Forces and frequency shifts in atomic-resolution dynamic-force microscopy

True atomic resolution in vacuum with a force microscope is now obtained routinely by using the frequency shift of an oscillating cantilever as the imaging signal. Here, a calculation is presented that relates the frequency shift to the forces between tip and sample for both large and small oscillation amplitudes. Also, the frequency versus distance data for van der Waals dominated tip-sample interactions is related to the geometry of the tip apex. Published frequency versus distance data are used to show that the apex of tips providing atomic resolution is faceted and not rounded. Further, an extended jump-to-contact criterion for large amplitudes is established.