Continuum model for the breathing oscillation of a spherical complex plasma

Abstract

A nonlinear equation of motion for the breathing oscillation of a spherical complex plasma is derived. A spherical complex plasma, or “dust ball,” is a three-dimensional arrangement of n identical charged particles interacting through a shielded Coulomb force (i.e., a Yukawa potential) with a Debye length λ and confined by a three-dimensional, isotropic, parabolic potential well for which the single-particle oscillation frequency is ω0. The dependence of the equilibrium radius R0 and small-amplitude breathing frequency ωbr on λ is computed. Exact analytical results are given for the continuum limit n→∞ (i.e., a spherical Yukawa fluid). The squared breathing frequency (ωbr∕ω0)2=3 for the unshielded Coulomb interaction (1∕λ→0), irrespective of n, and increases to (ωbr∕ω0)2=5 as 1∕λ→∞. The effects of a finite number of particles are modeled by assuming an inner cutoff for the Yukawa potential a distance a from any point in the complex plasma sphere. Three physical regimes are identified: a Coulomb regime where λ⪢R0 and corrections to the infinite-λ case are small, a nearest-neighbor regime, where R0⪢a>λ and nearest-neighbor interactions dominate, and a plasma regime where R0⪢λ≳a and continuum plasma theory is applicable. For 1∕λ→∞, (ωbr∕ω0)2∼a∕λ in the nearest-neighbor regime, while (ωbr∕ω0)2∼5 in the plasma regime.