calibcant/theory.tex: Increase linking to apparatus/cantilever-calib

diff --git a/src/apparatus/cantilever-calib.tex b/src/apparatus/cantilever-calib.tex
index d0a01e1b55af833027ebce12fabbd058af2e8502..68e1d6484a43cf61e917ba5391a914c39c8d430d 100644 (file)
--- a/src/apparatus/cantilever-calib.tex
+++ b/src/apparatus/cantilever-calib.tex
@@ -54,6 +54,7 @@ power spectral density ($\PSD_f$\index{PSD@\PSD!in frequency space})
 of a damped harmonic oscillator exposed to thermal noise
 \begin{equation}
   \PSD_f(V_p, f) = \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2} \;.
+  \label{eq:psd-Vp}
 \end{equation}
 In terms of the fit parameters $G_{1f}$\index{$G_{1f}$},
 $f_0$\index{$f_0$}, and $\beta_f$\index{$\beta_f$}, the expectation

diff --git a/src/calibcant/theory.tex b/src/calibcant/theory.tex
index 2b0f394f2c3482bbf84509b29e1127c5ba03989b..28ad001febc6187dadad22ec84b75ac76067a5cc 100644 (file)
--- a/src/calibcant/theory.tex
+++ b/src/calibcant/theory.tex
@@ -302,7 +302,7 @@ we have
                   = \sigma_p^2 \avg{x(t)^2} \;,
 \end{align}
 where $G_{1p}\equiv G_0/m_p^2=\sigma_p^2 G_1$.
-Plugging into the equipartition theorem yeilds
+Plugging into the equipartition theorem (\cref{eq:equipart_k}) yields
 \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}} \;.
@@ -347,7 +347,7 @@ The variance of the function $x(t)$ is then given by plugging into
      = \iOInf{f}{\frac{1}{2\pi}\PSD_f(x,f)2\pi\cdot}
      = \iOInf{f}{\PSD_f(x,f)} \;.
 \end{align}
-Therefore
+We can now extract \cref{eq:psd-Vp,eq:Vp-from-freq-fit}.
 \begin{align}
   \begin{split}
   \PSD_f(V_p, f) &= 2\pi\PSD(V_p,\omega)
@@ -361,7 +361,8 @@ Therefore
 %    = \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
+$G_{1f}\equiv G_{1p}/8\pi^3$.  Finally, we can generate
+\cref{eq:kappa}.
 \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}} \;.