Spin Tracking, Integration, and Effects of Quadrupole Nonlinearity

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Delve into the world of spin tracking, integration, and the impact of quadrupole nonlinearity in particle physics. Explore methods for pitch correction, amplitude measurement, and dealing with quad nonlinearity in simulations. Discover how to address amplitude dependence and tune, and compare different methods for analyzing spin tracking and integration reliability.

  • Spin Tracking
  • Integration
  • Quadrupole Nonlinearity
  • Particle Physics
  • Simulation
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  1. Spin tracking vs Integration and effect of quad nonlinearity D. Rubin February 7, 2019

  2. Pitch systematic y0 By 2/7/19 D. Rubin

  3. Pitch correction 3 ways to compute pitch correction in simulation Spin tracking Includes everything, but ppb precision requires many turns Integration along trajectory very good approximation far from resonances Measurement of vertical amplitude assumes quad linearity 2/7/19 D. Rubin

  4. Spin tracking using BMT Pitch correction vs vertical amplitude 2/7/19 D. Rubin

  5. Spin tracking and integration are in good agreement 2/7/19 D. Rubin

  6. Vertical amplitude (y0)/ v is not a good measure of angle at large amplitude 2/7/19 D. Rubin

  7. Quad nonlinearity => amplitude dependence of tune and And nonlinear dependence of on y0 2/7/19 D. Rubin

  8. We can correct for the amplitude dependence by measuring the vertical tune Alternatively, measure the angular distribution directly 2/7/19 D. Rubin

  9. 2/7/19 D. Rubin

  10. Some equations Only Ergives first-order contribution to the precession along the vertical direction: E field correction 3 ways to compute E-field contribution to 1. Spin tracking (BMT equation) 2. Integration a) Integration along trajectory (includes betatron oscillations) b) Integration along closed orbit ( ) Note that method 2b) is most nearly equivalent to the classic method, namely p c p 0 0 2 e x 2 2 (1 ) C n n E a 2 r 0 Compare the 3 methods in simulation to determine 1. If integration is a reliable proxy for spin tracking 2. The size of the contribution from finite betatron oscillation amplitude 3. Effect of quad nonlinearity 2/7/19 D. Rubin

  11. Distinct trajectories with common momentum offset - For trajectory compute a by spin tracking and by integration Is the Efield correction independent of the betatron amplitude? 2/7/19 D. Rubin

  12. Multiple points at each momentum correspond to different betatron amplitudes The spread at momentum zero is a measure of the accuracy of the simulation (since the E-field correction is nominally zero at the magic momentum) 2/7/19 D. Rubin

  13. Efield correction computed with spin tracking in good agreement with correction based on 2/7/19 D. Rubin

  14. along the trajectory is very nearly the same as along the closed orbit. (There is little dependence on betatron amplitude) 2/7/19 D. Rubin

  15. 2/7/19 D. Rubin

  16. The calculation of the E-field correction that assumes quad linearity, overestimates the effect at large momentum offset (where E-field does not increase linearly with displacement) 2/7/19 D. Rubin

  17. Replace n, R0 and with Qx and measure Qx for each momentum => The effect of quad nonlinearity can be corrected by measuring momentum dependence of horizontal tune 2/7/19 D. Rubin

  18. 2/7/19 D. Rubin

  19. 2/7/19 D. Rubin

  20. Comments Quad fields are based on an azimuthal slice of 3-D field map with no end effect details And perfect relative alignment of plates and absolute alignment about magic radius Perfect B-field Measurement of amplitude/momentum dependence of tunes would be very useful to diagnose quad fields and compensate nonlinearities. 2/7/19 D. Rubin

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