\section{Discussion}
\label{sec:calibcant:discussion}
+\subsection{Fitting with a Lorentzian}
+\label{sec:calibcant:lorentzian}
+
+It is popular to refer to the thermal power spectral density as a
+``Lorentzian''\citep{howard88,hutter93,roters96,levy02,florin95} even
+though \cref{eq:model-psd} differs from the classic
+Lorentzian\citep{mathworld-lorentzian}.
+\begin{equation}
+ L(x) = \frac{1}{\pi}\frac{\frac{1}{2}\Gamma}
+ {(x-x_0)^2 + \p({\frac{1}{2}\Gamma})^2} \;,
+ \label{eq:lorentzian}
+\end{equation}
+where $x_0$ sets the center and $\Gamma$ sets the width of the curve.
+It is unclear whether the references are due to uncertainty about the
+definition of the Lorentzian or to the fact that \cref{eq:model-psd}
+is also peaked and therefore \cref{eq:lorentzian} a potential
+substitute for \cref{eq:model-psd}. \citet{florian95}
+likely \cref{are} using \cref{eq:lorentzian}, as the slope of the
+fitted \PSD\ in their figure 2, has a slope at $f=0$.
+Using \cref{eq:model-psd}, the derivative would have been zero, as we
+can see by using the chain rule repeatedly,
+
+\begin{align}
+ \deriv{f}{\PSD_f}
+ &= \deriv{f}{}\p({\frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2}})
+ = \frac{-G_{1f}}{\p({(f_0^2-f^2)^2 + \beta_f^2 f^2})^2}
+ \deriv{f}{}\p({(f_0^2-f^2)^2 + \beta_f^2 f^2}) \\
+ &= \frac{-G_{1f}}{\p({(f_0^2-f^2)^2 + \beta_f^2 f^2})^2}
+ \p({2(f_0^2-f^2)\deriv{f}{}(f_0^2 - f^2) + 2\beta_f^2 f}) \\
+ &= \frac{-G_{1f}}{\p({(f_0^2-f^2)^2 + \beta_f^2 f^2})^2}
+ \p({-4f(f_0^2-f^2) + 2\beta_f^2 f}) \\
+ &= \frac{2G_{1f}f}{\p({(f_0^2-f^2)^2 + \beta_f^2 f^2})^2}
+ \p({2(f_0^2-f^2) - \beta_f^2}) \\
+ \left.\deriv{f}{\PSD_f}\right|_{f=0} &= 0 \;.
+ \label{eq:model-psd-df}
+\end{align}
+
+In order to avoid any uncertainty, we leave \cref{eq:model-psd}
+unnamed.
+
+\subsection{Peak frequency}
+\label{sec:calibcant:peak-frequency}
+
+Since we went through the trouble of calculating the derivative of
+$\PSD_f$ in \cref{eq:model-psd-df}, it's useful to also calculate the
+frequency of the resonant peak.
+\begin{align}
+ 0 &= \deriv{f}{\PSD_f}
+ = \frac{2G_{1f}f}{\p({(f_0^2-f^2)^2 + \beta_f^2 f^2})^2}
+ \p({2(f_0^2-f^2) - \beta_f^2})
+ = 2(f_0^2-f^2) - \beta_f^2 \\
+ f^2 &= f_0^2 - \frac{\beta_f^2}{2} \\
+ f &= \sqrt{f_0^2 - \frac{\beta_f^2}{2}} \;,
+ \label{eq:peak-frequency}
+\end{align}
+where we used $f\ne0$ during the large simplifying multiplication. We
+see that the peak frequency is actually shifed from $f_0$ depending on
+the damping term $\beta_f$. For overdamped cantilevers with large
+values of $\beta$, the peak frequency will not have a real solution.
+
\subsection{Propogation of errors}
\label{sec:calibcant:discussion:errors}
+Extracting cantilever spring constants with \cref{eq:kappa} is great,
+but the number you get is not much good if you can't estimate its
+accuracy. We can find the effect of measurement and fitting errors on
+the calculated $\kappa$ using Taylor expansions\citep{ku66}. To the
+first order,
+
+\begin{equation}
+ f(\vect{x}) \approx f_0 + \sum_i^n \p({\deriv{x_i}{f}(x_i - x_{i0})}) \;.
