Coordinates and Units

Each particle in the beam is described at fixed \(s\) by a set of 6 canonical phase space variables (x [m], px, y [m], py, t [m], pt). Coordinates x and y denote the horizontal and vertical displacement from the reference particle, respectively, and describe motion in the plane transverse to the velocity of the reference particle. The longitudinal coordinate t denotes the difference between the arrival time of the particle and the arrival time of the reference particle, multiplied by the speed of light \(c\).

The momenta conjugate to x, y, and t are denoted px, py, and pt, respectively. These variables are normalized by the magnitude of the momentum of the reference particle, and are therefore dimensionless. In a region of zero vector potential, for example, \(p_x = \Delta(\beta_x\gamma)/(\beta_0\gamma_0)\), where \(\beta_0\) and \(\gamma_0\) denote the relativistic factors associated with the reference velocity. In a region of zero scalar potential, pt denotes the deviation from the reference energy normalized by the design momentum times the speed of light, so that \(p_t = -\Delta(\gamma)/(\beta_0\gamma_0)\).

Unlike particles within the beam, the reference particle is described by a set of 8 phase space variables (x [m], px, y [m], py, z [m], pz, t [m], pt) that are specified in a global laboratory coordinate system (x,y,z). The momenta of the reference particle are normalized by \(mc\), so that \(p_x=\beta_x\gamma\), etc. A parameteric plot of the reference trajectory variables (x,z) allows the user to view the global geometry of the accelerator structure (footprint).