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A variational method is used to calculate the deflection and the fundamental and harmonic resonance frequencies of commercial V-shaped and rectangular atomic force microscopy cantilevers. The effective mass of V-shaped cantilevers is roughly half that calculated for the equivalent rectangular cantilevers. Damping by environmental gases, including air, nitrogen, argon, and helium, affects the frequency of maximum response and to a much greater degree the quality factor Q. Helium has the lowest viscosity, resulting in the highest Q, and thus provides the best sensitivity in noncontact force microscopy. Damping in liquids is dominated by an increase in effective mass of the cantilever due to an added mass of the liquid being dragged with that cantilever.
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Chen et al. (1994) studied this question.
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