Electron Bubble - Electron Repulsion From The Surface of Helium

Electron Repulsion From The Surface of Helium

Electrons are attracted to liquid helium due to the difference in dielectric constants between the gas and liquid phase of helium. The negative electron polarizes the helium at the surface, leading to an image charge which binds it to the surface. The electron is forbidden from entering the liquid for the same reason hydrogen atoms are stable: quantum mechanics. The electron and image charge form a bound state, just as an electron and proton do in a hydrogen atom, with a minimum average separation. In this case, the minimum energy is about 1 eV (a moderate amount of energy on an atomic scale).

When an electron is forced into liquid helium rather than floating on its surface, it forms a bubble rather than entering the liquid. The size of this bubble is determined by three main factors (ignoring small corrections): the confinement term, the surface tension term, and the pressure-volume term. The confinement term is purely quantum mechanical, since whenever an electron is tightly confined, its kinetic energy goes up. The surface tension term represents the surface energy of the liquid helium; this is exactly like water and all other liquids. The pressure-volume term is the amount of energy needed to push the helium out of the bubble.

Here E is the energy of the bubble, h is Planck's constant, m is the electron mass, R is the bubble radius, α is the surface energy, and P is the ambient pressure.