Download Overview Acceleration with RF fields Bunches Phase

Chapter 10
Acceleration and longitudinal
phase space
Rüdiger Schmidt (CERN) – Darmstadt TU - 2011, version E2.4
Beam optics essentials....
• Description for particle dynamics with transfer matrices
• Differential equation for particle dynamics
• Description of particle movement with the betatron function
• Betatron oscillation
• Beam size:
 x ( s) 
 x   x (s) und  z (s) 
 z   z (s)
• Working points: Q values
• Closed Orbit
• Dispersion
2
Overview
• Acceleration with RF fields
• Bunches
• Phase focusing in a Linear Accelerator
• Phase focusing in a Circular Accelerator
• Equation of motion for the longitudinal plane
• Synchrotron frequency
3
Principal machine components of an accelerator
Kreisbeschleuniger: Beschleunigung durch
vielfaches Durchlaufen durch (wenige)
Beschleunigungstrecken

4
Acceleration in a Cavity for T=0 (accelerating phase)
(100 MHz)

E(t )
2a
z
E(z)
E0
g
z
5
Acceleration in a Cavity for T=5ns (de-cellerating phase)
(100 MHz)

E(t )
2a
z
E(z)
g
z
-E0
6
Super conducting cavities (Cornell)
Cavity 200 MHz
Cavity 1300 MHz
Cavity 500 MHz
7
Super conducting cavity with 9 cells (XFEL, DESY)
8
Normal conducting cavity for LEP
9
Illustration for the electrical field in a cavity
10
Acceleration in time dependent field
Cavity
0

11
Acceleration with Cavities
Cavity 1
Cavity 2
z
A particle enters the cavity from the left. For acceleration, it needs to have the correct
phase in the electric field.
Assume that particle 1 travels at time t0 = 0 ns through cavity 1 – it will be accelerated by
1 MV. A particle that travels through the cavity at another time will be accelerated less,
or decelerated.
U(t)
Spannung
6
6
10 1 10
5
5 10
U ( t)
„decelleration"
0
t0 = 0
5
5 10
 10
6
6
1 10
8
1 10
8
 10
9
5 10
0
t
Zeit
9
5 10
8
1 10
8
10
12
Bunches
•
•
•
It is not possible to accelerate continuous beam in an RF field –
acceleration is always in bunches
The bunch length depends on several parameters, such as frequency and
voltage, and ranges from mm to m (in modern linacs possibly less than mm,
and micro bunching can happen)
Phase focusing is an essential mechanism to keep the particles in a bunch
U(t)
Spannung
6
6
10 1 10
5
5 10
U ( t)
0
5
5 10
 10
6
6
1 10
8
1 10
8
 10
9
5 10
0
t
Zeit
9
5 10
8
1 10
8
10
13
Phase focusing in a Linac– increasing field
z
Cavity 1
U(t)
Spannung
1.05 1.05
U ( t)
10
6
0.53
0
0.53
 1.05
1.05
2.5
 2.5
1.88 1.25 0.63
0
0.63 1.25 1.88
t
Zeit
2.5
3.13 3.75 4.38
5
5
14
Phase focusing in a Linac
z
Cavity 1
•
Cavity 2
We assume three particles, the velocity is much less than the speed of
light.
• A particle with nominal momentum
• A particle with more energy, and therefore higher velocity (blue)
• A particle with less energy, and therefore smaller velocity (green)
•
•
•
The red particle enters the cavity at t = 1.25 ns. It is assumed that the
electrical field increases (rising part of the RF field)
The green particle enters the cavity later at t = 1.55 ns and experiences a
higher field
The blue particle enters the cavity earlier at t = 0.95 ns, and experiences a
lower field
15
Phase focusing in a Linac
Assume that the difference in energy is large enough and the velocity is below
the speed of light.
Before entering cavity 1:
vblue > vred > vgreen
After exiting entering cavity 1:
vgreen > vred > vblue
The velocity of the green particle is largest and it will take over the other
two particles after a certain distance .
16
Phase focusing in a Linac– Synchrotron oscillations
z
Cavity 1
U(t)
10
6
U(t)
1.05 1.05
0.53
Spannung
Spannung
1.05 1.05
U ( t)
Cavity 2
0
U ( t)
10
6
0.53
0.53
0
0.53
 1.05
 1.05
1.05
5
5
2.5
0
t
Zeit
2.5
5
5
1.05
5
5
2.5
0
t
Zeit
2.5
5
5
17
Phase de-focusing – decreasing field
Cavity 1
Spannung
1.05
z
1.05
0.53
U ( t)
10
6
0
0.53
 1.051.05
0
0
0.63 1.25 1.88
2.5
3.13 3.75 4.38
5
5.63 6.25 6.88
t
Zeit
7.5
7.5
The particle with less energy and less velocity (green) arrives late at
t = 1.55 ns. It is accelerated less than the other particles. The velocity
difference between the particles increases, and the particles are de-bunching.
18
Phase de-focussing in a Linac
z
Cavity 1
1.05
1.05
0.53
Spannung
Spannung
1.05
Cavity 2
U ( t)
10
6
0
0.53
 1.051.05
1.05
0.53
U ( t)
10
6
0
0.53
0
0
1.5
3
4.5
t
Zeit
6
7.5
7.5
 1.051.05
0
0
1.5
3
4.5
t
Zeit
6
7.5
7.5
19
Phase focusing in a circular accelerator
Cavity
The particles with different momenta
are circulating on different orbits,
here shown simplified as a circle.
p0 + dp
p0
p0 - dp
L
p

