  We calculate the cross-section of ionization by free-electron impacts in high
or moderate density plasmas. We show that the so-called ionization potential
depression (IPD) strongly affects the magnitude of the cross-section in the
high-density domain. We use the well-known IPD formulas of Stewart-Pyatt and
Ecker-Kr\"oll. A more recent approach based on classical molecular dynamics
simulation is also investigated. The latter provides an alternative way to
calculate IPD values. At near-solid densities the effects of the free-electron
degeneracy should be investigated. The rates are then calculated within the
Fermi-Dirac statistics. We first use the semi-empirical formula of Lotz for
ionization cross-section. The results may differ significantly from measured
cross-sections or calculations with reliable atomic codes. Then, in a second
step, we propose a new formula that combines the Lotz formula and a polynomial
expansion in terms of the ratio of the energy of the incident electron and the
ionization energy. The coefficients of the polynomial expansion are adjusted to
fit the cross-section provided by robust atomic codes. A great advantage of the
new formula is that it allows a fully analytical calculation of the ionization
rate. Our results are compared to experiments measuring IPDs, cross-sections
and rate coefficients on aluminum at high and moderate densities and on Be-like
CNO ions.
