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% My working notes on thermal cantilever calibration.
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%     Copyright (C) 2008,2009 William Trevor King
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\begin{document}

\begin{center}
{\Large Thermal cantilever calibration} \\
{\it Equipartition theorem and power spectral density fitting}
\end{center}

\copytitle

\input{overview}
\input{setup_general}
\input{solve_highly_damped}
\input{solve_general}
\input{contour_integration}
\input{integrals}

\begin{thebibliography}{77} %77 is width of longest number of your reference?

% Original equipartition-based calibration method.
\bibitem{hutter93}
 J.\ Hutter and J.\ Bechhoefer.
 {\it Calibration of atomic-force microscope tips}.
 Rev.\ Sci.\ Instrum.\ {\bf 64}, 3342 (1993).
%and ?Rev. Sci. Instrum. 64 1868 1993

% Corrections to basic $kx^2 = kB T$
% due to higher order modes in rectangular cantilevers.
\bibitem{butt95}
 H.\ Butt and M.\ Jaschke.
 {\it Calculation of thermal noise in atomic force microscopy.}
 Nanotechnology {\bf 6}, 1-7 (1995).\\
 \url{http://www.iop.org/EJ/abstract/0957-4484/6/1/001/}

% Actually write down Lagrangian formula and give a decent derivation.
% Don't actually show or work out the integrals though...
\bibitem{roters96}
 A.\ Roters and D.\ Johannsmann.
 {\it Distance-dependent noise measurements in scanning force microscopy.}
 J.\ Phys.\ Condens.\ Matter {\bf 8}, 7561-7577 (1996).

% Defines the standard Lorentzian function
\bibitem{mathworld_lorentzian}
 Eric W.\ Weisstein.
 {\it Lorentzian Function.}
 MathWorld--A Wolfram Web Resource.
 \url{http://mathworld.wolfram.com/LorentzianFunction.html}

% Good review of calibration to 1005, with experimental comparison between
% resonance-shift, reference-spring, and thermal methods.
\bibitem{florin95}
 R. L'evy and M.\ Maaloum.
 {\it Measuring the spring constant of atomic force microscope cantilevers:
  thermal fluctuations and other methods.}
 Nanotechnology {\bf 13} 33-37 (2002).\\
 \url{http://stacks.iop.org/Nano/13/33/}

% Good review of thermal calibration to 2002,
% but not much on the derviation of the Lorentzian fit.
\bibitem{levy02}
 R. L'evy and M.\ Maaloum.
 {\it Measuring the spring constant of atomic force microscope cantilevers:
  thermal fluctuations and other methods.}
 Nanotechnology {\bf 13} 33-37 (2002).\\
 \url{http://stacks.iop.org/Nano/13/33/}

% Excellent review of thermal calibration to 2007,
% but nothing in the way of derivations.
% Compares thermal tune and Sader method with laser Doppler vibrometry.
\bibitem{ohler07}
 B.\ Ohler.
 {\it Cantilever spring constant calibration using laser Doppler vibrometry.}
 Rev.\ Sci.\ Instrum. {\bf 78} 063701 (2007).\\
 \url{http://link.aip.org/link/?RSI/78/063701}

% Higher mode adjustments for v-shaped cantilevers from simulation.
\bibitem{stark01}
 R.\ Stark, T.\ Drobek, and W.\ Heckl.
 {\it Thermomechanical noise of a free v-shaped cantilever for atomic-force microscopy.}
 Ultramicroscopy {\bf 86} 207-215 (2001).


% Tilted cantilever corrections (not needed? see Ohler/VEECO note)
\bibitem{hutter05}
 J.\ Hutter.
 {\it Comment on Tilt of Atomic Force Microscope Cantilevers:
      Effect on Spring Constant and Adhesion Measurements.}
 Langmuir {\bf 21} 2630 (2005).


\bibitem{cos_halfangle}
 S.\ Thornton and J.\ Marion.
 {Classical Dynamics of Particles and Systems}.
 5\sth Edition, Brooks/Cole, 2004. \\
 See Eq. D.16 on page 609.

\bibitem{four_deriv}
 Hmm, it is suprisingly difficult to find an `official' reference for this.
 I obviously need to get a spectral analysis book :p.
 See Wikipedia's currently excellent page (Feb 15th, 2008) \\
 \url{http://en.wikipedia.org/wiki/Fourier_transform#Functional_relationships} ,\\
 or derive it for yourself in about three lines :p.

\bibitem{parseval}
 W.\ Press, S.\ Teukolsky, W.\ Vetterling, and B.\ Flannery.
 {\it Numerical Recipies in C: The Art of Scientific Computing}.
 2\nd Edition, Cambridge University Press, 1992. \\
 \url{http://www.nrbook.com/a/bookcpdf.php}\\
 See Eq. 12.0.13 on page 498.

\bibitem{PSD}
 Press et al. See Eq. 12.0.14 on page 498.


\bibitem{wiener_khinchin}
 Press et al. See Eq. 12.0.12 on page 498.

\bibitem{wikipedia_wiener_khinchin}
 Wikipedia.
 {\it Wiener-Khinchin theorem}. \\
 \url{http://en.wikipedia.org/wiki/Wiener\%E2\%80\%93Khinchin_theorem}

\bibitem{tweezer_lab_notes}
 C.\ Grossman and A.\ Stout.
 {\it Optical Tweezers Advanced Lab}
 Fall 2005. See section 4.3.
 \url{http://chirality.swarthmore.edu/PHYS81/OpticalTweezers.pdf}

 \url{http://nlolab.swarthmore.edu/webstuff/Phys81_82/OpticalTweezersLabTheory.pdf}

\end{thebibliography}

\end{document}
