  Sympathetic laser cooling of ions stored within a linear-geometry, radio
frequency, electric-quadrupole trap has been investigated using computational
and theoretical techniques. The simulation, which allows 5 sample ions to
interact with 35 laser-cooled atomic ions, revealed an instability heating
mechanism, which can prevent ions below a certain critical mass from being
sympathetically cooled. This critical mass can however be varied by changing
the trapping field parameters thus allowing ions with a very large range of
masses to be sympathetically cooled using a single ion species. A theoretical
explanation of this instability heating mechanism is presented which predicts
that the cooling-heating boundary in trapping parameter space is a line of
constant $q_u$ (ion trap stability coefficient), a result supported by the
computational results. The threshold value of $q_u$ depends on the masses of
the interacting ions. A functional form of this dependence is given.
