Friction force on a vortex due to the scattering of superfluid excitations in helium II

Abstract

The longitudinal friction acting on a vortex line in superfluid ${}^{4}\mathrm{He}$ is investigated within a simple model based on the analogy between such vortex dynamics and that of the quantal Brownian motion of a charged point particle in a uniform magnetic field. The scattering of superfluid quasiparticle excitations by the vortex stems from a translationally invariant interaction potential which, expanded to first order in the vortex velocity operator, gives rise to vortex transitions between nearest Landau levels. The corresponding friction coefficient is shown to be, in the limit of elastic scattering (vanishing cyclotron frequency), equivalent to that arising from the Iordanskii formula. Proposing a simple functional form for the scattering amplitude, with only one adjustable parameter whose value is set in order to get agreement to the Iordanskii result for phonons, an excellent agreement is also found with the values derived from experimental data up to temperatures about 1.5 K. Finite values of the cyclotron frequency arising from recent theories are shown to yield similar results. The incidence of vortex-induced quasiparticle transitions on the friction process is estimated to be, in the roton dominated regime, about 50% of the value of the friction coefficient, $\ensuremath{\sim}8%$ of which corresponds to roton--phonon transitions and $\ensuremath{\sim}42%$ to roton ${R}^{+}\ensuremath{\leftrightarrow}{R}^{\ensuremath{-}}$ ones.