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Graduate Accelerator Physics Fall 2015
Accelerator Physics Particle Acceleration
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 5
Graduate Accelerator Physics Fall 2015
Bettor Phasor Diagram
Off crest, synchrotron phase
gi
cos
icVe
1 tancos
i
c gc
b
VV i e ii
bi
,g opti
tan sinc b sV i
cVs
s
Graduate Accelerator Physics Fall 2015
LCLS II
Subharmonic
Beam Loading
G. A. Krafft
Graduate Accelerator Physics Fall 2015
Subharmonic Beam Loading
• Under condition of constant incident RF power, there is a
voltage fluctuation in the fundamental accelerating mode when
the beam load is sub harmonically related to the cavity
frequency
• Have some old results, from the days when we investigated
FELs in the CEBAF accelerator
• These results can be used to quantify the voltage fluctuations
expected from the subharmonic beam load in LCLS II
Graduate Accelerator Physics Fall 2015
CEBAF FEL Results
Krafft and Laubach, CEBAF-TN-0153 (1989)
Bunch repetition
time τ1 chosen to
emphasize physics
Graduate Accelerator Physics Fall 2015
Model
• Single standing wave accelerating mode. Reflected power
absorbed by matched circulator.
• Beam current
• (Constant) Incident RF (β coupler coupling)
2 2 cosg cV ZP t
22
2
bc c c cc c
L c
d ZId V dV dVV
dt Q dt Q dt dt
b
l l
I t q t l I t l
Graduate Accelerator Physics Fall 2015
Analytic Method of Solution
• Green function
• Geometric series summation
• Excellent approximation
2ˆ ˆexp sin 1 1/ 42
c
c c c L
L
t tG t t t t Q
Q
/22cos cos
1c Lt n Q
c c L c
RP RV t t IQ e t
Q
/2
/ /22ˆ ˆcos cos cos 1
1 2
c L
c L c L
t n Q
Q Qcc c c c
RP Rq eV t t e t n e t n
Q D
Graduate Accelerator Physics Fall 2015
2
1
RP
L
RIQ
Q
cV
cV
Phasor Diagram of Solution
Graduate Accelerator Physics Fall 2015
Single Subharmonic Beam
0
0.2
0.4
0.6
0.8
1
1.2
0 200 400 600 800 1000 1200
Voltage Deviation 1
2
c
c
V
qR
Q
Graduate Accelerator Physics Fall 2015
Beam Cases
• Case 1
• Case 2
• Case 3
Beam Beam Pulse Rep. Rate Bunch Charge (pC) Average Current (µA)
HXR 1 MHz 145 145
Straight 10 kHz 145 1.45
SXR 1 MHz 145 145
Beam Beam Pulse Rep. Rate Bunch Charge (pC) Average Current (µA)
HXR 1 MHz 295 295
Straight 10 kHz 295 2.95
SXR 100 kHz 20 2
Beam Beam Pulse Rep. Rate Bunch Charge (pC) Average Current (µA)
HXR 100 kHz 295 295
Straight 10 kHz 295 2.95
SXR 100 kHz 20 2
Graduate Accelerator Physics Fall 2015
Case 1
Straight Current
HXR Current
SXR Current
Total Voltage cV t
t2 µsec
100 µsec
Graduate Accelerator Physics Fall 2015
Case 1
Volt
s
t/20 nsec
. . .
ΔE/E=10-4
Graduate Accelerator Physics Fall 2015
Case 2 Volt
s
t/20 nsec
. . . ΔE/E=10-4
Graduate Accelerator Physics Fall 2015
Case 3
t/100 nsec
Volt
s
ΔE/E=10-4
Graduate Accelerator Physics Fall 2015
Summaries of Beam Energies
• Case 1
• Case 2
• Case 3
Beam Minimum (kV) Maximum (kV) Form
HXR 0.690 1.814 Linear 10
Straight 1.574 1.574 Constant
SXR 0.753 1.876 Linear 10
Beam Minimum (kV) Maximum (kV) Form
HXR 0.631 1.943 Linear 100
Straight 1.550 1.550 Constant
SXR 0.788 1.911 Linear 10
Beam Minimum (kV) Maximum (kV) Form
HXR 0.615 1.222 Linear 100
Straight 0.908 0.908 Constant
SXR 0.618 1.225 Linear 100
For off-crest cavities, multiply by cos φ
Graduate Accelerator Physics Fall 2015
Summary
• Fluctuations in voltage from constant intensity subharmonic
beams can be computed analytically
• Basic character is a series of steps at bunch arrival, the step
magnitude being (R/Q)πfcq
• Energy offsets were evaluated for some potential operating
scenarios. Spread sheet provided that can be used to
investigate differing current choices
Graduate Accelerator Physics Fall 2015
Case I’
Zero Crossing
Left Deflection
Right Deflection
Total Voltage
cV t
t20 µsec
100 µsec
Graduate Accelerator Physics Fall 2015
Case 2 (100 kHz contribution minor)
Zero Crossing
Left Deflection
Right Deflection
Total Voltage
cV t
t2 µsec
100 µsec
Graduate Accelerator Physics Fall 2015
Case 3
Zero Crossing
Left Deflection
Right Deflection
Total Voltage cV t
t20 µsec
100 µsec
Graduate Accelerator Physics Fall 2015
Ring Stability and Tuning
For ring RF accelerating systems, the requirement of phase
stable operation introduces a lower limit on the coupling.
