In order to measure forces accurately with an AFM, it is important to
measure the cantilever spring constant. The force exerted on the
cantilever can then be deduced from it's deflection via Hooke's law
-$F=-\kappa x$.
+\begin{equation}
+ F=-\kappa x \;. \label{eq:hooke}
+\end{equation}
+
\nomenclature{$F$}{Force (newtons)}
\nomenclature{$\kappa$}{Spring constant (newtons per meter)}
\nomenclature{$x$}{Displacement (meters)}
\begin{align}
\kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
= \frac{2 \beta_f f_0^2 \sigma_p^2 k_BT}{\pi G_{1f}} \;.
+ \label{eq:kappa}
\end{align}
A calibration run consists of bumping the surface with the cantilever
tip to measure $\sigma_p$, measuring the buffer temperature $T$ with a
--- /dev/null
+\section{Conclusions}
+\label{sec:calibcant:conclusions}
--- /dev/null
+\section{Discussion}
+\label{sec:calibcant:discussion}
+
+\subsection{Propogation of errors}
+\label{sec:calibcant:discussion:errors}
+
+\subsection{Archiving experimental data}
+\label{sec:calibcant:discussion:data}
\chapter{Cantilever spring constant calibration}
\label{sec:calibcant}
-\section{Thermal calibration}
-\label{sec:calibcant:theory}
-\subsection{Overdamped case}
-\subsection{General solution}
-
-\section{Calibcant}
-\label{sec:calibcant:procedure}
-\subsection{Photodiode calibration}
-\subsection{Temperature measurements}
-\subsection{Thermal vibration}
-
-\section{Discussion}
-\label{sec:calibcant:discussion}
-\subsection{Propogation of errors}
-\subsection{Archiving experimental data}
-
-\section{Conclusions}
-\label{sec:calibcant:conclusions}
+\input{calibcant/overview}
+\input{calibcant/theory}
+\input{calibcant/procedure}
+\input{calibcant/discussion}
+\input{calibcant/conclusions}
--- /dev/null
+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}.
+
+The basic approach is to treat the cantilever as a simple harmonic
+oscillator (\cref{eq:hooke}) and use the equipartition theorem to
+connect the cantilever's thermal vibration with the temperature
+(\cref{eq:equipart}). The resulting calibration formula for the
+cantilever spring constant $\kappa$ is \cref{eq:kappa}, which I'll
+reproduce here for easy reference:
+\begin{align}
+ \kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
+ = \frac{2 \beta_f f_0^2 \sigma_p^2 k_BT}{\pi G_{1f}} \;.
+\end{align}
+where $\sigma_p$ is the photodiode sensitivity, $k_B$ is Boltzmann's
+constant, $T$ is the absolute temperature. The remaining
+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}
+
+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
+$\kappa\avg{x^2} = k_BT$ method.
+
+Roters and Johannsmann describe a similar approach to deriving the Lorentizian
+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.
--- /dev/null
+\section{Calibcant}
+\label{sec:calibcant:procedure}
+\subsection{Photodiode calibration}
+\subsection{Temperature measurements}
+\subsection{Thermal vibration}
--- /dev/null
+\section{Theory}
+\label{sec:calibcant:theory}
+% TODO: deprecated in favor of sec:cantilever-calib:intro
+
+Rather than computing the variance of $x(t)$ directly, we attempt to
+filter out noise by fitting the power spectral density (\PSD)%
+\nomenclature[PSDa]{$\PSD$}{Power spectral density in angular
+ frequency space}\index{PSD@\PSD}\nomenclature{$\omega$}{Angular
+ frequency (radians per second)} of $x(t)$ to the theoretically
+predicted \PSD\ for a damped harmonic oscillator (\cref{eq:model-psd})
+\begin{align}
+ \ddt{x} + \beta\dt{x} + \omega_0^2 x &= \frac{F_\text{thermal}}{m} \\
+ \PSD(x, \omega) &= \frac{G_1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} \;,
+\end{align}
+\index{Damped harmonic oscillator}
+where $G_1\equiv G_0/m^2$, $\omega_0$, and $\beta$ are used as the
+fitting parameters (see \cref{eq:model-psd}).%
+\index{$\beta$}\index{$\gamma$} The variance of $x(t)$ is then given
+by \cref{eq:DHO-var}
+\begin{equation}
+ \avg{x(t)^2} = \frac{\pi G_1}{2\beta\omega_0^2} \;,
+\end{equation}
+which we can plug into the equipartition theorem
+(\cref{eq:equipart}) yielding
+\begin{align}
+ \kappa = \frac{2 \beta \omega_0^2 k_BT}{\pi G_1} \;.
+\end{align}
+
+From \cref{eq:GO}, we find the expected value of $G_1$ to be
+\begin{equation}
+ G_1 \equiv G_0/m^2 = \frac{2}{\pi m} k_BT \beta \;. \label{eq:Gone}
+\end{equation}
+
+
+\section{Theoretical power spectral density for a damped harmonic oscillator}
+\label{sec:setup}
+
+Our cantilever can be approximated as a damped harmonic oscillator
+\begin{equation}
+ m\ddt{x} + \gamma \dt{x} + \kappa x = F(t) \;, \label{eq:DHO}
+ % DHO for Damped Harmonic Oscillator
+\end{equation}
+where $x$ is the displacement from equilibrium,
+ $m$ is the effective mass,
+ $\gamma$ is the effective drag coefficient,
+ $\kappa$ is the spring constant, and
+ $F(t)$ is the external driving force.
+During the non-contact phase of calibration,
+ $F(t)$ comes from random thermal noise.
