Tue: Transverse Motion
Guiding Particles Toward the Ideal Trajectory
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
- 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
Phase Space Distributions and Ellipses
slides
- 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”
Courant-Snyder Parameterization
slides
Envelope.R
- 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