+\end{equation}
+
+To applying this to \cref{eq:kappa}, we need the derivatives
+\begin{align}
+ \deriv{\sigma_p}{\kappa}
+ &= \deriv{\sigma_p}{}\p({\frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}})
+ = \frac{2\kappa}{\sigma_p} \\
+ \deriv{T}{\kappa} &= \frac{\kappa}{T} \\
+ \deriv{\avg{V_p(t)^2}}{\kappa} &= \frac{-\kappa}{\avg{V_p(t)^2}} \;,
+\end{align}
+where I have used $\avg{V_p(t)^2}$ directly to support alternative
+variance extraction models (\cref{sec:calibcant:vibration}).
+
+Our measurements of $\sigma_p$, $T$, and $\avg{V_p(t)^2}$ are
+independent and therefore uncorrelated. This lets us estimate
+standard deviation of $\kappa$ from the standard deviation of the
+measured parameters\citep{ku66}.
+
+\begin{align}
+ \sigma_\kappa &= \sqrt{
+ \p({\deriv{\sigma_p}{\kappa}})^2 \sigma_{\sigma_p}^2 +
+ \p({\deriv{T}{\kappa}})^2 \sigma_{T}^2 +
+ \p({\deriv{\avg{V_p(t)^2}}{\kappa}})^2 \sigma_{\avg{V_p(t)^2}}^2
+ } \\
+ &= \sqrt{
+ \frac{4\kappa^2}{\sigma_p^2} \sigma_{\sigma_p}^2 +
+ \frac{\kappa^2}{T^2} \sigma_{T}^2 +
+ \frac{\kappa^2}{\avg{V_p(t)^2}} \sigma_{\avg{V_p(t)^2}}^2
+ } \\
+ \frac{\sigma_\kappa}{\kappa} &= \sqrt{
+ 4\p({\frac{\sigma_{\sigma_p}}{\sigma_p}})^2 +
+ \p({\frac{\sigma_{T}}{T}})^2 +
+ \p({\frac{\sigma_{\avg{V_p(t)^2}}^2}{\avg{V_p(t)^2}}^2})
+ }
+\end{align}
+
+By repeating each experiment (surface bumps, temperature readings, and
+thermal vibrations) several times, we can estimate the statistical
+uncertainty in each parameter (\cref{fig:calibcant:statistics}).
+Values for $\sigma_p$ and $\avg{V_p^2}$ are quite sensitive to the
+location of the laser spot on the cantilever, so they can vary over
+large timescales as the microscope alignment drifts (e.g.~due to
+thermal expansion as the room warms up). However TODO
+
+For example, on recent calibration run\footnote{2013-02-07T08-20-46} I
+measured $\sigma_p=35.68\pm0.87\U{V/$\mu$m}$,
+$T=298.151\pm0.033\U{K}$, and $\avg{V_p^2}=96.90\pm0.99\U{mV$^2$}$,
+which gives $\kappa=54.1\pm2.7\U{mN/m}$. The uncertainty
+contributions from each term are
+\begin{align}
+ 4\p({\frac{\sigma_{\sigma_p}}{\sigma_p}})^2 &= 2.38\E{-3}\U{N$^2$/m$^2$} \\
+ \p({\frac{\sigma_{T}}{T}})^2 &= 1.29\E{-8}\U{N$^2$/m$^2$} \\
+ \p({\frac{\sigma_{\avg{V_p(t)^2}}^2}{\avg{V_p(t)^2}}^2})
+ &= 1.04\E{-4}\U{N$^2$/m$^2$}
+\end{align}
+The size of the thermal vibration is
+$\sqrt{\avg{V_p^2}}/\sigma_p\approx2.8\U{\AA}$ with forces on the
+order of $\kappa\sqrt{\avg{V_p^2}}/\sigma_p\approx15\U{pN}$.
+
+In this particular run, most of the uncertainty in $\kappa$ comes from
+$\sigma_{\sigma_p}$, with some from $\sigma_{\avg{V_p(t)^2}}$. To add
+uncertainty comparable to the photodiode sensitivity contribution, the
+temperature variance would have to be
+\begin{equation}
+ \sigma_T = \frac{\sigma_{\sigma_p}}{\sigma_p}\cdot T
+ \approx \frac{2\cdot 0.87}{35.68} \cdot 298.151\U{T}
+ \approx 15\U{K} \;.
+\end{equation}
+This is a large enough window that simply using room temperature (or
+even a hard-coded $300\U{K}$) should not introduce excessive error in
+the calculated $\kappa$.
