Exact linear dispersion relation for the Sigmund model of ion sputtering

The exact linear dispersion relation is obtained for the Sigmund model, a model which forms the basis of nearly all theories of the nanoscale patterns produced by ion bombardment of solid surfaces. It is demonstrated that the amplitude of a surface disturbance of short wavelength grows exponentially in time, and so the Sigmund model is ill defined unless one or more smoothing terms are added to the equation of motion. It is also shown that effective surface diffusion [M. A. Makeev and A.-L. Barabasi, Appl. Phys. Lett. 71, 2801 (1997)] cannot be the only physical mechanism that smoothes the solid surface, even at low temperatures.