searching for cesrta guide field nonlinearities in beam position spectra laurel hales mike billing...
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![Page 1: Searching for CesrTA guide field nonlinearities in beam position spectra Laurel Hales Mike Billing Mark Palmer](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d485503460f94a22df5/html5/thumbnails/1.jpg)
Searching for CesrTA guide field nonlinearities in beam
position spectra
Laurel HalesMike Billing
Mark Palmer
![Page 2: Searching for CesrTA guide field nonlinearities in beam position spectra Laurel Hales Mike Billing Mark Palmer](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d485503460f94a22df5/html5/thumbnails/2.jpg)
Goals
• Learn how to find and correct non-linear errors. • Correcting these errors will allow us to
– Withstand large amplitude oscillations without losing particles.
– Get a small vertical bunch size and avoid bunch shape distortions.
• We have two possible methods for finding non-linear errors. – Our first goal is to test these two methods using
simulations.– Then we can test them using the accelerator.
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The optics• Dipoles - Bend the beam.
• Quadrupoles - Focus the beam.
• Sextupoles - Compensate for the energy depended focusing due to the quadrupoles
• Errors in the optics can lead to:– Losing particles– Bunch shape distortions
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What is a BPM?
• Beam Position Monitor inside the beam-pipe.
• There are about 100 BPMs around CESR.
• The BPM can give you an x position and a y position for the beam
One BPM vs. Time
4 electrodes on the walls of the beam-pipe
Beam-pipe
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MIA
• Drive beam with a sinusoidal shaker
• Take position data: 100 BPMs ~ 1000 turns
• Create a matrix P= [position x history]
• Using Singular Value Decomposition to get: TP
Columns = spatial function around ring
(Diagonals) = Eigen values (λi) ~ amplitudes of the eigen components
Columns = time development of beam trajectory
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Our simulation
• Our simulation uses tracking codes from BMAD.
• In our simulation we give the particle bunch an initial amplitude and then track it as it circles freely.
• There is no damping.
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Sextupoles
• Sextupoles have a non-linear restoring force:
which can be solved for:
when we solve the above equation that gives us different multiples of ω because:
22
22
2
xkxdt
xd
tmBtxtxm
m coscos2
00
2
2cos1cos2
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1st Method
• The height of the different harmonics should be dependent on the driving amplitude (A).
22h :f2 A
33h :3f A
A1h :f
Τau matrix column
One of the principle components
Higher spectral component
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Results for Method 1
Change in oscillation magnitude for vertically driven simulation
0.0001
0.001
0.01
0.1
11 10 100
Initial displacement (mm)
Fv
2fv
3fv
4fv
Change in magnitude for horizontally driven simulation
1.00E-005
1.00E-004
1.00E-003
1.00E-002
1.00E-001
1.00E+000
1 10
Initial displacement (mm)
Ma
gn
itud
e/m
ax
ma
gn
itud
e
fh
2fh
3fh
4fh
The expected power law dependence is clearly shown in the vertically driven simulation.
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Machine data (horizontally driven)
Change in magnitude for horizonatally driven sample in unchanged lattice
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
0.01 0.1 1
Square root of the driving amplitude (au)
Mag
nitu
de (
au)
f h
2fh
3fh
4fh
Change in magnitude for horizontally driven sample in alternate lattice
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
0.01 0.1 1
Square root of the driving amplitude (au)
Ma
gn
itud
e (
au
)
fh
2fh
The horizontally driven data shows the power law relation between driving amplitude and the magnitude of the harmonic signals The line represents a linear dependence
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Change in magnitude for vertically driven sample in unchanged lattice
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
0.01 0.1 1
Square root of the driving amplitude (au)
Mag
nitu
de (
au)
fv
2fv
Machine data (vertically driven)
Change in magnitude for vertically drivien sample in alternate lattice
0.00001
0.0001
0.001
0.01
0.1
10.01 0.1 1
Square root of the driving amplitude (au)
Mag
nitu
de (
au)
fv
2fv
The vertically driven data also displays the power law relation. The line represents a linear dependence
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What are β and Φ?
