L_6_Acceleration_Rev2 - CASA

Report
Accelerator Physics
Particle 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
5 Cell Cavity
1 CEBAF Cavity
“Cells”
9 Cell Cavity
1 DESY Cavity
USPAS Accelerator Physics June 2013
Modern Jefferson Lab Cavities (1.497 GHz) are
optimized around a 7 cell design
Typical cell longitudinal dimension: λRF/2
Phase shift between cells: π
Cavities 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
USPAS Accelerator Physics June 2013
USPAS Accelerator Physics June 2013
Some Fundamental Cavity Parameters
• Energy Gain

d  mc2
  eE x t  , t  v


dt
• For standing wave RF fields and velocity of light particles



E  x , t   E  x  cos RF t       mc 2  e  Ez  0, 0, z  cos  z / RF    dz

=
eEz  2 / RF  e i  c.c.
Vc  eEz  2 / RF 
2
• Normalize by the cavity length L for gradient
Vc
E acc  MV/m  
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
Vc2
R

Q RF  stored energy 
• 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
USPAS Accelerator Physics June 2013
Unloaded Quality Factor
• As is usual in damped harmonic motion define a quality
factor by
Q
2  energy stored in oscillation 
energy dissipated in 1 cycle
• Unloaded Quality Factor Q0 of a cavity
Q0 
RF  stored energy 
heating power in walls
• Quantifies heat flow directly into cavity walls from AC
resistance of superconductor, and wall heating from
other sources.
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
1
1
total power lost
1
1




Qtot QL RF  stored energy  Qext Q0
• Coupling Factor
Q0
Q0

1 for present day SRF cavities, QL 
Qext
1 
• 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)
USPAS Accelerator Physics June 2013
Q0 vs. Gradient for Several 1300 MHz
Cavities
Q0
1011
1010
AC70
AC72
AC73
AC78
AC76
AC71
AC81
Z83
Z87
Courtesey: Lutz Lilje
109
0
10
20
Eacc [MV/m]
USPAS Accelerator Physics June 2013
30
40
Eacc vs. time
45
40
35
BCP
EP
10 per. Mov. Avg. (BCP)
10 per. Mov. Avg. (EP)
Eacc[MV/m]
30
25
20
15
10
5
Courtesey: Lutz Lilje
0
Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jan-03 Jan-04 Jan-05 Jan-06
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
 Quality Factor Q0:
Q0 
0W
Pdiss

Energy stored in cavity
Energy dissipated in cavity walls per radian
 Shunt impedance Ra:
Va2
Ra 
Pdiss

in ohms per cell
(accelerator definition); Va = accelerating voltage
Note: Voltages and currents will be represented as complex
quantities, denoted by a tilde. For example:

Vc  t   Re Vc  t  eit
where Vc  Vc
varying phase.

Vc  t   Vc ei  t 
is the magnitude of Vc
USPAS Accelerator Physics June 2013
and  is a slowly
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:
V c (t )  vc eit
 Impedance Z of the equivalent circuit: Z   1  1  iC 
 R iL

 Resonant frequency of the circuit:
1
W  CVc2
2
 Stored energy W:
USPAS Accelerator Physics June 2013
0  1/ LC
1
Equivalent Circuit for an rf Cavity
 Power dissipated in resistor R:
Pdiss
1 Vc2

2 R
 Ra  2 R
Va2
Ra 
Pdiss
 From definition of shunt impedance
Q0 


0W
Pdiss
 0CR
Quality factor of resonator:
Note:

   
Z  R 1  iQ0   0  
 0   

Wiedemann
16.13
1
For
  0 ,
USPAS Accelerator Physics June 2013

   0  
Z  R 1  2iQ0 


0



1
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.
dW
Stored energy W satisfies the equation: dt   Ptot
Total power being lost, Ptot, is: P  P  P  P
tot
diss
e
t
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
W
Q0 
Recall
0
Pdiss
t
 0
dW
0W