+\nomenclature{$\beta$}{Damped harmonic oscillator drag-acceleration
+ coefficient $\beta \equiv \gamma/m$}\index{$\beta$}%
+\nomenclature{$\gamma$}{Damped harmonic oscillator drag coefficient
+ $F_\text{drag} = \gamma\dt{x}$}\index{$\gamma$}%
+\index{damped harmonic oscillator}%
+\nomenclature{$\dt{s}$}{First derivative of the time-series $s(t)$
+ with respect to time. $\dt{s} = \deriv{t}{s}$}%
+\nomenclature{$\ddt{s}$}{Second derivative of the time-series $s(t)$
+ with respect to time. $\ddt{s} = \nderiv{2}{t}{s}$}%
+
+In the following analysis, we use the unitary, angular frequency Fourier transform normalization
+\begin{equation}
+ \Four{x(t)} \equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{x(t) e^{-i \omega t}}\;.
+\end{equation}
+\nomenclature{\Four{s(t)}}{Fourier transform of the time-series
+ $s(t)$. $s(f) = \Four{s(t)}$}\index{Fourier transform}
+
+We also use the following theorems (proved elsewhere):
+\begin{align}
+ \cos\left(\frac{\theta}{2}\right) &= \pm\sqrt{\frac{1}{2}[1+\cos(\theta)]}\;,
+ &\text{\citep{cos-halfangle}} \label{eq:cos-halfangle} \\
+ \Four{\nderiv{n}{t}{x(t)}} &= (i \omega)^n x(\omega) \;,
+ &\text{\citep{four-deriv}} \label{eq:four-deriv} \\
+% \Four{x*y} &= x(\omega) y(\omega), \label{eq:four-conv}
+% & \text{and} \\
+ \iInfInf{t}{\magSq{x(t)}} &= \iInfInf{\omega}{\magSq{x(w)}} \;.
+ &\text{(Parseval's)\citep{parseval}} \label{eq:parseval}
+\end{align}
+\index{cosine half-angle}
+\index{Parseval's theorem}
+%where $x*y$ denotes the convolution of $x$ and $y$,
+%\begin{equation}
+% x*y \equiv \iInfInf{\tau}{x(t-\tau)y(\tau)}.
+%\end{equation}
+As a corollary to Parseval's theorem, we note that the one sided power spectral density per unit time (\PSD) defined by
+\begin{align}
+ \PSD(x, \omega) &\equiv \normLimT 2 \left| x(\omega) \right|^2
+ &\text{\citep{PSD}} \label{eq:psd-def}
+\end{align}
+\index{PSD@\PSD}
+relates to the variance by
+\begin{align}
+ \avg{x(t)^2}
+ &= \iLimT{\magSq{x(t)}}
+ = \normLimT \iInfInf{\omega}{\magSq{x(\omega)}}
+ = \iOInf{\omega}{\PSD(x,\omega)} \;, \label{eq:parseval-var}
+\end{align}
+where $t_T$ is the total time over which data has been aquired.
+
+We also use the Wiener-Khinchin theorem,
+which relates the two sided power spectral density $S_{xx}(\omega)$
+to the autocorrelation function $r_{xx}(t)$ via
+\begin{align}
+ S_{xx}(\omega) &= \Four{ r_{xx}(t) } \;,
+ &\text{(Wiener-Khinchin)\citep{wiener-khinchin}} \label{eq:wiener_khinchin}
+\end{align}
+\index{Wiener-Khinchin theorem}
+where $r_{xx}(t)$ is defined in terms of the expectation value
+\begin{align}
+ r_{xx}(t) &\equiv \avg{x(\tau)\conj{x}(\tau-t)} \;,
+ &\text{\citep{wikipedia-wiener-khinchin}}
+\end{align}
+and $\conj{x}$ represents the complex conjugate of $x$.
+\nomenclature{$\conj{z}$}{Complex conjugate of $z$}
+
+
+\subsection{Fitting deflection voltage directly}
+
+In order to keep our errors in measuring $\sigma_p$ seperate from
+other errors in measuring $\avg{x(t)^2}$, we can fit the voltage
+spectrum before converting to distance.
+\begin{align}
+ \ddt{V_p}/\sigma_p + \beta\dt{V_p}/\sigma_p + \omega_0^2 V_p/\sigma_p
+ &= F_\text{thermal} \\
+ \ddt{V_p} + \beta\dt{V_p} + \omega_0^2 V_p
+ &= \sigma_p\frac{F_\text{thermal}}{m} \\
+ \ddt{V_p} + \beta\dt{V_p} + \omega_0^2 V_p
+ &= \frac{F_\text{thermal}}{m_p} \\
+ \PSD(V_p, \omega) &= \frac{G_{1p}}
+ { (\omega_0^2-\omega^2)^2 + \beta^2\omega^2 } \\
+ \avg{V_p(t)^2} &= \frac{\pi G_{1p}}{2\beta\omega_0^2}
+ = \frac{\pi \sigma_p^2 G_{1}}{2\beta\omega_0^2}
+ = \sigma_p^2 \avg{x(t)^2} \;,
+\end{align}
+where $m_p\equiv m/\sigma_p$, $G_{1p}\equiv G_0/m_p^2=\sigma_p^2 G_1$.
+Plugging into the equipartition theorem yeilds
+\begin{align}
+ \kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
+ = \frac{2 \beta\omega_0^2 \sigma_p^2 k_BT}{\pi G_{1p}} \;.
+\end{align}
+
+From \cref{eq:Gone}, we find the expected value of $G_{1p}$ to be
+\begin{equation}
+ G_{1p} \equiv \sigma_p^2 G_1 = \frac{2}{\pi m} \sigma_p^2 k_BT \beta \;.
+ \label{eq:Gone-p}
+\end{equation}
+
+
+\subsection{Fitting deflection voltage in frequency space}
+
+Note: the math in this section depends on some definitions from
+section \cref{sec:setup}.
