Collective Modes of Vortex Lattices

Abstract

In this paper, we study the dynamics of lattices of rectilinear vortices both in neutral systems (ideal classical fluids and He II) and in charged systems (type-II superconductors). The model we consider differs from those which have been studied until now in that it takes into account that a perturbation in a system propagates with finite velocity; this effect is taken into account by the introduction of a retardation time between a displacement of the core of a vortex and the relative modification of its velocity field. Two different forms for the retardation are considered: one for a neutral system, in which the propagation of a perturbation has wave character, and one for a charged system, in which this propagation seems to have diffusion character. We find that the stability of a vortex lattice is not modified by the presence of retardation. As regards the collective modes, we study only motions in which vortices do not bend. In a neutral system, the retardation completely modifies the spectrum ${\ensuremath{\omega}}_{R}(k)$ for very small wave numbers $k$: we find, indeed, ${\mathrm{lim}}_{k\ensuremath{\rightarrow}0}{\ensuremath{\omega}}_{R}(k)\ensuremath{\ne}o$; for larger wave vectors it introduces a damping of the collective modes. In a charged system, the collective modes become damped or aperiodic in the presence of retardation. In some materials this damping can be very large on the basis of our model.