Quantized Vortex Rings in Superfluid Helium, A Phenomenological Theory

Abstract

Rayfield and Reif (RR) and Careri, Cunsolo, and Mazzoldi (CCM) have done some experiments involving ion complexes moving through liquid helium in a constant uniform electric field. The results of RR give direct and compelling evidence that the ion complexes become attached to quantized vortex rings, the existence of which was conjectured by Feynman and Onsager. Their experiments, however, do not yield details concerning the interactions between ion complex and vortex ring. The results of CCM, on the other hand, exhibit a wealth of gross and fine structures, but they do not seem to be immediately understandable in simple terms. This paper attempts to understand all the results of RR and CCM in terms of a phenomenological theory of quantized vortex rings. Guided by experiments, we hypothesize that an ion complex can create a vortex ring only if its velocity $v$ and effective radius $R$ satisfy the condition $6\pi vR=\frac{\mathrm{nh}}{m}$, ($n=1, 2, 3, \dots$), where $h$ is Planck's constant and $m$ the helium mass. The number $6\pi$ is semiempirical. An experiment is suggested to test this hypothesis directly. After creation, the interaction of the vortex ring and the ion complex is described phenomenologically, with all parameters determined directly by experiments. This leads to a definite model that describes in detail the history of an ion complex in liquid helium. The determination of parameters makes extensive use of experimental data, but only one number from CCM. The model then reproduces quantitatively all the data of CCM.