### Longitudinal Dynamics I, II

```LONGITUDINAL DYNAMICS
Frank Tecker
based on the course by
Joël Le Duff
Many Thanks!
CAS on Advanced Level Accelerator Physics Course
Trondheim, 18-29 August 2013
CAS Trondheim, 18-29 August 2013
1
Summary of the 2 lectures:
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Acceleration methods
Accelerating structures
Phase Stability + Energy-Phase oscillations (Linac)
Circular accelerators: Cyclotron / Synchrotron
Dispersion Effects in Synchrotron
Longitudinal Phase Space Motion
Stationary Bucket
Injection Matching
Two more related lectures:
• Linear Accelerators I + II
• RF Cavity Design
– Maurizio Vretanar
- Erk Jensen
CAS Trondheim, 18-29 August 2013
2
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
dp
preferably along the direction of the initial momentum (z).
dt
= eEz
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.
CAS Trondheim, 18-29 August 2013
3
Electrostatic Acceleration
vacuum envelope
source
E
DV
Electrostatic Field:
Energy gain: W = e ΔV
Limitation: isolation 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)
CAS Trondheim, 18-29 August 2013
4
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
CAS Trondheim, 18-29 August 2013
B
5
Electrostatic acceleration limited by isolation possibilities => use RF fields
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
CAS Trondheim, 18-29 August 2013
6
Resonant RF 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)
CAS Trondheim, 18-29 August 2013
7
Some RF Cavity Examples
L = vT/2 (π mode)
Single Gap
L = vT (2π mode)
Multi-Gap
CAS Trondheim, 18-29 August 2013
8
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
CAS Trondheim, 18-29 August 2013
9
Transit time factor
The accelerating field varies during the passage of the particle
=> particle does not always see maximum field => effective acceleration smaller
Transit time factor
defined as:
energy gain of particle with v = b c
Ta =
maximum energy gain (particle with v ® ¥)
+¥
In the general case, the transit time factor is:
for
Ta =
E(s,r,t) = E1 (s,r) × E2 (t)
sö
æ
E
(s,r)
cos
w
çè RF ÷ø ds
ò-¥ 1
v
+¥
ò E (s, r) ds
1
-¥
Simple model
uniform field:
follows:
E1 ( s, r ) 
Ta = sin
w RF g w RF g
2v
2v
VRF
 const.
g
• 0 < Ta < 1
• Ta  1 for g  0, smaller ωRF
Important for low velocities (ions)
CAS Trondheim, 18-29 August 2013
10
-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
==>
CAS Trondheim, 18-29 August 2013
11
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.
CAS Trondheim, 18-29 August 2013
12
Energy Gain
In relativistic dynamics, total energy E and momentum p are linked by
(E = E0 +W )
2 2
2


p
c
E E0
2
Hence:
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.
CAS Trondheim, 18-29 August 2013
13
Velocity, Energy and Momentum
1
v
1
   1 2
c

Bet a
normalized velocity
electrons
normalized velocity
0.5
=> electrons almost reach the speed of light
very quickly (few MeV range)
protons
0
0
5
10
E_kinetic (MeV)
total energy
1 10
rest energy
1 10
total energy
1 10
rest energy
20
5
4
p = mv =

c
2
3
1
1 
2
Gamma
E m
1
  
2
E0 m0
v
1
Momentum
15
electrons
100
protons
10
E
E
b
c
=
b
= bg m0 c
2
c
c
1
0.1
1
CAS Trondheim, 18-29 August 2013
10
100
1 10
E _kineti c (MeV)
3
1 10
4
14
Summary: Relativity + Energy Gain
Newton-Lorentz Force
(
dp
F=
=e E+v´B
dt
Relativistics Dynamics
v
c
   1
p = mv =
1
2
g =
RF Acceleration
E
m
1
=
=
E0 m0
1- b2
E
E
b
c
=
b
= bg m0 c
2
c
c
2 2
2


E E0 p c
2
dE v dp
Ez = Eˆ z sinw RF t= Eˆ z sinf( t )
ò Eˆ z dz = Vˆ
W  eVˆ sin
(neglecting transit time factor)
dE dp dp
=v = =eEz
dz
dz dt
dE =dW =eEz dz
)
2nd term always perpendicular
to motion => no acceleration
W =e ò Ez dz
The field will change during the
passage of the particle through the
cavity
=> effective energy gain is lower
CAS Trondheim, 18-29 August 2013
15
Common Phase Conventions
1.
For circular accelerators, the origin of time is taken at the zero crossing of the RF
voltage with positive slope
2.
For linear accelerators, the origin of time is taken at the positive crest of the RF
voltage
Time t= 0 chosen such that:
1
2
E1
E2
   RF t
   RF t
2
1
E1 (t) = E0 sin (w RF t )
3.
