  Electron bunches with charge densities $\rho$ of the order of $10^2$ to
$10^3$ [nC/cm$^3$], energies between $20.$ and $100.$ [MeV], peak current
$>100$ [A], bunch lengths between 0.3 and 1.8 [cm], and bunch charge of 2.0 to
$20.$ [nC] are relevant to the design of Free Electron Lasers and future linear
colliders. In this paper we present the results of numerical simulations
performed with a particle in a cell (pic) code of an electron bunch in a drift
tube. The electron bunch has cylindrical symmetry with the $z$-axis oriented in
the direction of motion. The charge density distribution is constant in the
radial and Gaussian in the longitudinal direction, respectively. The electron
bunch experiences both a radial pinch in the middle of the pulse, corresponding
to the peak electron density, and a significant growth of the correlated
emittance. This behavior is explained, and an approximate scaling law is
identified. Comparisons of the results from the pic and PARMELA codes are
presented.
