21  Physics Models Used in Modelling Collisions

View:

21.1 Introduction

This chapter documents the current collision model used by the BEAMBEAM element in OPALX.

The present implementation is a reduced symmetric head-on collision model. It provides the interaction-window field solve and its associated mesh handling without yet introducing two independently tracked bunch objects.

The current model therefore uses:

  1. one physical bunch, represented by the actual particle container,
  2. one virtual bunch, reconstructed from the physical bunch by symmetry about the interaction point,
  3. an interaction-window field solve that is active only while the bunch traverses the interval defined by the element length L.

The model is entered when the head of the physical bunch reaches the beginning of the interaction window and is left once the tail of the bunch has passed the end of the window.

21.2 Symmetric Counter-Propagating Ansatz

Let z_IP denote the interaction-point position in the local beam coordinate system during the interaction-window solve. For a particle of the physical bunch with coordinates (x, y, z) and momentum (p_x, p_y, p_z), the virtual counter-propagating partner bunch is constructed by the symmetry transformation

\[ \begin{aligned} x' &= x, \\ y' &= y, \\ z' &= 2 z_{\mathrm{IP}} - z, \end{aligned} \qquad \begin{aligned} p'_x &= p_x, \\ p'_y &= p_y, \\ p'_z &= -p_z. \end{aligned} \tag{21.1}\]

This is the current fully symmetric head-on ansatz. It assumes that the two bunches are identical up to the longitudinal mirror transformation about the interaction point.

21.3 Mesh Model

Outside the interaction window, the longitudinal field mesh remains bunch-following. In software-engineering terms, this is a Lagrangian-in-z representation of the field mesh.

Inside the interaction window, the field mesh changes meaning:

  • in x and y, the mesh remains bunch-following,
  • in z, the mesh is frozen over the full interaction window.

The longitudinal field mesh therefore becomes Eulerian in z, while the physical bunch itself continues to move through that fixed window. The source chapter stresses this distinction:

  • physical bunch bounds remain the actual particle bounds,
  • field-domain bounds in z represent the fixed interaction window.

21.4 Scope of the Current Model

The present implementation should be interpreted as a first collision model for symmetric head-on beams. It is suitable for studying:

  • entry into and exit from the interaction window,
  • the switch from a bunch-following to a frozen longitudinal field mesh,
  • the self-field solve on that interaction-window mesh,
  • the symmetry-based reconstruction of the opposing bunch.

It is not yet a full two-bunch strong-strong model with two independently tracked particle populations.