Power-Law Distributions for a Trapped Ion Interacting with a Classical Buffer Gas

Classical collisions with an ideal gas generate non-Maxwellian distribution functions for a single ion in a radio frequency ion trap. The distributions have power-law tails whose exponent depends on the ratio of buffer gas to ion mass. This provides a statistical explanation for the previously observed transition from cooling to heating. Monte Carlo results approximate a Tsallis distribution over a wide range of parameters and have ab initio agreement with experiment.

Figure 1

Monte Carlo distributions for a single ${}^{136}\mathrm{Ba}^{+}$ ion cooled by six different buffer gases at 300 K ranging from $m_B=4$ (left) to $m_B=200$ (right). Note the evolution from Gaussian to power law (straight line) as the mass increases. The solid lines are Tsallis functions [Eq. (7)] with fixed $\sigma =0.0185\mathrm{cm}$ and the exponents of Table .

Figure 2

Predicted power-law ion lifetime $\tau \propto U^n$ where $U$ is the trap depth, $n$ is the Tsallis exponent of Table , and the parameters correspond to Fig. 1.

Figure 3

A heating and cooling diagram for a single collision in the case $m_b=m_i$ or $\alpha =0$ in Eq. (8). Heating occurs for $R>1$ and cooling for $R<1$. The explanation given in the text does not depend on the exact form of this result and requires only that there be a small phase space volume $\beta <1$ for $R>1$.