E2 (t) = E0 cos (w RF t )
I will stick to convention 1 in the following to avoid confusion
CAS Trondheim, 18-29 August 2013
16
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 .
eVs = eVˆ 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.
For a 2π mode,
the electric field
is the same in all
gaps at any given
time.
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)
CAS Trondheim, 18-29 August 2013
17
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
CAS Trondheim, 18-29 August 2013
18
Energy-phase Oscillations (1)
- Rate of energy gain for the synchronous particle:
dEs dps

 eE0 sins
dz
dt
- Rate of energy gain for a non-synchronous particle, expressed in
reduced variables,
w  W  Ws  E  Es
and
    s
dw  eE sin     sin    eE cos .
0
s
s
0
s
dz
:
small 
- Rate of change of the phase with respect to the synchronous one:



d
 RF dt   dt    RF 1  1    RF
2 v  vs 
dz
vs
 v vs 
 dz  dz s 
Since:


2
2
v  vs  c  s   c   s  w 3
2s
m0vs s
CAS Trondheim, 18-29 August 2013
19
Energy-phase Oscillations (2)
one gets:
RF
d

3 3 w
dz
m0vs s
Combining the two 1st order equations into a 2nd order equation gives:
d
2


s  0
2
dz
2
with
Stable harmonic oscillations imply:
hence:
2s 
eE0RF coss
3 3
m0vs s
W2s > 0 and real
coss  0
And since acceleration also means:
You finally get the result for
the stable phase range:
sin s  0
0  s  
2
CAS Trondheim, 18-29 August 2013
20
Longitudinal phase space
The energy – phase oscillations can be drawn in phase space:
DE, Dp/p
move
forward
reference
DE, Dp/p
acceleration
move
backward
deceleration
The particle trajectory in the
phase space (Dp/p, ) describes
its longitudinal motion.


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)
CAS Trondheim, 18-29 August 2013
21
Circular accelerators: Cyclotron
Used for protons, ions
RF generator, RF
B
= constant
RF = constant
Synchronism condition
 s   RF
2   vs TRF
g
Ion source
Cyclotron frequency
1.
Extraction
electrode
Ions trajectory
B
2.
qB

m0 
 increases with the energy
 no exact synchronism
if v  c    1
CAS Trondheim, 18-29 August 2013
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Cyclotron / Synchrocyclotron
TRIUMF 520 MeV cyclotron
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 )
CAS Trondheim, 18-29 August 2013
Allows to go beyond the
non-relativistic energies
23
Circular accelerators: The Synchrotron
B
R
E
1.
Constant orbit during acceleration
2.
To keep particles on the closed orbit,
B should increase with time
3.
 and RF increase with energy
RF cavity
RF
generator
Synchronism condition
w RF = h wr
Ts  h TRF
2 R
 h TRF
vs
CAS Trondheim, 18-29 August 2013
h integer,
harmonic number:
number of RF cycles
per revolution
24
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
If v≈c,
r
hence RF remain constant (ultra-relativistic e- )
CAS Trondheim, 18-29 August 2013
25
The Synchrotron
Energy ramping is simply obtained by varying the B field (frequency follows v):
p = eBr
Since:
Þ
dp
dt
= er B
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 s  2  R
VˆRF

B 

 s  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.
CAS Trondheim, 18-29 August 2013
26
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
CAS Trondheim, 18-29 August 2013
27
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
The “momentum compaction factor” is
defined as:
E
Circumference
2R
E+E
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
fr=revolution frequency
d fr
h=
dp
fr
Þ
p
CAS Trondheim, 18-29 August 2013

p df r
fr dp
28
Dispersion Effects in a Synchrotron (2)
ds0 = rdq
p dL
a=
L dp
s
s0
ds = ( r + x ) dq
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
Dx

R
CAS Trondheim, 18-29 August 2013
m
the average is
considered over
the bending
magnet only
29
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


 
fr   2
 p
dfr
dp
=h
fr
p
=0 at the transition energy
)
1
2 - 2
)
(
= 1- b
- 12
)
2 -1
g2
db
b
  12  

 tr  1

CAS Trondheim, 18-29 August 2013
30
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
above transition
  12  

CAS Trondheim, 18-29 August 2013
31
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 rapid change
of the RF phase, a ‘phase jump’.
CAS Trondheim, 18-29 August 2013
32
Synchrotron oscillations
Simple case (no accel.): B = const., below transition
   tr
The phase of the synchronous particle must therefore be 0 = 0.