L
p
20
RF-frequency and revolution frequency
A particle with nominal momentum travels around the accelerator. In order to be in
the same phase of the RF field during the next turn, the frequency of the RF field
must be a multiple of the revolution frequency:
𝝎𝑹𝑭 = 𝒉 ∗ 𝝎𝒓𝒆𝒗 , with h: integer number, so-called harmonic number
The maximum number of bunches is given by h.
T0
Spannung
1.05
1.05
0.53
U ( t)
10
6
0
0.53
 1.051.05
0
0
20
40
60
t
Zeit
80
100
Here: h = 8
100
21
Momentum Compaction Factor
A particle with different momentum travels on a different orbit with respect to
the orbit of a particle with nominal momentum.
The momentum compaction factor is the relative difference of the orbit
length:
 L p

/
L
p
It can be shown that the momentum compaction factor is given by:

1 D( s )
ds

L 0 ( s )
The relative change of the length of the
orbit for a particle with different momentum is:
L
p

L
p
22
Momentum of a particle and orbit length
Particles with larger energy with respect to the nominal energy:
• …travel further outside => larger path length => take more time for a turn
• …the speed is higher => take less time for a turn
Both effects need to be considered in order to calculate the revolution time
The change of the revolution time for a particle with a momentum different from
nominal momentum is given by:
𝛿T
1
= 𝛼− 2
𝑇0
𝛾
𝛼 … momentum compaction factor and 𝛾 =
1
1 −𝛽2
23
Phase focusing in a circular accelerator
z
z
First turn
• We assume three particles, the velocity is close to the speed of light
• A particle with nominal momentum
• A particle with more energy, and therefore higher velocity (blue)
• A particle with less energy, and therefore smaller velocity (green)
•
We assume that the three particles enter into the cavity at the same time
•
•
•
The red particle travels on the ideal orbit
The green particle has less energy, and travels on a shorter orbit
The blue particle has more energy, and travels on a longer orbit
24
Phase focusing in a circular accelerator– decreasing field
Cavity
Spannung
1.05
z
1.05
0.53
U ( t)
10
6
0
0.53
 1.051.05
0
0
0.63 1.25 1.88
2.5
3.13 3.75 4.38
t
Zeit
5
5.63 6.25 6.88
7.5
7.5
Next turn
• The particle with less energy (green) enters earlier into the cavity and is
accelerated more than the red particle
• The particle with larger energy (blue) enters later and is accelerated less.
It loses energy in respect to the red particle
25
Phase shift as a function of the energy deviation
After one turn the particle is delayed with respect to the particle
with nominal momenum by T
The phase difference between the two particles is :   HF  T
  h  rev  T  2    h 
With
T
T0
T
1 p
p 1 E
 (  2 ) 
und
 2
we get :
T0

