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Hong Qin and Ronald C. Davidson Plasma Physics Laboratory, Princeton University

Non-Abelian Courant-Snyder Theory for Coupled Transverse Dynamics. Hong Qin and Ronald C. Davidson Plasma Physics Laboratory, Princeton University US Heavy Ion Fusion Science Virtual National Laboratory www.princeton.edu/~hongqin/. How to make a smooth round beam?. Solenoid

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Hong Qin and Ronald C. Davidson Plasma Physics Laboratory, Princeton University

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  1. Non-Abelian Courant-Snyder Theory for Coupled Transverse Dynamics Hong Qin and Ronald C. Davidson Plasma Physics Laboratory, Princeton University US Heavy Ion Fusion Science Virtual National Laboratory www.princeton.edu/~hongqin/

  2. How to make a smooth round beam? • Solenoid • Final focus (NDCX) • How to match quadrupole with solenoid (NDCX-III)? • Skew-quadrupole

  3. Möbius Accelerator [Talman, PRL 95] • Round beam, one tune, one chromaticity • How? • Solenoid or skew-quadrupole • What is going on during the flip?

  4. Coupled transverse dynamics (2 degree of freedom) solenoidal, quadrupole, & skew-quadrupole

  5. Similar 2D problem – adiabatic invariant of gyromotion L. Spitzer suggested R. Kulsrud and M. Kruskal to look at a simpler problem first (1950s).

  6. + Particles dynamics in accelerators ( uncoupled, 1 degree of freedom)

  7. Courant-Snyder theory for uncoupled dynamics Courant-Snyder invariant Courant (1958) Envelope eq. Phase advance

  8. Courant-Snyder theory is the best parameterization • Provides the physics concepts of envelope, phase advance, emittance, C-S invariant, KV beam, … K. Takayama [82,83,92]

  9. Higher dimensions? 2D coupled transverse dynamics? 10 free parameters solenoidal, quadrupole, & skew-quadrupole

  10. Many ways [Teng, 71] to parameterize the transfer matrix Symplectic rotation form [Edward-Teng, 73]: Lee Teng uncoupled uncoupled CS transfer matrix No apparent physical meaning Have to define beta function from particle trajectories [Ripken, 70], [Wiedemann, 99]

  11. Can we do better? A hint from 1 DOF C-S theory

  12. Transfer matrix Original Courant-Snyder theory scalar Non-Abelian generalization

  13. Envelope equation Original Courant-Snyder theory scalar Non-Abelian generalization

  14. Phase advance rate Original Courant-Snyder theory Non-Abelian generalization

  15. Phase advance Original Courant-Snyder theory Non-Abelian generalization

  16. Courant-Snyder Invariant Original Courant-Snyder theory Non-Abelian generalization

  17. How did we do it? General problem

  18. Time-dependent canonical transformation Target Hamiltonian symplectic group

  19. Non-Abelian Courant-Snyder theory for coupled transverse dynamics Step I: envelope

  20. Step II: phase advance

  21. Application: Strongly coupled system Stability completely determined by phase advance Suggested by K. Takayama one turn map

  22. Application: Weakly coupled system Stability determined by uncoupled phase advance

  23. Application: Weakly coupled system Stability determined by uncoupled phase advance

  24. Numerical example – mis-aligned FODO lattice mis-alignment angle [Barnard, 96]

  25. Other appliations – exact invariant of magnetic moment [K. Takayama, 92] [H. Qin and R. C. Davidson, 06]

  26. Other applications • Globally strongly coupled beams. (many-fold Möbius accelerator). • 4D emittance [Barnard, 96]. • 4D KV beams. • …

  27. Other applications – symmetry group of 1D time-dependent oscillator ?! (2D) Wronskian (2D) Courant-Snyder symmetry (3D) Scaling (1D) [H. Qin and R. C. Davidson, 06]

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