This document provides an overview of nuclear magnetic resonance (NMR) techniques for analyzing hydrogen and fluorine-19 nuclei. It describes the magnetic properties of nuclei, how they interact with external magnetic fields to create a net magnetization, and how NMR instrumentation works to manipulate nuclei magnetization and measure signals. The chapter focuses on the theoretical principles behind NMR, including magnetic moments, spin states, Larmor frequencies, relaxation times, and chemical shifts. It also discusses choosing toothpaste and mineral oil samples containing hydrogen and fluorine-19 for experimental analysis.

1.
CHAPTER 1: The Physics of NMR
This thesis focuses on using the techniques of nuclear magnetic resonance (NMR) in
order to study and identify hydrogen and Fluorine­19 nuclei. The reason we choose to
analyze the nuclei of these two elements is for the simple fact that they carry an unpaired
proton. This spare proton will be the key in our ability to exploit the nuclei through NMR
for reasons that will be explored in the following introduction.
1.1 Magnetic Moment and Spin States
In a hydrogen and a Fluorine­19 nucleus, there is an uneven number of protons. In the
case of hydrogen, the most common of all nuclei, there is only one, while Fluorine­19 has
9. Because of this uneven number, there will an unpaired proton and some positive
charge.
This charge will make the nucleus develop a magnetic moment that will interact with
the external, uniform magnetic field . The magnetic moment, , of each nucleus will
act as a tiny magnet and feel compelled to align in the direction of the magnetic field. The
ratio between the nucleus’ magnetic moment and its spin angular momentum is known as
the gyromagnetic ratio, . This relationship is described in the equation below:
(Eq. 1.1)
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2. The values of the gyromagnetic ratio are unique to the nuclei of all elements. For a
hydrogen nucleus, consisting of a single proton, this value is 2.67 x 10​8​
rad/second­tesla
while that of a fluorine nucleus is approximately 2.52 x 10​8​
rad/second­tesla.
Though the magnetic moments of the nuclei will tend to align them along the magnetic
field, there are some that will anti­align. This is determined by the spin quantum number
m​ of the nuclei. Nuclei with ​m​ of +½, whose population we will call , will align with
the magnetic field while nuclei with an ​m​ of ­½ will anti­align with it, whose population
we call .
Between these two states emerges an energy difference that is shown in the diagram on
the following page.
Fig. 1.1 Hydrogen in an External Magnetic Field: As an external magnetic field, ,
increases, the energy difference between the aligned and anti­aligned states increases linearly.
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3. The population of the lower­energy nuclei will outnumber the population of the
higher­energy nuclei. It is this inequality between populations which is responsible for
the magnetization.
1.2 Net Magnetization of the Moments
As the population of nuclei in an external field collectively align and anti­align, they will
in effect create a net magnetization in the direction of the field, which in our case defines
the z­direction.
The expression for the magnetization in the z­direction reflects this difference:
(Eq. 1.2)
The population of nuclei that will adopt a positive spin compared to a negative spin is
described by Eq. 1.3, seen below. This expression is governed by the Boltzmann factor,
k, multiplied with the temperature T of the environment.
(Eq. 1.3)
For nuclei with no unpaired protons, the net magnetization is zero. But with ideal nuclei
such as hydrogen and Fluorine­19, an unpaired proton will yield a net magnetization.
The magnetization of the nuclei will not happen instantaneously but at an exponential
rate that is contained by the thermal equilibrium magnetization value. This magnetization
is derived from a hyperbolic tangent function:
(Eq. 1.4)
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4. To derive this simplification, we recall equation 1.3 and solve for the population of the
higher energy spins, .
(Eq. 1.5)
where
Recalling our expression (Equation 1.2) for the magnetization in the z­direction, we use
this definition of to give us the following:
(Eq. 1.6)
where
Then, we characterize the sum of both spin populations to a single quantity:
(Eq. 1.7)
We rewrite this with in terms of Eq. 1.5, then isolate :
Plugging Eq. 1.8 back into Eq. 1.6, we obtain a magnetization equation as:
By multiplying the numerator and denominator of our equation by , the result
becomes:
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5. The fractional term can be expressed as a hyperbolic tangent function:
We then include the substitution for and conclude the derivation for the net
magnetization (Eq. 1.4):
1.3 The Larmor Frequency and the Rotating Frame (RF)
As the magnetization is aligned along the z­axis due to the external magnetic field, it will
experience a precession, whose rate is characterized by the the Larmor, or resonance,
frequency.
(Eq. 1.8)
We determine what the Larmor frequency in our magnet will be for each nuclei of our
experimentation. For our magnetic field of around 0.5 T, the Larmor frequency for
hydrogen and F­19 nuclei we can calculate values we would expect:
For a Hydrogen nucleus:
For a Fluorine­19 nucleus:
5

