## 3 Results
### Application of Derived Expressions  
There is minimal experimental data available in the literature measuring \(D/L_{tip}\) for different cantilever types. There are, however, two sources of such data for relatively short FastScan cantilevers. The first was published by [PERSON] et al. [49] and the second provided by [PERSON] et al. [43]. Table 1 provides details of 11 cantilevers and includes dimensions, data on the height of the tips, as well as the \(D/L_{tip}\) ratio, the torque factors, \(T_{beam}\), \(T_{tri. }\) and \(T_{trap. }\), and the Euler correction factors, \(K\). The approach angle, \(\alpha\), used to calculate the torque corrections is \(12^{0}\). The details of the first four cantilevers (FSA1-4) are from [PERSON] et al. [49] and are for cantilevers from the same wafer. Dimensional data for cantilevers FSA5-7, FSB1-2, and FSC1-2 are provided by the authors and the spring constant data for cantilevers which have been previously reported by [PERSON] et al. [43]. A number of important insights into the cantilever geometry and spring constant could be determined from Table 1. The \(w/w_{0}\) ratio is less than 0.5 for all cantilever types which indicates that while they all have a trapezoidal plan-view shape, they are closer to a triangular shape than beam-shaped. This has interesting results for each of the Euler beam and torque correction factors discussed in this work. For the Euler beam correction factor, \(K\), we can see that if we simply assume a triangular shape, then \(K=1.5\) and the spring constant for the cantilevers will be underestimated by 13 to 20%, while for the torque correction factors if the beam shape was used compared to the triangular, the extreme limits for the plan-view shape, we see a difference of only 3%. This indicates that for the torque correction factor, the plan-view shape has a much weaker dependence than for the Euler beam formula. Therefore given that all cantilevers are closer to a triangular shape, either the triangular torque correction (Equation (16)) or the trapezoidal approximation (Equation (17)) would be suitable. However Table 1 does show that the torque factor, regardless of plan-view shape, for these cantilevers is significant and if not accounted for, could lead to errors of up to \(\sim\)11%. The variation in tip height, \(D\), is significant between probes of different types but also between probes of the same type. For example, the probes, FSA1-4, are from the same batch and show a difference in tip height from 4.55 to 7.03 \(\upmu\)m. This results in \(T_{trap. }\) values varying from 0.89 to 0.95 for these probes. The \(K\) correction factor is similar for probes of the same type but does show some change between types and reflects that the plan-view geometry for each cantilever model is generally consistent. Figure 5: Shows how a V-shaped cantilever, depicted in (**a**) with arm width \(w_{a}\) and overall base width \(w_{o}\), approximates a triangular cantilever with the same overall base width \(w_{o}\), depicted in (**b**), as the arm width of the V-shaped cantilever, \(w_{a}\) approaches \(w_{0}/2\). or \(2w_{a}=w_{o}\). What is also interesting to note is how much stiffer cantilever FSA6 is compared to the other FastScan A cantilevers. It is between 30 to 60% stiffer than the other FSA cantilevers despite having similar dimensions and tip height. The most likely reason is the difference between the length of the cantilever from the base to the tip apex, \(L_{tip}\). The spring constant values reported in Table 1 used methods that determine the spring constant of the cantilever at the tip apex and if we compare \(L_{tip}\) for cantilever FSA6 to the other cantilevers, we see that this dimension is on average 11 to 20% smaller. Given there is an inverse cubic relationship between cantilever spring constant and \(L_{tip}\) (refer to Equations (1) and (2)), then it is not surprising that FSA6 is significantly stiffer than the other FSA probes.