Accelerator Physicscasa.jlab.org/publications/viewgraphs/ODU2015/L_5...Graduate Accelerator Physics Fall 2015 Beam Cases • Case 1 • Case 2 • Case 3 Beam Beam Pulse Rep. Rate

Graduate Accelerator Physics Fall 2015

Accelerator Physics Particle Acceleration

G. A. Krafft

Old Dominion University

Jefferson Lab

Lecture 5


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


LCLS II

Subharmonic

Beam Loading

G. A. Krafft


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


CEBAF FEL Results

Krafft and Laubach, CEBAF-TN-0153 (1989)

Bunch repetition

time τ1 chosen to

emphasize physics


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


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


2

1

RP

L

RIQ

Q

cV

cV

Phasor Diagram of Solution


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


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


HXR 1 MHz 295 295


SXR 100 kHz 20 2


HXR 100 kHz 295 295


SXR 100 kHz 20 2


Case 1

Straight Current

HXR Current

SXR Current

Total Voltage cV t

t2 µsec

100 µsec


Case 1

Volt

s

t/20 nsec

. . .

ΔE/E=10-4


Case 2 Volt

s

t/20 nsec

. . . ΔE/E=10-4


Case 3

t/100 nsec

Volt

s

ΔE/E=10-4


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


HXR 0.631 1.943 Linear 100


SXR 0.788 1.911 Linear 10


HXR 0.615 1.222 Linear 100


SXR 0.618 1.225 Linear 100

For off-crest cavities, multiply by cos φ


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


Case I’

Zero Crossing

Left Deflection

Right Deflection

Total Voltage

cV t

t20 µsec

100 µsec


Case 2 (100 kHz contribution minor)

Zero Crossing

Left Deflection

Right Deflection

Total Voltage

cV t

t2 µsec

100 µsec


Case 3

Zero Crossing

Left Deflection

Right Deflection

Total Voltage cV t

t20 µsec

100 µsec


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


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


Robinson Damping*

*From Wiedemann


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


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


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


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


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


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 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


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


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


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


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


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


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


Skew Quadrupole

N N

S

S


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


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


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

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Accelerator Physicscasa.jlab.org/publications/viewgraphs/ODU2015/L_5...Graduate Accelerator Physics Fall 2015 Beam Cases • Case 1 • Case 2 • Case 3 Beam Beam Pulse Rep. Rate