AT basics

The Accelerator Toolbox is based on a fast and modular tracking engine.

Coordinate system

The AT coordinate system is based on a design trajectory along which the magnetic elements are aligned, and a reference particle:

By convention, the reference particle travels along the design trajectory at constant nominal energy and defines the phase origin of the RF voltage.

The 6-d coordinate vector \(\vec Z\) is:

\[\begin{split}\vec Z = \begin{pmatrix} x \\ p_x/p_0 \\ y \\ p_y/p_0 \\ \delta \\ \beta c\tau \end{pmatrix}\end{split}\]

\(p_0\) is the nominal momentum,

\(\delta=\dfrac{p-p_0}{p_0}\) is the relative momentum deviation.

AT works with relative path lengths: the 6^th coordinate \(\beta c\tau\) represents the path lengthening with respect to the reference particle. \(\tau\) is the delay with respect to the reference particle: the particle is late for \(\tau > 0\).

Tracking

All the AT processing is based on tracking. For instance, linear optics properties are deduced from a one-turn transfer matrix obtained by differentiating the tracking output on a small dimension grid centered on the closed orbit.

The tracking through each AT element is performed by an integrator, called its “passmethod”. There is no correlation between an element class and its associated passmethod. The passmethod must be explicitly specified by the PassMethod attribute of the element.

The tracking engine is written in C for the best performance. Each passmethod is dynamically loaded on-demand when encountering the first element requiring it.

See AT Passmethods for a description of the available passmethods,