+
+As yet another alternative, you could fit in frequency
+$f\equiv\omega/2\pi$ instead of angular frequency $\omega$. But we
+must be careful with normalization. Comparing the angular frequency
+and normal frequency unitary Fourier transforms
+\begin{align}
+ \Four{x(t)}(\omega)
+ &\equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{x(t) e^{-i \omega t}} \\
+ \Fourf{x(t)}(f) &\equiv \iInfInf{t}{x(t) e^{-2\pi i f t}}
+ = \iInfInf{t}{x(t) e^{-i \omega t}}
+ = \sqrt{2\pi}\cdot\Four{x(t)}(\omega=2\pi f) \;,
+\end{align}
+from which we can translate the \PSD
+\begin{align}
+ \PSD(x, \omega) &\equiv \normLimT 2 \magSq{ \Four{x(t)}(\omega) } \\
+ \begin{split}
+ \PSD_f(x, f) &\equiv \normLimT 2 \magSq{ \Fourf{x(t)}(f) }
+ = 2\pi \cdot \normLimT 2 \magSq{ \Four{x(t)}(\omega=2\pi f) } \\
+ &= 2\pi \PSD(x, \omega=2\pi f) \;.
+ \end{split}
+\end{align}
+\nomenclature{$t$}{Time (seconds)}
+\index{PSD@\PSD!in frequency space}
+The variance of the function $x(t)$ is then given by plugging into
+\cref{eq:parseval-var} (our corollary to Parseval's theorem)
+\begin{align}
+ \avg{x(t)^2} &= \iOInf{\omega}{\PSD(x,\omega)}
+ = \iOInf{f}{\frac{1}{2\pi}\PSD_f(x,f)2\pi\cdot}
+ = \iOInf{f}{\PSD_f(x,f)} \;.
+\end{align}
+Therefore
+\begin{align}
+ \begin{split}
+ \PSD_f(V_p, f) &= 2\pi\PSD(V_p,\omega)
+ = \frac{2\pi G_{1p}}{(4\pi f_0^2-4\pi^2f^2)^2 + \beta^2 4\pi^2f^2}
+ = \frac{2\pi G_{1p}}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2} \\
+ &= \frac{G_{1p}/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}}
+ = \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2}
+ \end{split} \\
+ \avg{V_p(t)^2} &= \frac{\pi G_{1f}}{2\beta_f f_0^2} \;.
+% = \frac{\pi G_{1p} / (2\pi)^3}{2\beta/(2\pi) \omega_0^2/(2\pi)^2}
+% = \frac{\pi G_{1p}}{2\beta\omega_0^2} = \avg{V_p(t)^2} % check!
+\end{align}
+where $f_0\equiv\omega_0/2\pi$, $\beta_f\equiv\beta/2\pi$, and
+$G_{1f}\equiv G_{1p}/8\pi^3$. Finally
+\begin{align}
+ \kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
+ = \frac{2 \beta_f f_0^2 \sigma_p^2 k_BT}{\pi G_{1f}} \;.
+\end{align}
+
+From \cref{eq:Gone}, we expect $G_{1f}$ to be
+\begin{equation}
+ G_{1f} = \frac{G_{1p}}{8\pi^3}
+ = \frac{\sigma_p^2 G_1}{8\pi^3}
+ = \frac{\frac{2}{\pi m} \sigma_p^2 k_BT \beta}{8\pi^3}
+ = \frac{\sigma_p^2 k_BT \beta}{4\pi^4 m} \;.
+ \label{eq:Gone-f}
+\end{equation}
+
+
+% TODO: re-integrate the following
+
+% \begin{split}
+% \PSD_f(V_p, f) =
+% 2\pi\PSD(V_p,\omega)
+% = \frac{2\pi G_{1p}}{(4\pi f_0^2-4\pi^2f^2)^2 + \beta^2 4\pi^2f^2}
+% = \frac{2\pi G_{1p}}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2} \\
+% &= \frac{G_{1p}/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}}
+% \end{split} \\
+
+% = \frac{\pi G_{1p} / (2\pi)^3}{2\beta/(2\pi) \omega_0^2/(2\pi)^2}
+% = \frac{\pi G_{1p}}{2\beta\omega_0^2} = \avg{V_p(t)^2} % check!
+
+%where $f_0\equiv\omega_0/2\pi$, $\beta_f\equiv\beta/2\pi$, and
+%$G_{1f}\equiv G_{1p}/8\pi^3$. Finally
+
+%From \cref{eq:Gone}, we expect $G_{1f}$ to be
+%\begin{equation}
+% G_{1f} = \frac{G_{1p}}{8\pi^3}
+% = \frac{\sigma_p^2 G_1}{8\pi^3}
+% = \frac{\frac{2}{\pi m} \sigma_p^2 k_BT \beta}{8\pi^3}
+% = \frac{\sigma_p^2 k_BT \beta}{4\pi^4 m} \;.
+% \label{eq:Gone-f}
+% \end{equation}
+
+\subsection{Highly damped case}
+
+For highly damped systems, the inertial term 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 server 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
+% ODHO stands for very Over Damped Harmonic oscillator
+\begin{align}
+ (i \gamma \omega + \kappa) x(\omega) &= F(\omega) \label{eq:ODHO-freq} \\
+ |x(\omega)|^2 &= \frac{|F(\omega)|^2}{\kappa^2 + \gamma^2 \omega^2} \;.
+ \label{eq:ODHO-xmag}
+\end{align}
+\index{Damped harmonic oscillator!extremely overdamped}
+We compute the \PSD\ by plugging \cref{eq:ODHO-xmag} into
+\cref{eq:psd-def}
+\begin{equation}
+ \PSD(x, \omega)
+ = \normLimT \frac{2\magSq{F(\omega)}}{\kappa^2 + \gamma^2\omega^2} \;.
