accelerator physics particle acceleration
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Accelerator Physics Particle Acceleration. G. A. Krafft, Alex Bogacz and Timofey Zolkin Jefferson Lab Colorado State University Lecture 6. RF Acceleration. Characterizing Superconducting RF (SRF) Accelerating Structures Terminology Energy Gain, R/Q , Q 0 , Q L and Q ext - PowerPoint PPT PresentationTRANSCRIPT
USPAS Accelerator Physics June 2013
Accelerator PhysicsParticle Acceleration
G. A. Krafft, Alex Bogacz and Timofey Zolkin
Jefferson Lab
Colorado State University
Lecture 6
USPAS Accelerator Physics June 2013
RF Acceleration
• Characterizing Superconducting RF (SRF) Accelerating Structures
– Terminology
– Energy Gain, R/Q, Q0, QL and Qext
• RF Equations and Control
– Coupling Ports
– Beam Loading
• RF Focusing
• Betatron Damping and Anti-damping
USPAS Accelerator Physics June 2013
Terminology
1 CEBAF Cavity
1 DESY Cavity
“Cells”
5 Cell Cavity
9 Cell Cavity
USPAS Accelerator Physics June 2013
Typical cell longitudinal dimension: λRF/2Phase shift between cells: πCavities usually have, in addition to the resonant structure in picture:
(1) At least 1 input coupler to feed RF into the structure(2) Non-fundamental high order mode (HOM) damping(3) Small output coupler for RF feedback control
Modern Jefferson Lab Cavities (1.497 GHz) areoptimized around a 7 cell design
USPAS Accelerator Physics June 2013
USPAS Accelerator Physics June 2013
Some Fundamental Cavity Parameters
• Energy Gain
• For standing wave RF fields and velocity of light particles
• Normalize by the cavity length L for gradient
2
, vd mc
eE x t tdt
2, cos 0,0, cos /
2 / c.c. = 2 /
2
RF z RF
iz RF
c z RF
E x t E x t mc e E z z dz
eE eV eE
accE MV/m cV
L
USPAS Accelerator Physics June 2013
Shunt Impedance R/Q
• Ratio between the square of the maximum voltage delivered by a cavity and the product of ωRF and the energy stored in a cavity
• Depends only on the cavity geometry, independent of frequency when uniformly scale structure in 3D
• Piel’s rule: R/Q ~100 Ω/cell
CEBAF 5 Cell 480 Ω
CEBAF 7 Cell 760 Ω
DESY 9 Cell 1051 Ω
2
stored energyc
RF
VR
Q
USPAS Accelerator Physics June 2013
• As is usual in damped harmonic motion define a quality factor by
• Unloaded Quality Factor Q0 of a cavity
• Quantifies heat flow directly into cavity walls from AC resistance of superconductor, and wall heating from other sources.
2 energy stored in oscillation
energy dissipated in 1 cycleQ
0
stored energy
heating power in wallsRFQ
Unloaded Quality Factor
USPAS Accelerator Physics June 2013
Loaded Quality Factor• When add the input coupling port, must account for the energy
loss through the port on the oscillation
• Coupling Factor
• It’s the loaded quality factor that gives the effective resonance width that the RF system, and its controls, seen from the superconducting cavity
• Chosen to minimize operating RF power: current matching (CEBAF, FEL), rf control performance and microphonics (SNS, ERLs)
0
1 1 total power lost 1 1
stored energytot L RF extQ Q Q Q
0 01 for present day SRF cavities, 1L
ext
Q QQ
Q
USPAS Accelerator Physics June 2013
0 10 20 30 40Eacc [MV/m]
Q0
AC70AC72AC73AC78AC76AC71AC81Z83Z87
1011
109
1010
Q0 vs. Gradient for Several 1300 MHz Cavities
Courtesey: Lutz Lilje
USPAS Accelerator Physics June 2013
Eacc vs. time
0
5
10
15
20
25
30
35
40
45
Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jan-03 Jan-04 Jan-05 Jan-06
Eac
c[M
V/m
]
BCP
EP
10 per. Mov. Avg. (BCP)
10 per. Mov. Avg. (EP)
Courtesey: Lutz Lilje
USPAS Accelerator Physics June 2013
RF Cavity Equations Introduction Cavity Fundamental Parameters RF Cavity as a Parallel LCR Circuit Coupling of Cavity to an rf Generator Equivalent Circuit for a Cavity with Beam Loading
• On Crest and on Resonance Operation • Off Crest and off Resonance Operation
Optimum Tuning Optimum Coupling
RF cavity with Beam and Microphonics Qext Optimization under Beam Loading and Microphonics
RF Modeling Conclusions
USPAS Accelerator Physics June 2013
Introduction Goal: Ability to predict rf cavity’s steady-state response and
develop a differential equation for the transient response
We will construct an equivalent circuit and analyze it
We will write the quantities that characterize an rf cavity and relate them to the circuit parameters, for
a) a cavity
b) a cavity coupled to an rf generator
c) a cavity with beam
USPAS Accelerator Physics June 2013
RF Cavity Fundamental Quantities
cV
Quality Factor Q0:
Shunt impedance Ra:
(accelerator definition); Va = accelerating voltage
Note: Voltages and currents will be represented as complex quantities, denoted by a tilde. For example:
where is the magnitude of and is a slowly varying phase.
