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\begin{document}
\title{Earth's Field NMR: first draft}
\author{Alisha Vira}
\affil{Affiliation not available}
\author{Emily A Kaplan}
\affil{Affiliation not available}
\date{\today}
\maketitle
\section{Aims} % Section titles are automatically converted to all-caps.
% Section numbering is automatic.
This experiment was designed to test the relationship between magnetization of protons, dependence on the polarizing field, and Larmor frequency by making a direct measurement of the gyromagnetic ratio and the field dependence of the precession frequency of protons in weak fields.
\section{Introduction}
In 1945, Edward Purcell, Robert Pound and Henry Torrey worked on the development of radar during World War II at MIT Radiation Laboratory. Purcell, who led the group, had been successful in producing and detecting radio frequency power and in absorbing such radio frequency power. Purcell's work led to the discovery of nuclear magnetic resonance (NMR). \\\\
Nuclear magnetic resonance is when nuclei in a magnetic field absorb and re-emit electromagnetic radiation. In order to measure NMR, an apparatus must be set at the resonance frequency, which depends on the strength of the magnetic field and the properties of the isotope of the atoms. \\\\
In this experiment, our apparatus (Fig.\ref{fig:TeachSpinApparatus}) studied the free precession of nuclear moments in the Earth's magnetic field and in a small applied external field by detecting the collective precession of the nuclear magnetic moments in a particular sample. \\ \\\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.42\columnwidth]{figures/Screen-Shot-2015-12-10-at-9.59.06-am/Screen-Shot-2015-12-10-at-9.59.06-am}
\caption{{\label{fig:TeachSpinApparatus} Earth's field nuclear magnetic resonance (EFNMR1-A) apparatus.%
}}
\end{center}
\end{figure}
The apparatus has two different field coils: Helmholtz coils and gradient coils. The Helmholtz coils produce a region of uniform magnetic field using two solenoid electromagnets on the same axis and can be used to cancel external magnetic fields. In this experiment, we will use the Helmhotz coils to provide the field for experiments on 2H nuclei. The gradient coils make the magnetic field more homogeneous, since there may be local fluctuations in Earth's magnetic field. There are three sets of gradient coils: x-component, y-component, and z-component. The z-component varies linearly in each direction. In this experiment, the gradient coils will allow for the measurement of the spin-lattice relaxation time and homogenize the local Earth's magnetic field.
\subsection{Varying the Polarization time}
When an external magnetic field $B$ is applied to a sample, the magnetization $M$ exponentially approaches the equilibrium magnetization. The magnetization does not assume the equilibrium value instantaneously. The relationship between the polarization time and the voltage is described by the growth rate. The growth rate of $M(t)$ towards $M_{\infty}$ is described by the following equation,
\begin{equation}
\label{eq:growthrate}
M(t)=M_{\infty}(1-e^\frac{t}{T_1})
\end{equation}
In the equation above, $M_{\infty}$ represents the equilibrium Curie value (Eq. \ref{eq:curielaw}). The time constant $T_1$ is known as the spin-lattice relaxation time. In other words, $T_1$ is the time it takes for the magnetization to exponentially approach $M_{\infty}$. Eq. \ref{eq:growthrate} is used to describe the relationship between the voltage verses a changing polarization time. The Curie law can derived using $M=n\mu\left$ where $\left$ represents the average value of $cos\theta$, measuring the alignment between the magnetic moment and the external field B, for all magnetic moments in the sample. The calculation of $\left$ using classical thermodynamics results in the Curie law. The Curie value can be calculated using the low field limit of the Brillouin function which is modeled by the following equation:
\begin{equation}
\label{eq:curielaw}
M=\frac{n\mu^2B}{3kT}
\end{equation}
where $\mu$ is the magnetic moment of each spin, $n$ is the number of magnetic moments per unit volume, and $B$ represents the magnetic field. \\ \\
\subsection{Varying the Magnetic Polarization}
In order to determine the relationship between magnetic field and degree of magnetization, we varied the applied magnetic field. When varying the magnetic field, the fractional populations are affected by the g-factor, or spectroscopic splitting factor. The fractional populations can be described by the lower state $N_1$ and upper state $N_2$. The difference between these two states multiplied by the magnetic moment describes the magnetization of the field. The following relationship, the Brillouin function, can be expressed in the equation below \cite{Kittel_1953},
\begin{equation}
\label{eq:tanh}
M=(N_1-N_2)\mu= N\mu\cdot\frac{e^x-e^{-x}}{e^x+e^{-x}}=N\mu \tanh{x}
\end{equation}
where $x \equiv \frac{\mu B}{k_{B}T}$.