• β(s) is the amplitude function. – β modulates the
amplitude of the oscillation of the particle beam
– The envelope of oscillation is defined as , where J is the Action of the beam.
• Φ defines the phase of the oscillation.– The phase increases
monotonically but not uniformly
• The Φ and β of the ring will change when a quadrupole strength is changed.
Jx ˆ
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2nd Method
• The sextupole magnets distort the phase space ellipse into a different shape.
• This distortion changes the equilibrium value of β(s)
• This change in β(s) is proportional to the driving amplitude:
x
x’
x
x’
With sextupoles
Without sextupoles
A
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2nd Method
• A change in β can create a change in phase.
• The phase of the entire ring is the tune. The tune shift from the β error is:
• We expect Q vs. A to have a parabolic relationship because:
2
1
s
s s
dss
A
2
Q
2
s
s
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Results for Method 2Tune Shift for Vertically Driven
Simulation
6.0000E-01
6.0200E-01
6.0400E-01
6.0600E-01
6.0800E-01
6.1000E-01
6.1200E-01
6.1400E-01
6.1600E-01
6.1800E-01
6.2000E-01
0 10 20 30 40
Initial displacement (mm)
Fra
ctio
n tu
ne (
vert
ical
)
Tune shift for horizontally driven simulation
5.3580E-01
5.3600E-01
5.3620E-01
5.3640E-01
5.3660E-01
5.3680E-01
5.3700E-01
5.3720E-01
5.3740E-01
5.3760E-01
5.3780E-01
0 5 10 15
Initial displacement (mm)
Fra
ctio
nal t
une
(hor
izon
tal)
The quadratic dependence is shown in the vertically driven simulation
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Machine dataTune shift in horizontally driven sample in
unaltered lattice
4.48E-001
4.49E-001
4.50E-001
4.51E-001
4.52E-001
4.53E-001
4.54E-001
0 0.1 0.2 0.3 0.4 0.5
Square root of amplitude
Fra
ctio
nal t
une
Tune shift in vertically driven sample in unaltered lattice
3.69E-001
3.70E-001
3.70E-001
3.71E-001
0 0.05 0.1 0.15
Square root of amplitude
Fra
ctio
na
l tu
ne
Tune shift in horizontally driven sample in altered lattice
4.44E-0014.46E-0014.48E-0014.50E-0014.52E-0014.54E-001
0 0.05 0.1 0.15 0.2
Square root of amplitude
Fra
ctio
na
l tu
ne
Tune shift in vertically driven sample in alternate lattice
3.69E-001
3.70E-001
3.70E-001
3.71E-001
0 0.05 0.1 0.15
Square root of amplitude
Fra
ctio
na
l tu
ne
The tune shift is large enough to see it in the data from the actual accelerator
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A resonance?
Tune shift for horizontally driven simulation
5.3550E-01
5.3600E-01
5.3650E-01
5.3700E-01
5.3750E-01
5.3800E-01
0 2 4 6 8 10 12
Initial amplitude
Hor
izon
tal t
une
shift
The horizontal data is not quite what we expected. This may be due to the fact that it is close to the 2(Qh)+3(Qv)+2(Qs)=3 or the 3(Qv)+3(Qs)=2 resonances.
Change in magnitude for horizontally driven simulation
1.00E-005
1.00E-004
1.00E-003
1.00E-002
1.00E-001
1.00E+000
1 10
Initial displacement (mm)
Osc
illat
ion
mag
nitu
de/m
ax
fh
2fh
3fh
4fh
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• We have shown that the magnitude for the signal heights of the different spectral components are dependent on the driving amplitude.
• We have also shown that there is a tune shift that is dependent on the driving amplitude.
• We have also shown that these effects can be detected in the signal from the particle accelerator.
Conclusions
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• We need to determine how changing the lattice effects the signals.
• From that data we can begin to figure out how we can use these methods to find non-linear errors.
Future plans