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Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University Jefferson Lab Lecture 14

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Page 1: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

Accelerator Physics

Closed Orbits and Chromaticity

G. A. Krafft

Old Dominion University

Jefferson Lab

Lecture 14

Page 2: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

Dipole Error

22

20

0

2

Kick at every turn. Solve a toy model:

exp / 2 sin

1 1/ 4 exp / 2

Geometric series summed

exp

BB

i

ih B B

i

B B B

ih B

kB d dk x s s s iL

B ds Q ds

x s k s s iL Q k s s iL

k k Q q k s s Q

x s k s

2

sin 1 cos cos sin/ 2

1 2 cos

cos / 2

2sin / 2

integer resonance

blows up when

B B B B

B

B B

ih

B B

B

k s s q k L k s s q k Ls Q

q k L q

k s s k Lx s

k k L

k L n

s’

Page 3: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

Closed Orbit Distortion

Perform summation over all kick sources

cos / 2

2sin / 2

bound oscillation generated by error

Source (dipole powering, quad displacement, etc.)

Oscillation can be observed a cond rrecte

B i Bico

i B B

k s s k Lx s

k k L

, 12

Using the real betatron motion

sin

the proper result is

cos

2s

d

in

s s

i

co i i

i

M s s s s

s sx s z

Page 4: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

Beta Measurement

If BPM close to steerer (there is little phase

advance between them), and the tune has been

measured, induce a closed orbit distortion to

measure the

cot2

i ico bpmx s

Page 5: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

Dipole Error Distribution

2

1 2

2

22

2

2

2

coscos

2sin 2sin

Angular stuff averages assuming independence of error distributions

/ 2

8sin

8sin

co i i i j i j

i j

ji

co i ii

u

u s s s s s s

s ss sds ds

su s

s N

2

For quad displacements replace

/x f

Page 6: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

Closed Orbit Correction

Suppose orbit does not go through center of

all BPMs. What do you do? (At CEBAF just

steer to BPM centers!)

Trim magnets added whose purpose is to bring CO

as close to zero as possible.

cosco i i

su s s

2sin

At BPM closed orbit reads

cos

2sin

Measure response matrix as trim magnets (index ) varied

cos

2sin

i

i

j i

j co j j i i

i

j i

j j k k jk k

i

s

j

s su u s s

k

s su s R

Page 7: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

Correction Algorithm

1

22

1

1

Desire . If have enough trims simply update

More sophisticated when less trims than BPMs, minimize

, ,

analogous to "least squares fitting" and gen

BPM

trims

j j

k kj j

N

BPM N

i

u u

R u

x u R

2

3/2

2 2

erally uses the same

types of computer algorithms, including Singular Value Decomposition

(SVD).

How many BPMs/trims?

Fourier Analyzing closed orbit equation

il sill

co l

l

Feu s s F s s e

l

Need enough to resolve the betatron orbit and distribute

uniformly in betatron phase

sds

Page 8: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

Quadrupole Field Errors

0

0 0

2

0

0

0

0

0

0

0

Error at location ; total strength 1/ focusing

cos sin sin1 0 1 0

11/ 2 1 1/ 2 1sin cos sin

cos sin sin 12

2 cos sin sin2

js u f Kdz

s s

M sf fs

s

ss mess

f

smess s

f

0

0 0

cos cos sin2

4 4

Add to get total

s

f

s sK dz

f f

Page 9: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

More Generally

0

0

22 2 2

0 02

2

0 0

2 22 2

0 0 02 2

00

0

Introduce normal betatron coordinates

, ,0 2

Fourier expand rhs

1 1

2 2

1

2 4

x K s x x K s x Kx

x s dsw s

s s

d ww Kw

d

F Kd Kds

d w d wF w w

d d

FKds

Metho

as a

d to

bove

Note measu : re

Page 10: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

Orbit Perturbation

1 2 2 1

1 2

Use Lagrange method of variation of parameters. If have

A solution to the inhomogeneous equation is

,

,

where and solve the homogeneous equation

with Wr

z

P z K z P z p z

P z p z G z z dz

G z z P z P z P z P z

P P

1 2 2 1

22 2 2

0 0 1 0 2 02

2

0 0

onskian 1

for normalized equation

, sin , cos

sin

z z

P z P z p z P z dz P z p z P z dz

d ww Kw P P

d

P K d

Page 11: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

Specific Case

2

0 0 0 0

0

define location 0 and suppose unperturbed orbit

displaced there with displacement

cos cos sin

Also must equal

cos

Evaluate the total tune shift by

perturbed

perturbed

a

w a a K d

w a

2

2

0 0 0 0 0

0

2

2

0 0 0 0

0

2

2

0 0 0 0 0 0

0

going around 1 turn

cos2 cos 2 cos sin 2

sin 2 2 cos sin 2

cos sin 2 cos cos 2 sin

1 1

4 4

K d

K d

K d

z K z dz

0

0

sin 2sin 2

now must avoid 1/2 integers!

z K z z dz

Page 12: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

Stop Bands

2

0 0 0 0

0

If error too large cannot solve for .