\\ \\
\subsection{Studying Larmor precession}
Larmor precession is when a proton in a magnetic field experiences a magnetic torque that aligns the proton magnetic moment with the field. Due to the angular momentum and spin, the proton's motion is a precession about the magnetic field. Taking $B_{e}= \text{Earth's magnetic field} \approx 43.3 \mu T$ and $\gamma= 2.675 \cdot 10^8 \frac{1}{s\cdot T}$, mathematically the precession frequency would be around,
\begin{equation}
\label{eq:precession}
f=\frac{\omega}{2\pi}= \frac{\gamma B_{e}}{2\pi} \approx 1.843 kHz
\end{equation}
\section{Methods}
%\textit{experimental set up- describe equipment used \\
%techniques used to generate data \\}
%Our setup consisted of a TeachSpin Earth's-Field NMR instrument, as well as two variable current inputs and an oscilloscope. The Earth's Field NMR instrument consisted of a solenoid, gradient coils, and Helmholtz coils, the magnetic field of which could all be varied separately. \textbf{(I will create and insert a more detailed figure of these)} A bottle of the substance for which we were performing NMR (in this case, water) was inserted into the center of the solenoid. The solenoid, with a coil constant of 15 mT/Amp, was used to apply a magnetic field to polarize the water. We could adjust the polarization time (time over which a field was applied) using the Earth's Field NMR instrument, and the magnetic field could be varied by using a variable current input. The field of the Helmholtz coils, which effectively artificially increased or decreased the Earth's magnetic field applied in the sample, was adjusted using a separate variable current input. The gradient coils changed the gradient of the magnetic field in the x, y, and z directions, but the gradient was not varied over the course of the experiment. A bandpass filter, part of the NMR instrument, was used to only let through the Larmor precession frequency and reduce and background noise such as 60 Hz fluctuations in the field due to current-carrying wires in the room. The oscilloscope, which was recording the variations in magnetization due to the Larmor precession, was triggered to start by the NMR instrument when the magnetic field from the solenoid stopped being applied. A simplified block diagram of the electronic setup for this is shown in Fig.~/ref{fig:block}.
Our setup consisted of a TeachSpin Earth's-Field NMR instrument, as well as two variable current inputs and an oscilloscope. The Earth's Field NMR instrument consisted of a solenoid, gradient coils, and Helmholtz coils, the magnetic fields of which could all be varied separately.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/nmr/nmr}
\caption{{\label{fig:coils} A more detailed view of the setup. 1- sample, 2 - solenoid, 3 - gradient coils, and 4 - Helmholtz coils.%
}}
\end{center}
\end{figure}
A bottle of the substance on which NMR was being performed could be inserted into the center of the solenoid (Fig. \ref{fig:coils}). In this experiment, $H_{2}O$ was used. The parts of the apparatus which were key to performing the NMR are the solenoid, Helmholtz coils, and gradient coils. \\\\ %All of these could be varied separately, as could the bandpass filter; they are discussed below.
A solenoid with a coil constant of 15mT/A was used to apply a magnetic field to polarize the water. The magnetic field of the solenoid, which was the polarizing field, could be varied using a variable current input. The field of the Helmholtz coils, which artificially increased or decreased the Earth's magnetic field applied to the sample, was adjusted using a separate variable current input. The gradient coils can be used to change the gradient of the magnetic field in the x, y, and z directions. In our experiment, the gradient coils were used to create a homogeneous field but were otherwise not adjusted over the course of the experiment. \\\\
A bandpass amplifier, part of the NMR instrument, was tuned to resonate at the Larmor precession frequency. The bandpass amplifier restricted the frequencies to those around the Larmor precession frequency, and reduced any background noise such as 60 Hz fluctuations in the field due to fluorescent lighting in the room, as well as amplified the precession frequency. \\\\
In order to measure the precession frequency, we adjusted the scale of the oscilloscope so that we could measure the frequency for 10 cycles. We used the frequency the oscilloscope measured for 10 cycles to determine the frequency of one cycle- the Larmor precession frequency (Fig. \ref{fig:screenshot}). In order to measure amplitude, we chose to measure the amplitudes of three cycles whose peaks fell on the points t=0 ms, t=50 ms, and t=100 ms (Fig. \ref{fig:cursor}). t=0 corresponded to the time when the polarization field is no longer applied and any artifacts from the electronics have died out. \\\\ %and measured the amplitude of the peak at each of those points as they were points which we could find easily on the oscilloscope.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/F0001TEK/F0001TEK}
\caption{{\label{fig:screenshot} Displays the method used to measure the frequency of the precession in Earth's magnetic field. Two cursors, shown vertically on the screen, were used to find the frequency for 10 cycles.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/Screen-Shot-2015-12-19-at-4.41.33-PM/Screen-Shot-2015-12-19-at-4.41.33-PM}
\caption{{\label{fig:cursor} Captures the method used to measure the different amplitudes at different times. The top left screenshot displays the horizontal cursor at t=0ms, the top right screenshot displays the horizontal cursor at t=50ms, and the bottom left screenshot displays the horizontal cursor at t=100ms.%
}}
\end{center}
\end{figure}
By separately changing the strengths of the polarization field and the polarization time, we could study the effects on precession frequency and magnetization. Keeping the polarization time constant at 4s and varying the current from 0.5A to 3.0A (corresponding to varying polarization field from 7.5 mT to 45 mT), we can determine dependence of precession frequency upon polarization field. We found the precession frequency to be $1.852 kHz \pm 0.018 kHz$ (Eq.~\ref{eq:precession}). We then kept the polarization time constant at 5s since this time produced a large enough amplitude but was still not very time consuming to take a lot of data. We also varied the field, measuring the amplitude of the precession data using the method described above. We then fit the data to a linear fit, so we can determine the relationship between polarization time and magnetization. We then kept the polarization field constant at 45 mT and varied polarization time, measuring amplitude and therefore magnetization. We fit the data to Eq.~\ref{eq:growthrate} to find the rate of polarization of the water molecules. We again kept field constant at 45 mT and varied the polarization time, measuring the amplitude to find the time at which saturation occurs. We could then record the amplitude seen when keeping the polarization at the saturation time (\sim{10}s, see Fig.~\ref{fig:measurepolarizationtime}) and varying the current. In this way, we can find the dependence of magnetization on magnetic field, as expressed in Eq. \ref{eq:tanh}.
