Force is calculated from Hooke's Law: F = -kx
F = force
k = spring constant
x = cantilever deflection

Three forces are generally present during contact-AFM operation.

Electrostatic Repulsion Force
Atoms first weakly attract each other as they are brought together. This increases until the atoms are so close their electron clouds begin to repel each other electrostatically. As the interatomic separation continues to decrease the electrostatic repulsion progressively weakens the attraction forces. When the distance between the atoms reaches a couple of angstroms (about the length of a chemical bond) the force goes to zero. The atoms are in contact when the total van der Waals force becomes positive (repulsive). When the cantilever pushes against the sample, it bends rather than forcing the tip atoms closer to the sample atoms because the slope of the van der Waals curve is very steep in the repulsive or contact regime.

Capillary Force
A capillary force arises when water wicks its way around the tip and applies a strong attractive force that holds it in contact with the surface. Its magnitude depends on the tip-to-sample separation. The magnitude and sign (repulsive or attractive) of the cantilever force depend on the deflection of the cantilever and its spring constant. Assuming the water layer is homogeneous, as long as the tip is in contact with the sample the capillary force should be constant.

Van der Waals Force
The cantilever exerts the variable force in contact AFM. The total force exerted by the tip on the sample is the sum of the capillary and cantilever forces and is balanced by the repulsive van der Waals force. The total force usually ranges from 10-8N to 10-6N.

The position of the cantilever is detected by bouncing a laser off its back onto a position-sensitive photodetector (PSPD). The system can detect sub-angstrom vertical movement of the cantilever tip because the ratio of the path length between the cantilever and the detector to the length of the cantilever itself produces a mechanical amplification. The computer stores the distance the scanner moves vertically at each (x,y) data point. This data point is used to form the topographic image of the sample surface.