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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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slide1

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
How to make a smooth round beam?
  • Solenoid
    • Final focus (NDCX)
    • How to match quadrupole with solenoid (NDCX-III)?
  • Skew-quadrupole
m bius accelerator
Möbius Accelerator

[Talman, PRL 95]

  • Round beam, one tune, one chromaticity
  • How?
    • Solenoid or skew-quadrupole
  • What is going on during the flip?
coupled transverse dynamics 2 degree of freedom
Coupled transverse dynamics (2 degree of freedom)

solenoidal, quadrupole,

& skew-quadrupole

slide5

Similar 2D problem – adiabatic invariant of gyromotion

L. Spitzer suggested R. Kulsrud and M. Kruskal to look at a simpler problem first (1950s).

slide6

+

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

slide7

Courant-Snyder theory for uncoupled dynamics

Courant-Snyder invariant

Courant (1958)

Envelope eq.

Phase advance

courant snyder theory is the best parameterization
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]

slide9

Higher dimensions? 2D coupled transverse dynamics?

10 free parameters

solenoidal, quadrupole,

& skew-quadrupole

many ways teng 71 to parameterize the transfer matrix
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]

slide12

Transfer matrix

Original Courant-Snyder theory

scalar

Non-Abelian generalization

slide13

Envelope equation

Original Courant-Snyder theory

scalar

Non-Abelian generalization

slide14

Phase advance rate

Original Courant-Snyder theory

Non-Abelian generalization

slide15

Phase advance

Original Courant-Snyder theory

Non-Abelian generalization

slide16

Courant-Snyder Invariant

Original Courant-Snyder theory

Non-Abelian generalization

time dependent canonical transformation
Time-dependent canonical transformation

Target Hamiltonian

symplectic group

application strongly coupled system stability completely determined by phase advance
Application: Strongly coupled system Stability completely determined by phase advance

Suggested by K. Takayama

one turn map

application weakly coupled system stability determined by uncoupled phase advance
Application: Weakly coupled system Stability determined by uncoupled phase advance
application weakly coupled system stability determined by uncoupled phase advance1
Application: Weakly coupled system Stability determined by uncoupled phase advance
numerical example mis aligned fodo lattice
Numerical example – mis-aligned FODO lattice

mis-alignment angle

[Barnard, 96]

other appliations exact invariant of magnetic moment
Other appliations – exact invariant of magnetic moment

[K. Takayama, 92]

[H. Qin and R. C. Davidson, 06]

other applications
Other applications
  • Globally strongly coupled beams. (many-fold Möbius accelerator).
  • 4D emittance [Barnard, 96].
  • 4D KV beams.
other applications symmetry group of 1d time dependent oscillator
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]