+ \label{eq:ODHO-psd-F}
+\end{equation}
+\index{PSD@\PSD}
+
+Because thermal noise is white (not autocorrelated + Wiener-Khinchin
+Theorem), we can denote the one sided thermal power spectral density
+per unit time by
+\begin{equation}
+ \PSD(F, \omega) = G_0
+ = \normLimT 2 \magSq{F(\omega)} \;. \label{eq:GOdef} % label O != zero
+\end{equation}
+
+Plugging \cref{eq:GOdef} into \cref{eq:ODHO-psd} we have
+\begin{equation}
+ \PSD(x, \omega) = \frac{G_0}{\kappa^2 + \gamma^2\omega^2} \;.
+\end{equation}
+This is the formula we would use to fit our measured \PSD, but let us
+go a bit farther to find the expected \PSD\ and thermal noise given
+$m$, $\gamma$ and $\kappa$.
+
+Integrating over positive $\omega$ to find the total power per unit
+time yields
+\begin{align}
+ \iOInf{\omega}{\PSD(x, \omega)}
+ = \iOInf{\omega}{\frac{G_0}{\kappa^2 + \gamma^2\omega^2}}
+ = \frac{G_0}{\gamma}\iOInf{z}{\frac{1}{\kappa^2 + z^2}}
+ = \frac{G_0 \pi}{2 \gamma \kappa} \;,
+\end{align}
+where the integral is solved in \cref{sec:integrals}.
+
+Plugging into our corollary to Parseval's theorem (\cref{eq:parseval-var}),
+\begin{equation}
+ \avg{x(t)^2} = \frac{G_0 \pi}{2 \gamma \kappa} \;. \label{eq:ODHO-var}
+\end{equation}
+
+Plugging \cref{eq:ODHO-var} into \cref{eq:equipart} we have
+\begin{align}
+ \kappa \frac{G_0 \pi}{2 \gamma \kappa} &= k_BT \\
+ G_0 &= \frac{2 \gamma k_BT}{\pi} \;.
+\end{align}
+
+So we expect $x(t)$ to have a power spectral density per unit time given by
+\begin{equation}
+ \PSD(x, \omega) = \frac{2}{\pi}
+ \cdot
+ \frac{\gamma k_BT}{\kappa^2 + \gamma^2\omega^2} \;.
+ \label{eq:ODHO-psd}
+\end{equation}
+\index{PSD@\PSD}
+
+\subsection{General form}
+
+The procedure here is exactly the same as the previous section. The
+integral normalizing $G_0$, however, becomes a little more
+complicated.
+
+Fourier transforming \cref{eq:DHO} and applying \cref{eq:four-deriv} we have
+\begin{align}
+ (-m\omega^2 + i \gamma \omega + \kappa) x(\omega) &= F(\omega)
+ \label{eq:DHO-freq} \\
+ (\omega_0^2-\omega^2 + i \beta \omega) x(\omega) &= \frac{F(\omega)}{m} \\
+ |x(\omega)|^2 &= \frac{|F(\omega)|^2/m^2}
+ {(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} \;,
+ \label{eq:DHO-xmag}
+\end{align}
+where $\omega_0 \equiv \sqrt{\kappa/m}$ is the resonant angular
+frequency and $\beta \equiv \gamma / m$ is the drag-aceleration
+coefficient.
+\index{Damped harmonic oscillator}\index{beta}\index{gamma}
+\nomenclature{$\omega_0$}{Resonant angular frequency (radians per second)}
+\index{$\omega_0$}
+
+We compute the \PSD\ by plugging \cref{eq:DHO-xmag} into \cref{eq:psd-def}
+\begin{equation}
+ \PSD(x, \omega)
+ = \normLimT \frac{2 |F(\omega)|^2/m^2}
+ {(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} \;.
+ \label{eq:DHO-psd-F}
+\end{equation}
+\index{PSD@\PSD}
+
+Plugging \cref{eq:GOdef} into \cref{eq:DHO-psd-F} we have
+\begin{equation}
+ \PSD(x, \omega) = \frac{G_0/m^2}{(\omega_0^2-\omega^2)^2 +\beta^2\omega^2}\;.
+ \label{eq:model-psd}
+\end{equation}
+Integrating over positive $\omega$ to find the total power per unit time yields
+\begin{align}
+ \iOInf{\omega}{\PSD(x, \omega)}
+ &= \frac{G_0}{2m^2}
+ \iInfInf{\omega}{\frac{1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}}
+ = \frac{G_0}{2m^2} \cdot \frac{\pi}{\beta\omega_0^2}
+ = \frac{G_0 \pi}{2m^2\beta\omega_0^2}
+ = \frac{G_0 \pi}{2m^2\beta \frac{\kappa}{m}} \\
+ &= \frac{G_0 \pi}{2m \beta \kappa} \;.
+\end{align}
+The integration is detailed in \cref{sec:integrals}.
+By the corollary to Parseval's theorem (\cref{eq:parseval-var}), we have
+\begin{equation}
+ \avg{x(t)^2} = \frac{G_0 \pi}{2m^2\beta\omega_0^2} \;. \label{eq:DHO-var}
+\end{equation}
+
+Plugging \cref{eq:DHO-var} into the equipartition theorem
+(\cref{eq:equipart}) we have
+\begin{align}
+ \kappa \frac{G_0 \pi}{2m \beta \kappa} &= k_BT \\
+ G_0 &= \frac{2}{\pi} k_BT m \beta \;. \label{eq:GO}
+\end{align}
+
+So we expect $x(t)$ to have a power spectral density per unit time given by
+\begin{equation}
+ \PSD(x, \omega) = \frac{2 k_BT \beta}
+ { \pi m \p[{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}] }\;.
