinal Phase Space Motion
• Stationary Bucket
• Injection Matching
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Bibliography
M. Conte, W.W. Mac Kay
An Introduction to the Physics of particle Accelerators
(World Scientific, 1991)
P. J. Bryant and K. Johnsen The Principles of Circular Accelerators and Storage Rings
(Cambridge University Press, 1993)
D. A. Edwards, M. J. Syphers An Introduction to the Physics of High Energy Accelerators
(J. Wiley & sons, Inc, 1993)
H. Wiedemann
Particle Accelerator Physics
(Springer-Verlag, Berlin, 1993)
M. Reiser
Theory and Design of Charged Particles Beams
(J. Wiley & sons, 1994)
A. Chao, M. Tigner
Handbook of Accelerator Physics and Engineering
(World Scientific 1998)
K. Wille
The Physics of Particle Accelerators: An Introduction
(Oxford University Press, 2000)
E.J.N. Wilson
An introduction to Particle Accelerators
(Oxford University Press, 2001)
And CERN Accelerator Schools (CAS) Proceedings
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Main Characteristics of an Accelerator
Newton-Lorentz Force
on a charged particle:
(
dp
F=
=e E+v´B
dt
)
2nd term always perpendicular
to motion => no acceleration
ACCELERATION is the main job of an accelerator.
• It provides kinetic energy to charged particles, hence increasing
their momentum.

• In order to do so, it is necessary to have an electric field E , preferably along the
direction of the initial momentum.
dp
= eEz
dt
BENDING is generated by a magnetic field perpendicular to the plane of the
particle trajectory. The bending radius  obeys to the relation :
p
 B
e
in practical units:
p [GeV/c]
B r [Tm] »
0.3
FOCUSING is a second way of using a magnetic field, in which the bending
effect is used to bring the particles trajectory closer to the axis, hence
to increase the beam density.
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Energy Gain
In relativistic dynamics, total energy E and momentum p are linked by
2 2
2


p
c
E E0
2
Hence:
(E = E0 +W )
W kinetic energy
dE  v dp
The rate of energy gain per unit length of acceleration (along z) is then:
dE dp dp
= v = =eEz
dz
dz dt
and the kinetic energy gained from the field along the z path is:
dW =dE =eEz dz
W =e ò Ez dz = eV
where V is just a potential.
Some relativistic relations:
E
E
p = mv = 2 b c = b = bg m0 c
c
c
E
m
1
g =
=
=
E0 m0
1- b2

Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
v
1
 1 2
c

5
Velocity and Energy
1
normalized velocity
electrons
Beta
v
1
   1 2
c

0.5
protons
=> electrons almost reach the speed of light
very quickly
0
0
5
10
E_kinetic (MeV)
15
20
1 10
5