p
p  E
2   h
E
2

E
For a phase change small compared to the revolution time
 
 ( 
1
)
2
d      2    h
1 E

 2
 (  2 ) 
dt
T0
 T0

E
26
Acceleration in a cavity: particle with nominal energy
It is assumed that the magnetic field increases. To keep a particle with nominal
energy on the nominal orbit the particle is accelerated, per turn by an
energy of:
dB( t )
W0  2    e 0    R 
dt
The energy is provided by the electrical
field in the cavity:
with U0 - maximum cavity Voltage
and  s  Phase for particle with
nominal energy
Spannung
W0  e0  U0  sin( s )
1.2
1.2
U0
W0
0.6
U ( t)
10
6
0
0.6
 1.2 1.2
2.5
 2.5
1
0.5
2
t
Zeit
3.5
5
5
27
Acceleration in a cavity: particle with a momentum different
from the particle with nominal momentum
A particle with differing energy enters at a different time (phase) into the cavity,
with an energy increase by:
W1  e0  U0  sin( s  (t ))
Spannung
1.2
1.2
U0
0.6
W1
U ( t)
10
6
W0
0
0.6
The difference of energies is given by:
 1.2 1.2
2.5
 2.5
  E  W - E0  e 0  U0  sin(  s  )  sin(  s )
1
0.5
2
t
Zeit
3.5
5
5
The change of energy for many turns (revolution T0):
(t) 
 e 0  U0

 sin( s  )  sin( s )
T0
T0
28
Acceleration in a cavity
If the difference with respect to the nominal phase is small:
   s
sin( s   )  sin( s )    cos( s )
and therefore:
 (t) 
e 0  U0
   cos( s )
T0
differentiation yields:
(t) 
e 0  U0 d( )

 cos( s )
T0
dt
29
Equation of motion
Mit
d() 2    h
1 
 2
 (  2 ) 
dt
  T0
 E
und (t) 
und
We get:
rev
e 0  U0 d(  )

 cos( s )
T0
dt
Change of phase due to the
change of energy
Change of energy when
travelling with a different
phase through a cavity
2

T0
  e 0  U0  h  cos( s )
1
(t)  rev

(


)
2
2
2    E

2
30
Solution of the equation of motion
The equation describes an harmonic oscillator:
(t)   2  ( t )  0
  E  W - E0
with the synchrotron frequency:
  rev  
e 0  U0  h  cos( s )
1

(


)
2
2
2    E

The energy difference between the nominal particle and particles with different
momentum is:
(t)   0  e i t
31
Synchrotron frequency
  rev  
e 0  U0  h  cos( s )
1

(


)
2
2
2    E

Synchrotron frequency
For ultra relativistic particles  >> 1 :
  rev  

1
0
2

e0  U0  h  cos( s )

2
2     E
cos( s ) must be negativ, therefore
For particles with:  

2
 s 
3
( falling edge)
2
1
0
2

cos( s ) must be positiv, therefore -

2
 s 

2
(rising edge )
32
Example
33
Synchrotron frequency of the model accelerator
35
Phase space and Separatrix
Synchrotron oscillations are for particles with small energy deviation. If the
energy deviation becomes too large, particle leave the bucket.
From K.Wille
36
RF off, de-bunching in ~ 250 turns, roughly 25 ms
LHC
2008
about 1000
turns
single turn
Courtesy
E. Ciapala
37
Attempt to capture, at exactly the wrong injection phase…
LHC
2008
Courtesy
E. Ciapala
38
Capture with corrected injection phasing
LHC
2008
Courtesy
E. Ciapala
39
Capture with optimum injection phasing, correct frequency
LHC
2008
Courtesy
E. Ciapala
40
RF buckets and bunches at LHC
RF Voltage
The particles
oscillate back and
forth in
time/energy
The particles are trapped in the RF voltage:
this gives the bunch structure
2.5 ns
E
time
LHC bunch spacing = 25 ns = 10 buckets  7.5 m
RF bucket
time
2.5 ns
450 GeV
RMS bunch length
RMS energy spread
11.2 cm
0.031%
7 TeV
7.6 cm
0.011%
41
Longitudinal bunch profile in SPS
Instabilities at low energy
(26 GeV)
a) Single bunches
Quadrupole mode developing
slowly along flat bottom. NB
injection plateau ~11 s
Pictures provided by T.Linnecar
Bunch profile during a
coast at 26 GeV
Bunch profile oscillations
on the flat bottom –
at 26 GeV
stable beam
42