1
- The particle is accelerated
- Below transition, an increase in energy means an increase in revolution
frequency
- The particle arrives earlier
– tends toward 0
V
RF
2
0
1
2
   RF t
- The particle is decelerated
- decrease in energy - decrease in revolution frequency
- The particle arrives later – tends toward 0
CAS Trondheim, 18-29 August 2013
33
Synchrotron oscillations (2)
VRF
2
t
0
1
Phase space picture
Dp
p

CAS Trondheim, 18-29 August 2013
34
Synchrotron oscillations (3)
   tr
Case with acceleration B increasing
VRF
1

   RF t
2
s
Phase space picture
s      s
Dp
p
stable region

unstable region
separatrix
CAS Trondheim, 18-29 August 2013
The symmetry of the
case B = const. is lost
35
Longitudinal Dynamics in Synchrotrons
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 s, 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
D =  - s
:
particle momentum :
Dp = p - ps
particle energy
:
DE = E – Es
azimuth angle
:
D =  - s
CAS Trondheim, 18-29 August 2013
36
First Energy-Phase Equation
v
D
s
R
fRF = h fr
Þ Df = -h Dq with q = ò w r dt
=> smaller RF phase
For a given particle with respect to the reference one:
d
d
1
d
1
D    h dt
Dr  D   
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 D   ps Rs 
 rs h rs dt
h rs
CAS Trondheim, 18-29 August 2013
37
Second Energy-Phase Equation
The rate of energy gained by a particle is:
dEeVˆsin  r
dt
2
The rate of relative energy gain with respect to the reference
particle is then:
æ Eö
2p D ç ÷ = eVˆ (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
ˆ sin f - sin f
2p ç
=
e
V
s
dt è w rs ÷ø
(
CAS Trondheim, 18-29 August 2013
)
38
Equations of Longitudinal Motion
DE   ps Rs d D   ps Rs 
 rs h rs dt
h rs
2 d  DE eVˆsin sin s 
dt  rs 
deriving and combining
d  Rs ps d   eVˆ sin sin 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 in the following…
CAS Trondheim, 18-29 August 2013
39
Small Amplitude Oscillations
Let’s assume constant parameters Rs, ps, s and :
  sin sin s   0
cos  s
2
s
with
hrs eVˆ cos s

2Rs ps
2
s
Consider now small phase deviations from the reference particle:
sin sin s  sin  s D sin s  cos s D
(for small D)
and the corresponding linearized motion reduces to a harmonic oscillation:
f + W Df = 0
2
s
where s is the synchrotron angular frequency
CAS Trondheim, 18-29 August 2013
40
Stability condition for ϕs
Stability is obtained when s is real and so s2 positive:
e VˆRF h h w s
W =
cos fs
2p Rs ps
Þ W2s > 0 Û
2
s
cos (s)
VRF

2
Stable in the region if
<
 0
h cos fs > 0
>
tr
0
acceleration
tr

3

2

>
<
tr
0
tr
 0
deceleration
CAS Trondheim, 18-29 August 2013
41
Large Amplitude Oscillations
For larger phase (or energy) deviations from the reference the
second order differential equation is non-linear:
2s




sin   sin s   0
cos s
(s as previously defined)
Multiplying by  and integrating gives an invariant of the motion:
2
2s
cos   sin s   I

2 coss
which for small amplitudes reduces to:
f
2
2
+W
2
s
( Df )
2
2
= I¢
(the variable is D, and s is constant)
Similar equations exist for the second variable : DEd/dt
CAS Trondheim, 18-29 August 2013
42
Large Amplitude Oscillations (2)
When  reaches -s the force goes
to zero and beyond it becomes non
restoring.
Hence -s 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:
2
2s
2s
cos   sin s    cos cos  s     s sin s 

2 coss
s
Second value m where the separatrix crosses the horizontal axis:
cosm  m sin s  cos  s     s sin s
Area within this separatrix is called “RF bucket”.
CAS Trondheim, 18-29 August 2013
43
Energy Acceptance
From the equation of motion it is seen that  reaches an extreme
when   0 , hence corresponding to   s .
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 s and plays an important role
for the capture at injection, and the stored beam lifetime.
CAS Trondheim, 18-29 August 2013
44
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 s
=180º (or 0°) which
correspond to no
acceleration . The RF
acceptance increases with
the RF voltage.