 W  W0e QL
dt
QL
W
Similarly define a “loaded” quality factor QL: QL  0
Ptot
Energy in the cavity decays exponentially with time constant:
Q
L  L
0
USPAS Accelerator Physics June 2013
Cavity with External Coupling (cont’d)
Equation
Ptot
Pdiss  Pe

0W
0W
suggests that we can assign a quality factor to each loss mechanism, such that
1
1
1


QL
Q0 Qe
where, by definition,
Qe 
0W
Pe
Typical values for CEBAF 7-cell cavities: Q0=1x1010, Qe QL=2x107.
USPAS Accelerator Physics June 2013
Cavity with External Coupling (cont’d)
 Define “coupling parameter”:

therefore
  is equal to:
Q0
Qe
1 (1   )

QL
Q0

Wiedemann
16.9
Pe
Pdiss
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.
USPAS Accelerator Physics June 2013
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:
I k (t )  i k eit
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
I k (t )  i k eit

Wiedemann
Fig. 16.1
I g (t )  ig eit
By definition,
Ik
Ig 
k
Z g  k 2Z0

R
R
R
 2
 Zg 
Z g k Z0

USPAS Accelerator Physics June 2013
Wiedemann
16.1
Generator Power
 When the cavity is matched to the input circuit, the power
dissipation in the cavity is maximized.
I g (t )  ig eit
max
diss
P
 Ig 
1
 Zg  
2
 2 
2
max
diss
or P
1

Ra I g2  Pg
16 
Wiedemann
16.6
 We define the available generator power Pg at a given
generator current I g to be equal to Pdissmax .
USPAS Accelerator Physics June 2013
Some Useful Expressions
 We derive expressions for W, Pdiss, Prefl, in terms of cavity
parameters
Q
Q V
0
W

Pg
0
Pdiss
1
Ra I g2
16 
Vc  I g ZTOT
ZTOT
0

 1
1

 
Z 
 Z g
2
c
 0 Ra
1
Ra I g2
16 
16  Q0 Vc2

Ra2  0 I g2
1
1


0 
ZTOT
(1


)

iQ



0


0



Q
1
 W  4 0
Pg
2
0

 
(1   ) 2  Q02 
 0
 
 0
For   0 
R
 a
2
W
4
Q0
2
(1   )  0
1

Q0    0 
1  2

 (1   )  0 
USPAS Accelerator Physics June 2013
2
Pg
Some Useful Expressions (cont’d)
W
4
Q0
2
(1   ) 0
1

Q0   0 
1  2

 (1   ) 0 
2
Pg
 Define “Tuning angle” :
  
  0
tan   QL   0   2QL
0
 0  

W=
4 Q0
1
Pg
2
2
(1+ ) 0 1+tan 
 Recall:
Pdiss 
0W
Q0

Pdiss 
4
1
Pg
(1   ) 2 1  tan 2 
USPAS Accelerator Physics June 2013
for   0
Wiedemann
16.12
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:
Prefl  Pg  Pdiss


4
1
Prefl  Pg 1 

2
2
(1


)
1

tan



.
1
0.9
0.8
On resonance:
4
Q0
W 
Pg
(1   ) 2  0
Pdiss
Dissipated and Reflected Power
4

Pg
2
(1   )
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2
 1  
Prefl  
 Pg
 1  
0
1
USPAS Accelerator Physics June 2013
2
3
4
5

6
7
8
9
10
Equivalent Circuit for a Cavity with
 Beam in the rf cavity isBeam
represented by a current generator.
 Equivalent circuit:
iC  C
dvC
,
dt
iR 
vC
,
RL / 2
vC  L
diL
dt
 Differential equation that describes the dynamics of the system:
 RL is the loaded impedance defined as:
USPAS Accelerator Physics June 2013
RL 
Ra
(1   )
Equivalent Circuit for a Cavity with
Beam (cont’d)
 Kirchoff’s law:
iL  iR  iC  ig  ib
 Total current is a superposition of generator current and beam
current and beam current opposes the generator current.
d 2vc 0 dvc
0 RL d
2