total energy
1 10
rest energy
1 10
3

c
2
Gamma
E m
1


2
E0 m0
v
1
4
1
1 
electrons
100
protons
10
2
1
0.1
1
10
100
E_kinetic (MeV)
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
1 10
3
1 10
4
6
Methods of Acceleration: Electrostatic
Electrostatic Field:
Energy gain: W=n e(V2-V1)
limitation : Vgenerator=ΣVi
 insulation problems
maximum high voltage (~ 10 MV)
used for first stage of acceleration:
particle sources, electron guns
x-ray tubes
750 kV Cockroft-Walton generator
at Fermilab (Proton source)
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Methods of Acceleration: Induction
From Maxwell’s Equations:
The electric field is derived from a scalar potential φ and a vector potential A
The time variation of the magnetic field H generates an electric field E
¶A
E = -Ñf ¶t
B = mH = Ñ ´ A
vacuum
pipe
beam
Bf
iron yoke
Example: Betatron
The varying magnetic field is used to guide
particles on a circular trajectory as well as
for acceleration.
Limited by saturation in iron
coil
E
beam
R
Bf
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
B
8
Methods of Acceleration: Radio-Frequency (RF)
Wideröe-type
structure
Cylindrical electrodes (drift tubes) separated by gaps and fed by a RF
generator, as shown above, lead to an alternating electric field polarity
Synchronism condition
L = v T/2
v = particle velocity
T = RF period
Similar for standing wave
cavity as shown (with v≈c)
D.Schulte
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The advantages of resonant cavities
- Considering RF acceleration, it is obvious that when particles get high
velocities the drift spaces get longer and one loses on the efficiency.
=> The solution consists of using a higher operating frequency.
- The power lost by radiation, due to circulating currents on the electrodes,
is proportional to the RF frequency.
=> The solution consists of enclosing the system in a cavity which resonant
frequency matches the RF generator frequency.
- The electromagnetic power is now
constrained in the resonant volume
- Each such cavity can be independently
powered from the RF generator
- Note however that joule losses will
occur in the cavity walls (unless made
of superconducting materials)
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The Pill Box Cavity
From Maxwell’s equations one can derive the wave
equations:
2
Ñ2 A - e0m0
¶A
=0
2
¶t
(A = E or H )
Solutions for E and H are oscillating modes, at
discrete frequencies, of types TMxyz (transverse
magnetic) or TExyz (transverse electric).
Ez
Hθ
Indices linked to the number of field knots in polar coordinates φ, r and z.
For l<2a the most simple mode, TM010, has the lowest
frequency, and has only two field components:
Ez = J 0 (kr) eiwt
i
Hq = - J1 (kr) eiwt
Z0
k=
2p
l
=
w
c
l = 2.62a Z0 = 377W
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The Pill Box Cavity (2)
The design of a pill-box cavity can be
sophisticated in order to improve its
performances:
- A nose cone can be introduced in order
to concentrate the electric field around
the axis
- Round shaping of the corners allows a
better distribution of the magnetic field
on the surface and a reduction of the
Joule losses.
It also prevents from multipactoring
effects.
A good cavity is a cavity which efficiently
transforms the RF power into accelerating
voltage.
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Multi-gap Accelerating Structures
L = vT/2 (π mode)
Single Gap
L = vT (2π mode)
Multi-Gap
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RF acceleration: Alvarez Structure
g
Used for protons, ions (50 – 200 MeV, f ~ 200 MHz)
L2
L1
L3
L4
RF generator
Synchronism condition
L5
LINAC 1 (CERN)
g  L
L  vs TRF   s RF
 RF
vs
 2
L
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Transit time factor
The accelerating field varies during the passage of the particle
=> particle does not see maximum field all the time => effective acceleration smaller
Defined as:
Ta =
energy gain of particle with v = b c
maximum energy gain (particle with v ® ¥)
+¥
In the general case, the transit time factor is:
for
E(s,r,t) = E1 (s,r) × E2 (t)
Ta =
æ
E
(s,r)
cos
çè w RF
1
ò-¥
sö
÷ø ds
v
+¥
ò E (s, r) ds
1
-¥
Simple model
uniform field:
follows:
VRF
E1 ( s, r ) 
 const.
g
w g w RF g
Ta = sin RF
2v
2v
• 0 < Ta < 1
• Ta  1 for g  0, smaller ωRF
Important for low velocities (ions)
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Important Parameters of Accelerating Cavities
Shunt Impedance R
V2
Pd =
R
Relationship between gap
voltage V and wall losses Pd
Quality Factor Q
Q=
wWs
Relationship between
stored energy Ws in the volume
and dissipated power on the walls
Pd
R V2
=
Q wWs
Filling Time τ
dW w
Pd = - s = Ws
dt Q
Exponential decay of the
stored energy Ws due to losses
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
t=
Q
w
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Disc loaded traveling wave structures
-When particles gets ultra-relativistic (v~c) the drift tubes become very long
unless the operating frequency is increased. Late 40’s the development of
radar led to high power transmitters (klystrons) at very high frequencies
(3 GHz).
-Next came the idea of suppressing the drift tubes using traveling waves.
However to get a continuous acceleration the phase velocity of the wave needs
to be adjusted to the particle velocity.
CLIC Accelerating Structures (30 & 11 GHz)
solution: slow wave guide with irises
==>
iris loaded structure
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The Traveling Wave Case
Ez = E0 cos (w RF t - kz )
k=
w RF
vj
wave number
z = v(t - t0 )
The particle travels along with the wave, and
k represents the wave propagation factor.
vφ = phase velocity
v = particle velocity
æ
ö
v
Ez = E0 cos ççw RF t - w RF t - f0 ÷÷
vj
è
ø
If synchronism satisfied:
v = vφ
and
Ez
= E0 cos f0
where Φ0 is the RF phase seen by the particle.
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
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Principle of Phase Stability (Linac)
Let’s consider a succession of accelerating gaps, operating in the 2π mode,
for which the synchronism condition is fulfilled for a phase s .
For a 2π mode,
the electric field
is the same in all
gaps at any given
time.
eVs = eV̂ sin F s
is the energy gain in one gap for the particle to reach the
next gap with the same RF phase: P1 ,P2, …… are fixed points.
If an energy increase is transferred into a velocity increase =>
M1 & N1 will move towards P1
=> stable
M2 & N2 will go away from P2
=> unstable
(Highly relativistic particles have no significant velocity change)
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A Consequence of Phase Stability
Transverse focusing fields at the entrance and defocusing at the exit of the cavity.
Electrostatic case: Energy gain inside the cavity leads to focusing
RF case:
Field increases during passage => transverse defocusing!
Longitudinal phase stability means :
V
t
The divergence of the field is
zero according to Maxwell :
0
.E  0 
E z
z
0
defocusing
RF force
E x E z

