Concepts

Reference Trajectory

ImpactX is an s-based beam dynamics code with space charge. Particles are tracked using the longitudinal accelerator lattice position \(s\) as the independent dynamical variable. Particle phase space coordinates are specified relative to a nominal reference trajectory.

The reference trajectory plays an important role in ImpactX. In addition to specifying the nominal machine orbit, it specifies the relationship between the local beam coordinate system and the global lab coordinate system. A reference trajectory is favorable instead of, e.g., differences to the beam centroid, because:

  1. the reference trajectory is the ideal single-particle orbit used as part of the optics design, so it is better that it can be computed independently of the beam distribution,

  2. differences between the beam centroid and the reference particle are important when investigating the effects of misalignments and errors for beamline designs,

  3. the fields are usually specified relative to the reference trajectory, and the dynamics becomes more nonlinear as one moves away from the reference trajectory, so knowing how far the beam particles are form the reference trajectory is important for accuracy,

  4. we want to keep track of the reference trajectory in global coordinates, and that global information is not available in the particle data alone, so it would need to be specified in some other way.

The reference values of \(z=ct\) and \(s\) should be identical until reaching a bending element. (If the lattice contains no bending elements, then they should coincide.) Also, the reference value of \(ct\) coincides with the value of \(s\) in the ultrarelativistic limit. More generally, the derivative \(ds/d(ct) = \beta\), where the relativistic \(\beta = \sqrt{1-\frac{1}{p_t^2}}\).

Collective Effects

Collective effects from space charge of the particle beam are solved between steps in \(s\). One can set the number of slice steps through each lattice element for the application of the space charge push.

Note

Currently, space charge kicks are calculated in 3D. A 2D space-charge solver for purely transversal effects will be added in the future.