+ \label{eq:DHO-psd}
+\end{equation}
+\index{PSD@\PSD}
+
+As expected, the general form \cref{eq:DHO-psd} reduces to the
+extremely overdamped form \cref{eq:ODHO-psd}. Plugging in for
+$\beta\equiv\gamma/m$ and $\omega_0\equiv\sqrt{\kappa/m}$,
+\begin{align}
+ \lim_{m\rightarrow 0} \PSD(x, \omega)
+ &= \lim_{m\rightarrow 0} \frac{2 k_BT \gamma}
+ { \pi m^2 \p[{(\kappa/m-\omega^2)^2 + \gamma^2/m^2\omega^2}] }
+ = \lim_{m\rightarrow 0} \frac{2 k_BT \gamma}
+ { \pi \p[{(\kappa-m\omega^2)^2 + \gamma^2\omega^2}] } \\
+ &= \frac{2}{\pi}
+ \cdot
+ \frac{\gamma k_BT}{\kappa^2 + \gamma^2\omega^2} \;.
+\end{align}
calibrating an AFM cantilever, so I work it out here with all the
gory details :p.
-The testq subdirectory contains some python scripts I used to test my
-algebra and get a better feel for what was going on. The dot
-subdirectory is a preliminary data-flow map of the cantilever
-calibration procedure.
+The test subdirectory contains some Python scripts I used to test my
+algebra and get a better feel for what was going on.
\chapter{Cantilever Calibration}
\label{sec:cantilever-calib}
-\input{cantilever-calib/overview}
-\input{cantilever-calib/setup_general}
-\input{cantilever-calib/solve_highly_damped}
-\input{cantilever-calib/solve_general}
\input{cantilever-calib/contour_integration}
\input{cantilever-calib/integrals}
+++ /dev/null
-In this appendix, we derive the formulas presented in
-\cref{sec:cantilever-calib:intro} for calibrating an AFM cantilever.
-You should read that section to understand the goal of this appendix
-and familiarize yourself with the notation we will be using.
-
-\subsection{Related papers}
-
-Various corrections taking into acount higher order 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 $\kappa\avg{x^2} = k_BT$ method.
-
-Roters and Johannsmann describe a similar approach to deriving the Lorentizian
-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.
-
-\section{Methods}
-% TODO: deprecated in favor of sec:cantilever-calib:intro
-
-Rather than computing the variance of $x(t)$ directly, we attempt to
-filter out noise by fitting the power spectral density (\PSD)%
-\nomenclature[PSDa]{$\PSD$}{Power spectral density in angular
- frequency space}\index{PSD@\PSD}\nomenclature{$\omega$}{Angular
- frequency (radians per second)} of $x(t)$ to the theoretically
-predicted \PSD\ for a damped harmonic oscillator (\cref{eq:model-psd})
-\begin{align}
- \ddt{x} + \beta\dt{x} + \omega_0^2 x &= \frac{F_\text{thermal}}{m} \\
- \PSD(x, \omega) &= \frac{G_1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} \;,
-\end{align}
-\index{Damped harmonic oscillator}
-where $G_1\equiv G_0/m^2$, $\omega_0$, and $\beta$ are used as the
-fitting parameters (see \cref{eq:model-psd}).%
-\index{$\beta$}\index{$\gamma$} The variance of $x(t)$ is then given
-by \cref{eq:DHO-var}
-\begin{equation}
- \avg{x(t)^2} = \frac{\pi G_1}{2\beta\omega_0^2} \;,
-\end{equation}
-which we can plug into the equipartition theorem
-(\cref{eq:equipart}) yielding
-\begin{align}
- \kappa = \frac{2 \beta \omega_0^2 k_BT}{\pi G_1} \;.
-\end{align}
-
-From \cref{eq:GO}, we find the expected value of $G_1$ to be
-\begin{equation}
- G_1 \equiv G_0/m^2 = \frac{2}{\pi m} k_BT \beta \;. \label{eq:Gone}
-\end{equation}
-
-
-\subsection{Fitting deflection voltage directly}
-
-In order to keep our errors in measuring $\sigma_p$ seperate from
-other errors in measuring $\avg{x(t)^2}$, we can fit the voltage
-spectrum before converting to distance.
-\begin{align}
- \ddt{V_p}/\sigma_p + \beta\dt{V_p}/\sigma_p + \omega_0^2 V_p/\sigma_p
- &= F_\text{thermal} \\
- \ddt{V_p} + \beta\dt{V_p} + \omega_0^2 V_p
- &= \sigma_p\frac{F_\text{thermal}}{m} \\
- \ddt{V_p} + \beta\dt{V_p} + \omega_0^2 V_p
- &= \frac{F_\text{thermal}}{m_p} \\
- \PSD(V_p, \omega) &= \frac{G_{1p}}
- { (\omega_0^2-\omega^2)^2 + \beta^2\omega^2 } \\
- \avg{V_p(t)^2} &= \frac{\pi G_{1p}}{2\beta\omega_0^2}
- = \frac{\pi \sigma_p^2 G_{1}}{2\beta\omega_0^2}
- = \sigma_p^2 \avg{x(t)^2} \;,
-\end{align}
-where $m_p\equiv m/\sigma_p$, $G_{1p}\equiv G_0/m_p^2=\sigma_p^2 G_1$.
-Plugging into the equipartition theorem yeilds
-\begin{align}
- \kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
- = \frac{2 \beta\omega_0^2 \sigma_p^2 k_BT}{\pi G_{1p}} \;.
-\end{align}
-
-From \cref{eq:Gone}, we find the expected value of $G_{1p}$ to be
-\begin{equation}
- G_{1p} \equiv \sigma_p^2 G_1 = \frac{2}{\pi m} \sigma_p^2 k_BT \beta \;.
