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