Analysis of the Frequency Shift versus Force Gradient of a Dynamic AFM Quartz Tuning Fork Subject to Lennard-Jones Potential Force.

A self-sensing and self-actuating quartz tuning fork (QTF) can be used to obtain its frequency shift as function of the tip-sample distance. Once the function of the frequency shift versus force gradient is acquired, the combination of these two functions results in the relationship between the force gradient and the tip-sample distance. Integrating the force gradient once and twice elucidates the values of the interaction force and the interatomic potential, respectively. However, getting the frequency shift as a function of the force gradient requires a physical model which can describe the equations of motion properly. Most papers have adopted the single harmonic oscillator model, but encountered the problem of determining the spring constant. Their methods of finding the spring constant are very controversial in the research community and full of discrepancies. By circumventing the determination of the spring constant, we propose a method which models the prongs and proof mass as elastic bodies. Through the use of Hamilton's principle, we can obtain the equations of motion of the QTF, which is subject to Lennard-Jones potential force. Solving these equations of motion analytically, we get the relationship between the frequency shift and force gradient.