Nucleation of vortex rings by negative ions in liquid helium at low temperatures

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Abstract
The critical velocity is calculated for the formation of vortex rings by negative ions in liquid helium. The system is described by a Lagrangian, from which the Hamiltonian is derived. Care is taken to ensure that the equation of continuity is satisfied. This is done by defining the surface of the core by a contour of constant A, where A is a component of the vector potential. The model which is analysed is that proposed by Schwarz and Jang (1973) in which a vortex ring is formed encircling the ion. The new definition of the core leads to a critical velocity if the vortex is treated as a classical object. However, there is a potential barrier which stops the classical vortex from entering the fluid from the surface of the ion. The vortex is then treated as a quantum object in a potential well, and account taken of its zero-point motion, and its energy levels. The critical velocity for the formation of vortex rings occurs when the energy of the bare ion equals the energy of one of these energy levels. These critical velocities are calculated as a function of pressure and compared with the values obtained from experiment.