Collective pinning in a film of finite thickness

The theory of collective pinning, which successfully explains the two-dimensional pinning of flux lines in thin superconducting films, is extended to films of finite thicknessd in order to describe the transition from two- to three-dimensional pinning. After short summaries of the collective pinning idea and of the (nonlocal) elastic properties of the flux-line lattice, the displacement correlation function of a weakly pinned flux-line lattice is calculated for a film. The transverse and longitudinal correlation lengths, which determine the correlation volume and the pinning forcej _c B, are obtained. The condition for the dimensional crossover is given.