## 1 Introduction
### Euler Beam Equation  
The standard Euler beam equation, relating the spring constant to the cantilever dimensions and material properties, for a beam-shaped cantilever is expressed in Equation (1)  
\[k=E_{c}t^{3}w/(4L^{3}) \tag{1}\]  
where \(k\) is the spring constant for the cantilever, \(E_{c}\) is the Young's modulus for the cantilever, \(w\) is the width of the cantilever, \(t\) is the thickness, and \(L\) is the length of the cantilever. For AFM cantilevers, it is important to note that the length of the cantilever is often defined as the distance from the base to the tip apex. This is appropriate when applying Euler beam equations to AFM cantilevers since the tip apex is where the force is applied. Therefore, to correct the spring constant of the cantilever from the beam end to the position of the tip apex, the following formula must be applied:  
\[k_{tip}=k_{end}\left(L/L_{tip}\right)^{3} \tag{2}\]  
where \(k_{tip}\) is the spring constant at the tip apex position, \(k_{end}\) is the spring constant at the very end of the cantilever, and \(L_{tip}\) is the length of the cantilever from the base to the tip. Equation (2) is for a beam-shaped cantilever geometry, but it has been shown to be a good approximation for all cantilever geometries [14; 38]. When using Equation (1), the cantilever dimensions and Young's modulus are the primary sources of uncertainty. Since the thickness of the cantilever is a cubed term (and often less than 1 \(\upmu\)m), in Equation (1), it contributes significant uncertainty to the calculation. To determine this dimension with an uncertainty of 5% or less requires careful edge-on imaging with an experienced scanning electron microscopy (SEM) operator. The length of the standard commercially available cantilevers is typically between 100 to 200 \(\upmu\)m, and the width between 20 to 50 \(\upmu\)m, therefore, these dimensions can be measured with good accuracy using SEM, or a sufficiently powerful optical microscope. The majority of AFM cantilevers are composed of silicon or silicon nitride. Provided that the crystal orientation is known, Young's modulus could be calculated to high-precision in the case of silicon. However, it is well established that Young's modulus of silicon nitride can vary depending on the conditions of the manufacturing process. [PERSON] and [PERSON] [50] reviewed the Young's modulus that different research groups reported for silicon nitride films produced by chemical vapour deposition (CVD) and found a reported range of 66 GPa [51] to 373 GPa [52]. AFM cantilevers often have their back coated with reflective metals, such as gold or aluminium, to improve the reflection of the laser typically used to measure the motion of the cantilever. This is the standard optical lever arrangement implemented in many commercial AFMs. While the thicknesses of these metallic coatings are small (50 to 100 nm), Young's modulus of gold and aluminium is sufficiently reduced compared to silicon and silicon nitride where these coatings can decrease the effective Young's modulus of the overall cantilever by as much as 10 to 20% [46]. Fortunately, simple expressions have been derived by [PERSON] et al. [53] to account for these types of coatings. There have been a number of papers devoted to deriving Euler beam equations to determine the spring constant of cantilevers that are not beam-shaped, but these are primarily focussed on V-shaped cantilevers [14; 54]. A landmark theoretical paper that concentrated on deriving Euler beam formulas for a number of plan-view geometries was authored by [PERSON] and [PERSON] [1]. In the [PERSON] and [PERSON] article, general analytical formulas for trapezoidal, triangular, and V-shaped cantilevers were derived. In this work, we use [PERSON] and [PERSON]'s research as the basis to determine a simple correction factor to Equation (1) that could determine the spring constant for trapezoidal-shaped cantilevers.