  A simple model of a trapped ion cloud cooled by collisions in a buffer gas in
a Paul trap is presented. It is based on the customary decomposition of the ion
motion in micro- and macro- (or secular) motions and a statistical treatment of
hard-sphere collisions and ion trajectories. The model also relies on the
evidence that the effective trapping area in real Paul traps is limited to a
certain radius, where the harmonics of the potential of order >2 become non
negligible. The model yields analytical formulae for the properties of the ion
cloud and equilibration times, which are in good agreement for a wide range of
parameters with the results of a numerical simulation, whose reliability has
been verified with experimental observations. When the confining potential is
efficient enough to suppress evaporation from the trap, the model yields an
effective temperature for the ions T$_{eff}$=2T/(1-m$_{g}$/m), where T is the
temperature of the buffer gas, m and m$_{g}$ are the masses of the ions and gas
molecules respectively. The so-called rf heating effect, responsible for
T$_{eff}$>T, is interpreted in light of the model as the result of an
incomplete cooling of the ion motion, limited to the macromotion, while the net
effect of the micromotion is to double the average ion kinetic energy for
m/m$_{g}$ >>1. For m/m$_{g}$ <=1, the incomplete cooling is not sufficient to
overcome the thermal agitation of the cloud to which the micromotion
participates; the ions are therefore led out of the trap.
