Dynamics of quartz tuning fork force sensors used in scanning probe microscopy

We have performed an experimental characterization of the dynamics of oscillating quartz tuning forks which are being increasingly used in scanning probe microscopy as force sensors. We show that tuning forks can be described as a system of coupled oscillators. Nevertheless, this description requires knowledge of the elastic coupling constant between the prongs of the tuning fork, which has not yet been measured. Therefore, tuning forks have usually been described within the single oscillator or the weakly coupled oscillators approximation that neglects the coupling between the prongs. We propose three different procedures to measure the elastic coupling constant: an opto-mechanical method, a variation of the Cleveland method and a thermal noise based method. We find that the coupling between the quartz tuning fork prongs has a strong influence on the dynamics and the measured motion is in remarkable agreement with a simple model of coupled harmonic oscillators. The precise determination of the elastic coupling between the prongs of a tuning fork allows us to obtain a quantitative relation between the resonance frequency shift and the force gradient acting at the free end of a tuning fork prong.