 0 vc 
ig  ib 

2
dt
QL dt
2QL dt
 Assume that vc , ig , ib
have a fast (rf) time-varying
component and a slow varying component:
vc  Vc eit
ig  I g eit
ib  I b eit
where  is the generator angular frequency and
are complex quantities.
USPAS Accelerator Physics June 2013
Vc , I g , Ib
Equivalent Circuit for a Cavity with
Beam (cont’d)
 Neglecting terms of order
arrive at:
d 2Vc dI 1 dVc
,
,
2
dt
dt QL dt
we
dVc 0
0 RL

(1  i tan )Vc 
( I g  Ib )
dt 2QL
4QL
where  is the tuning angle.
 For short bunches:
beam current.
| Ib | 2I 0
where I0 is the average
Wiedemann 16.19
USPAS Accelerator Physics June 2013
Equivalent Circuit for a Cavity with
Beam (cont’d)
dVc 0
R

(1  i tan )Vc  0 L ( I g  Ib )
dt 2QL
4QL
 At steady-state:
or
RL / 2
RL / 2
Ig 
Ib
(1  i tan  )
(1  i tan  )
R
R
Vc  L I g cos ei  L I b cos ei
2
2
Vc 
or
Vc  Vgr cos ei  Vbr cos ei
or
Vc 
RL


V

I
g


 gr

2


R
L
V  
Ib 
br



2

Vg

Vb
are the generator and beam-loading
voltages on resonance
and
Vg

Vb



are the generator and beam-loading voltages.
USPAS Accelerator Physics June 2013
Equivalent Circuit for a Cavity with
Beam (cont’d)
 Note that:
| Vgr |
2 
1 
Pg RL  2 Pg RL for large 
| Vbr | RL I 0
Wiedemann
16.16
Wiedemann
16.20
USPAS Accelerator Physics June 2013
Equivalent Circuit for a Cavity with
Im(VBeam
)
g
Vg  Vgr cos e
i
Vgr cos ei
Vb  Vbr cos ei
Re(Vg )
Vgr

As  increases the magnitude of both Vg and Vb
decreases while their phases rotate by .
USPAS Accelerator Physics June 2013
Equivalent Circuit for a Cavity with
Beam (cont’d)
Vc  Vg  Vb
 Cavity voltage is the superposition of the generator and
beam-loading voltage.
 This is the basis for the vector diagram analysis.
USPAS Accelerator Physics June 2013
Example of a Phasor Diagram

Vb
Vbr
Vg
b
I acc
Vc
Ib
I dec
Wiedemann
Fig. 16.3
USPAS Accelerator Physics June 2013
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
Ib
Vbr

Vc
Va  Vgr  Vbr
where Va is the accelerating voltage.
USPAS Accelerator Physics June 2013
Vgr
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
 2 
Va  Pg Ra 
 1  
From:
| Vgr |
2 
1 
| Vbr | RL I 0
Va  Vgr  Vbr

K  
1 



 
2
4  Q0 
K 
W
1


 Pg
(1   ) 2  0 

Pg RL
2

Pdiss
4 
K 

1 
 Pg
(1   ) 2 

I 0Va  I 0 Ra Pdiss

2 
I 0Va
K 


2 K 1 

Pg
1 



2
Prefl  Pg  Pdiss  I 0Va  Prefl
USPAS Accelerator Physics June 2013
(   1)  2 K  
 P

g
2
(   1)
More Useful Equations (cont’d)
 For  large,
Pg
1
(Va  I 0 RL ) 2
4 RL
Prefl
1
(Va  I 0 RL ) 2
4 RL
 For Prefl=0 (condition for matching) 
VaM
RL  M
I0
and
Pg
M
0
M
a
I V
4
 Va
I0 

