Commensurate and Incommensurate Vortex States in Superconductors with   Periodic Pinning Arrays
  As a function of applied field, we find a rich variety of ordered and partially-ordered vortex lattice configurations in systems with square or triangular arrays of pinning sites. We present formulas that predict the matching fields at which commensurate vortex configurations occur and the vortex lattice orientation with respect to the pinning lattice. Our results are in excellent agreement with recent imaging experiments on square pinning arrays [K. Harada et al., Science 274, 1167 (1996)]. 
C. Reichhardt, C.J. Olson, and Franco Nori