chapter 10
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Chapter 10
Rüdiger Schmidt (CERN) – Darmstadt TU - 2011, version E2.4
Acceleration and longitudinal phase space
2
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:
• Working points: Q values
• Closed Orbit
• Dispersion
(s)s und (s)s zzzxxx )()(
3
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
4
Kreisbeschleuniger: Beschleunigung durch vielfaches Durchlaufen durch (wenige) Beschleunigungstrecken
Principal machine components of an accelerator
5
2a
Acceleration in a Cavity for T=0 (accelerating phase)
z
zE0
E(z)
)(tE
g
(100 MHz)
6
2a
Acceleration in a Cavity for T=5ns (de-cellerating phase)
z
)(tE
z
-E0
E(z) g
(100 MHz)
7
Super conducting cavities (Cornell)
Cavity 200 MHz
Cavity 500 MHz
Cavity 1300 MHz
8
Super conducting cavity with 9 cells (XFEL, DESY)
9
Normal conducting cavity for LEP
10
Illustration for the electrical field in a cavity
11
Acceleration in time dependent field
0
Cavity
12
Acceleration with Cavities
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.
zCavity 1 Cavity 2
1 10 8 5 10 9 0 5 10 9 1 10 81 106
5 105
0
5 105
1 106 U(t)
Zeit
Spa
nnun
g 106
106
U t( )
10 810 8 t
t0 = 0
„decelleration"
13
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
1 10 8 5 10 9 0 5 10 9 1 10 81 106
5 105
0
5 105
1 106 U(t)
Zeit
Spa
nnun
g 106
106
U t( )
10 810 8 t
14
Phase focusing in a Linac– increasing field
zCavity 1
2.5 1.88 1.25 0.63 0 0.63 1.25 1.88 2.5 3.13 3.75 4.38 51.05
0.53
0
0.53
1.05U(t)
Zeit
Spa
nnun
g
1.05
1.05
U t( )
106
52.5 t
15
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
16
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 .
Phase focusing in a Linac– Synchrotron oscillations
z
Cavity 1 Cavity 2
17
5 2.5 0 2.5 51.05
0.53
0
0.53
1.05U(t)
Zeit
Spa
nnun
g
1.05
1.05
U t( )
106
55 t5 2.5 0 2.5 51.05
0.53
0
0.53
1.05U(t)
Zeit
Spa
nnun
g
1.05
1.05
U t( )
106
55 t
18
Phase de-focusing – decreasing field
zCavity 1
0 0.63 1.25 1.88 2.5 3.13 3.75 4.38 5 5.63 6.25 6.88 7.51.05
0.53
0
0.53
1.05
Zeit
Spa
nnun
g
1.05
1.05
U t( )
106
7.50 t
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.
19
Phase de-focussing in a Linac
z
Cavity 1 Cavity 2
0 1.5 3 4.5 6 7.51.05
0.53
0
0.53
1.05
Zeit
Spa
nnun
g
1.05
1.05
U t( )
106
7.50 t0 1.5 3 4.5 6 7.51.05
0.53
0
0.53
1.05
Zeit
Spa
nnun
g
1.05
1.05
U t( )
106
7.50 t
20
Phase focusing in a circular accelerator
Cavity
The particles with different momenta are circulating on different orbits, here shown simplified as a circle.
pp
LL
p0p0 - dp
p0 + dp
21
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:
, with h: integer number, so-called harmonic number
The maximum number of bunches is given by h.
