Transverse resistive wall effects on the dynamics of a bunched electron beam

In the wigglers of future free-electron lasers, the electron beam will be required to travel over a length of 10 m or more in pipes with small diameters. Transverse resistive wall effects could lead to beam breakup during this transport. To investigate this possibility, the equation of motion for a bunched beam is solved analytically. Results show that a steady-state solution is reached for times larger than the diffusion time. This solution can either oscillate or grow exponentially with the length of the pipe, depending on the relative magnitudes of the resistive wall effect and the focusing force in the wiggler. The magnitude of the resistive wall effect depends on the pipe radius b (it increases as 1/b2) but is independent of the thickness and conductivity of the pipe. The thickness and conductivity affect only the time required to reach the steady-state solution. The possibility of a significant transient is also discussed