+
+\begin{figure}
+ \begin{center}
+ \includegraphics[width=0.8\textwidth]{figures/calibcant/statistics.png}
+ \caption{Estimating the statistical uncertainty of the calculated
+ cantilever spring constant $\kappa$ through repeated
+ measurements.\label{fig:calibcant:statistics}}
+ \end{center}
+\end{figure}
+
\subsection{Archiving experimental data}
\label{sec:calibcant:discussion:data}
+
+TODO
\section{Calibcant}
\label{sec:calibcant:procedure}
+A calibration run based on \cref{eq:kappa} consists of bumping the
+surface with the cantilever tip to measure $\sigma_p$, measuring the
+buffer temperature $T$ with a thermocouple, and measuring thermal
+vibration when the tip is far from the surface to extract the fit
+parameters $G_{1f}$, $f_0$, and $\beta_f$. I've written the
+\calibcant\ package to carry out this calibration procedure, building
+on packages in the \pyafm\ stack (\cref{fig:pyafm:stack}).
+
\subsection{Photodiode calibration}
\label{sec:calibcant:bump}
+To calculate the photodiode sensitivity $\sigma_p$, we need surface
+bumps with a clearly delimited contact slope. The \calibcant\ package
+uses \imint{python}|AFM.move_just_onto_surface|
+(\cref{sec:pyafm:pyafm}) to position the cantilever tip a configurable
+distance off the surface (\cref{sec:pyafm:h5config}). Then
+\calibcant\ uses the \pypiezo\ component (\cref{sec:pyafm:pypiezo}) to
+ramp the tip towards the surface a configurable distance before
+returning the tip to its original position. The cantilever deflection
+during this approach--retract cycle is analyzed to measure $\sigma_p$
+(\cref{fig:calibcant:bump}).
+
+\begin{figure}
+ \begin{center}
+ \includegraphics[width=0.8\textwidth]{figures/calibcant/bump.png}
+ \caption{Measuring the photodiode sensitivity $\sigma_p$ by
+ bumping the cantilever tip on the substrate surface. This is
+ the first bump from the 2013-02-07T08-20-46
+ calibration.\label{fig:calibcant:bump}}
+ \end{center}
+\end{figure}
+
+The retraction data is analyzed using a similar approach to \pypiezo's
+surface detaction algorithm to extract the slope of the contact
+region. Where \pypiezo\ uses a bilinear model
+(\cref{eq:bilinear-surface}), \calibcant\ uses a limited linear model.
+
+\begin{equation}
+ d(z) = \begin{cases}
+ d_\text{rail} & z \le z_\text{rail} \\
+ d_\text{kink} + \sigma_p (z - z_\text{kink})
+ & z_\text{rail} \le z \le z_\text{kink} \\
+ d_\text{kink} & z \ge z_\text{kink}
+ \end{cases}
+ \label{eq:limited-linear-surface}
+\end{equation}
+
+The fitted parameters are the surface contact point $(z_\text{kink},
+d_\text{kink})$ and the contact slope $\sigma_p$. The clipping
+deflection $d_\text{rail}$ is the deflection ADC's maximum measureable
+voltage ($2^{16}\U{bits}$ for our 16-bit ADCs).
+
+By explicitly modeling the clipping voltage, we avoid the need for
+manual intervention when the configured approach distance is too large
+for the cantilever geometry and a bump pushes too hard. With short
+cantilevers, even small tip deflection distances can generate large
+laser deflection angles (\cref{fig:afm-schematic}), leading to
+unmeasurable deflection voltages. One of the unfolding pulls in
+\cref{fig:pyafm:labview-comparison:many} exhibits this effect,
+although it was recorded using a different stack.
+
\subsection{Temperature measurements}
\label{sec:calibcant:temperature}
+After a series of surface bumps have been made to measure $\sigma_p$,
+the stepper motor is used to move the cantilever a configurable
+distance from the surface (generally $\sim30\U{$\mu$m}$). While the
+cantilever settles down after the jarring stepper motion, we measure
+the buffer temperature using \pypid\ (\cref{sec:pyafm:pypid}), a
+Melcor Series MTCA Thermoelectric Cooler Controller\citep{melcor}, and
+a type E thermocouple\footnote{Part number 5TC-TT-E-30-72 from OMEGA
+ Engineering Inc.\citep{omega}. Breaking down the product number,
+ it's a five pack of thermocouples (5TC) with perfluoroalkoxy
+ insulation (TT), type E metals (chromel--constantan), number 30 AWG
+ wires ($0.255\U{mm}$ diameter), in a $72\U{inch}$ length.}. The
+thermocouple is inserted through one of the ports in the AFM fluid
+cell, so the thermocouple tip is in the buffer less than $TODO\U{mm}$
+from the cantilever tip.
