By Bernard Beauzamy

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17) into Eq. 18) yields an explicit, secondorder differential equation relating the displacement of the cantilever to the buckling force, ∂2 12L δx T ( x ) = ht Fx . 19) will form the basis of the modeling of the response of the triangular cantilever to Fx . 2 Slope The slope of the cantilever at a point x along its length, θ xT ( x ), is obtained by integrating Eq. 20) which diverges at x = L. 3 Angular Spring Constant The cantilever angular spring constant, k θx ( x ), is the ratio of the force acting T on its tip to the resultant slope of the cantilever at the point x, Fx .

16) into Eq. 17) gives ξ z (x) = 3(2L − x ) x . 2 Bending Due to Fz The function ξ z ( x ) is used by the software of the AFM to calibrate the tip– sample force using the specifications of the particular cantilever in use. 6 Numerical Example: Si We can now present numerical examples for the response of the Si cantilever to a force acting in the zˆ direction. 2 shows the slope of the cantilever, θz ( x ), as a function of the distance x from its base for Fz = 10 (solid line), 20 (dashed line), and 30 nN (dotted line).

11) Replacing k in Eq. 11) with its value given by Eq. 10), yields the value of f in terms of L and w, f = 1 2π E w . 13) 51 52 3 Cantilever Conversion Tables and c2 = E . 14) Selecting a given cantilever material dictates the values of c1 and c2 . 16) f = c1 and k = c2 respectively. 18) and which involve the parameter set { L, w, k, f }. The solution of these two equations gives the value of each parameter in terms of the other two. 3, where each parameter is expressed in terms of the other two parameters.