= Solution
Assume a conventional finite-radius tapping tip whose apex radius and shank are not negligible compared with a $10\,\mathrm{nm}$ trench. \b[The recorded depression is shallower and tip-limited], because the tip contacts the sides before its apex reaches the sharp bottom. Its apparent shape is the accessible envelope of the tip and specimen, not a faithful geometric section.
At a descending edge the separation initially increases, so the oscillation amplitude tends to rise above its setpoint until the controller moves downward. At an ascending edge it tends to fall, until the controller moves upward. For a narrow trench these transients can overlap or partly cancel. With ideal tracking the amplitude remains at its setpoint even when the height is wrong because of tip geometry. Thus a small amplitude error is not proof that the trench bottom was reached.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2018/paper-1-tapping-response.png]
{title=Original illustrative tapping-mode responses to narrow and wide depressions; heights are tip-accessible envelopes and amplitude errors depend on the stated finite-bandwidth model}
Width alone does not determine a unique trace: depth, tip radius, cone angle, scan speed and feedback bandwidth are also needed. The plotted traces illustrate the common finite-tip interpretation rather than infer those missing quantities from the drawing.
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