Analytical calculation of the space charge potential and the temperature of stored ions in an rf quadrupole trap

The use of a stored ion cloud in an rf quadrupole trap has shown its possibilities as a high-performance frequency standard. The most important source of systematic error of such a standard is the second-order Doppler shift, an effect of special relativity, which produces a relative frequency shift approximated by the ratio of the mean kinetic energy to the rest mass energy of an ion. Assuming that the ion cloud can be characterized by a temperatureT, the determination ofT is of paramount importance in evaluating the second-order Doppler shift. In this paper, an analytical expression of the ion-cloud temperature is established taking space-charge effects into account and involving only physical parameters which can be easily measured. The validity of this approximate expression is discussed and it is compared to numerically established results published elsewhere.