+
+\Cref{eq:kappa} depends on the \emph{absolute} temperature, so labs
+without easy access to a thermocouple can probably get away with
+estimating the buffer temperature. Errors of $5\U{K}$ from an actual
+temperature of around $300\U{K}$ will be within 1.7\% of the actual
+value. The effect of this error on $\kappa$ will be modest, but see
+\cref{sec:calibcant:discussion:errors} for a full discussion.
+
\subsection{Thermal vibration}
\label{sec:calibcant:vibration}
+
+After the temperature measurements are complete, we measure the
+cantilever's thermal vibration without moving the piezo
+(\cref{fig:calibcant:vibration}). The parameters controlling these
+vibrations are configurable (with \hFconfig,
+\cref{sec:pyafm:h5config}), but the default values are:
+
+\begin{description}
+ \item[frequency] The sampling frequency, which defaults to
+ $50\U{kHz}$. This value gives a Nyquist frequency of $25\U{kHz}$,
+ which is well above our resonant cantilever frequencies
+ ($\sim5\U{kHz}$ in the buffer).
+ \item[sample time] The acquisition time in seconds. This is rounded
+ up as required so the number of samples will be an integer power
+ of two for efficient Fourier transformation. It defaults to
+ $1\U{s}$.
+ \item[model] The vibration model. This between fitting methods for
+ extracting the variance $\avg{V_p(t)^2}$. By default,
+ \cref{eq:psd-Vp} is used, but you can add the constant offset
+ (discussed below) or use the na\"{\i}ve
+ $\avg{V_p(t)^2}=\sum(V_p^2)/N$.
+ \item[minimum fit frequency] The low-frequency end of the
+ \PSD\ usually has a good deal of noise due to detector drift or
+ background (non-cantilever) vibrations. This parameter allows
+ you to select a window of the \PSD\ for fitting that excludes the
+ troublesome low-frequency region. It defaults to $500\U{Hz}$.
+ \item[maximum fit frequency] For completeness, you can also set a
+ high-frequency cutoff, although I've never had to use this
+ parameter.
+ \item[chunk-size, overlap, \ldots] Assorted parameters for Fourier
+ transforms used to compute the \PSD.
+\end{description}
+
+\begin{figure}
+ \begin{center}
+ \subfloat[][]{\label{fig:calibcant:vibration:offset}
+ \includegraphics[width=0.8\textwidth]{figures/calibcant/vibration.png}}
+ \caption{\protect\subref{fig:calibcant:vibration:offset}Measuring
+ the cantilever's thermal vibration. The top panel shows the raw
+ time series data in bins, the middle panel shows the
+ distribution of bin values with a Gaussian fit, and the bottom
+ panel shows the $\PSD_f(V_p,f)$ with a fit following
+ \cref{eq:psd-Vp-offset}. The constant offset $P_{0f}$, drawn as
+ the horizontal line in the third panel, accounts for white noise
+ in the measurement circuit. Only data in the blue region was
+ used when computing the best fit. This is the first vibration
+ from the 2013-02-07T08-20-46 calibration, yielding a fitted
+ variance $\avg{V_p(t)^2}=96.90\pm0.99\U{mV$^2$}$.
+ \label{fig:calibcant:vibration}}
+ \end{center}
+\end{figure}
+
+\begin{figure}
+ \ContinuedFloat
+ \begin{center}
+ \subfloat[][]{\label{fig:calibcant:vibration:no-offset}
+ \includegraphics[width=0.8\textwidth]{figures/calibcant/vibration-no-offset.png}}
+ \caption{\protect\subref{fig:calibcant:vibration:no-offset}This is
+ the same data as in
+ \protect\subref{fig:calibcant:vibration:offset} fit with
+ \cref{eq:psd-Vp}, yielding a fitted variance
+ $\avg{V_p(t)^2}=120.92\pm0.90\U{mV$^2$}$. The third panel is
+ very similar to figure 2 in \citet{florin95}, but they do not go
+ into further detail on the method or model. They may be fitting
+ their data to \cref{eq:lorentzian}, see
+ \cref{sec:calibcant:lorentzian}. Another similar figure is in
+ \citet{hutter93}}.
