The effect of pressure on statics, dynamics and stability of multielectron bubbles

The effect of pressure and negative pressure on the modes of oscillation of a multi-electron bubble in liquid helium is calculated. Already at low pressures of the order of 10-100 mbar, these effects are found to significantly modify the frequencies of oscillation of the bubble. Stabilization of the bubble is shown to occur in the presence of a small negative pressure, which expands the bubble radius. Above a threshold negative pressure, the bubble is unstable. Typeset using REVTEX 1 Multielectron bubbles (MEBs) in liquid helium are fascinating entities expected to display novel resonant behavior, the possibility of superconductivity, and sufficient electron density for Wigner crystallization and quantum melting. MEBs are bubbles inside the helium, containing only electrons that form a curved two-dimensional electron gas (2DEG) on the spherical surface of the bubble, the width of the thin spherical shell that conforms to the helium surface being of order 5-20 Å [1,2], compared to 70-80 Å on a flat surface [3]. New efforts to trap and localize MEBs for long periods of time [4] have led to further consideration of the long term stability of bubbles. In this letter we discuss the use of pressure, and in particular negative pressure to stabilize MEBs against dynamic instabilities. The radius of the bubble depends on the enclosed charge. In a simplified model, valid for large bubbles, the radius is given by RC = [e N/(16πσε)] where e is the electron charge, N is the number of electrons in the bubble, σ is the surface tension of helium and ε is the dielectric constant of helium [1]. A single electron bubble has a radius of 17.2 Å [5] while, for example, a bubble of 10 electrons has a theoretical radius of 1 micron. There is some question about the static stability of an MEB: since the energy of the bubble, defined as the sum of the electrostatic and the surface tension energy, is proportional to N [1], clearly the energy of two bubbles with N/2 electrons is lower than that of a single bubble with N electrons. Evidently fissioning is hindered by a formation barrier, since MEBs have been observed [6]. Gravitational fields may flatten very large bubbles and lead to instability [1]. Dexter and Fowler [7] showed that two-electron bubbles are unstable. Salomaa and Williams [2] considered the dynamic stability against fissioning off of single electrons from large bubbles and found stability against this decay mode for bubbles with N greater than 15-20. It is straightforward to show that a positive pressure radially stabilizes a bubble, although angular modes can be unstable. With increasing negative pressure the bubble is first absolutely stabilized and then explodes, as we shall describe. Salomaa and Williams also considered the dynamic instability due to one of the surface oscillation modes, or ripplons, being soft (zero frequency) and found that this mode may be stabilized by the anharmonicity in the bubble’s radial oscillation that results in a radius larger than RC. 2 In this communication we study pressure related effects on the frequency of the modes of oscillation, on the equilibrium bubble radius, and on the stability of MEBs; we show the counterintuitive result that positive pressures can destabilize all higher angular modes, while negative pressures have a window of stabilization. We neglect gravity so that the MEBs are spherical. The frequencies of the modes of oscillation of a charged droplet were first calculated by Rayleigh [8], and in the case of a charged bubble by Plesset and Prosperetti [9]. We first set up a Lagrangian formalism to calculate the spherical ripplon modes. We then consider the effect of pressure on the static and dynamic properties of the bubble. The surface of the bubble is described by a function R(θ, φ) that gives the distance of the surface from the geometrical center of the bubble, in the direction given by the two spherical angles θ, φ . This function can be written as R(θ, φ) = Rb + u(θ, φ),where Rb is the angle-averaged radius of the bubble, and u(θ, φ) describes the deformation of the surface from a sphere. This deformation can be expanded in a series of spherical harmonic deformations Ylm(θ, φ) with amplitude Qlm u(θ, φ) = ∞