6. We consider dealing with this non­stationary magnetization to be undesirable and decide
to make it stationary by introducing a rotating frame (RF). By setting the frequency of
this frame to the Larmor frequency, it will allow us to view the precessing magnetization
as though it is stationary, making its analysis easier.
1.4 ​ ​Application of the First Pulse
The new x and y axes can be designated as x’ and y’. The application of the field, in
the x­y direction and applied perpendicular to causes a torque on the magnetization.
This process is represented in the figure below where a small solenoid coil is situated
between our external, uniform magnet with its field, , pointed in the z­direction.
Fig. 1.2: Representation of and in the uniform magnet where
This solenoid is given a current that creates the field perpendicular to ​ which we call
. The introduction of this new field creates a torque, , on the magnetization that will
flip it from the z­direction onto the x­y plane.
(Eq. 1.9)
6

7. As the magnetization is flipped along the direction of ​, it magnetization will continue
to precess about the z­direction. The solenoid will pick up this flux over time, which will
induce a voltage. The instrument then receives this voltage, or emf, as a signal as is
defined in the equation below, where N is the number of turns on the solenoid.
(Eq. 1.10)
This torque essentially translates into the application of our first pulse or signal, called
the pulse.
At its moment of reaching the plane, the frame stops being transmitted and the
magnetization is left alone. Below is a basic representation of the change:
Fig. 1.3: Magnetization before and after the first pulse
7

8. Once this occurs, the magnetization will want to return to equilibrium and re­align with
the z­axis. The time that this will take is characterized by the spin­lattice relaxation time,
or .
We prevent the magnetization from returning to equilibrium by switching the RF is
turned back on. This time, it is left on for twice the length of time as it was before. Unlike
the magnetization before the RF, this magnetization will not last long and will start to
precess about the stationary z­axis, or the direction of. The magnetization will decay
along the x’ and y’ plane by the rates as follows:
(Eq. 1.11,1.12)
T​2​ is known as the spin­spin relaxation time, which characterizes the decay of the signal.
The second signal hits the system and flips the magnetization about the x’­axis to close
back at the y’­axis.
Figure 1.4: Magnetization before the second pulse (spreading out) and after the second pulse (closing
in)
8

9. The total time that it takes for the magnetization to regroup is called the free induction
decay (FID) time. The FID is ultimately the key to identifying the nuclei that is being
tested. As they engage in the process of relaxation, they return back to their original state
until they are hit with the next series of pulses. The nuclei emit back electromagnetic
signals whose frequencies are related to the energy E.
1.5 Interpreting Chemical Shifts
Chapter 3 will introduce the fact that we were forced to proceed with experimentation
using the NMR instrument from the chemistry department for our detection of
Fluorine­19. Because of this, the mechanics of of chemical shift had to be applied to this
thesis.
The chemical shift, measure in parts per million (ppm), of a nucleus is defined as the
variation between its resonant frequency and the external magnetic field.
(Eq. 1.13)
For our experimentation, we expect the chemical shift for Fluorine­19 to be ­100 ppm, as
this is the accepted value. We will hope to match this value with the NMR instrument
from the chemistry department to prove that there was were Fluorine­19 nuclei in the
toothpaste sample.
9

10.
CHAPTER 2: METHODOLOGY
2.1 Choosing Samples
As this thesis was focused on identifying Fluorine­19 and Hydrogen nuclei, it was
imperative that the samples chosen for experimentation were sure to contain heavy traces
of each nuclei.
AquaFresh toothpaste was chosen because of its expected abundance of fluorine. Crest,
however, was also considered for this same reason, but AquaFresh was ultimately chosen
for convenience. Mineral oil was ideal to test because of its density in protons, which was
better suited for NMR testing.
Fig. 2.1 AquaFresh vs. Crest Toothpaste sample to use in NMR testing
Many of the popular solvents in NMR experimentation contain fluorine, which would
have interfered with the testing for fluorine in our sample. Therefore, we chose to
dissolve the toothpaste in distilled water.
10