+ \end{center}
+\end{figure}
+
+\Cref{eq:psd-Vp} rolls off for large frequencies, but the measured
+\PSD\ levels out (\cref{fig:calibcant:vibration:no-offset}). I
+attribute this to background white noise in the measurement circuit,
+and not due to cantilever oscillation. To avoid artificially
+inflating the estimated $\avg{V_p(t)^2}$, I created an alternative
+model for $\PSD_f(V_p,f)$ that adds a frequency-independent offset
+$P_{0f}$.
+
+\begin{equation}
+ \PSD_f(V_p, f) = \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2} + P_{0f} \;.
+ \label{eq:psd-Vp-offset}
+\end{equation}
+
+Plots of \cref{eq:psd-Vp-offset} fits look better than
+\cref{eq:psd-Vp} fits (\cref{fig:calibcant:vibration}), but the
+significance on the variance calculated with
+\cref{eq:Vp-from-freq-fit} depends on the amount of background noise
+in the vibration data. With over an order of magnitude difference
+between the power of the damped harmonic oscillator peak and the
+background noise, the effect of $P_{0f}$ will be small. With noisier
+setups, removing the white background noise can lead to a significant
+difference. The fitted variance $\avg{V_p(t)^2}$ of my
+2013-02-07T08-20-46 data shifts from $120.92\pm0.90\U{mV$^2$}$ using
+\cref{eq:psd-Vp} to $96.90\pm0.99\U{mV$^2$}$ using
+\cref{eq:psd-Vp-offset}, a 20\% decrease. The calculated spring
+constant increases from $43.3\pm{2.1}$ to $54.1\pm2.7\U{mN/m}$, a 25\%
+increase. Changes of this magnitude are important for accurate
+unfolding force calibration.
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+@string{NEURON = "Neuron"}
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+@string{ALWeisenhorn = "Weisenhorn, A.~L."}
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@string{MWen = "Wen, M."}
project = "Cantilever Calibration"
}
+@article{ berg-sorensen05,
+ author = KBergSorensen #" and "# HFlyvbjerg,
+ title = {The colour of thermal noise in classical Brownian motion: a
+ feasibility study of direct experimental observation},
+ year = 2005,
+ month = feb,
+ day = 1,
+ journal = NJP,
+ volume = 7,
+ number = {1},
+ pages = {38},
+ doi = {10.1088/1367-2630/7/1/038},
+ url = {http://stacks.iop.org/1367-2630/7/i=1/a=038},
+ eprint = {http://iopscience.iop.org/1367-2630/7/1/038/pdf/1367-2630_7_1_038.pdf},
+ abstract = {One hundred years after Einstein modelled Brownian
+ motion, a central aspect of this motion in incompressible fluids
+ has not been verified experimentally: the thermal noise that
+ drives the Brownian particle, is not white, as in Einstein's
+ simple theory. It is slightly coloured, due to hydrodynamics and
+ the fluctuation--dissipation theorem. This theoretical result from
+ the 1970s was prompted by computer simulation results in apparent
+ violation of Einstein's theory. We discuss how a direct
+ experimental observation of this colour might be carried out by
+ using optical tweezers to separate the thermal noise from the
+ particle's dynamic response to it. Since the thermal noise is
+ almost white, very good statistics is necessary to resolve its
+ colour. That requires stable equipment and long recording times,
+ possibly making this experiment one for the future only. We give
+ results for experimental requirements and for stochastic errors as
+ functions of experimental window and measurement time, and discuss
+ some potential sources of systematic errors.},
+}
+
@article { bedard08,
author = SBedard #" and "# MMGKrishna #" and "# LMayne #" and "#
SWEnglander,
rationalized at the molecular level."
}
+@article{ martin87
+ author = YMartin #" and "# CCWilliams #" and "# HKWickamasinghe,
+ title = {Atomic force microscope---force mapping and profiling on a
+ sub 100-\AA scale},
+ year = 1987,
+ month = may,
+ day = 15,
+ journal = JAP,
+ volume = 61,
+ number = 10,
+ pages = {4723--4729},
+ doi = {10.1063/1.338807},
+ url = {http://jap.aip.org/resource/1/japiau/v61/i10/p4723_s1}
+ abstract = {A modified version of the atomic force microscope is
+ introduced that enables a precise measurement of the force between
+ a tip and a sample over a tip-sample distance range of 30--150
+ \AA. As an application, the force signal is used to maintain the
+ tip-sample spacing constant, so that profiling can be achieved
+ with a spatial resolution of 50 \AA. A second scheme allows the
+ simultaneous measurement of force and surface profile; this scheme
+ has been used to obtain material-dependent information from
+ surfaces of electronic materials.},
+}
+
@article { butt95,
author = HJButt #" and "# MJaschke,
title = "Calculation of thermal noise in atomic force microscopy",
folding reaction."
