For short electron bunches in accelerators, the radial ion density, due to
residual-gas ionization faithfully reproduces the radial electron bunch dis
tribution for time scales similar to the electron bunch length. If the elec
tron bunch length is sufficiently long, however, the ions focus and, even f
or radially uniform electron beams, tend to form a very nonuniform equilibr
ium distribution. This ion distribution, in turn, leads to nonlinear focusi
ng forces on the electron bunch itself. In this paper, we find the equilibr
ium distribution when the electron distribution is uniform, and calculate t
he 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 resid
ual gas pressure and electron bunch length.