1. Searching for CesrTA guide field nonlinearities in beam position spectra Laurel Hales Mike Billing Mark Palmer

2. 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.

3. 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

4. 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

5. 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: Columns = spatial function around ring (Diagonals) = Eigen values (λ i ) ~ amplitudes of the eigen components Columns = time development of beam trajectory

6. 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.

7. 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:

8. 1 st Method The height of the different harmonics should be dependent on the driving amplitude (A). Τau matrix column One of the principle components Higher spectral component

9. Results for Method 1 The expected power law dependence is clearly shown in the vertically driven simulation.

10. Machine data (horizontally driven) 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

11. Machine data (vertically driven) The vertically driven data also displays the power law relation. The line represents a linear dependence

12. 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.

13. 2 nd 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 With sextupoles Without sextupoles

14. 2 nd 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:

15. Results for Method 2 The quadratic dependence is shown in the vertically driven simulation

16. Machine data The tune shift is large enough to see it in the data from the actual accelerator

17. A resonance? 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.

18. 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

19. 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