Chapter 2 Tue: Transverse Motion

2.1 Guiding Particles Toward the Ideal Trajectory


  • 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
    • Small-angle bending element
    • Thin lens quadrupole
    • Solenoid magnet
    • (Accelerating cavity – time focusing)
  • The notion of a beam envelope
  • Separation of beam properties from beam line properties

2.2 Phase Space Distributions and Ellipses


  • Comments on particle sources
  • 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
    • see Some Notes on Ellipses
  • 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”

2.3 Courant-Snyder Parameterization



  • 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