Computed energy and spatial statistical properties of stored ions cooled by a buffer gas

We describe a three-dimensional model, based on the temporal invariance of the statistical properties, which allows us to calculate the spatial and energy probability densities of strongly confined ions in the presence of a buffer gas. This model is then applied to a population of ${\mathrm{Cs}}^{+}$ ions for which we compute these densities within the whole stability diagram and for various gases. In the case of helium, in particular, we show that these laws can be described by Gaussian functions, and we present the evolution of the temporal average of the spatial dispersions and ionic temperatures at the working point.