}
+@article{ howard87,
+ author = JHoward #" and "# AJHudspeth,
+ title = {Mechanical relaxation of the hair bundle mediates
+ adaptation in mechanoelectrical transduction by the
+ bullfrog's saccular hair cell.},
+ journal = PNAS,
+ year = 1987,
+ month = may,
+ volume = 84,
+ number = 9,
+ pages = {3064--3068},
+ issn = {0027-8424},
+ url = {http://www.ncbi.nlm.nih.gov/pubmed/3495007},
+ keywords = {Acclimatization},
+ keywords = {Animals},
+ keywords = {Electric Conductivity},
+ keywords = {Electric Stimulation},
+ keywords = {Hair Cells, Auditory},
+ keywords = {Membrane Potentials},
+ keywords = {Microelectrodes},
+ keywords = {Physical Stimulation},
+ keywords = {Rana catesbeiana},
+ keywords = {Saccule and Utricle},
+ abstract = {Mechanoelectrical transduction by hair cells of the
+ frog's internal ear displays adaptation: the electrical response
+ to a maintained deflection of the hair bundle declines over a
+ period of tens of milliseconds. We investigated the role of
+ mechanics in adaptation by measuring changes in hair-bundle
+ stiffness following the application of force stimuli. Following
+ step stimulation with a glass fiber, the hair bundle of a saccular
+ hair cell initially had a stiffness of approximately equal to
+ $1\U{mN/m}$. The stiffness then declined to a steady-state level
+ near $0.6\U{mN/m}$ with a time course comparable to that of
+ adaptation in the receptor current. The hair bundle may be modeled
+ as the parallel combination of a spring, which represents the
+ rotational stiffness of the stereocilia, and a series spring and
+ dashpot, which respectively, represent the elastic element
+ responsible for channel gating and the apparatus for adaptation.},
+ language = {eng},
+}
+
+@article{ howard88,
+ author = JHoward #" and "# AJHudspeth,
+ title = {Compliance of the Hair Bundle Associated with Gating of
+ Mechanoelectrical Transduction Channels in the Bullfrog's Saccular
+ Hair Cell},
+ year = 1988,
+ month = may,
+ journal = NEURON,
+ volume = 1,
+ pages = {189--199},
+ doi = {10.1016/0896-6273(88)90139-0},
+ url = {http://www.cell.com/neuron/retrieve/pii/0896627388901390},
+ eprint = {http://download.cell.com/neuron/pdf/PII0896627388901390.pdf},
+ note = {Initial thermal calibration paper as cited by
+ \citet{florin95}. This is not an AFM paper, but it uses the
+ equipartition theorem to calculate the spring constant of hair
+ fibers by measuring their tip displacement variance. The
+ discussion occurs in the \emph{Manufacture and Calibration of
+ Fibers} section on pages 197--198. Actual details are scarce, but
+ I believe this is the original source of the ``Lorentzian'' and
+ ``10\% accuracy'' ideas that have haunted themal calibration ever
+ since.},
+}
+
+@article{ florin94,
+ author = ELFlorin #" and "# VMoy #" and "# HEGaub,
+ title = {Adhesion forces between individual ligand-receptor pairs},
+ year = 1994,
+ month = apr,
+ day = 15,
+ journal = SCI,
+ volume = 264,
+ number = 5157,
+ pages = {415--417},
+ doi = {10.1126/science.8153628},
+ url = {http://www.sciencemag.org/content/264/5157/415.abstract},
+ eprint = {http://www.sciencemag.org/content/264/5157/415.full.pdf},
+ abstract ={The adhesion force between the tip of an atomic force
+ microscope cantilever derivatized with avidin and agarose beads
+ functionalized with biotin, desthiobiotin, or iminobiotin was
+ measured. Under conditions that allowed only a limited number of
+ molecular pairs to interact, the force required to separate tip
+ and bead was found to be quantized in integer multiples of
+ $160\pm20$ piconewtons for biotin and $85\pm15$ piconewtons for
+ iminobiotin. The measured force quanta are interpreted as the
+ unbinding forces of individual molecular pairs.},
+}
+
@article { florin95,
author = ELFlorin #" and "# MRief #" and "# HLehmann #" and "# MLudwig #"
and "# CDornmair #" and "# VMoy #" and "# HEGaub,
force is knowledg e of the spring constant of the cantilevers. Here, we
compare different techniqu es that allow for the in situ measurement of