11. 2.2 Preparing the Samples for Testing
To prepare the samples for experimentation, they was required to be completely
dissolved to the point where they was clear and no solid particulates were visible. For the
heavy mineral oil, this was not an issue since it was provided in the kit. However, we had
to create a toothpaste solution that was free of any solids from scratch.
2.2.1 Heavy Mineral Oil Sample Preparation
As stated above, the heavy mineral oil was already provided for us in the TeachSpin kit.
We used a pipette to transfer the oil from the container into the test tube. The total
amount of sample for our testing was 5mm. It was important that the amount be around
this number because ​, the field from the RF, was best uniform over half of its length
of 12mm. From the middle of the O­ring to the center of the sample needed to be a
distance of 39mm.
Fig. 2.2 The test tube with the stopper and o­ring containing the heavy mineral oil sample
11

12. 2.2.2 Toothpaste Sample Preparation
One ounce of the AquaFresh toothpaste sample was measured with a precise balance. The
sample was originally dissolved in about 50 mL of distilled water. This amount was
desirable both because the sample could be less concentrated throughout the greater
amount of water and some of the liquid can be lost in the filtration process. The liquid
was placed inside a flat, wide beaker, in which a glass stick was used to mix and dissolve
the sample well. Even after this procedure, there was still a good amount of solid
particulates floating around the liquid. Two further methods of filtration had to be used to
obtain a clear liquid.
The first filtration method consisted of passing the liquid around five times through a
small aperture clogged by wool glass. This was successful in ridding the liquid of the
larger particulates.
Fig. 2.3: The first filtration method we used was running the liquid through a funnel stuffed
with glass wool to filter out the larger solids in the liquid
12

13. The second method was vacuum filtration by passing the liquid into another container
through a narrow tube while a vacuum interrupted the flow and sucked out all the smaller
particulates of the sample. This produced an adequately clear liquid solution that was
ideal for testing.
2.3 Setting up the Mainframe and Magnet
The minimal technology necessary for this thesis comprised of the PS­2 Mainframe, the
permanent magnet, and an oscilloscope.
The mainframe comprised of four components: the 15 MHz Receiver, 21 MHz
Synthesizer, and Pulse Programmer, pictured below:
Fig. 2.4: The mainframe and its four components. This thesis only required us to use the first three ­ the
receiver, synthesizer, and pulse programmer.
13

14. With the Synthesizer, the signal will interact with the magnetization the nuclei is given
with the magnetic field. The pulse programmer can control the duration of the signal and
how often it is repeated. The receiver was capable of amplifying and analyzing the signal,
or the emf , that was returned from the sample.
Fig. 2.5: An image of the entire setup showing the mainframe, magnet, and oscilloscope.
The magnet, which is connected to the mainframe as shown in the picture above, contains
a small slot where the narrow test tube containing the sample was inserted.
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15.
Fig. 2.6: Overhead view of magnet interior. The test tube containing the sample is placed in the middle
slot. The fine and coarse tuning capacitors are on its sides.
All of these components were connected to an oscilloscope which allowed us to view the
interaction between the pulse from the spectrometer and the nuclei of the sample on a
screen.
The mainframe parameters of the single­pulse experiment for heavy mineral oil were as
follows:
Length of A pulse (A_len) 0.02 μs
Length of B pulse (B_len) 10 μs
Frequency 21.400 MHz
P period 0.2 μs
Filter TC 0.75
Table 2.1: Ideal mainframe settings for our testing of heavy mineral oil
To get a reading of the desired 5eV peak, the fine tuning capacitors on both sides of the
sample needed to be twisted with the non­magnetic screw. Screwing in the tuning
15

16. capacitors excessively, however, had the unfortunate consequence of getting one of the
capacitors stuck in place and permanently compressed. This lead to the instrument
yielding no spike to indicate a pulse received.
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17.
CHAPTER 3: RESULTS
Partway through the experimentation, we experienced technical issues with the NMR
instrument in the physics department. We were able to test for the detection of hydrogen
heavy mineral oil, but were unable to test for detection of Fluorine­19 nuclei in
toothpaste. For the latter, we had to proceed using the NMR instrument from the LMU
chemistry department.
3.1 Hydrogen Nuclei Detection Test Results
In testing the mineral oil with our NMR instrument, we were able to produce a peak at
the frequency of our settings in the section before of 21.4 MHz. We measured this peak at
around 5eV in height.
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18.
Fig. 3.1: A picture of our FID results for mineral oil on the oscilloscope
This voltage measured the decrease in magnetization in the x­y plane as the spins
returned to equilibrium at the z­axis. In detecting this magnetization at this frequency,
which was our assumed gyromagnetic ratio for hydrogen, we were able to confirm that
there were large traces of hydrogen nuclei in the heavy mineral oil.
3.2 Fluorine­19 Detection Test Results
Before we could test the toothpaste with our NMR instrument, it experienced technical
difficulties and we were unable to use it effectively again. Therefore, we were forced to
test our solution with the NMR instrument belonging to the LMU chemistry department.
This instrument had a magnetic field strength of 1.9T, considerably more powerful than
the one in the physics department.
18