0 
x
z
E x
0
x
External focusing (solenoid, quadrupole) is then necessary
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Energy-Phase Equations
Rate of energy gain for the synchronous particle:
Rate of energy gain for a non-synchronous particle, expressed in reduced
= - s
variables w = W -W = E - E and
s
s
j f f
Rate of change of the phase with respect to the synchronous one:
Since:
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Energy Phase Oscillations
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Longitudinal phase space
DE, Dp/p
move
forward
reference
DE, Dp/p
acceleration
move
backward
deceleration
The particle trajectory in the
phase space (Dp/p, f) describes
its longitudinal motion.
f
f
Emittance: phase space area including
all the particles
NB: if the emittance contour correspond
to a possible orbit in phase space, its
shape does not change with time
(matched beam)
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The Capture Problem
- Previous results show that at ultra-relativistic energies (γ>> 1) the longitudinal
motion is frozen. Since this is rapidly the case for electrons, all traveling wave
structures can be made identical (phase velocity=c).
- Hence the question is: can we capture low kinetic electrons energies (γ< 1), as
they come out from a gun, using an iris loaded structure matched to c ?
Ez = E0 sin f (t)
The electron entering the structure, with velocity v < c, is not synchronous
with the wave. The path difference, after a time dt, between the wave and
the particle is:
dz = (c - v)dt
Since
f = w RF t - kz
one gets
with propagation factor
lg
dz =
df =
df
w RF
2p
c
and
k=
w RF
vj
df 2p
=
c (1- b )
dt lg
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
=
w RF
c
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The Capture Problem (2)
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Bunching with a Pre-buncher
A long bunch coming
from the gun enters
an RF cavity.
The reference particle
is the one which has no
velocity change. The
others get accelerated
or decelerated, so the
bunch gets an energy
and velocity modulation.
After a distance L
bunch gets shorter:
bunching effect.
This short bunch can
now be captured more
efficiently by a TW
structure (vϕ=c).
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
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Bunching with a Pre-buncher (2)
The bunching effect is a space modulation caused by a velocity modulation,
similar to the phase stability phenomenon. Let’s look at the particles in the
vicinity of the reference and use a classical approach.
Energy gain as a function of cavity crossing time:
æ1
2ö
DW = D ç m0 v ÷ = m0 v0 Dv = eV0 sin f » eV0f
è2
ø
Dv =
eV0f
m0 v0
Perfect linear bunching will occur after a time delay τ, corresponding to a
distance L, when the path difference is compensated between a particle and
the reference one:
f
Dv t = Dz = v0 Dt = v0
w RF
Since L = v0 τ one gets:
(assuming the reference particle
enters the cavity at time t=0)
2v0W
L=
eV0w RF
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Bunch compression
At ultra-relativistic energies (γ>> 1) the longitudinal motion is frozen. This is
rapidly the case for electrons.
For example for linear colliders, you need very short bunches (few 100-50µm).
Solution: introduce energy/time correlation with a magnetic chicane.
long.
phase
space
N.Walker
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Bunch compression (2)
before
after
Longitudinal phase space evolution for a bunch compressor (PARMELA code simulations)
Introducing correlated energy spread increases total energy spread in the
bunch. => chromatic effects (depend on relative energy spread ΔE/E)
Solution: compress at low energy before further acceleration
=> absolute energy spread constant but relative is decreased
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Circular accelerators: Cyclotron
Used for protons, ions
RF generator, RF
B
B
= constant
RF = constant
Synchronism condition
 s   RF
2   vs TRF
g
Ion source
Cyclotron frequency
1.
Extraction
electrode
Ions trajectory
2.
qB