Even if the synchrotron motion is nominally phase stable,
there may be an additional requirement on the detuning
angle.
Because the impedance is different at the two synchrotron
sideband frequencies when not crested, an energy
difference between the two sidebands is deposited in the
fundamental mode
This energy difference can lead to instability
We’ll follow Wiedemann’s argument
Graduate Accelerator Physics Fall 2015
Robinson Stability Criterion
Phase stability if (modified Wiedemann 16.69)
Total induced voltage must satisfy (for stability)
Current limit
In terms of beam power
In terms of coupling
2s
sin sin cosc s brV V
1beam dissP P
0 cosL b cI R V
2opt
Graduate Accelerator Physics Fall 2015
Robinson Damping*
*From Wiedemann
Graduate Accelerator Physics Fall 2015
Synchrotron Sidebands
Current as phase changes
Phase motion from synchrotron oscillation
Linearized
02 cosh bI t I h t
0 sin st t
0 0 0
0 0 0 0
2 cos 2 sin sin
2 cos cos cos
h b b s
b b s s
I t I h t I h t t
I h t I h t h t
Wiedemann
16.77
0 1
Graduate Accelerator Physics Fall 2015
Induced Voltage
Voltage induced by a line
Total induced voltage
Linearized
0 0cos sinh h r h r hV t ZI Z I h t Z I h t
0 sin st t
0 0 0
0 0 0 0
2 cos 2 sin sin
2 cos cos cos
h b b s
b b s s
I t I h t I h t t
I h t I h t h t
Wiedemann
16.79
0 1
Graduate Accelerator Physics Fall 2015
Busch’s Theorem
For cylindrical symmetry magnetic field described by a vector potential:
Conservation of Canonical Momentum gives Busch’s Theorem:
1ˆ, , is nearly constant
0, 0,,
2 2
z
z z
r
A A z r B rA z rr r
B r z r B r z rA z r B
2
22
for particle with 0 where 0, 0
2 2
z
czLarmor
P mr qrA const
B P
qr Bmr
Beam rotates at the Larmor frequency which implies coupling
Graduate Accelerator Physics Fall 2015
Radial Equation
2 2
2
2 2
2 2
2 2 22
2
thin lens focal length
1 weak compared to quadrupole for high
4
L z L
L
z
z
z
dmr mr qr B mr
dt
kc
e B dz
f m c
If go to full ¼ oscillation inside the magnetic field in the “thick” lens case, all particles
end up at r = 0! Non-zero emittance spreads out perfect focusing!
