An atomic-force-microscope cantilever at is calibrated from its thermal motion alone

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Electronics, Instrumentation & Communication · Scientific programming for physicists

Scientific Computing for Physicists, Ch Spectra with error bars: the fast transform, convolution and spectral estimation

Problem

An atomic-force-microscope cantilever at T=295 KT=295\,\mathrm{K} is calibrated from its thermal motion alone. Its tip deflection xx is recorded at fs=200 kHzf_s=200\,\mathrm{kHz} for 10 s10\,\mathrm{s}. The cantilever behaves as one harmonic oscillator of stiffness k=mω02k=m\omega_0^2, with its resonance near 20 kHz20\,\mathrm{kHz} and a full width at half maximum Γ=400 Hz\Gamma=400\,\mathrm{Hz}, so equipartition gives ⟨12kx2⟩=12kBT\langle\tfrac12kx^2\rangle=\tfrac12k_BT. ((a)) The thermal peak of the one-sided deflection PSD, with the detector's flat noise floor subtracted, has area 5.0×10−20 m25.0\times10^{-20}\,\mathrm{m^2}. Explain why that area is ⟨x2⟩\langle x^2\rangle, and find the stiffness kk. ((b)) The PSD is estimated by Welch's method with untapered, non-overlapping segments, treated as independent. Choose the shortest power-of-two segment length MM whose frequency resolution is no coarser than Γ/10\Gamma/10, so that the peak spans at least ten bins. State the resolution, the number of whole segments KK the record provides, and the relative uncertainty of each bin of the estimate. ((c)) The detector adds a flat floor of 1.0×10−26 m2 Hz−11.0\times10^{-26}\,\mathrm{m^2\,Hz^{-1}} from 00 to the Nyquist frequency. If the whole PSD were integrated without subtracting that floor, by what fraction would the stiffness come out wrong, and in which direction?

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