 M
M 
V
I
0 
 a
USPAS Accelerator Physics June 2013
2
Example
 For Va=20 MV/m, L=0.7 m, QL=2x107 , Q0=1x1010 :
I0 = 0
I0 = 100 A
I0 = 1 mA
3.65 kW
4.38 kW
14.033 kW
Pdiss
29 W
29 W
29 W
I0Va
0W
1.4 kW
14 kW
Prefl
3.62 kW
2.951 kW
~ 4.4 W
Power
Pg
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
Ib  2 I 0ei b
Vc  Vcei c

and set  c  0
The generator power can then be expressed as:
2
2
 
 
Vc2 (1   ) 
I 0 RL
 I 0 RL

Pg 
1

cos


tan


sin


b
b 

RL 4  
Vc
Vc


 


USPAS Accelerator Physics June 2013
Wiedemann
16.31
Off Crest and Off Resonance Operation
 Condition for optimum tuning:
tan  
I 0 RL
sin b
Vc
 Condition for optimum coupling:
opt  1 
I 0 Ra
cos b
Vc
 Minimum generator power:
Pg ,min 
Vc2 opt
Ra
USPAS Accelerator Physics June 2013
Wiedemann
16.36
RF Cavity with Beam and Microphonics
The detuning is now: tan   2QL
 f0   f m
tan  0  2QL
f0
 f0
f0
where  f0 is the static detuning (controllable)
Probability Density
Medium  CM Prototype, Cavity #2, CW @ 6MV/m
400000 samples
10
8
6
4
0.25
2
0
-2
-4
-6
-8
-10
90
95
100
105
Time (sec)
110
115
120
Probability Density
Frequency (Hz)
and  f m is the random dynamic detuning (uncontrollable)
0.2
0.15
0.1
0.05
0
-8
-6
-4
-2
0
2
Peak Frequency Deviation (V)
USPAS Accelerator Physics June 2013
4
6
8
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
2

 
 
V (1   )  Itot RL
I tot RL

Pg 
1

cos


tan


sin


tot 
tot  

RL 4  
Vc
Vc
 
 


2
c
tan   2QL
f
f0
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:


f
2
 opt  (b  1)   2Q0
 b tan tot 
f
0


I R
where b  tot a cos tot
Vc
2
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.
USPAS Accelerator Physics June 2013
Qext Optimization (cont’d)
2
2

 
 
V (1   )  Itot RL
I tot RL

Pg 
cos tot    tan  
sin tot  
1 
RL 4  
Vc
Vc
 
 


2
c
tan   2QL
where:
 f0   fm
f0
 To minimize generator power with respect to tuning:
 f0  

f0
b tan 
2Q0
2



Vc2 (1   ) 

f


2
m
Pg 
(1  b   )  2Q0
 
RL 4  
f
0  



USPAS Accelerator Physics June 2013
Qext Optimization (cont’d)
 Condition for optimum coupling:
2
 opt

 fm 
2
 (b  1)   2Q0

f
0 

Pgopt
2



Vc2 

f

b  1  (b  1) 2   2Q0 m  
2 Ra 
f0  



and
 In the absence of beam (b=0):
and
2
 opt

 fm 
 1   2Q0

f
0 

Pgopt
2


V 
f 

1  1   2Q0 m  
2 Ra 
f0  



2
c
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?
USPAS Accelerator Physics June 2013
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.
USPAS Accelerator Physics June 2013
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
USPAS Accelerator Physics June 2013
RF System Model
USPAS Accelerator Physics June 2013
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.
USPAS Accelerator Physics June 2013
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 Focussing
In 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)

1 
 A  
c t
2 
2



2
2
 A 2 A
   2 
c
c
USPAS Accelerator Physics June 2013
For cylindrically symmetrical accelerating mode, functional form
can only depend on r and z
Az r , z   Az 0 z   Az1 z r 2  ...
 r , z   0 z   1 z r 2  ...
Maxwell’s Equations give recurrence formulas for succeeding
approximations
2
2n 
2
Azn 
d Az ,n 1
2


