From 944da0511a7073aa6ef280b272d1fb0029854c1e Mon Sep 17 00:00:00 2001 From: "W. Trevor King" Date: Wed, 27 Jun 2012 15:13:49 -0400 Subject: [PATCH] Use kappa for spring constants (k is for rates). --- src/apparatus/cantilever-calib.tex | 20 ++++++------ src/cantilever-calib/README | 2 +- src/cantilever-calib/overview.tex | 8 ++--- src/cantilever-calib/setup_general.tex | 4 +-- src/cantilever-calib/solve_general.tex | 21 ++++++------ src/cantilever-calib/solve_highly_damped.tex | 34 +++++++++++--------- 6 files changed, 46 insertions(+), 43 deletions(-) diff --git a/src/apparatus/cantilever-calib.tex b/src/apparatus/cantilever-calib.tex index 9db022b..caa2a67 100644 --- a/src/apparatus/cantilever-calib.tex +++ b/src/apparatus/cantilever-calib.tex @@ -4,14 +4,14 @@ 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 -$F=-kx$. +$F=-\kappa x$. \nomenclature{$F$}{Force (newtons)} -\nomenclature{$k$}{Spring constant (newtons per meter)} +\nomenclature{$\kappa$}{Spring constant (newtons per meter)} \nomenclature{$x$}{Displacement (meters)} The basic idea is to use the equipartition theorem\citep{hutter93}, \begin{equation} - \frac{1}{2} k \avg{x^2} = \frac{1}{2} k_BT \;, \label{eq:equipart} + \frac{1}{2} \kappa \avg{x^2} = \frac{1}{2} k_BT \;, \label{eq:equipart} \end{equation} where $k_B$ is Boltzmann's constant, $T$ is the absolute temperature, and $\avg{x^2}$ denotes the expectation value of $x^2$ as measured @@ -23,12 +23,12 @@ over a very long interval $t_T$, \begin{equation} \avg{A} \equiv \iLimT{A} \;. \end{equation}} -Solving the equipartition theorem for $k$ yields +Solving the equipartition theorem for $\kappa$ yields \begin{equation} - k = \frac{k_BT}{\avg{x^2}} \;, \label{eq:equipart_k} + \kappa = \frac{k_BT}{\avg{x^2}} \;, \label{eq:equipart_k} \end{equation} so we need to measure (or estimate) the temperature $T$ and variance -of the cantilever position $\avg{x^2}$ in order to estimate $k$. +of the cantilever position $\avg{x^2}$ in order to estimate $\kappa$. To find $\avg{x^2}$, the raw photodiode voltages $V_p(t)$ are converted to distances $x(t)$ using the photodiode sensitivity @@ -37,9 +37,9 @@ while the tip is in contact with the surface) via \begin{equation} x(t) = \frac{V_p(t)}{\sigma_p} \;. \label{eq:x-from-Vp} \end{equation} -By keeping $V_p$ and $\sigma_p$ separate in our calculation of $k$, we -can gauge the relative importance errors in each parameter and -calculate the uncertainty in our estimated $k$. +By keeping $V_p$ and $\sigma_p$ separate in our calculation of +$\kappa$, we can gauge the relative importance errors in each +parameter and calculate the uncertainty in our estimated $\kappa$. In order to filter out noise in the measured value of $\avg{V_p^2}$ we fit the measured cantilever deflection to the expected theoretical @@ -64,7 +64,7 @@ the expectation value for $V_p$ is given by Combining \cref{eq:equipart_k,eq:x-from-Vp,eq:Vp-from-freq-fit}, we have \begin{align} - k &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}} + \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} diff --git a/src/cantilever-calib/README b/src/cantilever-calib/README index 368f355..5b275f7 100644 --- a/src/cantilever-calib/README +++ b/src/cantilever-calib/README @@ -1,6 +1,6 @@ I didn't have a good understanding of the theory behind thermally calibrating an AFM cantilever, so I work it out here with all the -gorey details :p. +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 diff --git a/src/cantilever-calib/overview.tex b/src/cantilever-calib/overview.tex index 4acb896..389e155 100644 --- a/src/cantilever-calib/overview.tex +++ b/src/cantilever-calib/overview.tex @@ -10,7 +10,7 @@ Various corrections taking into acount higher order modes 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 $k\avg{x^2} = k_BT$ method. +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}. %, @@ -56,7 +56,7 @@ by \cref{eq:DHO-var} which we can plug into the equipartition theorem (\cref{eq:equipart}) yielding \begin{align} - k = \frac{2 \beta \omega_0^2 k_BT}{\pi G_1} \;. + \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 @@ -86,7 +86,7 @@ spectrum before converting to distance. 