m0 
 increases with the energy
 no exact synchronism
if v  c    1
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Cyclotron / Synchrocyclotron
TRIUMF 520 MeV cyclotron
Vancouver - Canada
Synchrocyclotron: Same as cyclotron, except a modulation of RF
B
= constant
RF decreases with time
 RF
= constant
The condition:
qB
 s (t )   RF (t ) 
m0  (t )
Allows to go beyond the
non-relativistic energies
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Circular accelerators: The Synchrotron
B
Synchronism condition
R
Ts  h TRF
2 R
 h TRF
vs
E
h integer,
harmonic number:
number of RF cycles
per revolution
RF cavity
RF
generator
1.
2.
RF and  increase with energy
To keep particles on the closed orbit, B should increase
with time
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Circular accelerators: The Synchrotron
The synchrotron is a synchronous accelerator since there is a synchronous RF
phase for which the energy gain fits the increase of the magnetic field at each
turn. That implies the following operating conditions:
^
E
B
Bending
magnet
e V sin 
Energy gain per turn
   s  cte
Synchronous particle
 RF  h r
RF synchronism
(h - harmonic number)
  cte R  cte
Constant orbit
B  P  B
e
Variable magnetic field
R=C/2π
injection
extraction

bending
radius
If v≈c,
r
hence RF remain constant (ultra-relativistic e- )
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Circular accelerators: The Synchrotron
LEAR (CERN)
Low Energy Antiproton Ring
EPA (CERN)
Electron Positron Accumulator
© CERN Geneva
© CERN Geneva
Examples of different
proton and electron
synchrotrons at CERN
PS (CERN)
Proton Synchrotron
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
© CERN Geneva
34
The Synchrotron
Energy ramping is simply obtained by varying the B field (frequency follows v):
p = eBr
Þ
dp
= er B
dt
Since:
Þ (Dp)turn = er BTr =
2 p er RB
v
E 2 = E02 + p2 c2 Þ DE = vDp
( DE )turn = ( DW ) s =2p er RB=eVˆ sinf s
Stable phase φs changes during energy ramping
B
sin f s  2  R
VˆRF

B 

fs  arcsin  2  R
ˆ 
V
RF 

• The number of stable synchronous particles is equal to the harmonic
number h. They are equally spaced along the circumference.
• Each synchronous particle satisfies the relation p=eB. They have the
nominal energy and follow the nominal trajectory.
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The Synchrotron
During the energy ramping, the RF frequency
increases to follow the increase of the
revolution frequency :
wr =
2
f
(t)
v(t)
1
ec
r
Hence: RF
=
=
B(t)
h
2p Rs 2p Es (t) Rs
Since
E 2 = (m0 c2 )2 + p2 c2
( using
w RF
h
= w (B, Rs )
p(t) = eB(t)r, E = mc2 )
the RF frequency must follow the variation
of the B field with the law
ü
fRF (t)
c ì
B(t)
=
í
ý
2
2
2
h
2p Rs î (m0 c / ecr ) + B(t) þ
2
This asymptotically tends towards
compared to m0 c 2 / (ecr )
which corresponds to
v ®c
fr ®
c
2p Rs
1
2
when B becomes large
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Dispersion Effects in a Synchrotron
If a particle is slightly shifted in
momentum it will have a different
orbit and the length is different.
cavity
E
Circumference
E+E
2R
The “momentum compaction factor” is
defined as:
a=
dL
dp
L
Þ
p
p dL
a=
L dp
If the particle is shifted in momentum it
will have also a different velocity.
As a result of both effects the revolution
frequency changes:
p=particle momentum
R=synchrotron physical radius
fr=revolution frequency
d fr
h=
dp
fr
Þ