2
in ohms per cellaa
diss
VR
P
00
Energy stored in cavity
Energy dissipated in cavity walls per radiandiss
WQ
P
c cV V
Re i ti tc c c cV t V t e V t V e
USPAS Accelerator Physics June 2013
Equivalent Circuit for an rf Cavity
Simple LC circuit representing an accelerating resonator.
Metamorphosis of the LC circuit into an accelerating cavity.
Chain of weakly coupled pillbox cavities representing an accelerating cavity. Chain of coupled pendula as its mechanical analogue.
USPAS Accelerator Physics June 2013
Equivalent Circuit for an rf Cavity An rf cavity can be represented by a parallel LCR circuit:
Impedance Z of the equivalent circuit:
Resonant frequency of the circuit:
Stored energy W:
11 1
Z iCR iL
0 1/ LC
21
2 cW CV
( ) i tc cV t v e
USPAS Accelerator Physics June 2013
Equivalent Circuit for an rf Cavity Power dissipated in resistor R:
From definition of shunt impedance
Quality factor of resonator:
Note: For
21
2c
diss
VP
R
2a
adiss
VR
P 2aR R
00 0
diss
WQ CR
P
1
00
0
1Z R iQ
1
00 0
0
1 2Z R iQ
,
Wiedemann16.13
USPAS Accelerator Physics June 2013
Cavity with External Coupling Consider a cavity connected to an rf
source A coaxial cable carries power from an rf
source to the cavity The strength of the input coupler is
adjusted by changing the penetration of the center conductor
There is a fixed output coupler, the transmitted power probe, which picks up power transmitted through the cavity
USPAS Accelerator Physics June 2013
Cavity with External Coupling (cont’d)
Consider the rf cavity after the rf is turned off.Stored energy W satisfies the equation:
Total power being lost, Ptot, is:
Pe is the power leaking back out the input coupler. Pt is the power coming out the transmitted power coupler. Typically Pt is very small Ptot Pdiss + Pe
Recall
Similarly define a “loaded” quality factor QL:Energy in the cavity decays exponentially with time constant:
tot diss e tP P P P
0L
tot
WQ
P
tot
dWP
dt
00
diss
WQ
P
0
00
L
t
Q
L
dW WW W e
dt Q
0
LL
Q
USPAS Accelerator Physics June 2013
Cavity with External Coupling (cont’d)
Equation
suggests that we can assign a quality factor to each loss mechanism, such that
where, by definition,
Typical values for CEBAF 7-cell cavities: Q0=1x1010, Qe QL=2x107.
0 0
tot diss eP P P
W W
0
1 1 1
L eQ Q Q
0e
e
WQ
P
USPAS Accelerator Physics June 2013
Cavity with External Coupling (cont’d)
Define “coupling parameter”:
therefore
is equal to:
It tells us how strongly the couplers interact with the cavity. Large implies that the power leaking out of the coupler is large compared to the power dissipated in the cavity walls.
0
e
Q
Q
0
1 (1 )
LQ Q
e
diss
P
P
Wiedemann16.9
USPAS Accelerator Physics June 2013
( ) kki tI t i e
Cavity Coupled to an rf Source The system we want to model:
Between the rf generator and the cavity is an isolator – a circulator connected to a load. Circulator ensures that signals coming from the cavity are terminated in a matched load.
