From a5fc3e4e0002085465d8497eb1e4cb05845b8f55 Mon Sep 17 00:00:00 2001 From: "W. Trevor King" Date: Sat, 4 May 2013 22:04:27 -0400 Subject: [PATCH] calibcant: flesh out calibcant chapter Figures, references, theory, etc. --- src/apparatus/afm.tex | 13 +- src/calibcant/conclusions.tex | 12 + src/calibcant/discussion.tex | 151 +++++++++ src/calibcant/overview.tex | 32 +- src/calibcant/procedure.tex | 178 +++++++++++ src/calibcant/theory.tex | 8 +- ...13-02-07T08-20-46-calibcant-data-offset.h5 | Bin 0 -> 3481992 bytes .../2013-02-07T08-20-46-calibcant-data.h5 | Bin 0 -> 3472896 bytes src/figures/calibcant/bump.png | Bin 0 -> 292127 bytes src/figures/calibcant/statistics.png | Bin 0 -> 285390 bytes src/figures/calibcant/stepper.png | Bin 0 -> 133707 bytes src/figures/calibcant/vibration-no-offset.png | Bin 0 -> 299852 bytes src/figures/calibcant/vibration.png | Bin 0 -> 297941 bytes src/pyafm/frameworks.tex | 8 +- src/pyafm/stack.tex | 13 +- src/root.bib | 298 +++++++++++++++++- 16 files changed, 660 insertions(+), 53 deletions(-) create mode 100644 src/figures/calibcant/2013-02-07T08-20-46-calibcant-data-offset.h5 create mode 100644 src/figures/calibcant/2013-02-07T08-20-46-calibcant-data.h5 create mode 100644 src/figures/calibcant/bump.png create mode 100644 src/figures/calibcant/statistics.png create mode 100644 src/figures/calibcant/stepper.png create mode 100644 src/figures/calibcant/vibration-no-offset.png create mode 100644 src/figures/calibcant/vibration.png diff --git a/src/apparatus/afm.tex b/src/apparatus/afm.tex index f026d5c..52e2b27 100644 --- a/src/apparatus/afm.tex +++ b/src/apparatus/afm.tex @@ -17,12 +17,13 @@ incident on a position sensitive photodetector\citep{meyer88} cantilever is known\citep{levy02}, the force applied to the sample can be calculated using Hooke's law (\cref{eq:sawsim:hooke}). -The substrate is mounted on a three dimensional piezoelectric actuator -so that the tip may be positioned on the surface with sub-nanometer -resolution (although signal drift and piezo hysteresis can cause -larger errors in the positioning accuracy). Our tubular piezo has a -range of TODO in the horizontal directions and a range of TODO in the -vertical (\cref{fig:piezo-schematic}). +The substrate is mounted in a fluid cell\citep{drake89,radmacher92} on +a three dimensional piezoelectric actuator so that the tip may be +positioned on the surface with sub-nanometer resolution (although +signal drift and piezo hysteresis can cause larger errors in the +positioning accuracy). Our tubular piezo has a range of TODO in the +horizontal directions and a range of TODO in the vertical +(\cref{fig:piezo-schematic}). \begin{figure} \begin{center} diff --git a/src/calibcant/conclusions.tex b/src/calibcant/conclusions.tex index e08291c..b677595 100644 --- a/src/calibcant/conclusions.tex +++ b/src/calibcant/conclusions.tex @@ -1,2 +1,14 @@ \section{Conclusions} \label{sec:calibcant:conclusions} + +Thermal cantilever calibration has been common practice for many years +now\citep{hutter93,florin95}, starting with estimates of thermal +noise\citep{martin87}. However, explicit derivations and discussion +has been limited. While this is likely done because the underlying +theory is ``obvious'', it makes it more likely that corner cases slip +by the notice of calibration experts\citep{hutter93-erratum} or +incorrect formula are used during the fitting +(\cref{sec:calibcant:lorentzian}). By centralizing calibration +procedures in an open package, \calibcant\ should both reduce the +effort needed to calibrate AFM cantilevers and improve the quality of +the calibration. diff --git a/src/calibcant/discussion.tex b/src/calibcant/discussion.tex index ff78883..9173778 100644 --- a/src/calibcant/discussion.tex +++ b/src/calibcant/discussion.tex @@ -1,8 +1,159 @@ \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 diff --git a/src/calibcant/overview.tex b/src/calibcant/overview.tex index b297d77..6e94c02 100644 --- a/src/calibcant/overview.tex +++ b/src/calibcant/overview.tex @@ -1,8 +1,9 @@ The most common method for calibrating cantilevers for atomic force -microscopes is via thermal vibration. In