p
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
p dfr
fr dp
37
Dispersion Effects in a Synchrotron (2)
ds0 = rdq
p dL
a=
L dp
s
ds = ( r + x ) dq
s0
The elementary path difference
from the two orbits is:
definition of dispersion Dx
p  dp
p
x
d
x


dl ds - ds0 x Dx dp
=
= =
ds0
ds0
r r p
leading to the total change in the circumference:
dL = ò dl =
C
x
ò r ds
1 Dx (s)
a= ò
ds0
L C r(s)
0
=
ò
Dx dp
ds0
r p
With ρ=∞ in
straight sections
we get:
< >m means that
a=
Dx
m
R
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
the average is
considered over
the bending
magnet only
38
Dispersion Effects in a Synchrotron (3)
bc
fr =
2p R
Þ
dfr d b dR db
dp
=
=
-a
fr
b
R
b
p
definition of momentum
compaction factor
(
E0
dp d b d 1 - b
p = mv = bg
Þ
=
+
c
p
b
1- b2
(
dfr  1
dp
  2   
fr  
 p
dfr
dp
=h
fr
p
=0 at the transition energy
)
1
2 - 2
)
- 12
(
= 1- b
)
2 -1
g2
db
b
  12  

 tr  1

Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
39
Phase Stability in a Synchrotron
From the definition of  it is clear that an increase in momentum gives
- below transition (η > 0) a higher revolution frequency
(increase in velocity dominates) while
- above transition (η < 0) a lower revolution frequency (v  c and longer path)
where the momentum compaction (generally > 0) dominates.
Stable synchr. Particle
for  < 0
>0
  12  

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40
Crossing Transition
At transition, the velocity change and the path length change with
momentum compensate each other. So the revolution frequency there is
independent from the momentum deviation.
Crossing transition during acceleration makes the previous stable
synchronous phase unstable. The RF system needs to make a ‘phase jump’.
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41
Synchrotron oscillations
Simple case (no accel.): B = const., below transition
   tr
The phase of the synchronous particle must therefore be f0 = 0.
f1
- The particle is accelerated
- Below transition, an increase in energy means an increase in revolution
frequency
- The particle arrives earlier – tends toward f0
VRF
f2
f0
f1
f2
f  RF t
- The particle is decelerated
- decrease in energy - decrease in revolution frequency
- The particle arrives later – tends toward f0
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42
Synchrotron oscillations (2)
VRF
f2
ft
f0
f1
Phase space picture
Dp
p
f
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43
Synchrotron oscillations (3)
   tr
Case with acceleration B increasing
VRF
1
f
f  RF t
2
fs
Phase space picture
fs  f    fs
Dp
p
stable region
f
unstable region
separatrix
The symmetry of the
case B = const. is lost
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44
Longitudinal Dynamics
It is also often called “synchrotron motion”.
The RF acceleration process clearly emphasizes two coupled
variables, the energy gained by the particle and the RF phase
experienced by the same particle. Since there is a well defined
synchronous particle which has always the same phase fs, and the
nominal energy Es, it is sufficient to follow other particles with
respect to that particle.
So let’s introduce the following reduced variables:
revolution frequency :
Dfr = fr – frs
particle RF phase
Df = f - fs
:
particle momentum :
Dp = p - ps
particle energy
:
DE = E – Es
azimuth angle
:
D =  - s
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45
First Energy-Phase Equation
v
D
s
R
fRF = h fr
Þ Df = -h Dq with q = ò w r dt
particle ahead arrives earlier
=> smaller RF phase
For a given particle with respect to the reference one:
df
Dr  d D    1 d Df    1
dt
h dt
h dt
Since:
ps æ dw r ö
h=
w rs çè dp ÷ø s
one gets:
2 2
2
=
+
p
E E0
c
2
and
DE = vs Dp = w rs Rs Dp
DE  ps Rs dDf  ps Rs f
 rs hrs dt
h rs
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
46
Second Energy-Phase Equation
The rate of energy gained by a particle is:
dE eVˆsin f  r
dt
2
The rate of relative energy gain with respect to the reference
particle is then:
æ Eö
2p D ç ÷ = eV̂ (sin f - sin fs )
èwr ø
Expanding the left-hand side to first order:
D ( ETr )
d
@ EDTr + Trs DE = DE Tr + Trs DE = (Trs DE )
dt
leads to the second energy-phase equation:
d æ DE ö
2p ç
= eV̂ sin f - sin f s
÷
dt è w rs ø
(
)
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
47
Equations of Longitudinal Motion
DE  ps Rs dDf  ps Rs f
 rs hrs dt
h rs
2 d  DE eVˆsin f sinf s 
dt   rs 
deriving and combining
d  Rs ps df   eVˆ sin f sin f s  0
dt  hrs dt  2
This second order equation is non linear. Moreover the parameters
within the bracket are in general slowly varying with time.
We will study some cases later…
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
48
Small Amplitude Oscillations
Let’s assume constant parameters Rs, ps, s and :
f  sin f sin f s  0
cosf s
2
s
with
hrs eVˆ cosf s