the absolute value of the spring co nstant of cantilevers.",
- note = "Good review of calibration to 1995, with experimental comparison
- between resonance-shift, reference-spring, and thermal methods.",
+ note = {Good review of calibration to 1995, with experimental
+ comparison between resonance-shift, reference-spring, and
+ thermal methods. They incorrectly cite \citet{hutter93} as
+ being published in 1994.},
project = "Cantilever Calibration"
}
@article { hutter93,
author = JHutter #" and "# JBechhoefer,
title = "Calibration of atomic-force microscope tips",
- collaboration = "",
year = 1993,
journal = RSI,
volume = 64,
publisher = AIP,
doi = "10.1063/1.1143970",
url = "http://link.aip.org/link/?RSI/64/1868/1",
- keywords = "ATOMIC FORCE MICROSCOPY; CALIBRATION; QUALITY FACTOR; PROBES;
- RESONANCE; SILICON NITRIDES; MICA; VAN DER WAALS FORCES",
- note = "Original equipartition-based calibration method (thermal
- calibration).",
+ keywords = {atomic force microscopy; calibration; quality factor; probes;
+ resonance; silicon nitrides; mica; van der waals forces},
+ note = {Original equipartition-based calibration method (thermal
+ calibration), after the brief mention in \citet{howard88}.
+ This is the first paper I've found that works out the theory
+ in detail, although they punt to page 431 of \citet{heer72}
+ instead of listing a formula for their ``Lorentzian''. The
+ experimental data uses high-$Q$ cantilevers in air, and their
+ figure 2 shows clear water-layer snap-off. There is a
+ published erratum\citep{hutter93-erratum}.},
project = "Cantilever Calibration"
}
+@article{ hutter93-erratum,
+ author = JHutter #" and "# JBechhoefer,
+ title = "Erratum: Calibration of atomic-force microscope tips",
+ year = 1993,
+ month = nov,
+ journal = RSI,
+ volume = 64,
+ number = 11,
+ pages = 3342,
+ publisher = AIP,
+ doi = "10.1063/1.1144449",
+ url = "http://rsi.aip.org/resource/1/rsinak/v64/i11/p3342_s1",
+ note = {V.~Croquette pointed out that they should calibrate the
+ response of their optical-detection electronics.},
+ project = "Cantilever Calibration",
+}
+
+@book{ heer72,
+ author = CVHeer,
+ title = {Statistical mechanics, kinetic theory, and stochastic processes},
+ year = 1972,
+ publisher = AcP,
+ address = {New York},
+ numpages = 602,
+ isbn = {0-123-36550-3},
+ language = {English},
+ keywords = {Statistical mechanics.; Kinetic theory of gases.; Stochastic processes.},
+}
+
@article { hyeon03,
author = CHyeon #" and "# DThirumalai,
title = "Can energy landscape roughness of proteins and {RNA} be measured
ISSN = "1079-7114",
doi = "10.1103/PhysRevLett.56.930",
URL = "http://www.ncbi.nlm.nih.gov/pubmed/10033323",
+ eprint = {http://prl.aps.org/pdf/PRL/v56/i9/p930_1},
language = "eng",
note = "Original AFM paper.",
}
+@article{ drake89,
+ author = BDrake #" and "# CBPrater #" and "# ALWeisenhorn #" and "#
+ SAGould #" and "# TRAlbrecht #" and "# CQuate #" and "#
+ DSCannell #" and "# HHansma #" and "# PHansma,
+ title = {Imaging crystals, polymers, and processes in water with the
+ atomic force microscope},
+ year = 1989,
+ month = mar,
+ day = 24,
+ journal = SCI,
+ volume = 243,
+ number = 4898,
+ pages = {1586--1589},
+ doi = {10.1126/science.2928794},
+ url = {http://www.sciencemag.org/content/243/4898/1586.abstract},
+ eprint = {http://www.sciencemag.org/content/243/4898/1586.full.pdf},
+ abstract ={The atomic force microscope (AFM) can be used to image
+ the surface of both conductors and nonconductors even if they are
+ covered with water or aqueous solutions. An AFM was used that
+ combines microfabricated cantilevers with a previously described
+ optical lever system to monitor deflection. Images of mica
+ demonstrate that atomic resolution is possible on rigid materials,
+ thus opening the possibility of atomic-scale corrosion experiments
+ on nonconductors. Images of polyalanine, an amino acid polymer,
+ show the potential of the AFM for revealing the structure of
+ molecules important in biology and medicine. Finally, a series of
