6 Design and Principles of Synchrotrons and Circular Colliders
247
the results [55]. The outcome of the analysis allows to answer the most important
questions for the design of a machine such as:
• Stability of particle motion
• Dynamic aperture
• Specifications for the properties of machine elements
• Optimization or the particle stability
In general the results of these studies are used in an iterative procedure to improve
and optimize the design of the machine.
6.6.4.1 Techniques
A requirement for all techniques employed for particle tracking is that the associated
maps must be symplectic. To solve the equation of motion, most programs use
explicit canonical integration techniques, e.g.:
• Thin lens tracking (most common since they are automatically symplectic and
fast)
• Ray tracing (accuracy by slicing into large number of steps, but time consuming)
• Symplectic integration (see [54] and references therein)
6.6.4.2 Analysis of Tracking Data
Some of the analysis techniques are discussed in the chapter on non-linear dynamics
in this handbook in more detail and some are mentioned here for completeness:
• Taylor maps using Truncated Power Series Algebra (TPSA, [54])
• Lie algebraic maps [54, 56]
• Normal form analysis
The results of the analysis include non-linear resonances and distortion, nonlinear tuneshift with amplitude and an evaluation of the long term stability. In all
cases the interpretation of the results requires a careful analysis of the range where
the data is meaningful to avoid wrong conclusions. Typical problems are numerical
effects which can lead to unphysical features.
6.6.5 Multi Particle Tracking Codes
Multi particle tracking codes are used when we are concerned with the behaviour of
an ensemble of particle. The calculations largely rely on techniques developped for
single particle dynamics. Typical applications are the simulation of:
• Space charge effects, mutual interaction of particles within the same beam.
247
the results [55]. The outcome of the analysis allows to answer the most important
questions for the design of a machine such as:
• Stability of particle motion
• Dynamic aperture
• Specifications for the properties of machine elements
• Optimization or the particle stability
In general the results of these studies are used in an iterative procedure to improve
and optimize the design of the machine.
6.6.4.1 Techniques
A requirement for all techniques employed for particle tracking is that the associated
maps must be symplectic. To solve the equation of motion, most programs use
explicit canonical integration techniques, e.g.:
• Thin lens tracking (most common since they are automatically symplectic and
fast)
• Ray tracing (accuracy by slicing into large number of steps, but time consuming)
• Symplectic integration (see [54] and references therein)
6.6.4.2 Analysis of Tracking Data
Some of the analysis techniques are discussed in the chapter on non-linear dynamics
in this handbook in more detail and some are mentioned here for completeness:
• Taylor maps using Truncated Power Series Algebra (TPSA, [54])
• Lie algebraic maps [54, 56]
• Normal form analysis
The results of the analysis include non-linear resonances and distortion, nonlinear tuneshift with amplitude and an evaluation of the long term stability. In all
cases the interpretation of the results requires a careful analysis of the range where
the data is meaningful to avoid wrong conclusions. Typical problems are numerical
effects which can lead to unphysical features.
6.6.5 Multi Particle Tracking Codes
Multi particle tracking codes are used when we are concerned with the behaviour of
an ensemble of particle. The calculations largely rely on techniques developped for
single particle dynamics. Typical applications are the simulation of:
• Space charge effects, mutual interaction of particles within the same beam.