Collective Modes in Vortex Ring Beams

Abstract

The existence of quantized vortex rings was conjectured by Feynman¹ in his discussion of superfluid helium, but it was not until several years later that Rayfield and Reif² first identified a charged vortex ring in He II. Since then many experiments dealing with the individual properties of vortex rings were performed and a report on these can be found in a recent review article.³ In this work we are interested in investigating phenomena arising from the mutual interactions among charged vortices. We seek plasmalike effects that can be found in a gas made up of individual charged vortex rings. The conjecture of such a phenomenon was first made by Careri.* The initial work on the gaslike properties of charged vortices, referred to as dc experiments, was reported earlier.⁵ There it was shown that a single pulse of charged vortices does not smear out as would a pulse of, say, electrons in a vacuum, but in fact compresses longitudinally until it reaches a steady-state distribution. The final width of this “bell”-shaped distribution was found to be rather insensitive to the initial width of the pulse as well as the energy. Hasegawa and Varma⁶ explained this behavior by a simple one-dimensional theory which assumed that the vortices acted as particles in a gas. The theory includes the Coulomb interaction among the vortices as well as pressure due to fluctuations. They assume that the vortices are, in fact, not monoenergetic but that there exists a distribution whose average value is equal to the accelerating energy given to the beam.† The subsequent work, referred to as the ac experiments and reported here, involves the study of collective modes of oscillation of these vortex beams which also arise from the above interactions. Again Hasegawa and Varma⁶ take a one-dimensional picture but use the Vlasov equation, instead of the fluid equation used earlier, to write a phase-space distribution function f (x, p, t). This is solved and a wave dispersion relation is found which shows that an instability in the wave would occur if k was below a critical wave number k _c . Due to this instability the wave is expected to grow as a function of the propagation distance.