Intrinsic dissipation in atomic force microscopy cantilevers

In this paper we build a practical modification to the standard Euler-Bernoulli equation for flexural modes of cantilever vibrations most relevant for operation of AFM in high vacuum conditions. This is done by the study of a new internal dissipation term into the Euler-Bernoulli equation. This term remains valid in ultra-high vacuum, and becomes particularly relevant when viscous dissipation with the fluid environment becomes negligible. We derive a compact explicit equation for the quality factor versus pressure for all the flexural modes. This expression is used to compare with corresponding extant high vacuum experiments. We demonstrate that a single internal dissipation parameter and a single viscosity parameter provide enough information to reproduce the first three experimental flexural resonances at all pressures. The new term introduced here has a mesoscopic origin in the relative motion between adjacent layers in the cantilever.