CAS Trondheim, 18-29 August 2013
45
Stationnary Bucket - Separatrix
This is the case sins=0 (no acceleration) which means s=0 or  . The
equation of the separatrix for s=  (above transition) becomes:
2


 2s cos   2s
2
2



 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

p s Rs 

h rs
and introducing the expression
for s leads to the following
equation for the separatrix:
C -eVˆ E s
f
f
W =±2
sin = ±Wbk sin
c 2p hh
2
2
CAS Trondheim, 18-29 August 2013
46
Stationnary Bucket (2)
Setting = in the previous equation gives the height of the bucket:
eVˆ Es
C
W bk  2 c 2 h
This results in the maximum energy acceptance:
DEmax
The area of the bucket is:
Since:
one gets:
2
0
w rs
-eVˆRF Es
=
Wbk = b s 2
2p
phh
2
Abk  2 0 W d

 sin 2 d  4
C -eVˆ E s
Abk = 8Wbk = 16
c 2p hh
CAS Trondheim, 18-29 August 2013
W bk  A8bk
47
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


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
CAS Trondheim, 18-29 August 2013
48
Bunch Matching into a Stationnary Bucket
A particle trajectory inside the separatrix is described by the equation:
2
2



 s cos  sin s I
2 cos s
W
The points where the trajectory
crosses the axis are symmetric with
respect to s= 
Wbk
fˆ
Wb

0
m
s= 
2
2-m
2


 2s cos   I
2

2


 2s cos   2s cos  m
2
   s 2cos m  cos 
W = ±Wbk cos
2
jm
2
- cos
cos(f ) = 2 cos2
CAS Trondheim, 18-29 August 2013
2
j
2
f
2
-1
49
Bunch Matching into a Stationnary Bucket (2)
Setting    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
fˆ

W b  A8bk cos 2m
or:
2
f m æ DE ö
fˆ
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 (m close to , fˆ small)
will require a bigger RF acceptance, hence a higher voltage
For small oscillation amplitudes the equation of the ellipse reduces to:
2
Abk ˆ 2
W=
f -( Df )
16
2
2
æ 16W ö æ Df ö
+ç
=1
çè
÷
÷
ˆ
ˆ
Abkf ø è f ø
Ellipse area is called longitudinal emittance
CAS Trondheim, 18-29 August 2013
Ab =
p
16
Abk fˆ
2
50
Capture of a Debunched Beam with Fast Turn-On
CAS Trondheim, 18-29 August 2013
51
Capture of a Debunched Beam with Adiabatic Turn-On
CAS Trondheim, 18-29 August 2013
52
Potential Energy Function
The longitudinal motion is produced by a force that can be derived from
a scalar potential:
2
d   F  
2
dt
F  U

U  0 F d    cos   sin s F 0
cos  s

2
s
The sum of the potential
energy and kinetic energy is
constant and by analogy
represents the total energy
of a non-dissipative system.
CAS Trondheim, 18-29 August 2013
53
Hamiltonian of Longitudinal Motion
Introducing a new convenient variable, W, leads to the 1st order
equations:
W  2  DE  2 Rs Dp
  rs 
d
h rs
 1
W
dt
2 ps Rs
dW eVˆsin sin s 
dt
The two variables ,W are canonical since these equations of
motion can be derived from a Hamiltonian H(,W,t):
d H

dt W
dW  H
dt

h rs 2
H ,W, t eVˆcos  cos  s    s sin  s  1
4 Rs ps W
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54
Though there are many physical processes that can damp the
longitudinal oscillation amplitudes, one is directly generated by the
acceleration process itself. It will happen in the synchrotron, even
ultra-relativistic, when ramping the energy but not in the ultrarelativistic electron linac which does not show any oscillation.
As a matter of fact, when Es varies with time, one needs to be more
careful in combining the two first order energy-phase equations in
one second order equation:
The damping coefficient is
proportional to the rate of
energy variation and from the
definition of s one has:

Es

 2 s
Es
s
 
d E   2E D
s s
dt s
Es  Es  2s Es D  0
Es 


    2sEs D  0
Es
CAS Trondheim, 18-29 August 2013
55
So far it was assumed that parameters related to the acceleration
process were constant. Let’s consider now that they vary slowly with
respect to the period of longitudinal oscillation (adiabaticity).
For small amplitude oscillations the hamiltonian reduces to:
ˆ
h rs 2
e
V
H( ,W,t)  cos  s D 2  1
2
4 Rs ps W
with
W Wˆ cosst
 
D  Dˆ sinst
Under adiabatic conditions the Boltzman-Ehrenfest theorem states
that the action integral remains constant:
I W d const.
(W,  are canonical variables)
Since:
the action integral becomes:
d H
h rs

 1
W
dt W
2 Rs ps
d
h rs
2
I  W dt   1
dt
W
dt
2 Rs ps 
CAS Trondheim, 18-29 August 2013
56
Previous integral over one period:
2
ˆ
h
I   rs W  const.
2Rs ps s
2
ˆ
2
W
 dt   W
s
From the quadratic form of the hamiltonian one gets the relation:
2 ps Rss ˆ
Wˆ 
D
h rs
Finally under adiabatic conditions the long term evolution of the
oscillation amplitudes is shown to be:
1/ 4



1/ 4
ˆ
D  

Es

2 ˆ
 E s RsV cos s 
Wˆ or DEˆ  E1s/4
Wˆ ×Dfˆ =invariant
CAS Trondheim, 18-29 August 2013
57
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
CAS Trondheim, 18-29 August 2013
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