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 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, ∼8 % of which corresponds to roton-phonon transitions and ∼42 % to roton R + ↔ R − ones.

Damping of vortex waves in a superfluid

“…kq denote scattering amplitudes depending on the momentum of the heat bath scatterers and the wave vector kẑ of the vortex wave. Recalling that the vortex coordinate can be written as r = r 0 (z) + r ′ (z) + zẑ and taking into account that r 0 (z) and r ′ (z) commute, the exponential factor in (17) can be factorized as e −i(kz−qz)z e −i(k−q)•r ′ (z) e −i(k−q)•r 0 (z) . Since the amplitude of the vortex wave was assumed to be very small, it is tempting to expand the last two exponentials retaining only first order terms in r ′ (z) and r 0 (z).…”

“…In fact, only the phonon drag force arising from r 0 (z) could be analyzed by Fetter [10], but we shall see that all sources of friction acting on r ′ (z) can be studied. To this aim, let us set r 0 (z) = 0 in (17) while retaining only the first order term in r ′ (z). Then, using the second-quantized expression for r ′ (z) (cf ( 12)), performing the integral in z and recalling the above-mentioned periodic boundary conditions, the interaction (17) reads…”

“…We showed that memory effects could represent up to ∼ 10% of the D value as the number of heat bath scatterers is increased, that is, such effects are found to be increasing with temperature and impurity concentration. The scattering amplitudes leading to such results reads as [11,17],…”

Feynman–Vernon model of a moving thermal environment

“…This is an approximation, since in general there will be some uncertainty -e.g. due to thermal fluctuations -influencing the measurement process and the initial conditions (14) should be correspondingly replaced by a distribution with finite x-and v-width.…”

“…Such a type of environments are met in some condensed matter systems and the formalism presented here can be used as a starting point for studying e.g. the effective dynamics of vortices in superconductors, where the superfluid or the normal component of the fluid represents an environment with its characteristic velocity [13][14][15].…”