mThe stopping of energetic ions by ultrathin films is of interest, as such films are mesoscopic structures which comprise the delamination limit of crystals, and as they are of practical interest because of their connection to microelectronics. A matter of special interest is the effect of the number of target layers on the stopping. Here we show that the stopping cross-section, S( N, u), of a system of N layers for projectiles with velocity u is a linear function of N-' for small N. Such behavior would be expected for N + m, in which case the constant term would be the bulk crystalline cross section at the specified projectile velocity and the coefficient of the linear term would be the lowest-order surface correction. Surprisingly, the constant coefficient from a small N fit is reasonably consistent (within limits of imprecision) with the crystalline cross section. This linear scaling of S with N-' for small N tracks the scaling behavior of the cohesive energy of the films. We discuss the physics and provide some illustrations. 0 1995 John Wiley & Sons, Inc.