Download Slide 1 - CLASSE Cornell

Searching for CesrTA guide
field nonlinearities in beam
position spectra
Laurel Hales
Mike 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 nonlinear 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?
One BPM vs. Time
Beam-pipe
• 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
4 electrodes on the walls of
the 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: P  T
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:
d 2x
 2 x  k 2 x 2
2
dt
which can be solved for:
xt   x0 cos t    Bm cosm  t 
2
m 0
when we solve the above equation that gives
us different multiples of ω because:
1  cos 2
cos  
2
2
1st Method
• The height of the
different harmonics
should be
dependent on the
driving amplitude
(A).
fh : 1  A
2fh : 2  A2
3fh : 3  A3
Τau matrix column
One of the principle
components
Higher spectral
component
Results for Method 1
Change in magnitude for horizontally driven
simulation
Magnitude/max magnitude
1.00E+000
1
10
1.00E-001
fh
1.00E-002
2fh
3fh
1.00E-003
4fh
1.00E-004
1.00E-005
Initial displacement (mm)
The expected power law dependence is clearly shown in the vertically driven
simulation.
Machine data (horizontally driven)
Change in magnitude for horizontally
driven sample in alternate lattice
Change in m agnitude for horizonatally
driven sam ple in unchanged lattice
1
0. 1
1
Magnitude (au)
0. 1
fh
0. 01
2f h
0. 001
3f h
4f h
0. 0001
0. 00001
Square root of the driving
amplitude (au)
0. 000001
Magnitude (au)
1
0. 01
0.01
0.1
0.1
1
0.01
fh
0.001
2fh
0.0001
0.00001
0.000001
Square root of the driving amplitude
(au)
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
Machine data (vertically driven)
Change in magnitude for vertically
drivien sample in alternate lattice
Change in m agnitude for vertically
driven sam ple in unchanged lattice
1
0.01
0.1
0.1
0.01
0.001
0.0001
0.00001
0.000001
Square root of the driving amplitude
(au)
1
fv
2f v
Magnitude (au)
Magnitude (au)
1
0.01
0.1
1
0.1
0.01
fv
2fv
0.001
0.0001
0.00001
Square root of the driving
amplitude (au)
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 xˆ   J , 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.
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:

A

x’
x
Without sextupoles
x’
x
With sextupoles
2nd Method
• A change in β can create a change in phase.
 s   
s2
s1
ds
 s 
• The phase of the entire ring is the tune. The tune
shift from the β error is:
   s   2

 
Q 
2   s  
• We expect Q vs. A to have a parabolic
relationship because:

A

Results for Method 2
Tune Shift for Vertically Driven
Simulation
6.2000E-01
5. 3780E -01
Fraction tune (vertical)
Fractional tune (horizontal)
Tune shift for horizontally
driven simulation
5. 3760E -01
5. 3740E -01
5. 3720E -01
5. 3700E -01
5. 3680E -01
5. 3660E -01
5. 3640E -01
5. 3620E -01
5. 3600E -01
5. 3580E -01
6.1800E-01
6.1600E-01
6.1400E-01
6.1200E-01
6.1000E-01
6.0800E-01
6.0600E-01
6.0400E-01
6.0200E-01
6.0000E-01
0
5
10
Initial displacement (mm)
15
0
10
20
30
Initial displacement (mm)
The quadratic dependence is shown in the vertically driven simulation
40
Machine data
Tune shift in vertically driven sample in
unaltered lattice
Fractional tune
Fractional tune
Tune shift in horizontally driven sample in
unaltered lattice
4.54E-001
4.53E-001
4.52E-001
4.51E-001
4.50E-001
4.49E-001
3.71E-001
3.70E-001
3.70E-001
3.69E-001
0
4.48E-001
0
0.1
0.2
0.3
0.4
4.54E-001
4.52E-001
4.50E-001
4.48E-001
4.46E-001
4.44E-001
0.1
0.15
Square root of amplitude
0.15
Tune shift in vertically driven sample in
alternate lattice
Fractional tune
Fractional tune
Tune shift in horizontally driven sample in altered
lattice
0.05
0.1
Square root of amplitude
Square root of amplitude
0
0.05
0.5
0.2
3.71E-001
3.70E-001
3.70E-001
3.69E-001
0
0.05
0.1
0.15
Square root of amplitude
The tune shift is large enough to see it in the data from the actual accelerator
A resonance?
Change in magnitude for horizontally
driven simulation
Tune shift for horizontally driven
simulation
Oscillation
magnitude/max
1
10
1.00E-001
fh
1.00E-002
2fh
3fh
1.00E-003
4fh
1.00E-004
1.00E-005
Horizontal tune shift
1.00E+000
5.3800E-01
5.3750E-01
5.3700E-01
5.3650E-01
5.3600E-01
5.3550E-01
0
Initial displacement (mm)
2
4
6
8
10
12
Initial amplitude
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.
Conclusions
• 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.
Future plans
• 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.