19. Our results showed that a chemical shift of approximately ­167.32ppm occurred when the
toothpaste solution was tested.
Fig. 3.2: Our presumed chemical shift (ppm) of AquaFresh toothpaste solution. The spike happens at
­167.32 ppm, an incredibly small value.
We initially believed this to have been a result of Fluorine­19 nuclei being detected, but
compared it to the reading the researchers found of ­100 ppm. This made us doubt the
accuracy of this value and prompted us to re­test the toothpaste solution with the same
instrument in an effort to get a better reading.
19

20.
Fig. 3.3: Our second set of results showed a spike occur only at ­76.55 ppm, but nothing else to indicate
a detection of Fluorine­19
Our second test yielded completely different results. We not only failed to reproduce the
spike of our first test, but we failed to get any spike for the toothpaste solution. The only
spike occurred at ­76.55 ppm, which was to be expected as it is the chemical shift
standard of trifluoroacetic acid of the solution. With no other shift recorded, this told us
that the spike we saw in our first reading was the result of an unknown factor.
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21. 3.3 Possible Error in Sample Preparation
Our method of sample preparation may have been the primary reason for the instrument
being unable to detect any Fluorine­19 nuclei in our sample. The researchers in Sichuan
University dehydrated ordinary, store­bought toothpaste at 378 K for two hours before
cooling it in a vacuum drier. The internal standard stock solution was prepared by
weighing about 1.273 g of CFOONa then dissolving it in 1000 mL of de­ionizer water.
The suspension in the liquid was sonicated, or vibrated, for 10 minutes to obtain complete
dissolution.
This may have proved to be a far more effective method than the two methods of
filtration we used. The dissolution for our liquid was not perfect as we were still able to
spot tiny, but noticeable, particulates floating around the liquid sample.
Another problem we encountered was that this shift was incredibly small and difficult to
detect with our NMR instrument. We needed a more sensitive instrument to detect the
small amount of NaFl in the AquaFresh toothpaste (0.015%).
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22.
CONCLUSION
Upon the completion of this thesis, I was successful in detecting traces of hydrogen
nuclei in heavy mineral oil using the TeachSpin system with the NMR instrument and
magnet. I was unable to use the same system to detect Fluorine­19, but use the more
powerful NMR instrument of the chemistry Department. However, even with such a
sensitive tool, there were no results produced that indicated a detection of Fluorine­19 in
the AquaFresh toothpaste liquid sample, something that could be attributed to erroneous
sample preparation.
In all, the NMR instrument of the physics department was found to be operational and
reliable in its results. With the instrument running, I am hopeful that future students
interested in completing a senior thesis in NMR will be able use my thesis as a
foundation or resource for theirs.
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23.
BIBLIOGRAPHY
1. Hornak, Dr. Joseph​. The Basics of MRI​. Henrietta, NY: Interactive Learning Service,
2010. N. pag. Print.
2. Reichert, Jonathan​. Teach Spin Instruction Manual​. Knoxville, TN: The University of
Tennessee (2009)
3. Richards, S. A., and J. C. Hollerton. ​Essential Practical NMR for Organic chemistry​.
Chichester, West Sussex, U.K.: John Wiley, 2011. Print.
4. Vaughn, Joseph, Dr. "NMR Sample Preparation Instructions." ​FSU Dept. of Chem. &
Biochem.​ Florida State University, n.d. Web. 01 Feb. 2016.
5. "Magnetic Resonance Spectroscopy." MIT, Cambridge, MA. Web. 20 Jan. 2016.
6. Deng, Dongli, Deng, Pengchi, Wang, Xiaoyan and Hou, Xiandeng (2009) “Direct
Determination of Sodium Fluoride and Sodium Monofluorophospate in Toothpaste by
Quantitative ​19​
F­NMR: A Green Analytical Method”.Spectroscopy Letters, 42: 6, 334 ­
340
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