deletions | additions       diff --git a/subsection_Varying_the_Polarization_Time__.tex b/subsection_Varying_the_Polarization_Time__.tex  index d4dd34b..237cca9 100644  --- a/subsection_Varying_the_Polarization_Time__.tex  +++ b/subsection_Varying_the_Polarization_Time__.tex 
...
\subsection{Varying the Polarization Time}  We first measured the Larmor frequency, as described in the Methods, and found a value of $1.842\pm0.014~kHz$, which is very close as the manual says that the precession frequency should be approximately 2 kHz. We expect magnetization, and therefore voltage, to change exponentially with changing polarization time according to Equation~\ref{eq:growthrate} \textbf{(cite manual)}. By plotting data for three different times after the polarization field was no longer applied, as shown in Fig.~\ref{fig:measurepolarizationtime}, Fig.~\ref{fig:measurepolarizationtime} and discussed in the Methods,  we could fit the data to Equation~\ref{eq:growthrate} and obtain values for the spin-lattice relaxation time ${T_1}$, which should be the same for our three different curves. We found ${T_1}=2.15\pm0.05 s$. \subsection{Varying the Current}  We also varied current, keeping all other values constant, and measured precession frequency. We found there to be no dependence, as expected and predicted by Equation~\ref{eq:precession}. We expect the magnetization to vary according to Equation~\ref{eq:tanh} when we artificially increase Earth's magnetic field, using a constant polarization time (which was in this case 10 seconds), as seen in Fig.~\ref{fig:polarizationtime10s}. Our data was initially plotted against voltage, not magnetization, but Eq.~\ref{eq:tanh} allows us to find magnetization as weknow or  can calculate $\mu$ using \begin{equation}  \label{eq:mu}  \mu=\gamma\hbar\sqrt{I(I+1)}  \end{equation}  where $\gamma$ is the known gyromagnetic constant for protons and $I$ is the nuclear spin quantum number. We could calculate $N$ for approximately  125 mL of water, the volume that was used in this experiment. \subsection{Studying Larmor Precession}  We examined the relationship between precession frequency and magnetic field as discussed in the Methods section. we We  can see by Eq.~\ref{eq:precession} that precession frequency should vary linearly with magnetic field, which is what we see in Fig.~\ref{fig:precession}. The slope of the line is the gyromagnetic constant, and the intercept represents the frequency without any extra applied field, agreeing within uncertainty with the value we previously found for precession frequency.