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

TL;DR

This paper investigates the phase transition from a charged particle cloud to a crystal in a Paul trap, revealing power-law behavior of metastable state lifetime and scaling of critical damping with particle number.

Contribution

It demonstrates that the lifetime of metastable states follows a power law near the transition and identifies how the critical damping scales with particle number, independent of particle count.

Findings

Metastable lifetime follows a power law with respect to damping constant.

Critical damping scales logarithmically with the number of particles.

Critical exponent depends on trap control parameter q, but not on particle number or shape.

Abstract

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.