- \label{eq:Gone-p}
-\end{equation}
-
-
-\subsection{Fitting deflection voltage in frequency space}
-
-Note: the math in this section depends on some definitions from
-section \cref{sec:setup}.
-
-As yet another alternative, you could fit in frequency
-$f\equiv\omega/2\pi$ instead of angular frequency $\omega$. But we
-must be careful with normalization. Comparing the angular frequency
-and normal frequency unitary Fourier transforms
-\begin{align}
- \Four{x(t)}(\omega)
- &\equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{x(t) e^{-i \omega t}} \\
- \Fourf{x(t)}(f) &\equiv \iInfInf{t}{x(t) e^{-2\pi i f t}}
- = \iInfInf{t}{x(t) e^{-i \omega t}}
- = \sqrt{2\pi}\cdot\Four{x(t)}(\omega=2\pi f) \;,
-\end{align}
-from which we can translate the \PSD
-\begin{align}
- \PSD(x, \omega) &\equiv \normLimT 2 \magSq{ \Four{x(t)}(\omega) } \\
- \begin{split}
- \PSD_f(x, f) &\equiv \normLimT 2 \magSq{ \Fourf{x(t)}(f) }
- = 2\pi \cdot \normLimT 2 \magSq{ \Four{x(t)}(\omega=2\pi f) } \\
- &= 2\pi \PSD(x, \omega=2\pi f) \;.
- \end{split}
-\end{align}
-\nomenclature{$t$}{Time (seconds)}
-\index{PSD@\PSD!in frequency space}
-The variance of the function $x(t)$ is then given by plugging into
-\cref{eq:parseval-var} (our corollary to Parseval's theorem)
-\begin{align}
- \avg{x(t)^2} &= \iOInf{\omega}{\PSD(x,\omega)}
- = \iOInf{f}{\frac{1}{2\pi}\PSD_f(x,f)2\pi\cdot}
- = \iOInf{f}{\PSD_f(x,f)} \;.
-\end{align}
-Therefore
-\begin{align}
- \begin{split}
- \PSD_f(V_p, f) &= 2\pi\PSD(V_p,\omega)
- = \frac{2\pi G_{1p}}{(4\pi f_0^2-4\pi^2f^2)^2 + \beta^2 4\pi^2f^2}
- = \frac{2\pi G_{1p}}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2} \\
- &= \frac{G_{1p}/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}}
- = \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2}
- \end{split} \\
- \avg{V_p(t)^2} &= \frac{\pi G_{1f}}{2\beta_f f_0^2} \;.
-% = \frac{\pi G_{1p} / (2\pi)^3}{2\beta/(2\pi) \omega_0^2/(2\pi)^2}
-% = \frac{\pi G_{1p}}{2\beta\omega_0^2} = \avg{V_p(t)^2} % check!
-\end{align}
-where $f_0\equiv\omega_0/2\pi$, $\beta_f\equiv\beta/2\pi$, and
-$G_{1f}\equiv G_{1p}/8\pi^3$. Finally
-\begin{align}
- \kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
- = \frac{2 \beta_f f_0^2 \sigma_p^2 k_BT}{\pi G_{1f}} \;.
-\end{align}
-
-From \cref{eq:Gone}, we expect $G_{1f}$ to be
-\begin{equation}
- G_{1f} = \frac{G_{1p}}{8\pi^3}
- = \frac{\sigma_p^2 G_1}{8\pi^3}
- = \frac{\frac{2}{\pi m} \sigma_p^2 k_BT \beta}{8\pi^3}
- = \frac{\sigma_p^2 k_BT \beta}{4\pi^4 m} \;.
- \label{eq:Gone-f}
-\end{equation}
-
-
-% TODO: re-integrate the following
-
-% \begin{split}
-% \PSD_f(V_p, f) =
-% 2\pi\PSD(V_p,\omega)
-% = \frac{2\pi G_{1p}}{(4\pi f_0^2-4\pi^2f^2)^2 + \beta^2 4\pi^2f^2}
-% = \frac{2\pi G_{1p}}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2} \\
-% &= \frac{G_{1p}/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}}
-% \end{split} \\
-
-% = \frac{\pi G_{1p} / (2\pi)^3}{2\beta/(2\pi) \omega_0^2/(2\pi)^2}
-% = \frac{\pi G_{1p}}{2\beta\omega_0^2} = \avg{V_p(t)^2} % check!
-
-%where $f_0\equiv\omega_0/2\pi$, $\beta_f\equiv\beta/2\pi$, and
-%$G_{1f}\equiv G_{1p}/8\pi^3$. Finally
-
-%From \cref{eq:Gone}, we expect $G_{1f}$ to be
-%\begin{equation}
-% G_{1f} = \frac{G_{1p}}{8\pi^3}
-% = \frac{\sigma_p^2 G_1}{8\pi^3}
-% = \frac{\frac{2}{\pi m} \sigma_p^2 k_BT \beta}{8\pi^3}
-% = \frac{\sigma_p^2 k_BT \beta}{4\pi^4 m} \;.
-% \label{eq:Gone-f}
-% \end{equation}
-
+++ /dev/null
-\section{Theoretical power spectral density for a damped harmonic oscillator}
-\label{sec:setup}
-
-Our cantilever can be approximated as a damped harmonic oscillator
-\begin{equation}
- m\ddt{x} + \gamma \dt{x} + \kappa x = F(t) \;, \label{eq:DHO}
- % DHO for Damped Harmonic Oscillator
-\end{equation}
-where $x$ is the displacement from equilibrium,
- $m$ is the effective mass,
- $\gamma$ is the effective drag coefficient,
- $\kappa$ is the spring constant, and
- $F(t)$ is the external driving force.
-During the non-contact phase of calibration,
- $F(t)$ comes from random thermal noise.
