Stability of a Lattice of Superfluid Vortices

Landau's superfluid hydrodynamics is applied to the vibration spectrum of a lattice of rectilinear vortices in both charged and neutral superfluid systems. The resulting vortex dynamics is identical with that of classical hydrodynamics: each vortex moves with the local superfluid velocity at its core. The only mode considered is one in which the vortices move without bending. This mode is unstable for all lattice structures in a neutral system (liquid helium II); in a charged system (type-II superconductors) the mode is unstable for a square lattice but stable for a triangular lattice. The corresponding long-wavelength dispersion relation is $\omega =\left(\frac{\mathrm{eB}}{\mathrm{mc}}\right)q^2\lambda d{\left(\frac{\sqrt{3}}{32\pi }\right)}^{\frac{1}{2}}$, where $B$ is the magnetic induction, $\lambda$ is the London penetration depth, $q$ is the wave number, and $d$ is the lattice spacing. An elasticity theory of the lattice vibrations in the charged system is shown to predict identical results. These calculations agree qualitatively with those of de Gennes and Matricon but disagree with those of Abrikosov, Kemoklidze, and Khalatnikov; the discrepancies are discussed in detail.