Vanishing Cantilever Calibration Error with Magic Ratio Atomic Force Microscopy

An analysis is presented of the error propagation through a simplified contact model for atomic force microscopy (AFM) static force spectroscopy, one that is meant to be intuitive and pedagogically informative rather than maximally realistic. It is shown that an important dimensionless ratio appears in several critical locations in the closed form error equation, and that under common calibration methods there exists a ``magic'' ratio where an important class of systematic calibration errors cancel out completely. This argument is extended to more complex contact models and the presence of a magic ratio is demonstrated using real data. A method is shown to identify and operate at optimal conditions in commercial AFMs. Targeting this magic ratio provides a simple, comprehensive guideline for cantilever selection, system calibration, and imaging parameter selection. Analysis shows that an important dimensionless ratio appears in several critical locations in the closed form error equation, and that under common calibration methods there exists a ``magic'' ratio where an important class of systematic calibration errors cancel out completely. Targeting this magic ratio provides a simple, comprehensive guideline for cantilever selection, system calibration, and imaging parameter selection.