-\nomenclature{$\beta$}{Damped harmonic oscillator drag-acceleration
- coefficient $\beta \equiv \gamma/m$}\index{$\beta$}%
-\nomenclature{$\gamma$}{Damped harmonic oscillator drag coefficient
- $F_\text{drag} = \gamma\dt{x}$}\index{$\gamma$}%
-\index{damped harmonic oscillator}%
-\nomenclature{$\dt{s}$}{First derivative of the time-series $s(t)$
- with respect to time. $\dt{s} = \deriv{t}{s}$}%
-\nomenclature{$\ddt{s}$}{Second derivative of the time-series $s(t)$
- with respect to time. $\ddt{s} = \nderiv{2}{t}{s}$}%
-
-In the following analysis, we use the unitary, angular frequency Fourier transform normalization
-\begin{equation}
- \Four{x(t)} \equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{x(t) e^{-i \omega t}}\;.
-\end{equation}
-\nomenclature{\Four{s(t)}}{Fourier transform of the time-series
- $s(t)$. $s(f) = \Four{s(t)}$}\index{Fourier transform}
-
-We also use the following theorems (proved elsewhere):
-\begin{align}
- \cos\left(\frac{\theta}{2}\right) &= \pm\sqrt{\frac{1}{2}[1+\cos(\theta)]}\;,
- &\text{\citep{cos-halfangle}} \label{eq:cos-halfangle} \\
- \Four{\nderiv{n}{t}{x(t)}} &= (i \omega)^n x(\omega) \;,
- &\text{\citep{four-deriv}} \label{eq:four-deriv} \\
-% \Four{x*y} &= x(\omega) y(\omega), \label{eq:four-conv}
-% & \text{and} \\
- \iInfInf{t}{\magSq{x(t)}} &= \iInfInf{\omega}{\magSq{x(w)}} \;.
- &\text{(Parseval's)\citep{parseval}} \label{eq:parseval}
-\end{align}
-\index{cosine half-angle}
-\index{Parseval's theorem}
-%where $x*y$ denotes the convolution of $x$ and $y$,
-%\begin{equation}
-% x*y \equiv \iInfInf{\tau}{x(t-\tau)y(\tau)}.
-%\end{equation}
-As a corollary to Parseval's theorem, we note that the one sided power spectral density per unit time (\PSD) defined by
-\begin{align}
- \PSD(x, \omega) &\equiv \normLimT 2 \left| x(\omega) \right|^2
- &\text{\citep{PSD}} \label{eq:psd-def}
-\end{align}
-\index{PSD@\PSD}
-relates to the variance by
-\begin{align}
- \avg{x(t)^2}
- &= \iLimT{\magSq{x(t)}}
- = \normLimT \iInfInf{\omega}{\magSq{x(\omega)}}
- = \iOInf{\omega}{\PSD(x,\omega)} \;, \label{eq:parseval-var}
-\end{align}
-where $t_T$ is the total time over which data has been aquired.
-
-We also use the Wiener-Khinchin theorem,
-which relates the two sided power spectral density $S_{xx}(\omega)$
-to the autocorrelation function $r_{xx}(t)$ via
-\begin{align}
- S_{xx}(\omega) &= \Four{ r_{xx}(t) } \;,
- &\text{(Wiener-Khinchin)\citep{wiener-khinchin}} \label{eq:wiener_khinchin}
-\end{align}
-\index{Wiener-Khinchin theorem}
-where $r_{xx}(t)$ is defined in terms of the expectation value
-\begin{align}
- r_{xx}(t) &\equiv \avg{x(\tau)\conj{x}(\tau-t)} \;,
- &\text{\citep{wikipedia-wiener-khinchin}}
-\end{align}
-and $\conj{x}$ represents the complex conjugate of $x$.
-\nomenclature{$\conj{z}$}{Complex conjugate of $z$}
+++ /dev/null
-\subsection{General form}
-
-The procedure here is exactly the same as the previous section. The
-integral normalizing $G_0$, however, becomes a little more
-complicated.
-
-Fourier transforming \cref{eq:DHO} and applying \cref{eq:four-deriv} we have
-\begin{align}
- (-m\omega^2 + i \gamma \omega + \kappa) x(\omega) &= F(\omega)
- \label{eq:DHO-freq} \\
- (\omega_0^2-\omega^2 + i \beta \omega) x(\omega) &= \frac{F(\omega)}{m} \\
- |x(\omega)|^2 &= \frac{|F(\omega)|^2/m^2}
- {(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} \;,
- \label{eq:DHO-xmag}
-\end{align}
-where $\omega_0 \equiv \sqrt{\kappa/m}$ is the resonant angular
-frequency and $\beta \equiv \gamma / m$ is the drag-aceleration
-coefficient.
-\index{Damped harmonic oscillator}\index{beta}\index{gamma}
-\nomenclature{$\omega_0$}{Resonant angular frequency (radians per second)}
-\index{$\omega_0$}
-
-We compute the \PSD\ by plugging \cref{eq:DHO-xmag} into \cref{eq:psd-def}
-\begin{equation}
- \PSD(x, \omega)
- = \normLimT \frac{2 |F(\omega)|^2/m^2}
- {(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} \;.
- \label{eq:DHO-psd-F}
-\end{equation}
-\index{PSD@\PSD}
-
-Plugging \cref{eq:GOdef} into \cref{eq:DHO-psd-F} we have
-\begin{equation}
- \PSD(x, \omega) = \frac{G_0/m^2}{(\omega_0^2-\omega^2)^2 +\beta^2\omega^2}\;.
