Universal critical phenomena of the cloud $\rightarrow$ crystal phase transition in the Paul trap: Powerlaws

$N$ charged particles, simultaneously stored in a radio-frequency (rf) Paul trap, exhibit deterministic heating. Depending on the damping ($\gamma$) imparted to the system, these particles can exist in multiple phases, the most commonly found being the cloud and crystal phases. With a small $\gamma$, the particles exhibit gas-like behavior, where the heating and cooling equilibrate and a stable cloud results. For larger $\gamma$, the damping overcomes the heating and the particles are forced into the crystalline state. We explore the cloud $\rightarrow$ crystal transition as a critical phenomenon. We find that the transition occurs at a critical value $\gamma_c$ of the damping constant $\gamma$. We find that as a function of $N$, $\gamma_c$ scales approximately like an iterated log law. We also present a universal power law, $\bar\tau_m \sim (\gamma - \gamma_c)^{-\beta}$, $\gamma > \gamma_c$, $\beta > 0$, independent of both $N$ and the Paul trap parameter $a$, depending only on the Paul trap parameter $q$, that describes the number of cycles necessary for the system to crystallize as a function of $\gamma-\gamma_c$.