Equivalent circuit:
RF Generator + Circulator Coupler Cavity Coupling is represented by an ideal transformer of turn ratio 1:k
USPAS Accelerator Physics June 2013
Cavity Coupled to an rf Source
20
kg
g
II
k
Z k Z
( ) kki tI t i e
( )g gi tI t i e
By definition,
( )g gi tI t i e
20
gg
R R RZ
Z k Z
Wiedemann
16.1
WiedemannFig. 16.1
USPAS Accelerator Physics June 2013
Generator Power When the cavity is matched to the input circuit, the power
dissipation in the cavity is maximized.
We define the available generator power Pg at a given generator
current to be equal to Pdissmax .
gI
2
max max 21 1or
2 2 16g
diss g diss a g g
IP Z P R I P
( )g gi tI t i e
Wiedemann16.6
USPAS Accelerator Physics June 2013
Some Useful Expressions We derive expressions for W, Pdiss, Prefl, in terms of cavity
parameters 20 0
200 0
2 22 2 0
1
1
00
0
02
0 2 2 00
0
0
02
0
161 1
16 16
1 1
(1 )2
14
(1 )
For
4 1
(1 )1
cdiss
ca
g a ga g a g
c g TOT
TOTg
aTOT
g
Q Q VP
W Q VR
P R IR I R I
V I Z
ZZ Z
RZ iQ
QW P
Q
QW
2
0 0
0
2(1 )
gPQ
USPAS Accelerator Physics June 2013
Some Useful Expressions (cont’d)
Define “Tuning angle” :
Recall:
022
0 0 0
0
4 1
(1 )1 2
(1 )
g
QW P
Q
0 00
0 0
0g2 2
0
tan 2 for
Q4 1= P
(1+ ) 1+tan
L LQ Q
W
0
0
2 2
4 1
(1 ) 1 tan
diss
diss g
WP
Q
P P
Wiedemann16.12
USPAS Accelerator Physics June 2013
Some Useful Expressions (cont’d). Optimal coupling: W/Pg maximum or Pdiss = Pg
which implies Δω = 0, β = 1
this is the case of critical coupling
. Reflected power is calculated from energy conservation:
. On resonance:
Dissipated and Reflected Power
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10
02
0
2
2
4
(1 )
4
(1 )
1
1
g
diss g
refl g
QW P
P P
P P
2 2
4 11
(1 ) 1 tan
refl g diss
refl g
P P P
P P
USPAS Accelerator Physics June 2013
Equivalent Circuit for a Cavity with Beam Beam in the rf cavity is represented by a current generator.
Equivalent circuit:
Differential equation that describes the dynamics of the system:
RL is the loaded impedance defined as:
, i , / 2
C C LC R C
L
dv v dii C v L
dt R dt
(1 )a
L
RR
USPAS Accelerator Physics June 2013
Equivalent Circuit for a Cavity with Beam (cont’d) Kirchoff’s law:
Total current is a superposition of generator current and beam current and beam current opposes the generator current.
Assume that have a fast (rf) time-varying component and a slow varying component:
where is the generator angular frequency and are complex quantities.
L R C g bi i i i i
2
20 002 2
c c Lc g b
L L
d v dv R dv i i
dt Q dt Q dt
, ,c g bv i i
i tc c
i tg g
i tb b
v V e
i I e
i I e
, ,c g bV I I
USPAS Accelerator Physics June 2013
Equivalent Circuit for a Cavity with Beam (cont’d)
Neglecting terms of order we arrive at:
where is the tuning angle.
For short bunches: where I0 is the average beam
current.
2
2
1, ,c c
L
d V dI dV
dt dt Q dt
0 0(1 tan ) ( )2 4
c Lc g b
L L
dV Ri V I I
dt Q Q
0| | 2bI I
Wiedemann 16.19
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Equivalent Circuit for a Cavity with Beam (cont’d)
0 0(1 tan ) ( )2 4
c Lc g b
L L
dV Ri V I I
dt Q Q
At steady-state:
are the generator and beam-loading voltages on resonance and are the generator and beam-loading voltages.
