We first measured the Larmor frequency, as described in the Methods, and found a value of \(1.852\pm0.018~kHz\), 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 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 \({T_1}=2.15\pm0.05 s\).
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 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 \[\label{eq:mu} \mu=\gamma\hbar\sqrt{I(I+1)}\] 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.
We examined the relationship between precession frequency and magnetic field as discussed in the Methods section. 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 \(\frac{\gamma}{2\pi}\), where \(\gamma\) is the gyromagnetic ratio for protons. Our slope was \(42.1\pm0.7~\frac{rad}{s\cdot \mu T}\), which agrees with the know value \(\frac{\gamma}{2\pi}=42.5~\frac{rad}{s\cdot \mu T}\). The intercept represents the frequency without any extra applied field, which we found to be \(1847\pm7~Hz\), agreeing with the value of \(1852\pm18~Hz\) we previously found for precession frequency.