= Small-amplitude AFM frequency shift
{title2=$\Delta\omega/\omega_0\simeq-F'(z_0)/(2c)$}
At equilibrium separation $z_0$, expanding the tip force gives $m_{\rm eff}\ddot x=-[c-F'(z_0)]x$. Thus $\omega/\omega_0=\sqrt{1-F'(z_0)/c}$, and Taylor expansion gives the displayed result for $|F'|\ll c$. The displacement convention takes positive $z$ away from the surface. A positive gradient of an attractive force softens the resonance; a large-amplitude oscillation samples more than the local gradient.
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