/ 2 / 2
(1 tan ) (1 tan )
or cos cos2 2
or cos cos
or
L Lc g b
i iL Lc g b
i ic gr br
c g b
R RV I I
i i
R RV I e I e
V V e V e
V V V
2
2
Lgr g
Lbr b
RV I
RV I
g
b
V
V
USPAS Accelerator Physics June 2013
Equivalent Circuit for a Cavity with Beam (cont’d)
Note that:
0
2| | 2 for large
1
| |
gr g L g L
br L
V P R P R
V R I
Wiedemann16.16
Wiedemann16.20
USPAS Accelerator Physics June 2013
Equivalent Circuit for a Cavity with Beam
As increases the magnitude of both Vg and Vb decreases while their phases rotate by .
cos igrV e
grV
Im( )gV
Re( )gV
cos
cos
ig gr
ib br
V V e
V V e
USPAS Accelerator Physics June 2013
Equivalent Circuit for a Cavity with Beam (cont’d)
Cavity voltage is the superposition of the generator and beam-loading voltage.
This is the basis for the vector diagram analysis.
c g bV V V
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cV
bI
brVbV
gV
decI
accI
b
Example of a Phasor Diagram
WiedemannFig. 16.3
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On Crest and On Resonance Operation Typically linacs operate on resonance and on crest in order
to receive maximum acceleration. On crest and on resonance
where Va is the accelerating voltage.
grVcVbrV bI
a gr brV V V
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More Useful Equations
We derive expressions for W, Va, Pdiss, Prefl in terms of and the loading parameter K, defined by: K=I0/2 Ra/Pg
From:
0
2| |
1
| |
gr g L
br L
a gr br
V P R
V R I
V V V
0
2
20
2
2
0 0
0
2
0 2
21
1
41
(1 )
41
(1 )
22 1
1
( 1) 2
( 1)
a g a
g
diss g
a a diss
a
g
refl g diss a refl g
KV P R
Q KW P
KP P
I V I R P
I V KK
P
KP P P I V P P
USPAS Accelerator Physics June 2013
More Useful Equations (cont’d)
For large,
For Prefl=0 (condition for matching)
20
20
1( )
4
1( )
4
g a LL
refl a LL
P V I RR
P V I RR
0
2
0 0
0
and
4
Ma
L M
M Ma a
g M Ma
VR
I
I V V IP
V I
USPAS Accelerator Physics June 2013
Example For Va=20 MV/m, L=0.7 m, QL=2x107 , Q0=1x1010 :
Power
I0 = 0 I0 = 100 A I0 = 1 mA
Pg 3.65 kW 4.38 kW 14.033 kW
Pdiss 29 W 29 W 29 W
I0Va 0 W 1.4 kW 14 kW
Prefl 3.62 kW 2.951 kW ~ 4.4 W
USPAS Accelerator Physics June 2013
Off Crest and Off Resonance Operation Typically electron storage rings operate off crest in order
to ensure stability against phase oscillations. As a consequence, the rf cavities must be detuned off
resonance in order to minimize the reflected power and the required generator power.
Longitudinal gymnastics may also impose off crest operation operation in recirculating linacs.
We write the beam current and the cavity voltage as
The generator power can then be expressed as:
02
and set 0
b
c
ib
ic c c
I I e
V V e
2 220 0(1 )
1 cos tan sin4
c L Lg b b
L c c
V I R I RP
R V V
Wiedemann16.31
USPAS Accelerator Physics June 2013
Off Crest and Off Resonance Operation
Condition for optimum tuning:
Condition for optimum coupling:
Minimum generator power:
0tan sinLb
c
I R
V
0opt 1 cosa
bc
I R
V
2opt
,minc
ga
VP
R
Wiedemann
16.36
USPAS Accelerator Physics June 2013
RF Cavity with Beam and Microphonics
The detuning is now: 0 0
0 0
0
0tan 2 tan 2
where is the static detuning (controllable)
and is the random dynamic detuning (uncontrollable)
m
L L
m
f f fQ Q
f f
f
f
-10
-8-6
-4
-20
2
4
68
10
90 95 100 105 110 115 120
Time (sec)
Fre
quency (Hz)
Probability DensityMedium CM Prototype, Cavity #2, CW @ 6MV/m
400000 samples
0
0.05
0.1
0.15
0.2
0.25
-8 -6 -4 -2 0 2 4 6 8
Peak Frequency Deviation (V)
Pro
bab
ility
Den
sity
USPAS Accelerator Physics June 2013
Qext Optimization under Beam Loading and Microphonics
Beam loading and microphonics require careful optimization of the external Q of cavities.
