## 3 Results
### Application of Derived Expressions
The same analysis for the FastScan B cantilevers gives a thickness of 0.29 \(\upmu\)m using our equation compared to 0.30 \(\upmu\)m, which is the default value quoted by the manufacturer for FastScan B probes. This is a 3.3% difference. The same analysis for the FastScan C cantilevers gives a thickness of 0.31 \(\upmu\)m using our equation compared to 0.30 \(\upmu\)m, which is the default value for the thickness quoted by the manufacturer for FastScan C probes. This is a 3.3% difference, suggesting sufficient overall accuracy for successful practical application. [PERSON] et al. [43] also demonstrated that the spring constant calibration of smaller cantilevers is more challenging than standard larger AFM cantilevers. This is particularly true for the FastScan A model, which is the smallest and stiffest of the FastScan cantilever models. [PERSON] et al. [43] showed that the thermal noise method, often built-in to modern AFM instruments, overestimated the spring constant of FastScan A cantilevers significantly (30 to 70%). This was attributed to the reduced thermal noise profile for these types of probes, and this effect has been observed by other research groups [24]. Therefore, we have applied the Sader indirect method [16; 34] to the calibration of the spring constant of the FastScan A cantilevers reported in this work. In the Sader indirect method [16], typically the spring constant of one cantilever is accurately measured and then the term \(E_{c}t^{3}\) is determined using an appropriate Euler beam equation, based on the cantilever plan-view geometry. This \(E_{c}t^{3}\) value is then assumed to be constant for all cantilevers of the same type and subsequent cantilevers can then be calibrated with only the length and width of the cantilever requiring measurement. In our case, we use a slightly different approach. We have already determined an average effective Young's modulus, \(E_{c}\), for FastScan cantilevers of \(179\pm 29\) GPa. We, therefore, determine an average cantilever thickness, \(t\), for the FastScan A probes from the data presented in Table 1, which gives \(0.60\pm 0.04\)\(\upmu\)m, with the error quoted being one standard deviation in the data. Combining this with \(E_{c}\) gives us an \(E_{c}t^{3}\) value of \(3.87\pm 0.97\times 10^{-8}\) Nm for our FastScan A probes. The uncertainty on the \(E_{c}t^{3}\) term is approximately 25% and has contributions from the uncertainty on the \(E_{c}\) term and the uncertainty on the average thickness for the FastScan A probes, \(t\), which is a cubed term. We have used this \(E_{c}t^{3}\) value to then calculate the spring constant of FastScan A cantilevers FSA1-7. The Sader indirect method [16] results are reported in Table 3 and compared to the more accurate spring constant values cantilevers as reported by [PERSON] et al. [49] and [PERSON] et al. [43]. The agreement between the accurate spring constant values and the Sader indirect method is good with all Sader indirect results being within 2 to 15% of the accurate values. Therefore, the \(E_{c}t^{3}\) value determined in this work could be used by other groups to calibrate the spring constant of the FastScan A cantilevers with the proviso that the uncertainty of the spring constants using this method could be as high as 25% given the uncertainty on the \(E_{c}t^{3}\) term calculated in this work.