Inherent coupling of lateral forces in nc-AFM using a qPlus sensor

Many of the best AFM experiments (or at least in my opinion) have been performed with the qPlus sensor. A sensor made from a watch tuning fork, with a metal tip glued on:

An Omicron qPlus sensor.

An Omicron qPlus sensor.

When someone asked me about the possibility of using thermal tuning to measure the spring constant of a qPlus sensor, I thought perhaps the length with affect the calculations. In short, after lots of maths, I found out that there was relatively little change to the spring constant, but the tip length causes the apex to move at an angle relative to the surface. This can lead to inaccurate force spectroscopy measurements:

In positions where lateral forces are significant (in the direction of the sensor's lateral motion, blue arrow) significant errors are introduced into force spectroscopy measurements.

In positions where lateral forces are significant (in the direction of the sensor’s lateral motion, blue arrow) significant errors are introduced into force spectroscopy measurements.

This work was published in Beilstein Jounrnal of Nanotechnology under the title: “Calculation of the effect of tip geometry on noncontact atomic force microscopy using a qPlus sensor“.
(The paper is open access).