Derive expressions for the optimum setting of cavity parameters when operating under
a) heavy beam loading
b) little or no beam loading, as is the case in energy recovery linac cavities
and in the presence of microphonics.
USPAS Accelerator Physics June 2013
Qext Optimization (cont’d)2 22 (1 )
1 cos tan sin4
c tot L tot Lg tot tot
L c c
V I R I RP
R V V
0
tan 2 L
fQ
f
where f is the total amount of cavity detuning in Hz, including static detuning and microphonics. Optimizing the generator power with respect to coupling gives:
where Itot is the magnitude of the resultant beam current vector in the cavity and tot is the phase of the resultant beam vector with respect to the cavity voltage.
2
20
0
( 1) 2 tan
where cos
opt tot
tot atot
c
fb Q b
f
I Rb
V
USPAS Accelerator Physics June 2013
Qext Optimization (cont’d)
where:
To minimize generator power with respect to tuning:
0
0
tan 2 mL
f fQ
f
2 22 (1 )1 cos tan sin
4c tot L tot L
g tot totL c c
V I R I RP
R V V
00
0
tan2
ff b
Q
222
00
(1 )(1 ) 2
4c m
gL
V fP b Q
R f
USPAS Accelerator Physics June 2013
Qext Optimization (cont’d) Condition for optimum coupling:
and In the absence of beam (b=0):
and
2
20
0
222
00
( 1) 2
1 ( 1) 22
mopt
opt c mg
a
fb Q
f
V fP b b Q
R f
2
00
22
00
1 2
1 1 22
mopt
opt c mg
a
fQ
f
V fP Q
R f
USPAS Accelerator Physics June 2013
Problem for the Reader
Assuming no microphonics, plot opt and Pgopt as function
of b (beam loading), b=-5 to 5, and explain the results.
How do the results change if microphonics is present?
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Example
ERL Injector and Linac:
fm=25 Hz, Q0=1x1010 , f0=1300 MHz, I0=100 mA, Vc=20 MV/m, L=1.04 m, Ra/Q0=1036 ohms per cavity
ERL linac: Resultant beam current, Itot = 0 mA (energy recovery)
and opt=385 QL=2.6x107 Pg = 4 kW per cavity.
ERL Injector: I0=100 mA and opt= 5x104 ! QL= 2x105 Pg = 2.08 MW per cavity!
Note: I0Va = 2.08 MW optimization is entirely dominated by beam loading.
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RF System Modeling
To include amplitude and phase feedback, nonlinear effects from the klystron and be able to analyze transient response of the system, response to large parameter variations or beam current fluctuations
• we developed a model of the cavity and low level controls using SIMULINK, a MATLAB-based program for simulating dynamic systems.
Model describes the beam-cavity interaction, includes a realistic representation of low level controls, klystron characteristics, microphonic noise, Lorentz force detuning and coupling and excitation of mechanical resonances
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RF System Model
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RF Modeling: Simulations vs. Experimental Data
Measured and simulated cavity voltage and amplified gradient error signal (GASK) in one of CEBAF’s cavities, when a 65 A, 100 sec beam pulse enters the cavity.
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Conclusions
We derived a differential equation that describes to a very good approximation the rf cavity and its interaction with beam.
We derived useful relations among cavity’s parameters and used phasor diagrams to analyze steady-state situations.
We presented formula for the optimization of Qext under beam loading and microphonics.
We showed an example of a Simulink model of the rf control system which can be useful when nonlinearities can not be ignored.
