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.

我们研究了在靠近液体表面的金属尖端的不均匀电场中，超流体氦表面捕获的电荷的空间分布。电荷的静电压力会导致液体表面变形，从而导致“小丘”（称为“泰勒锥”）或“凹痕”，具体取决于尖端位于表面上方还是下方。我们使用有限元模拟来计算表面轮廓以及尖端附近的相应电荷密度。典型的电场E处于几kV / cm的范围内，最大平衡表面变形的高度大约为（但略小于）液体⁴的毛细管长度He（0.5 mm）和小丘或酒窝中的基本电荷的最大数密度受电流体动力学不稳定性的限制，约为10 ¹³  m ^-2。这些结果可用于根据测得的表面轮廓确定液氦表面的电荷密度。他们还暗示，体氦表面的不均匀电场不允许人们将电子密度增加到实质上超过均匀场的极限，因此对于达到预期表面态电子显示出偏离电子的密度状态不可行。经典行为。讨论了一些替代解决方案。