We can find the relationship between the field in which the precession occurs and the precession frequency by applying a field such that it seems as though the Earth's field has increased. This is done by using Helmholtz coils to apply a field which artificially increases Earth's field. We increase the current in the coils from 0 A to 0.17 A, corresponding to increasing the field from $45.08~\mu T$ to $58.4~\mu T$ (as the coils have a coil constant of $89~\mu T/A$ and this field is applied in addition to Earth's field of $43.3 \mu T$). We tune the bandpass filter frequency range as we go along because the amplitude of the precession on the oscilloscope will be greatest when the resonant frequency of the circuit within the instrument coincides with the Larmor precession frequency. So, as our precession frequency changes with magnetic field, we must adjust the resonant frequency of the circuit to match the Larmor precession frequency. We optimized our signal by using the coarse and fine tuners of the bandpass filter. Once the resonant frequency has been optimized, we can record the precession frequency. We go through this process each time we change the field, and then plot precession frequency vs field (shown in Fig.~\ref{fig:precession}).
\section{Results}
\subsection{Varying the Polarization time}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.91\columnwidth]{figures/Screen-Shot-2015-12-10-at-1.11.01-pm/nmr-plot2}
\caption{{\label{fig:measurepolarizationtime} Displays the voltage verses the polarization time. The data was fitted using Eq.~\ref{eq:growthrate}. Therefore, the exponent for each curve represents the inverse of spin-lattice relaxation time, $T_1$, and the saturation values represent $M_{\infty}$ (shown in Table \ref{table:slopevalues}).Each curve represents a different location at which the amplitude of the oscillating decay was measured, as discussed in the Methods section. For instance, the blue curve was measured by placing the cursor at t=0 ms, the orange curve was measured by placing the cursor at t=50 ms, and the green curve was measured by placing the cursor at t=100 ms.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/Screen-Shot-2015-12-08-at-3.17.36-pm/Screen-Shot-2015-12-19-at-5.44.29-pm}
\caption{{\label{fig:linearized} Plots the linearized data from Fig. \ref{fig:measurepolarizationtime}. The data was linearized with respect to Eq. \ref{eq:growthrate}. Again, each line has the same physical meaning. The slope of each line is the inverse of the relaxation time, within uncertainty.
The information gained from each line is presented below:
\begin{table}
\label{table:slopevalues}
\begin{tabular}{ c c c}
\textit{color of the curve} & \textit{$T_1$} & \textit{$M_{\infty}$}
blue & 0.456 & 5.06
orange & 0.465 & 4.91
green & 0.476 & 4.75
\end{tabular}
\end{table}%
}}
\end{center}
\end{figure}
\subsection{Varying the Polarization Field}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Screen-Shot-2015-12-10-at-3.46.05-pm/Screen-Shot-2015-12-10-at-3.46.05-pm}
\caption{{\label{fig:polarizationtime10s} Generated by varying the magnetic field with a constant polarization time of 10s. The data was fitted using the fit described in Eq. \ref{eq:tanh}. In addition, each curve represents the different magnetization measured at constant cursor position on the oscilloscope. The blue curve was measured by placing the cursor at t=0 ms, the orange curve was measured by placing the cursor at t=50 ms, and the green curve was measured by placing the cursor at t=100 ms.
