Transverse force in a relativistic beam moving along a curved trajectory

When a relativistic beam moves along a curved trajectory it loses energy on coherent radiation. The effect of this radiation on the beam dynamics is described by the so call coherent synchrotron radiation (CSR) longitudinal wake field. Among different models of the CSR wake, the simplest is the one-dimensional (1D) one where the beam is treated as a line-charge. The self-field of the beam on a curved trajectory also creates a transverse component of the force. Unfortunately, there is a confusion in the literature as to whether a 1D model can be worked out for the transverse force inside the beam. In this paper, we show how such a 1D model can be consistently derived for a general curvilinear beam orbit if one formulates the equations of motion for the beam particles in Hamiltonian form and uses a renormalized transverse force. Unfortunately, this scheme cannot be applied to the classical problem of relativistic beam passing through a single bending magnet, because the scalar potential in the initial conditions for the Hamiltonian variables cannot be defined in the 1D model. In this case, one should use the 3D transverse force, for which we derive analytical expressions. For the steady state, our calculations show an excellent agreement with computer simulations. We also calculate the transient effects for this force at the entrance to and the exit from the bend. Our results provide a new way to accommodate the transverse force into the existing and new simulation codes which is important for many applications of the high-current, small-emittance relativistic beams.

Figure 1

A bending magnet with the beam nominal trajectory shown by the blue line. The beam shown by red is at its initial position at the entrance to the magnet.

Figure 2

Transverse force for the parameter set A. The solid black and blue lines are calculated using Eqs. (42) and (43) and the dots show the results of the numerical calculations with PyCSR3D. Left panel: dependence of $F_{\perp }$ versus $z$ for $r=0$ (on axis) and $r={\sigma }_{\perp }$, respectively (the values on the axis are larger than at $r={\sigma }_{\perp }$). Right panel: dependence of $F_{\perp }$ versus $x$ for $z=0$. The force is normalized by $Q\kappa /{\sigma }_z$ where $Q$ is the charge of the beam.

Figure 3

Transverse force for the parameter set B. The solid black and blue lines are calculated using Eq. (44) with $x=0$ (on axis) and $x={\sigma }_x$, respectively. The dots show the results of the numerical calculations with PyCSR3D for the same offsets. The force is normalized by $Q\kappa /{\sigma }_z$ where $Q$ is the charge of the beam.

Figure 4

Transient transverse force on the axis of the beam at the entrance to the bend. The numbers near each curve show the distance from the entrance.

Figure 5

Transient transverse force on the axis of the beam after the exit from the bend. The numbers near each curve show the distance from the exit.