## 3 Results
### Application of Derived Expressions
While the absolute difference between the \(L_{tip}\) for FSA6 and the other FSA cantilevers is only a few microns, this becomes more significant to the spring constant the shorter the cantilever becomes. Therefore, methods that determine the spring constant at the tip apex have an advantage. For methods that do not determine the spring constant at the tip apex, such as those that rely on measuring the resonant frequency of the cantilever, care must be taken to accurately determine \(L\) and \(L_{tip}\) and then use Equation (2) to account for this effect. Indeed, the smaller AFM cantilevers become the more important this effect will be and should be noted by all AFM users intending to perform accurate force measurements using small cantilevers. To apply the analysis in Sections 3.1 and 3.2, we use the results from Table 1 and Equation (6) to determine Young's modulus of each of the cantilevers in Table 1. We report the effective Young's modulus, but also Young's modulus of silicon nitride the cantilevers are composed of. To determine Young's modulus for the silicon nitride required for correcting the effective Young's modulus values for the metal coatings on the back of the cantilevers using equations derived by [PERSON] et al. [53]. This data is displayed in Table 2, and for reference shows the spring constant values for each cantilever as reported in Table 1.  
The results in Table 2 show an average effective Young's modulus and silicon nitride Young's modulus of 179 \(\pm\) 29 and 200 \(\pm\) 34 GPa, respectively for all the cantilevers, with the error quoted representing one standard deviation in the data. The average Young's modulus for the silicon nitride is in excellent agreement with the value provided to the authors by the manufacturer, which is 200 GPa. The values also fit within the range determined by [PERSON] et al. [50] who reported Young's modulus for silicon nitride in the cantilevers they analysed to be, on average, 202 \(\pm\) 45 GPa. [PERSON] [60] \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**AFM Probe**} & \multirow{2}{*}{**Spring Constant (N/m)**} & \multicolumn{2}{c}{**Effective Young’s**} & **Silicon Nitride Young’s** \\  & & **Modulus of Cantilever** & **Modulus of Cantilever** \\  & & **(GPa)** & **(GPa)** \\ \hline FSA1 & 15.81 & 201.3 & 229.2 \\ \hline FSA2 & 14.24 & 210.1 & 240.6 \\ \hline FSA3 & 17.05 & 151.9 & 168.3 \\ \hline FSA4 & 11.89 & 216.4 & 238.9 \\ \hline FSA5 & 21.4 & 155 & 171 \\ \hline FSA6 & 32.3 & 161 & 178 \\ \hline FSA7 & 18.5 & 132 & 143 \\ \hline FSB1 & 2.24 & 161 & 177 \\ \hline FSB2 & 2.05 & 164 & 181 \\ \hline FSC1 & 0.938 & 204 & 230 \\ \hline FSC2 & 0.916 & 212 & 241 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Shows the spring constant, effective Young’s modulus of the cantilevers, and [PERSON]’s modulus of the silicon nitride the cantilevers are composed of which has been corrected for the thin metal coatings on the back of the cantilevers using equations derived by [PERSON] et al. [53]. also reported [PERSON]'s modulus for micromechanical cantilevers made from silicon nitride equal to \(194\pm 14\) GPa, which is in good agreement with our average value for silicon nitride. This suggests the effective application of the method and correction factors derived in this work. Further analysis can reveal how robust Equation (6) is if we use the 179 GPa result and default values for \(k\), \(L_{tip}\), \(w_{0}\) provided by the manufacturer for each type of FastScan probe, and then determine the thickness using Equation (6). If the cantilever thickness we calculate is within 10% of the default value quoted by the manufacturer, then our Equation (6) can be considered acceptably accurate. Default values for each cantilever type are provided in Table S1 of the Supplementary material. Our analysis reveals that for FastScan A cantilevers, we obtain a thickness measurement of 0.54 \(\upmu\)m and the default value provided by the manufacturer for the thickness is 0.58 \(\upmu\)m. This is a difference of 6.9% compared to the default value.