%We can then find the actual value of the magnetization by finding a conversion factor, which will be discussed in section \ref{sec:Analysis}.%
}}
\end{center}
\end{figure}
\subsection{Studying Larmor precession}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.56\columnwidth]{figures/saturation-fit/saturation-fit}
\caption{{\label{fig:precession} Data points were obtained by measuring the corresponding frequency for a varying magnetic field. The fitted red line was obtained by fitting the simple function $f_{resonance}=a+bH$ where $a=1847 \pm 7$ and $b=42.1 \pm 0.7$. The slope of this line is $\frac{\gamma}{2\pi}$, where $\gamma$ represents the gyromagnetic ratio for protons. Using this method, we experimentally determined $\gamma$=$(2.65\pm0.04) \cdot 10^8~\frac{1}{s\cdot T}$.%
}}
\end{center}
\end{figure}
\section{Analysis} \label{sec:Analysis}
%\textit{meaning of graphs}
%
%\textit{-equations and how they relate to our graphs/agreement with graphs} \\
%
%\textit{-Precession frequency in Earth's field}\\
%\textit{-gyromagnetic constant for protons}\\
%\textit{-polarization relaxation time}\\
%\textit{-magnetization--is it on order quoted in manual--they have a number for 1 ml of water or something similar}\\
\subsection{Varying the Polarization Time}
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 reasonably accurate.
\subsection{Varying the Polarization Field}
We also varied polarizing field by varying current to the solenoid, keeping all other values constant, and measured precession frequency. We found there to be no dependence of precession frequency on polarization field, 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}. We plotted our data against magnetization by using Eq.~\ref{eq:tanh} to find magnetization as we 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. We found $\tilde{\chi}_\nu^2 = 0.45$ for t=0 ms, $\tilde{\chi}_\nu^2 = 0.38$ for t=50 ms, and $\tilde{\chi}_\nu^2 = 0.99$ for t=100 ms, possibly indicating that we overestimated our error for t=0 ms and t=50 ms.
\subsection{Studying Larmor Precession}
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 where the only magnetic field is Earth's field, which we found to be $1847\pm7~Hz$, agreeing with the value of $1852\pm18~Hz$ we previously found for precession frequency. We found $\tilde{\chi}_\nu^2 = 6.21$ for the fit in Fig.~\ref{fig:precession}, which could indicate that our error estimates are too small for some reason, such as there being some error in the way we did our measurements, since we know that the data should match a linear fit.
\section{Discussion}
All of our data was taken from the oscilloscope, which means that we were quite limited in our methods of data collection. For example, as described in our Methods, we found the precession frequency by averaging the frequency over ten cycles and dividing by ten. Although this method is more accurate, we determined that we could only measure frequency to a precision of 18 Hz, and likewise we could only measure the amplitude to a precision of 0.08 V. Had we been using different equipment or an oscilloscope with finer resolution, we might have obtained more precise results. As such, our main sources of error were likely due to the limitations of using the oscilloscope. For example, finding exact values of peak or dips in the data was not always possible. There were occasionally points where the exponential decay may not have quite flattened out yet, but the oscilloscope resolution was too coarse to measure a difference in amplitude between the points we were using to take data.
According to the \href{http://physics.nist.gov/cgi-bin/cuu/Value?gammap}{NIST website}, the gyromagnetic ratio has a value of $(2.675221900 \pm 0.000000018) \cdot 10^8 \frac{1}{s\cdot T}$, which agrees with our measured value of $(2.65 \pm 0.04)\cdot 10^8 \frac{1}{s\cdot T}$. This does speak somewhat to the accuracy of our experiment, as the gyromagnetic ratio is a constant.
%Additionally, we estimated that we were using 125 mL of water, as that is the volume of the bottle which was used, but we cannot be sure that we had exactly 125 mL of water.
\section{Conclusion}
NMR techniques were successfully used to characterize the relationships between magnetization, dependence on the polarizing field, and Larmor precession frequency. The relationships found all followed the relationships expected from theory. We also determined values for Larmor precession frequency $1852\pm18~Hz$ (corresponding to a local Earth's magnetic field of $43.3\pm 0.3\mu T$), spin-lattice relaxation time $2.15\pm0.05~s$, and gyromagnetic ratio for a proton $(2.65\pm0.04) \cdot 10^8~\frac{1}{s\cdot T}$, agreeing with NIST's value of $(2.675221900 \pm 0.000000018) \cdot 10^8 \frac{1}{s\cdot T}$. The main source of our uncertainty is likely due to the the resolution of the oscilloscope.
\begin{acknowledgments}
We would like to thank Professor Nat Fortune for setting up the equipment, guiding us through the lab process, and helping us understand the physics behind Earth's NMR.
%We also would like to express our gratitude to Professor Nat Fortune for helping us when we were confused about our apparatus. Finally, we want to thank Professor Will Williams for helping us to understand the physics behind Franck Hertz experiment.
\end{acknowledgments}
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