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General equation of spin motion in a accelerator ring.

General equation of spin motion in a accelerator ring. Perturbations from betatron and synchrotron motion and machine imperfections. Field on the design orbit. For EDM ring only vertical field on the orbit:. Perturbation part of spin precession:. main EDM term.

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General equation of spin motion in a accelerator ring.

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  1. General equation of spin motion in a accelerator ring. Perturbations from betatron and synchrotron motion and machine imperfections Field on the design orbit For EDM ring only vertical field on the orbit:

  2. Perturbation part of spin precession: main EDM term All type of motion can be included here, that is in general case: dispersion function betatron motion closed orbit

  3. Forced momentum oscillations: Spin resonances at n0=nm Spin tune modulation

  4. Problem 1 Resonances at the same position n0=nm compete with EDM resonance Competing resonance is caused by the vertical dispersion. Tolerances on the vertical dispersion: • Source of the vertical dispersion: • quadrupole rolls • quadrupole vertical misalignment • dipole rolls tolerances should be derived MAD simulations

  5. Problem 2 Competing resonances caused by the tune modulation. wk – imperfection resonance creates modulation resonances at Tolerance on vertical closed orbit. DEPOL calculation with strength:

  6. Problem 3 Coherence time. Defined by <wy> Averaging should leave the contribution mainly from even multipoles k2n Define tolerances on nonlinear field and on the beam properties (emittances). Higher dispersion components should be taken into account too:

  7. Required sextupole tolerance for 1000s coherence

  8. General formula from vertical precession component (from BMT equation) spin tune change

  9. Momentum spread contribution For Tcoh=1000s, Dnsp= <wy>=2e-11, requires d ~3e-06

  10. x’ and y’ contribution Then for 20p e-06 95% normalized emittance: Dnsp = 1.1e-07

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