USPAS Accelerator Physics June 2013
RF FocussingIn any RF cavity that accelerates longitudinally, because of Maxwell Equations there must be additional transverse electromagnetic fields. These fields will act to focus the beam and must be accounted properly in the beam optics, especially in the low energy regions of the accelerator. We will discuss this problem in greater depth in injector lectures. Let A(x,y,z) be the vector potential describing the longitudinal mode (Lorenz gauge)
tcA
1
2
22
2
22
cA
cA
USPAS Accelerator Physics June 2013
...,
...,2
10
210
rzzzr
rzAzAzrA zzz
For cylindrically symmetrical accelerating mode, functional form can only depend on r and z
Maxwell’s Equations give recurrence formulas for succeeding approximations
12
2
21
22
1,2
2
2
1,2
2
2
2
nn
n
nznz
zn
cdz
dn
Acdz
AdAn
USPAS Accelerator Physics June 2013
Gauge condition satisfied when
nzn
c
i
dz
dA
in the particular case n = 0
00
c
i
dz
dAz
Electric field is
t
A
cE
�
1
USPAS Accelerator Physics June 2013
So the magnetic field off axis may be expressed directly in terms of the electric field on axis
00,0 zz A
c
i
dz
dzE
102
2
20
2
4,0 zzz
z AAcdz
AdzE
c
i
And the potential and vector potential must satisfy
zEc
rirAB zz ,0
2 2 1
USPAS Accelerator Physics June 2013
And likewise for the radial electric field (see also )
dz
zdErzrE z
r
,0
2 2 1
Explicitly, for the time dependence cos(ωt + δ)
tzEtzrE zz cos,0,,
tdz
zdErtzrE z
r cos,0
2,,
tzE
c
rtzrB z sin,0
2,,
0 E
USPAS Accelerator Physics June 2013
USPAS Accelerator Physics June 2013
USPAS Accelerator Physics June 2013
USPAS Accelerator Physics June 2013
Motion of a particle in this EM field
B
c
VEe
dt
md
V
''
'sin'2
''
'cos'
'
2
'
z
- z
dz
ztzGc
zxz
ztdz
zdGzx
zz
zz
xx
USPAS Accelerator Physics June 2013
The normalized gradient is
2
0,
mc
zeEzG z
and the other quantities are calculated with the integral equations
z
zzz dzzt
z
zGzz ''cos
'
'
z
dzztzGz ''cos'
z
zzz cz
dz
c
zzt
'
'lim 0
0
USPAS Accelerator Physics June 2013
These equations may be integrated numerically using the cylindrically symmetric CEBAF field model to form the Douglas model of the cavity focussing. In the high energy limit the expressions simplify.
z
a zz
x
z
a z
x
dzztzz
zGzxazax
dzzz
zzaxzx
''cos''
'
2
'
'''
''
2
USPAS Accelerator Physics June 2013
Transfer Matrix
E
EI
L
E
E
TG
G
21
4
21
For position-momentum transfer matrix
dzczzG
dzczzGI
/sinsin
/coscos
222
222
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Kick Generated by mis-alignment
E
EG
2
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Damping and AntidampingBy symmetry, if electron traverses the cavity exactly on axis, there is no transverse deflection of the particle, but there is an energy increase. By conservation of transverse momentum, there must be a decrease of the phase space area. For linacs NEVER use the word “adiabatic”
0
Vtransverse dt
md
x xz z
USPAS Accelerator Physics June 2013
Conservation law applied to angles
, 1
/ /
x y z
x x z x y y z y
zx x
z
zy y
z
zz z
zz z
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Phase space area transformation
zx x
z
zy y
z
dx d z dx dz z
dy d z dy dz z
Therefore, if the beam is accelerating, the phase space area after the cavity is less than that before the cavity and if the beam is decelerating the phase space area is greater than the area before the cavity. The determinate of the transformation carrying the phase space through the cavity has determinate equal to
Det zcavity
z
Mz z
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By concatenation of the transfer matrices of all the accelerating or decelerating cavities in the recirculated linac, and by the fact that the determinate of the product of two matrices is the product of the determinates, the phase space area at each location in the linac is
0
00
000
yz
zy
xz
zx
ddyzz
zddy
ddxzz
zddx
Same type of argument shows that things like orbit fluctuations are damped/amplified by acceleration/deceleration.
USPAS Accelerator Physics June 2013
Transfer Matrix Non-Unimodular
edeterminat daccumulateby normalize and
)before! (as part" unimodular" track theseparatelycan
detdetdet
!unimodular det
212
2
1
1
21
MPMPM
M
M
M
M
MMP
MPM
MMP
MMM
tot
tottot
tot
USPAS Accelerator Physics June 2013
ENERGY RECOVERY WORKSGradient modulator drive signal in a linac cavity measured without energy recovery (signal level around 2 V) and with energy recovery (signal level around 0).
GASK
-0.5
0
0.5
1
1.5
2
2.5
-1.00E-04 0.00E+00 1.00E-04 2.00E-04 3.00E-04 4.00E-04 5.00E-04
Time (s)
Vo
lta
ge
(V
)
Courtesy: Lia Merminga