2Rs ps
2
s
Consider now small phase deviations from the reference particle:
sin f sin f s  sin f s Df sin f s  cosf s Df
(for small Df)
and the corresponding linearized motion reduces to a harmonic oscillation:
f + W Df = 0
2
s
where s is the synchrotron angular frequency
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
49
Stability condition for ϕs
Stability is obtained when s is real and so s2 positive:
e V̂RF h h w s
W =
cos fs
2p Rs ps
2
s
cos (fs)
Þ W2s > 0 Û
VRF

2
Stable in the region if
<
 0
h cos fs > 0
tr
>
0
acceleration
f
3

2

tr
>
0
tr
<
 0
tr
deceleration
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
50
Large Amplitude Oscillations
For larger phase (or energy) deviations from the reference the
second order differential equation is non-linear:
2s



f
sin f  sin fs   0
cos fs
(s as previously defined)
Multiplying by f and integrating gives an invariant of the motion:
f2
2s
cos f  f sin fs   I

2 cos fs
which for small amplitudes reduces to:
f
2
2
+W
2
s
( Df )
2
2
= I¢
(the variable is Df, and fs is constant)
Similar equations exist for the second variable : DEdf/dt
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
51
Large Amplitude Oscillations (2)
When f reaches -fs the force goes
to zero and beyond it becomes non
restoring.
Hence -fs is an extreme amplitude
for a stable motion which in the
f
phase space(
, Df ) is shown as
Ws
closed trajectories.
Equation of the separatrix:
f2
2s
2s
cos f  f sin fs    cos f cos  fs     fs sin fs 

2 cos fs
s
Second value fm where the separatrix crosses the horizontal axis:
cos fm  fm sin fs  cos  fs     fs sin fs
Area within this separatrix is called “RF bucket”.
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
52
Energy Acceptance
From the equation of motion it is seen that f reaches an extreme
when f  0 , hence corresponding to f  fs.
Introducing this value into the equation of the separatrix gives:
2
fmax
= 2W2s {2 + ( 2fs - p ) tan fs }
That translates into an acceptance in energy:
G (f s ) = éë 2cosf s +( 2f s -p ) sinf s ùû
This “RF acceptance” depends strongly on fs and plays an important role
for the capture at injection, and the stored beam lifetime.
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
53
RF Acceptance versus Synchronous Phase
The areas of stable motion
(closed trajectories) are
called “BUCKET”.
As the synchronous phase
gets closer to 90º the
buckets gets smaller.
The number of circulating
buckets is equal to “h”.
The phase extension of the
bucket is maximum for fs
=180º (or 0°) which
correspond to no
acceleration . The RF
acceptance increases with
the RF voltage.
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
54
Potential Energy Function
The longitudinal motion is produced by a force that can be derived from
a scalar potential:
2
d f  Ff 
2
dt
Ff   U
f
U   0 Ff df   s cosf f sin f s F 0
cosf s
f
2
The sum of the potential
energy and kinetic energy is
constant and by analogy
represents the total energy
of a non-dissipative system.
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
55
Hamiltonian of Longitudinal Motion
Introducing a new convenient variable, W, leads to the 1st order
equations:
W 2  DE 2 Rs Dp
  rs 
df
h rs
1