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} - k &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}} + \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} @@ -147,7 +147,7 @@ Therefore 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} - k &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}} + \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} diff --git a/src/cantilever-calib/setup_general.tex b/src/cantilever-calib/setup_general.tex index fc9abe2..36c7811 100644 --- a/src/cantilever-calib/setup_general.tex +++ b/src/cantilever-calib/setup_general.tex @@ -3,13 +3,13 @@ Our cantilever can be approximated as a damped harmonic oscillator \begin{equation} - m\ddt{x} + \gamma \dt{x} + k x = F(t) \;, \label{eq:DHO} + 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, - $k$ is the spring constant, and + $\kappa$ 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. diff --git a/src/cantilever-calib/solve_general.tex b/src/cantilever-calib/solve_general.tex index 17028ae..5d86c4b 100644 --- a/src/cantilever-calib/solve_general.tex +++ b/src/cantilever-calib/solve_general.tex @@ -6,15 +6,16 @@ complicated. Fourier transforming \cref{eq:DHO} and applying \cref{eq:four-deriv} we have \begin{align} - (-m\omega^2 + i \gamma \omega + k) x(\omega) &= F(\omega) + (-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 $\omega_0 \equiv \sqrt{k/m}$ is the resonant angular frequency -and $\beta \equiv \gamma / m$ is the drag-aceleration coefficient. +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{$\omega_0$} @@ -40,8 +41,8 @@ Integrating over positive $\omega$ to find the total power per unit time yields \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{k}{m}} \\ - &= \frac{G_0 \pi}{2m \beta k} \;. + = \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 @@ -52,7 +53,7 @@ By the corollary to Parseval's theorem (\cref{eq:parseval-var}), we have Plugging \cref{eq:DHO-var} into the equipartition theorem (\cref{eq:equipart}) we have \begin{align} - k \frac{G_0 \pi}{2m \beta k} &= k_BT \\ + \kappa \frac{G_0 \pi}{2m \beta \kappa} &= k_BT \\ G_0 &= \frac{2}{\pi} k_BT m \beta \;. \label{eq:GO} \end{align} @@ -66,14 +67,14 @@ So we expect $x(t)$ to have a power spectral density per unit time given by 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 $\omega_0\equiv\sqrt{k/m}$, +$\beta\equiv\gamma/m$ 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[{(k/m-\omega^2)^2 + \gamma^2/m^2\omega^2}] } + { \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[{(k-m\omega^2)^2 + \gamma^2\omega^2}] } \\ + { \pi \p[{(\kappa-m\omega^2)^2 + \gamma^2\omega^2}] } \\ &= \frac{2}{\pi} \cdot - \frac{\gamma k_BT}{k^2 + \gamma^2\omega^2} \;. + \frac{\gamma k_BT}{\kappa^2 + \gamma^2\omega^2} \;. \end{align} diff --git a/src/cantilever-calib/solve_highly_damped.tex b/src/cantilever-calib/solve_highly_damped.tex index b9d07d7..d8a6bbc 100644 --- a/src/cantilever-calib/solve_highly_damped.tex +++ b/src/cantilever-calib/solve_highly_damped.tex @@ -12,8 +12,8 @@ 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 + k) x(\omega) &= F(\omega) \label{eq:ODHO-freq} \\ - |x(\omega)|^2 &= \frac{|F(\omega)|^2}{k^2 + \gamma^2 \omega^2} \;. + (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} @@ -21,13 +21,14 @@ 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)}}{k^2 + \gamma^2\omega^2} \;. + = \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 +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 @@ -35,29 +36,30 @@ we can denote the one sided thermal power spectral density per unit time by Plugging \cref{eq:GOdef} into \cref{eq:ODHO-psd} we have \begin{equation} - \PSD(x, \omega) = \frac{G_0}{k^2 + \gamma^2\omega^2} \;. + \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 $k$. +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 +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}{k^2 + \gamma^2\omega^2}} - = \frac{G_0}{\gamma}\iOInf{z}{\frac{1}{k^2 + z^2}} - = \frac{G_0 \pi}{2 \gamma k} \;, + = \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 k} \;. \label{eq:ODHO-var} + \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} - k \frac{G_0 \pi}{2 \gamma k} &= k_BT \\ + \kappa \frac{G_0 \pi}{2 \gamma \kappa} &= k_BT \\ G_0 &= \frac{2 \gamma k_BT}{\pi} \;. \end{align} @@ -65,7 +67,7 @@ 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}{k^2 + \gamma^2\omega^2} \;. + \frac{\gamma k_BT}{\kappa^2 + \gamma^2\omega^2} \;. \label{eq:ODHO-psd} \end{equation} \index{PSD@\PSD} -- 2.26.2