0 20 40 60 80 1001.05
0.53
0
0.53
1.05
Zeit
Spa
nnun
g
1.05
1.05
U t( )
106
1000 t Here: h = 8
0T
22
Momentum Compaction Factor
pp
LL /
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:
dsssD
L1
0
)()(
It can be shown that the momentum compaction factor is given by:
The relative change of the length of theorbit for a particle with different momentum is: p
pLL
From beam optics
23
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:
momentum compaction factor and
24
Phase focusing in a circular accelerator
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
z
25
Phase focusing in a circular accelerator– decreasing field
zCavity
0 0.63 1.25 1.88 2.5 3.13 3.75 4.38 5 5.63 6.25 6.88 7.51.05
0.53
0
0.53
1.05
Zeit
Spa
nnun
g
1.05
1.05
U t( )
106
7.50 t
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
26
Phase shift as a function of the energy deviation
EE
Th
Tdtd
EEh
EE
pp
pp
Thh rev
HF
)(
)(
)(
20
20
22
220
0
12
time revolution the to compared small change phase a For
12
:get we1 und 1TT With
T2T
T : is particles two the between difference phase The
T by momenum nominal with
particle the to respect withdelayed is particle the turn one After
27
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:
dttdBRe2W 00
)(
The energy is provided by the electrical
field in the cavity:
energy nominal withparticle for Phase and
Voltage cavity maximum - with 0
000
s
s
U
UeW
)sin( 0W0U
2.5 1 0.5 2 3.5 51.2
0.6
0
0.6
1.2
Zeit
Spa
nnun
g
1.2
1.2
U t( )
106
52.5 t
28
2.5 1 0.5 2 3.5 51.2
0.6
0
0.6
1.2
Zeit
Spa
nnun
g
1.2
1.2
U t( )
106
52.5 t
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:
The difference of energies is given by:
The change of energy for many turns (revolution T0):
))(sin( tUeW s001
)sin()sin( ss000 UeE-WE
)sin()sin( ss0
00
0 TUe
T(t)
0W0U
1W
29
Acceleration in a cavity
If the difference with respect to the nominal phase is small:
)cos()sin()sin( sss
s
)cos( s0
00
TUe(t)
and therefore:
differentiation yields:
)cos()(s
0
00
dtd
TUe(t)
30
Equation of motion
We get:
T2 und
0rev
E1
Th2
dtd Mit
20
2
)()(
)cos()(s
0
00
dtd
TUe(t) und
1E2
hUe(t) 22s00
2rev
)()cos(
Change of phase due to the change of energy
Change of energy when travelling with a different phase through a cavity
31
Solution of the equation of motion
The equation describes an harmonic oscillator:
with the synchrotron frequency:
0t(t) 2 )(
1E2
hUe22
s00rev )()cos(
The energy difference between the nominal particle and particles with different momentum is:
e(t) ti
0
0E-WE
32
Synchrotron frequency
1E2
hUe22
s00rev )()cos(
For ultra relativistic particles >> 1 :
edge) falling2
32
therefore negativ, be must
2 200
rev
()cos(
)cos(
ss
s
EhUe
For particles with:
012
012
) edge rising22
- therefore positiv, be must ()cos( ss
Synchrotron frequency
33
Example
35
Synchrotron frequency of the model accelerator
36
Phase space and Separatrix
From K.Wille
Synchrotron oscillations are for particles with small energy deviation. If the energy deviation becomes too large, particle leave the bucket.
37
Courtesy E. Ciapala
single turn
about 1000 turns
RF off, de-bunching in ~ 250 turns, roughly 25 ms
LHC 2008
38
Courtesy E. Ciapala
Attempt to capture, at exactly the wrong injection phase…
LHC 2008
39
Courtesy E. Ciapala
Capture with corrected injection phasing
LHC 2008
40
Courtesy E. Ciapala
Capture with optimum injection phasing, correct frequency
LHC 2008
41
RF buckets and bunches at LHC
E
time
RF Voltage
time
LHC bunch spacing = 25 ns = 10 buckets 7.5 m
2.5 ns
The particles are trapped in the RF voltage:this gives the bunch structure
RMS bunch length 11.2 cm 7.6 cmRMS energy spread 0.031% 0.011%
450 GeV 7 TeV
The particles oscillate back and forth in time/energy
RF bucket
2.5 ns
42
Longitudinal bunch profile in SPS
Instabilities at low energy (26 GeV)
a) Single bunchesQuadrupole mode developing slowly along flat bottom. NB injection plateau ~11 s
Bunch profile oscillations on the flat bottom – at 26 GeV
Bunch profile during a coast at 26 GeV
stable beam
Pictures provided by T.Linnecar
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