W
dt
2 ps Rs
dW eVˆsinf sinf s 
dt
The two variables f,W are canonical since these equations of
motion can be derived from a Hamiltonian H(f,W,t):
df H

dt W
dW   H
dt
f
h rs 2
H f,W, t eVˆcos f cos f s f f s sin f s 1
4 Rs ps W
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
56
Stationnary Bucket - Separatrix
This is the case sinfs=0 (no acceleration) which means fs=0 or  . The
equation of the separatrix for fs=  (above transition) becomes:
2

f
 2s cos f  2s
2
2

f
f
 22s sin 2
2
2
Replacing the phase derivative by the canonical variable W:
W
0
with C=2Rs

W  2 DE   2
Wbk
2
 rs
f
ps Rs 
f
h rs
and introducing the expression
for s leads to the following
equation for the separatrix:
C -eV̂ E s
f
f
W =±2
sin = ±Wbk sin
c 2p hh
2
2
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
57
Stationnary Bucket (2)
Setting f= in the previous equation gives the height of the bucket:
eVˆ E s
C
W bk  2 c 2 h
This results in the maximum energy acceptance:
DEmax
The area of the bucket is:
2
0
w rs
-eV̂RF Es
=
Wbk = b s 2
2p
phh
2
Abk  2 0 W df
f
Since:
 sin 2 df  4
one gets:
C -eV̂ E s
Abk = 8Wbk = 16
c 2p hh
W bk  A8bk
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
58
Effect of a Mismatch
Injected bunch: short length and large energy spread
after 1/4 synchrotron period: longer bunch with a smaller energy spread.
W
W
f
f
For larger amplitudes, the angular phase space motion is slower
(1/8 period shown below) => can lead to filamentation and emittance growth
W.Pirkl
stationary bucket
accelerating bucket
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
59
Bunch Matching into a Stationnary Bucket
A particle trajectory inside the separatrix is described by the equation:
2
2

f

 s cosf f sin f s I
2 cosf s
W
The points where the trajectory
crosses the axis are symmetric with
respect to fs= 
Wbk
f̂
Wb

0
fm
fs= 
2
2-fm
2

f
 2s cosf  I
2
f
2

f
 2s cosf  2s cos f m
2
f   s 2cosf m  cosf 
W = ±Wbk cos
2
jm
2
- cos
cos(f ) = 2 cos2
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
2
j
2
f
2
-1
60
Bunch Matching into a Stationnary Bucket (2)
Setting f   in the previous formula allows to calculate the bunch height:
W b = W bk cos
fm
2
=W bk sin
æ DE ö
çè
÷ø =
Es b
f̂
f
W b  A8bk cos 2m
or:
2
f m æ DE ö
f̂
æ DE ö
çè
÷ø cos 2 = çè
÷ø sin 2
E s RF
E s RF
This formula shows that for a given bunch energy spread the proper
matching of a shorter bunch (fm close to , f̂ small)
will require a bigger RF acceptance, hence a higher voltage
For small oscillation amplitudes the equation of the ellipse reduces to:
2
Abk 2
W=
f̂ -( Df )
16
2
2
æ 16W ö æ Df ö
+ç
=1
çè
÷
÷
Abkf̂ ø è f̂ ø
Ellipse area is called longitudinal emittance
Ab =
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
p
16
Abk f̂
2
61
Effect of a Mismatch
Injected bunch: short length and large energy spread
after 1/4 synchrotron period: longer bunch with a smaller energy spread.
W
W
f
f
For larger amplitudes, the angular phase space motion is slower
(1/8 period shown below) => can lead to filamentation and emittance growth
W.Pirkl
stationary bucket
accelerating bucket
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
62
Capture of a Debunched Beam with Fast Turn-On
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
63
Capture of a Debunched Beam with Adiabatic Turn-On
Longitudinal Dynamics, CAS Granada, 28 Oct-9 Nov 2012
64