Critically Charged Superfluid 4 He Surface in Inhomogeneous Electric Fields

We have studied the spatial distribution of charges trapped at the surface of superfluid helium in the inhomogeneous electric field of a metallic tip close to the liquid surface. The electrostatic pressure of the charges generates a deformation of the liquid surface, leading to a “hillock” (called “Taylor cone”) or “dimple”, depending on whether the tip is placed above or below the surface. We use finite element simulations for calculating the surface profile and the corresponding charge density in the vicinity of the tip. Typical electric fields E are in the range of a few kV/cm, the maximum equilibrium surface deformations have a height on the order of (but somewhat smaller than) the capillary length of liquid 4He (0.5 mm), and the maximum number density of elementary charges in a hillock or dimple, limited by an electrohydrodynamic instability, is some 1013 m−2. These results can be used to determine the charge density at a liquid helium surface from the measured surface profile. They also imply that inhomogeneous electric fields at a bulk helium surface do not allow one to increase the electron density substantially beyond the limit for a homogeneous field, and are therefore not feasible for reaching a density regime where surface state electrons are expected to show deviations from the classical behavior. Some alternative solutions are discussed.