x
y
Graduate Accelerator Physics Fall 2015
Larmor’s Theorem
This result is a special case of a more general result. If go to frame that rotates with the
local value of Larmor’s frequency, then the transverse dynamics including the
magnetic field are simply those of a harmonic oscillator with frequency equal to the
Larmor frequency. Any force from the magnetic field linear in the field strength is
“transformed away” in the Larmor frame. And the motion in the two transverse
degrees of freedom are now decoupled. Pf: The equations of motion are
2
2
2 2
2
2 2
2
2
2-D Harmonic Oscillator
z
L L z L z
L
dmr mr qr B
dt
mr qA cons P
dmr mr mr mr qr B qr B
dt
mr P
dmr mr mr
dt
mr P
Graduate Accelerator Physics Fall 2015
Transfer Matrix
For solenoid of length L, transfer matrix is
Decoupled matrix in rotating coordinate system (Eq. 17.34)
Matrix from Rotating Coordinates (Eq. 17.36, corrected)
1
sol end from rot dec to rot endM M M M M M
cos / / sin / 0 0
/ sin / cos / 0 0
0 0 cos / / sin /
0 0 / sin / cos /
L z z L L z
L z L z L z
dec
L z z L L z
L z L z L z
L v v L v
v L v L vM
L v v L v
v L v L v
cos / 0 sin / 0
/ sin / cos / / cos / sin /
sin / 0 cos / 0
/ cos / sin / / sin / cos /
L z L z
L z L z L z L z L z L z
from rot
L z L z
L z L z L z L z L z L z
L v L v
v L v L v v L v L vM
L v L v
v L v L v v L v L v
Graduate Accelerator Physics Fall 2015
To/from rotating coordinates
cos / sin /
sin / cos /
cos / sin /
sin / cos /
L z L z
L z L z
L LL z L z
z z
L LL z L z
z z
to rot
v z x z z v y z z v
w z x z z v y z z v
dv dx dyy z v x z v
dz dz v dz v
dw dx dyy z v x z v
dz dz v dz v
v z x z
v z xM
w z
w z
cos / 0 sin / 0
/ sin / cos / / cos / sin /
sin / 0 cos / 0
/ cos / sin / / sin / cos /
L z L z
L z L z L z L z L z L z
L z L z
L z L z L z L z L z L z
z v z v x z
z v z v z v v z v z v x z
y z z v z v y z
y z v z v z v v z v z v y z
x z
/ cos / sin /
/ sin / cos /
cos / 0 sin / 0
/ sin / cos / / cos /
L z L z L z
L z L z L z
L z L z
L z L z L z L z L
from rot
v y z v z v v z
y z v x z v z v w z
x z v z z v z v
x z v z v z v z v v z vM
y z w z
y z w z
sin /
sin / 0 cos / 0
/ cos / sin / / sin / cos /
z L z
L z L z
L z L z L z L z L z L z
v z
z v v z
z v z v w z
v z v z v v z v z v w z
Graduate Accelerator Physics Fall 2015
1
1 0 0 0 1 0 0 0
0 1 / 0 0 1 / 0
0 0 1 0 0 0 1 0
/ 0 0 1 / 0 0 1
z L z L
end end
z L z L
v vM M
v v
1
1 0 0 0
0 1 / 00
0 0 1 0
/ 0 0 1
L z
to rot end
L z
vM z M
v
Fringe effect by conservation cannonical momentum
Match to Boundary Conditions at z = 0
Graduate Accelerator Physics Fall 2015
Total Solenoid Transfer
2 2
2 2
2 2
2 2
cos 1/ sin 2 1/ 2 sin 2 2 / sin
/ 4 sin 2 cos / 2 sin 1/ 2 sin 2
1/ 2 sin 2 2 / sin cos 1/ sin 2
/ 2 sin 1/ 2 sin 2 / 4 sin 2 cos
sol
S S
S SM
S S
S S
Wiedemann 17.39 / 2 /L z L zL v S v
1 0 0 0
1/ 1 0 0
0 0 1 0
0 0 1/ 1
sol
sol
sol
fM
f
Thin Lens
Graduate Accelerator Physics Fall 2015
Easy Calculation that Works
cos 2sin / 0 0
cos 0 sin 0sin cos 0 0
0 cos 0 sin 2
sin 0 cos 0 0 0 cos 2sin /
0 sin 0 cos0 0 sin cos
2
sol
S
S
MS
S
Wiedemann points out the following simple calculation is OK
Works because
Explanation hard to follow
1
cos 0 sin 0
0 cos 0 sin
sin 0 cos 0
0 sin 0 cos
to rot end
end from rot
M M I
M M
Graduate Accelerator Physics Fall 2015
Skew Quadrupole
N N
S
S
Graduate Accelerator Physics Fall 2015
Equations
0
0
Bx y
B
By x
B
0
0
Bx y x y
B
Bx y x y
B
Focusing in x + y, defocusing in x - y
Graduate Accelerator Physics Fall 2015
Transfer Matrix
1cos sin 0 0
sin cos 0 0
10 0 cosh sinh
0 0 sinh coshafter
Bk
B
kL kLx y xk
x y k kL kL
x ykL kL
kx y
k kL kL
1cos sin 0 0
1 0 1 0 1 0 1 0
0 1 0 1 sin cos 0 0 0 1 0 11 1
1 0 1 0 1 1 0 1 02 20 0 cosh sinh
0 1 0 1 0 1 0 1
0 0 sinh cosh
before
skew
y
x y
x y
x y
kL kLk
k kL kLM
kL kLk
k kL kL
Graduate Accelerator Physics Fall 2015
In terms of the usual variables
/ /
1
2 / /
cos cosh sin sinh
after before
x xC S k C S k
x xkS C kS C
y yC S k C S k
y ykS C kS C
C kL kL S kL kL
,
1 0 0
0 1 1/ 0
0 0 1
1/ 0 0 1
skew thin
L
fM
L
f
Thin Lens