2
2
Az ,n 1
dz
c
2
2
d


2
2n  n  n21   2 n1
dz
c
USPAS Accelerator Physics June 2013
Gauge condition satisfied when
dAzn
i
  n
dz
c
in the particular case n = 0
dAz 0
i
  0
dz
c
Electric field is



1 A
E   
c t
USPAS Accelerator Physics June 2013
And the potential and vector potential must satisfy
d 0 i 
E z 0, z   

Az 0
dz
c
d Az 0 
i

Ez 0, z  
 2 Az 0  4 Az1
2
c
dz
c
2
2
So the magnetic field off axis may be expressed directly in terms
of the electric field on axis
i r
 B  2rAz1 
E z 0, z 
2 c
USPAS Accelerator Physics June 2013

And likewise for the radial electric field (see also   E  0)
r dE z 0, z 
 Er  2r1 z   
2
dz
Explicitly, for the time dependence cos(ωt + δ)
Ez r, z, t   Ez 0, z cost   
r dE z 0, z 
Er r , z, t   
cos t   
2
dz
B r , z , t   
r
2c
E z 0, z sin t   
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




  V 
d mV
 e E   B 
dt
c


 z  x z       x   
xz ' dGz '



cost z '    
z 
dz'
2
dz'
 

z z 'xz '
 z  z '

- 

G z 'sin t z '   


2c
USPAS Accelerator Physics June 2013
The normalized gradient is
eE z  z ,0 
G z  
2
mc
and the other quantities are calculated with the integral equations
z
 z        Gz'cost z'   dz'

Gz'
 z  z z      z    
cost z'   dz'
 z'
 z
z
z
z0
dz'
t z   lim

z0     c
 z'c
z
 z
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 ' x z '
xz   xa   
dz'
 z ' z z '
a
z
 x   
x  z ' G  z '
z  a   
 xa  
cost z '   dz'
2
 z   
2  z ' z z '
a
z
USPAS Accelerator Physics June 2013
Transfer Matrix
For position-momentum transfer matrix
L 
 EG
1 

2E
 

T
EG 
I

  4 1  2 E 



I  cos2    G 2 z  cos2 z / c dz


 sin 2    G 2 z sin 2 z / c dz

USPAS Accelerator Physics June 2013
Kick Generated by mis-alignment

E G
 
2E
USPAS Accelerator Physics June 2013
Damping and Antidamping
By 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”



d mVtransverse
0
dt
  z  x  z      x  
USPAS Accelerator Physics June 2013
Conservation law applied to angles
 x ,  y   z  1
x  x / z
x  y   y / z
y
     z   
x  z  
 x   
  z  z  z 
     z   
y  z 
 y   
  z  z  z 
USPAS Accelerator Physics June 2013
Phase space area transformation
     z   
dx  d x   
dx  d x  z  
  z  z  z 
     z   
dy  d y   
dy  d y  z  
  z  z  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
     z   
Det  M cavity  
  z  z  z 
USPAS Accelerator Physics June 2013
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 z 0
dx  d x z  
dx  d x 0
 z  z z 
 0 z 0
dy  d y z  
dy  d y 0
 z  z z 
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
M tot  M 1  M 2
M
PM  
det M
PM 
unimodular!
M tot
M1
M2
PM tot  

 PM 1   PM 2 
det M tot det M 1 det M 2
 can separatelytrack the" unimodular part"(as before!)
and normalizeby accumulated determinate
USPAS Accelerator Physics June 2013
ENERGY RECOVERY WORKS
Gradient 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
2.5
2
Voltage (V)
1.5
1
0.5
0
-1.00E-04
0.00E+00
1.00E-04
2.00E-04
3.00E-04
4.00E-04
5.00E-04
-0.5
Time (s)
Courtesy: Lia Merminga
USPAS Accelerator Physics June 2013

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