Nonlinear focusing by residual-gas ionization in long-pulse electron linacs

For short electron bunches in accelerators, the radial ion density due to residual-gas ionization faithfully reproduces the radial electron bunch distribution for time scales similar to the electron bunch length. If the electron bunch length is sufficiently long, however, the ions focus and, even for radially uniform electron beams, tend to form a very nonuniform equilibrium distribution. This ion distribution, in turn, leads to nonlinear focusing forces on the electron bunch itself. In this paper, we find the equilibrium distribution when the electron distribution is uniform, and calculate the emittance growth for axial slices in the electron bunch later in time. A regime is found in which the emittance growth is quadratic with both residual gas pressure and electron bunch length.