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During the 1986 Project FIRE (First International Satellite Cloud Climatology Project Regional Experiment) field campaign, four 22° halo-producing cirrus clouds were studied jointly from a groundbased polarization lidar and an instrumented aircraft. The lidar data show the vertical cloud structure and the relative position of the aircraft, which collected a total of 84 slides by impaction, preserving the ice crystals for later microscopic examination. Although many particles were too fragile to survive impaction intact, a large fraction of the identifiable crystals were columns and radial bullet rosettes, with both displaying internal cavitations, and radial plate-column combinations. Particles that were solid or displayed only a slight amount of internal structure were relatively rare, which shows that the usual model postulated by halo theorists, i.e., the randomly oriented, solid hexagonal crystal, is inappropriate for typical cirrus clouds. With the aid of new ray-tracing simulations for hexagonal hollow ended column and bullet-rosette models, we evaluate the effects of more realistic ice-crystal structures on halo formation and lidar depolarization and consider why the common halo is not more common in cirrus clouds.