Indicates breakdown of approximation and next level needed

cos cos sin

is inserted in the equation for the perturbation

cos

perturbedw a a K d

2

0 0 0 0 0

0

2

2 2 2

0 0 0 0

0 0

2

201 0 0 0

0

2 cos 2 cos sin

cos sin sin

sin 2 sin 22

Second term oscillatory and tends to average t

K d

K K d d

I F K d

200

1 200

o zero

for 2 for 1/ 22

nFI

nF

Page 13: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

0 00 0

2 2

0 0 0 0

222 20

2

0 0

2 22 2

222 20

2,

0 0

22 2

1

2

16

Integer Resonance

16

i ii i

n

ini i

d d d d

I k k

e e e e d d

I k k

e e e

2

222 20

2, 1/2

0 0

2 1/2 2 1/22 2

1/ 2 Integer Resonance

16

in

n

i n i ni i

e d d

I k k

e e e e d d

Page 14: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

2 222

* 2 20

2

0 0

22 2

2, 2 0

22 2

2, 2 1 0

22 22 2 2

0 0 2 0

22 22 2 2

0 2 0

0 2

2

2

48

48

cos 2 1 2 2 48

2 2 48

/ 2 / 4

1 1

2 2

ij ij

j j j

n n

n n

n

n

n

i n

n

F F F K e d K e d

I F F

I F F

F F F

F F F

F F

F z K z e

2

2

similarly

1 1

2 2

z

i n z

n

dz

F z K z e dz

Page 15: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

Chromaticity

0

2 2 2

2 2 2

2 2

Defined by

2 /

Thin lens FODO system

1 0 1 1 0 1 1 0

1/ 2 1 0 1 1/ 1 0 1 1/ 2 1

1 / 2 /

/ / 2 1 /

for one period is

cos 1 /

Suppose particle has a mom

p p

L L

f f f

L f L L f

L L f f L f

L f

0

00 0 0

0

entum error /

/ 1 /1 /

p p

ff p p f p p

p p

Page 16: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

Tune Shift

22

0 02

0

2

0 0 0 2

0 0

0 0

0 0 0

0

0

cos 1 1 /2

cos sin sin cos

2 1 cossin 2 tan

sin 2

tan / 2

2 /

This is per period. Total ring chromaticity "proportional"

to the number of periods

Lp p

f

L p

f p

p p

p p

p p

.

Page 17: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

More Sophisticated

2 2

0

0

0

2 2

0 0

0 0

/2

/

/

expand to second order

/ /2

/ /

Final terms give geometric aberations; ig

x

x

x

mx kx kx p p x y

y ky ky p p mxy

x x D p p y y

mx kx kx p p mx D p p x y

y ky ky p p mD p p y mx y

nor for now

tune change

2

2

x x x

y y x

k

k mD dz

k mD dz

Page 18: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

General Formula for Chromaticity

2 2

1

4

1

4

use sextupoles to zero out

for no sextupoles

1

4

1

4

works for thin lens

/ / 1 / / 1

/ 1 / 1

1 1

/

x x x

y y x

x x

y y

k mD dz

k mD dz

kdz

kdz

f L f L f L f LL L

f L f L

kdzf f L L

Page 19: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

0

2

1

4

1

4 4

/1 1tan / 2

2 2/ 1

x xkdz

k dz k dz kdz

f L

f L

Page 20: Accelerator Physics Closed Orbits and Chromaticity · Graduate Accelerator Physics Fall 2015 Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University

Graduate Accelerator Physics Fall 2015

Chromaticity Correction

0

0

0 1 1 1 2 2 2

0 1 1 1 2 2 2

0 2 0 2

1

1 1 2 2 1

0 1 0 1

2

2

1

4

1

4

Thin sextupoles

10

4

10

4

sextupole strength

4

4

x x x x

y y y x

x x x x x x sext

y y x y x y sext

x y x x

s

x x y x y

x y y x

s

x

mD dz

mD dz

m D m D l

m D m D l

m l

m l

1 2 2 1x y x y