this chapter, I'll derive -the theory behind this procedure and introduce my software for -performing this calibration automatically. For a quick overview of -the theory, see \cref{sec:cantilever-calib:intro}. +microscopes is via thermal vibration\citep{florin95}. In this +chapter, I'll derive the theory behind this procedure and introduce +my \calibcant\ package for performing this calibration automatically. +For a quick overview of the theory, +see \cref{sec:cantilever-calib:intro}. The basic approach is to treat the cantilever as a simple harmonic oscillator (\cref{eq:hooke}) and use the equipartition theorem to @@ -20,32 +21,19 @@ parameters---$G_{1f}$, $f_0$, and $\beta_f$---come from fitting the thermal vibration of the cantilever when it is far from the surface. \subsection{Related papers} +\label{sec:calibcant:survey} In reality, the cantilever motion is more complicated than a pure simple harmonic oscillator. Various corrections taking into acount higher order vibrational modes\citep{butt95,stark01} and cantilever tilt\citep{hutter05} have been proposed and reviewed\citep{florin95,levy02,ohler07}, but we will focus here on the -derivation of Lorentzian noise in damped simple harmonic oscillators -that underlies all frequency-space methods for improving the basic +derivation of noise in damped simple harmonic oscillators that +underlies all frequency-space methods for improving the basic $\kappa\avg{x^2} = k_BT$ method. -Roters and Johannsmann describe a similar approach to deriving the Lorentizian +Roters and Johannsmann describe a similar approach to deriving the power spectral density\citep{roters96}. %, %as do % see Gittes 1998 for more thermal noise details -% see Berg-Sorenson for excellent overdamped treament. - -\emph{WARNING}: It is popular to refer to the power spectral density -as a ``Lorentzian''\citep{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} \;, -\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. In order to avoid any -uncertainty, we will leave \cref{eq:model-psd} unnamed. +% see Berg-Sorensen05 for excellent overdamped treament. diff --git a/src/calibcant/procedure.tex b/src/calibcant/procedure.tex index 32ce4fa..cd012ec 100644 --- a/src/calibcant/procedure.tex +++ b/src/calibcant/procedure.tex @@ -1,11 +1,189 @@ \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. diff --git a/src/calibcant/theory.tex b/src/calibcant/theory.tex index 28ad001..720cc5e 100644 --- a/src/calibcant/theory.tex +++ b/src/calibcant/theory.tex @@ -99,10 +99,10 @@ and $\conj{x}$ represents the complex conjugate of $x$. For highly damped systems, the inertial term in \cref{eq:DHO} becomes insignificant ($m \rightarrow 0$). This model is commonly used for -optically trapped beads\citep{TODO}. Because it is simpler and -solutions are more easily available\citep{grossman05,TODO}, it will -serve to outline the general approach before we dive into the general -case. +optically trapped beads\citep{bechhoefer02}. Because it is simpler +and solutions are more easily +available\citep{bechhoefer02,grossman05}, it will serve to outline the +general approach before we dive into the general case. Fourier transforming \cref{eq:DHO} with $m=0$ and applying \cref{eq:four-deriv} we have diff --git a/src/figures/calibcant/2013-02-07T08-20-46-calibcant-data-offset.h5 b/src/figures/calibcant/2013-02-07T08-20-46-calibcant-data-offset.h5 new file mode 100644 index 0000000000000000000000000000000000000000..c6eb9f8a694eec3401c908d127171ce099da3b51 GIT binary patch literal 3481992 zcmeF434l)3|NkFLOobFmN=>2fWGSR9DMQ6%YoeISHq0=?4AVUOdhfH$zT3$%WX~2t zen^%QDWwH%^ewa~ZTNnZ{;xat9Lq2>^yeFT9^TLAp5>n9b3W&DKA&@+IX5HX(T-KG zsdJ4%`-FrTDjKR6UCOUN#V7ra`-`GvSf&pU7v%&^7j#Z}{i2*f`)4rJ7V+1M@{8wt zs>frG8OSNimsH$lI8;f$Y9%9r^dH56h{xJTY8{Rey_WK}S;(1WMs!kA@`&g`aWXt; z#PGrKLkxz2=ENi|&J>%N6q_7vGAG9tWwFE$BoselxVT|Zbi!#dLCW^lJ*HL`o78#QWRgleEFgqLnuePs8DVIIg37$76on~zxL7c zw47JtNrOSvf`9v`r1>1Bch+DN^R%q_JV2ndP_b%9g{}l&{&*v~v@pvIWvK>hy zjPa*TPeV-X;Dp#gCu52x6PcUJ8_J+PyFc-$Hm9ZBH5GCz%PB6$p`yG42lNY>ulhH7 zxaoH~jS%@-t!q(qCG!msVRgM8FW12VqQf#@x(Lr{s23!H2HQ@(fihpA2(R0%$IJC_ 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