Bubbles Containing 4, 6 and 12 Electrons in Liquid Helium-3

We have performed calculations of the properties of bubbles in liquid helium containing small numbers of electrons. We use an iterative approach to estimate the energy of the electrons inside a bubble of given shape, and then vary the shape of the bubble to find the minimum energy. For helium-3 we show that at zero applied pressure bubbles containing 4 electrons are unstable against breakup into single electron bubbles, but are stable at pressures between -0.23 and -0.15 bars. Bubbles with 6 electrons are stable between -0.17 and -0.05 bars and bubbles with 12 electrons are stable over the pressure range -0.1 to 0.08 bars.