Critical exponents for the cloud-crystal phase transition of charged particles in a Paul Trap

It is well known that charged particles stored in a Paul trap, one of the most versatile tools in atomic and molecular physics, may undergo a phase transition from a disordered cloud state to a geometrically well-ordered crystalline state (the Wigner crystal). In this paper we show that the average lifetime $\bar\tau_m$ of the metastable cloud state preceding the cloud $\rightarrow$ crystal phase transition follows a powerlaw, $\bar\tau_m \sim (\gamma-\gamma_c)^{-\beta}$, $\gamma>\gamma_c$, where $\gamma_c$ is the critical value of the damping constant $\gamma$ at which the cloud $\rightarrow$ crystal phase transition occurs. The critical exponent $\beta$ depends on the trap control parameter $q$, but is independent of the number of particles $N$ stored in the trap and the trap control parameter $a$, which determines the shape (oblate, prolate, or spherical) of the cloud. For $q=0.15,0.20$, and $0.25$, we find $\beta=1.20\pm 0.03$, $\beta=1.61\pm 0.09$, and $\beta=2.38\pm 0.12$, respectively. In addition we find that for given $a$ and $q$, the critical value $\gamma_c$ of the damping scales approximately like $\gamma_c=C \ln [ \ln (N)] + D$ as a function of $N$, where $C$ and $D$ are constants. Beyond their relevance for Wigner crystallization of nonneutral plasmas in Paul traps and mini storage rings, we conjecture that our results are also of relevance for the field of crystalline beams.