An analysis is provided of the change in topography and consequent apparent change in sputtering yields of surfaces as a result of ion sputtering using Yamamura et al.'s model for the angular dependence of the sputtering yield for monatomic primary ions. This is extended to provide a general scheme for any monatomic primary ion and any amorphous elemental solid target. Details are provided of the topographic changes for surfaces with an undulating cosine profile sputtered with primary ions normal to, and at 45° to, the average surface and also for spherical particles. Smoothing occurs at normal incidence and both smoothing and roughening can occur at 45°. Predictions show how the sputtering rate of the material varies with the depth sputtered as a result of the topographical changes, the rate either increasing or decreasing. The effect of roughening usually increases the sputtering rate with values increasing up to four times. The effect of sample rotation is a fast smoothing. Conditions are described for the optimal profiling of particles with a layered structure and calculated profiles are evaluated. The effect of rotation on particles is discussed in relation to different instrumental geometries. In this work, recoil mixing, redeposition, diffusion and the topographic effects at the size of the individual ion cascade are ignored. © Crown copyright 2011. Reproduced by permission of the Controller of HMSO and the Queen's printer for Scotland.