## 4 Discussion  
In this work, we have derived a simple correction factor to the Euler beam equation for beam-shaped cantilevers which allows the spring constant of trapezoidal plan-view shaped cantilevers to be determined (provided the material properties and dimensions are known). The derivation for this correction factor, \(K\), is based on the analytical work by [PERSON] and [PERSON] [1]. We have also derived simple expressions for the torque correction factor for triangular plan-view shaped cantilevers, as well as an approximation for the torque correction for trapezoidal plan-view shaped cantilevers. The torque corrections were based on the work by [PERSON] et al. [55] and [PERSON] [40]. Analysis of the spring constant and dimensions of 11 Bruker FastScan AFM probes revealed that the \(K\) correction factor was between 1.2 and 1.31, which is a significant correction given that \(K=\)1 for a beam-shaped cantilever and \(K=\) 1.5 for a triangular-shaped cantilever. The torque correction factors for the FastScan cantilevers were significant, and if not accounted for could introduce errors as high as \(\sim\)11%. However, the torque corrections did show a much weaker dependence on the plan-view geometry, and there was, at maximum, only a 3% difference between the torque correction factors for a beam-shaped cantilever and a triangular-shaped cantilever for the same cantilever. However, given \(w/w_{0}\) is less than 0.5 for all FastScan cantilevers studied in this work we would expect the triangular torque correction, or the trapezoidal approximation, to be suitable for Bruker FastScan cantilevers. It is also important to note that as AFM cantilevers become smaller the \(D/L_{tip}\) ratio will increase, and the plan-view shape of the cantilever will play a more important role in determining which torque correction factor is the most appropriate to use. Close analysis of the different dimensions for the shortest cantilever type (FastScan A) indicated that the \(L_{tip}\) dimension is crucial in contributing to the variability in the spring constant for these types of AFM probes. This is due to the inverse cubic relationship between \(L_{tip}\) and the cantilever spring constant (refer to Equations (1) and (2)) and indicates that the smaller AFM cantilevers become the more significant \(L_{tip}\) will become. This will impact methods that do not calibrate the spring constant of the cantilever at the tip apex such as those based on the resonant frequency of the cantilever. For these techniques, the precise measurement of \(L\) and \(L_{tip}\) will be required. We also applied the derived correction factors to determine Young's modulus of 11 Bruker FastScan AFM probes. Our results gave an average Young's modulus for the silicon nitride the cantilevers are composed of equal to \(200\pm 34\) GPa which is comparable to the results reported by other research groups [50; 60], and equal to that quoted to us by the manufacturer. We also applied the [PERSON] indirect [16] method to calibrate the FastScan A type cantilevers and found good agreement between the [PERSON] indirect [16] method and the more precise results reported by [PERSON] et al. [43] and [PERSON] et al. [49] for these probes. This will be useful for AFM users as research has shown that some standard methods, such as the thermal noise method, struggle to provide accurate results for these types of AFM probes. \begin{table}
\begin{tabular}{c c c} \hline \hline
**AFM Probe** & **Spring Constant (N/m)** & **Sader Indirect Method (N/m)** \\ \hline FSA1 & 15.81 & 15.5 \\ \hline FSA2 & 14.24 & 14.1 \\ \hline FSA3 & 17.05 & 19.0 \\ \hline FSA4 & 11.89 & 11.9 \\ \hline FSA5 & 21.4 & 19.1 \\ \hline FSA6 & 32.3 & 27.3 \\ \hline FSA7 & 18.5 & 19.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Shows a comparison between the spring constant of the FastScan A cantilevers determined using the methods described in [PERSON] et al. [43] and [PERSON] et al. [49] and the Sader indirect method [16].