- \label{eq:model-psd}
-\end{equation}
-Integrating over positive $\omega$ to find the total power per unit time yields
-\begin{align}
- \iOInf{\omega}{\PSD(x, \omega)}
- &= \frac{G_0}{2m^2}
- \iInfInf{\omega}{\frac{1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}}
- = \frac{G_0}{2m^2} \cdot \frac{\pi}{\beta\omega_0^2}
- = \frac{G_0 \pi}{2m^2\beta\omega_0^2}
- = \frac{G_0 \pi}{2m^2\beta \frac{\kappa}{m}} \\
- &= \frac{G_0 \pi}{2m \beta \kappa} \;.
-\end{align}
-The integration is detailed in \cref{sec:integrals}.
-By the corollary to Parseval's theorem (\cref{eq:parseval-var}), we have
-\begin{equation}
- \avg{x(t)^2} = \frac{G_0 \pi}{2m^2\beta\omega_0^2} \;. \label{eq:DHO-var}
-\end{equation}
-
-Plugging \cref{eq:DHO-var} into the equipartition theorem
-(\cref{eq:equipart}) we have
-\begin{align}
- \kappa \frac{G_0 \pi}{2m \beta \kappa} &= k_BT \\
- G_0 &= \frac{2}{\pi} k_BT m \beta \;. \label{eq:GO}
-\end{align}
-
-So we expect $x(t)$ to have a power spectral density per unit time given by
-\begin{equation}
- \PSD(x, \omega) = \frac{2 k_BT \beta}
- { \pi m \p[{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}] }\;.
- \label{eq:DHO-psd}
-\end{equation}
-\index{PSD@\PSD}
-
-As expected, the general form \cref{eq:DHO-psd} reduces to the
-extremely overdamped form \cref{eq:ODHO-psd}. Plugging in for
-$\beta\equiv\gamma/m$ and $\omega_0\equiv\sqrt{\kappa/m}$,
-\begin{align}
- \lim_{m\rightarrow 0} \PSD(x, \omega)
- &= \lim_{m\rightarrow 0} \frac{2 k_BT \gamma}
- { \pi m^2 \p[{(\kappa/m-\omega^2)^2 + \gamma^2/m^2\omega^2}] }
- = \lim_{m\rightarrow 0} \frac{2 k_BT \gamma}
- { \pi \p[{(\kappa-m\omega^2)^2 + \gamma^2\omega^2}] } \\
- &= \frac{2}{\pi}
- \cdot
- \frac{\gamma k_BT}{\kappa^2 + \gamma^2\omega^2} \;.
-\end{align}
+++ /dev/null
-\subsection{Highly damped case}
-
-For highly damped systems, the inertial term 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 server 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
-% ODHO stands for very Over Damped Harmonic oscillator
-\begin{align}
- (i \gamma \omega + \kappa) x(\omega) &= F(\omega) \label{eq:ODHO-freq} \\
- |x(\omega)|^2 &= \frac{|F(\omega)|^2}{\kappa^2 + \gamma^2 \omega^2} \;.
- \label{eq:ODHO-xmag}
-\end{align}
-\index{Damped harmonic oscillator!extremely overdamped}
-We compute the \PSD\ by plugging \cref{eq:ODHO-xmag} into
-\cref{eq:psd-def}
-\begin{equation}
- \PSD(x, \omega)
- = \normLimT \frac{2\magSq{F(\omega)}}{\kappa^2 + \gamma^2\omega^2} \;.
- \label{eq:ODHO-psd-F}
-\end{equation}
-\index{PSD@\PSD}
-
-Because thermal noise is white (not autocorrelated + Wiener-Khinchin
-Theorem), we can denote the one sided thermal power spectral density
-per unit time by
-\begin{equation}
- \PSD(F, \omega) = G_0
- = \normLimT 2 \magSq{F(\omega)} \;. \label{eq:GOdef} % label O != zero
-\end{equation}
-
-Plugging \cref{eq:GOdef} into \cref{eq:ODHO-psd} we have
-\begin{equation}
- \PSD(x, \omega) = \frac{G_0}{\kappa^2 + \gamma^2\omega^2} \;.
-\end{equation}
-This is the formula we would use to fit our measured \PSD, but let us
-go a bit farther to find the expected \PSD\ and thermal noise given
-$m$, $\gamma$ and $\kappa$.
-
-Integrating over positive $\omega$ to find the total power per unit
-time yields
-\begin{align}
- \iOInf{\omega}{\PSD(x, \omega)}
- = \iOInf{\omega}{\frac{G_0}{\kappa^2 + \gamma^2\omega^2}}
- = \frac{G_0}{\gamma}\iOInf{z}{\frac{1}{\kappa^2 + z^2}}
- = \frac{G_0 \pi}{2 \gamma \kappa} \;,
-\end{align}
-where the integral is solved in \cref{sec:integrals}.
-
-Plugging into our corollary to Parseval's theorem (\cref{eq:parseval-var}),
-\begin{equation}
- \avg{x(t)^2} = \frac{G_0 \pi}{2 \gamma \kappa} \;. \label{eq:ODHO-var}
-\end{equation}
-
-Plugging \cref{eq:ODHO-var} into \cref{eq:equipart} we have
-\begin{align}
- \kappa \frac{G_0 \pi}{2 \gamma \kappa} &= k_BT \\
- G_0 &= \frac{2 \gamma k_BT}{\pi} \;.
-\end{align}
-
-So we expect $x(t)$ to have a power spectral density per unit time given by
-\begin{equation}
- \PSD(x, \omega) = \frac{2}{\pi}
- \cdot
- \frac{\gamma k_BT}{\kappa^2 + \gamma^2\omega^2} \;.
- \label{eq:ODHO-psd}
-\end{equation}
-\index{PSD@\PSD}
apparatus/main,%
% sawsim/main,% currently empty
pyafm/main,%
- cantilever-calib/main,%
+ calibcant/main,%
packaging/main,%
figures/main,%
root}
projects / thesis.git / commitdiff