June 19, Wednesday: Transverse motion

Transverse Motion

Guiding Particles Toward the Ideal Trajectory (J. Kunz)

[slides]

• The ideal trajectory and motion relative to it
• Comments on time scales, adiabatic changes and separation of motion
• Tracing particle trajectories through a focusing system
  • Drift space
  • Thin lens quadrupole
• Hill's equation
• Intro to simple "optical" matrix representation

Whiteboard time

Phase Space Distributions and Ellipses (T. Mohayai)

[slides]

• Particle distributions and transverse phase space
• Phase space area – emittance
  • phase space distributions and second moments
  • general equation/properties of phase space ellipse
  • invariance of the emittance in linear system
• Courant-Snyder parameters and emittance
• Matrix formalism of ellipses and of particle motion
• Propagation of ellipse parameters through elements
• The development of a “beam envelope”

Courant-Snyder Parameterization

[slides]

• Hill’s equation and analytical treatment of transverse motion
• Envelope function and phase advance
• Single-particle vs. “beam” emittance
• Correspondence between analytical, matrix forms
• Adiabatic damping and normalized emittance

See also:

Lecture notes by Linda Spentzouris:

• Transverse dynamics: magnetic fields, equations for transverse motion, machine tune and the Twiss parameters, transfer maps
• Emittance: particle distributions, phase space, high energy beams, transverse emittance from Hill's equation, normalized emittance