searching for cesrta guide field nonlinearities in beam position spectra laurel hales mike billing...
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Searching for CesrTA guide field nonlinearities in beam
position spectra
Laurel HalesMike Billing
Mark Palmer
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.
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
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
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
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.
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
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
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.
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
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
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
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
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
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
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
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
• 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
• 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