+ ten images of the polymerization of fibrin, the basic component of
+ blood clots, illustrate the potential of the AFM for revealing
+ subtle details of biological processes as they occur in real
+ time.},
+}
+
+@article{ radmacher92,
+ author = MRadmacher #" and "# RWTillamnn #" and "# MFritz #" and "# HEGaub,
+ title = {From molecules to cells: imaging soft samples with the
+ atomic force microscope},
+ year = 1992,
+ month = sep,
+ day = 25,
+ journal = SCI,
+ volume = 257,
+ number = 5078,
+ pages = {1900--1905},
+ doi = {10.1126/science.1411505},
+ url = {http://www.sciencemag.org/content/257/5078/1900.abstract},
+ eprint = {http://www.sciencemag.org/content/257/5078/1900.full.pdf},
+ abstract ={Since its invention a few years ago, the atomic force microscope has become one of the most widely used near-field microscopes. Surfaces of hard sample are imaged routinely with atomic resolution. Soft samples, however, remain challenging. An overview is presented on the application of atomic force microscopy to organic samples ranging from thin ordered films at molecular resolution to living cells. Fundamental mechanisms of the image formation are discussed, and novel imaging modes are introduced that exploit different aspects of the tip-sample interaction for local measurements of the micromechanical properties of the sample. As examples, images of Langmuir-Blodgett films, which map the local viscoelasticity as well as the friction coefficient, are presented.},
+}
+
@article{ williams86,
author = CCWilliams #" and "# HWickramasinghe,
title = "Scanning thermal profiler",
volume = 49,
number = 23,
pages = "1587--1589",
- abstract = "A new high‐resolution profilometer has been demonstrated
- based upon a noncontacting near‐field thermal probe. The thermal
+ abstract = "A new high-resolution profilometer has been demonstrated
+ based upon a noncontacting near-field thermal probe. The thermal
probe consists of a thermocouple sensor with dimensions
approaching 100 nm. Profiling is achieved by scanning the heated
sensor above but close to the surface of a solid. The conduction
pages = "4723--4729",
abstract = "A modified version of the atomic force microscope is
introduced that enables a precise measurement of the force between
- a tip and a sample over a tip‐sample distance range of 30--150
+ a tip and a sample over a tip-sample distance range of 30--150
\AA. As an application, the force signal is used to maintain the
- tip‐sample spacing constant, so that profiling can be achieved
+ tip-sample spacing constant, so that profiling can be achieved
with a spatial resolution of 50 \AA. A second scheme allows the
simultaneous measurement of force and surface profile; this scheme
has been used to obtain material-dependent information from
investigation of four different adaptive single-loop
controllers.},
}
+
+@article{ ku66,
+ author = HHKu,
+ title = {Notes on the use of propagation of error formulas},
+ year = 1966,
+ month = oct,
+ journal = JRNBS:C,
+ volume = {70C},
+ number = 4,
+ pages = {263--273},
+ publisher = NBS,
+ issn = {0022-4316},
+ url = {http://nistdigitalarchives.contentdm.oclc.org/cdm/compoundobject/collection/p13011coll6/id/78003/rec/5},
+ eprint = {http://nistdigitalarchives.contentdm.oclc.org/utils/getfile/collection/p13011coll6/id/78003/filename/print/page/download},
+ keywords = {Approximation; error; formula; imprecision; law of
+ error; products; propagation of error; random; ratio; systematic;
+ sum},
+ abstract = {The ``law of propagation of error'' is a tool that
+ physical scientists have conveniently and frequently used in their
+ work for many years, yet an adequate reference is difficult to
+ find. In this paper an expository review of this topic is
+ presented, particularly in the light of current practices and
+ interpretations. Examples on the accuracy of the approximations
+ are given. The reporting of the uncertainties of final results is
+ discussed.},
+}
projects / thesis.git / commitdiff
