**Bubbles With Electrons in Excited States**

The electron in an electron bubble is normally in the 1S ground state with a wave function *ψ *given by:

where R is the radius of the bubble and C_1S is the normalization constant. The electron energy is h²/8mR².

The electron can be excited to the 1P state through application of light from a carbon dioxide laser. The wave function then becomes:

where k = 4.4934 / R and C_1P is another normalization constant.

When the electron is in this state, the shape of the electron bubble changes which results in a further change in the wave function. The equilibrium shape of the bubble is determined by a balance between surface tension pulling the surface in and the outward pressure exerted by the electron. This outward pressure is

with ∇*ψ* evaluated at the surface.

A calculation of the bubble shape gives this result[1]:

One can perform similar calculations for the other quantum states:

We can also calculate the negative pressures at which these different bubble explode [2]:

State |
P_c (bars) |

1S | -1.89 |

2S | -1.33 |

1P | -1.63 |

2P | -1.22 |

1D | -1.49 |

These critical pressures have been measured for the 1S and the 1P states.[3]

The shapes shown above correspond to mechanical equilibrium, i.e., the shape that a bubble will have after the electron has been excited and the bubble shape has evolved to reach a state of minimum energy. It is interesting to consider how this equilibrium shape is reached. At high temperatures (“high” here means greater than about 1.5 K) there is a high density of rotons. These rotons scatter from the surface of the bubble and damp its motion. Thus, the bubble smoothly relaxes from it original spherical shape to the new equilibrium shape. At lower temperatures, the damping of the surface becomes very small, and when the bubble reaches the equilibrium shape the inertia of the moving liquid causes the shape to continue to evolve [1]

The way that the shape of the bubble evolves with time at low temperature after optical excitation can be calculated using time-dependent density functional methods.[4, 5] These results show that what happens to the bubble depends on the pressure * *in the liquid. For the bubble overshoots the equilibrium shape but then relaxes back to it. To see a movie of this time evolution *click here*** **. But above a critical pressure the bubble breaks into two parts

*click here***.**

The calculations just mentioned are for a bubble which initially is perfectly spherical. Because of thermal fluctuations, the initial shape will never be a perfect sphere. This may affect how the shape evolves, and it is hard to perform a simulation to allow for this.

What happens to the electron when a bubble breaks into two (there is only one electron!) is something we are still trying to understand.

- ^H.J. Maris,
*On the Fission of Elementary Particles and Electrons in Liquid Helium*, J. Low Temp. Phys.,**120**, 173 (2000). - ^H.J. Maris and D. Konstantinov,
*Bubbles in Liquid Helium Containing Electrons in Excited States*, J. Low Temp. Phys.**121**, 615 (2000). - ^D. Konstantinov and H.J. Maris
*Detection of Excited State Electron Bubbles in Superfluid Helium*, Phys. Rev. Lett.,**90**, 025302 (2003). - ^D. Mateo, M. Pi, and M. Barranco,
*Evolution of the Excited Electron Bubble in Liquid 4He and the Appearance of Fission-like Processes*, Phys. Rev. B**81**, 174510 (2010). - ^D. Jin, W. Guo, W. Wei and H.J. Maris,
*Electrons in Superfluid Helium-4*, J. Low Temp. Phys.**158**, 307 (2010).