Post on 20-Jul-2020
Graduate Accelerator Physics Fall 2015
Accelerator Physics
Closed Orbits and Chromaticity
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 14
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’
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
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
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
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
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
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
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
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
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
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
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
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
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
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
.
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
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
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
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