deletions | additions       diff --git a/subsection_Varying_the_Polarization_Time__.tex b/subsection_Varying_the_Polarization_Time__.tex  index 899b604..89c0081 100644  --- a/subsection_Varying_the_Polarization_Time__.tex  +++ b/subsection_Varying_the_Polarization_Time__.tex 
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We first measured the Larmor frequency, as described in the Methods, and found a value of $1.852\pm0.018~kHz$. According to Equation~\ref{eq:precession}, a magnetic field of $43.5 \mu T$ corresponds to a precession frequency equal to 1852Hz, which is within agreement of our measured field of $43.3\pm 0.3 \mu T$. We expect magnetization, indicated by an amplitude on the oscilloscope, to change exponentially with changing polarization time according to Eq.~\ref{eq:growthrate} \cite{TeachSpin}. By plotting data for three different times after the polarization field was no longer applied, as shown in Fig.~\ref{fig:measurepolarizationtime}, we could fit the data to Eq.~\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 ${T_1}=2.15\pm0.05 s$. For the fit at t=0 s, we found our reduced chi square value $\tilde{\chi}_\nu^2 = 1.57$, at t=50 ms $\tilde{\chi}_\nu^2 = 2.71$, and at t=100 ms $\tilde{\chi}_\nu^2 = 1.85$, meaning that our fits our reasonable accurate.  \subsection{Varying the Magnetic Polarization} Polarization Field}  We also varied magnetic field by varying current to the solenoid, keeping all other values constant, and measured precession frequency. We found there to be no dependence, as expected and predicted by Eq.~\ref{eq:precession}. We expect the magnetization to vary according to Eq.~\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 we can calculate $\mu$ using   \begin{equation}