Pulsed NMR in Extracting Spin-Spin and Spin-Lattice Relaxation Times of Mineral Oil and Glycerol

Kent Lee, Dean Henze, Patrick Smith, and Janet Chao University of San Diego (Dated: March 2, 2013) Using TeachSpin’s PS1-A nuclear magnetic resonance (NMR) spectrometer, we preformed inversion recovery and spin echo experiments on samples of both glycerol and mineral oil with the goal of determining spin-lattice and spin-spin relaxation times, T1 and T2. In glycerol, T1 was found to be 41.4 ms with an uncertainty of 1.9 ms while T2 was found to be 47.6 ms with an uncertainty of 4.3 ms. In mineral oil, T1 was found to be 20.6 ms with an uncertainty of 1.0 ms while T2 was found to be 19.7 ms with an uncertainty of 1.0 ms. T2 for mineral oil agree with measurements documented by other universities, however, T1 for mineral oil and both relaxation times for glycerol do not agree with measurements documented by other universities.

I. INTRODUCTION II. THEORY

A. Quantum Interpretation Since Bloch and Purcell’s discovery of nuclear magnetic resonance (NMR) in 1946, interest blossomed in the field of NMR theory and application. Improved understand- Protons and neutrons, the constituents of the nucleus ing and accelerated development in the field of NMR has of an atom, are much like electrons in that they are both particles with spin 1 , represented as I = 1 . Both also re- made it invaluable not only in the field of physics, but 2 2 also in the fields of chemistry and medicine. From help- side in discrete energy levels when placed in an external ing to understand the classical and quantum properties of magnetic field and both have magnetic moments. Be- nuclei to chemical structure analysis to soft-tissue imag- cause the nucleus is composed of protons and neutrons, ing, this form of non-invasive, non-destructive technology the nucleus also has angular momentum and a magnetic has contributed to the furthering of countless studies. In moment. In a given nucleus with angular momentum fact, the impact of NMR discovery is so significant that I and magnetic moment µ aligned along the spin axis, it is a widely taught subject both in undergraduate and there is a proportionality that exists, given by graduate courses. In teaching NMR theory, it is important to not only µ = γ~I, (1) explain the theory, but also show experimental applica- tions of the theory in an actual spectrometer to promote a where γ is known as the gyromagnetic ratio, with di- thorough understanding of NMR theory. As such, Teach- mensions of radians per second-tesla. For a proton, Spin has developed several NMR instruments designed γ = 2.675 × 108 s−1T−1. for teaching NMR spectroscopy and NMR theory. It is For a given nucleus of total spin I, it can exist in any of through one of TeachSpin’s products that the following the (2I +1) sublevels mI , where mI = (I,I −1,I −2, ...). exploration of NMR spin-spin and spin-lattice relaxation In the absence of a magnetic field, these sublevels are times was conducted. The goal was to learn more about degenerate. However, when the nucleus is placed in a NMR theory by applying it to two sample liquids, glyc- magnetic field B0, the energies of these sublevels become erol and mineral oil. Specifically, both spin-spin and spin- different, with the energies of each sublevel being given lattice relaxation times for both glycerol and mineral oil by were determined. The significance of T1 and T2 is that they are both characteristic of the chemical being ana- E lyzed. = −γB0mI , (2) ~ The theory behind NMR will first be discussed both from a quantum mechanical perspective and classically with the energy difference between adjacent energy sub- in Sec. II A and Sec. II B. The theory section will also levels, where ∆mI = ±1, being given by include an introduction to π and π/2 pulses in Sec. II C and spin-lattice and spin-spin relaxation times in Sec. ∆E II D and Sec. II E. After discussing the theory, the ex- = γB0 = ω0, (3) perimental design of the apparatus used and the method ~ of determining both relaxation times will be introduced where ω0 is known as the Larmor frequency. The splitting in Sec. III A and Sec. III B and III C respectively. The of nuclear energy levels is illustrated in Fig. 1. results of the experiment will then be presented in Sec. For simplicity, the case of a proton is considered (as in IV, followed by a discussion of the results in Sec. V. the ion H+, where the nucleus is simply a proton). In 2

to the Larmor frequency of the sample, with B1 B0, the probability to be in each state becomes dependent on the length of time that B1 is applied. Thus, by applying a rotating field B1 at the same frequency as the energy difference between the two states, the probability to be in each state vary in a time dependent manner and are predictable. At a time equal to nπ/2 of a cycle of time dependent perturbation (where n is an odd integer), the probability of being in the spin up state is equal to the probability of being in the spin down state; the two states are superimposed. As seen in the Stern-Gerlach experiment, probing a superimposed state yields polarization onto an orthogonal axis. Thus, since the spin states are in the z axis (due to the static magnetic field in the z axis), FIG. 1. When nuclei are placed in a magnetic field, splitting the superimposed states become polarized onto the +x of the energy levels occurs, forcing one energy level to become two. axis, which is the basis of the π/2 pulse discussed later. Therefore, in a nucleus with nonzero angular momentum (I) and nonzero magnetic moment (µ), the simulta- this experiment, we will be studying the NMR of a co- neous presence of a static magnetic field B0 and a per- valently bonded hydrogen atom. It is important to note turbing magnetic field B1 operating at the Larmor fre- that, in a free hydrogen atom, there will be no NMR due quency yields equally populated spin states if applied for to an unpaired electron. However, since the NMR of a the proper duration of time. Because B1 rotates at the covalently bonded hydrogen has paired electron spin, it Larmor frequency, it is able to promote spins from one 1 state to another, given that the energy difference is the can be treated as a proton. The spin of a proton is 2 , 1 Larmor frequency. Once the presence of B1 is removed, so the total nuclear spin (I) is I = 2 and there are only two sublevels, m = − 1 and m = + 1 . In the absense however, the population of the states will decay back I 2 I 2 to that of thermal equilibrium, with higher population of a magnetic field B0, the two sublevels are degenerate. When a B is applied to the proton, the sublevels split in the lower energy state. Since spontaneous emission 0 is forbidden by selection rules, this decay occurs due to in energy, with the state with spin parallel to B0 being lower in energy. Thus, defining +z by the direction of spin interactions with both the spins of other nuclei in- 1 side the molecule and spins of nuclei in other molecules. B0, mI = + 2 is the lower energy state. Given a population of protons in thermal equlibirum, the population The characteristic decay times due to these interactions 1 are known as spin-lattice and spin-spin relaxation times distribution in the lower energy spin up state (mI = + 2 ) will be greater than the population distribution in the respectively and will be discussed later. 1 higher energy spin down state (mI = − 2 ) because of the Boltzmann distribution

−E/kT N(E) = N0e , (4) where N(E) is the number of particles in energy state E, N0 is the normalization factor, T is the temperature in Kelvins, and k is the Boltzmann constant [1]. Since the energy of each state is known, from Eq. 2, the population distribution in this two state system is given by

mI γB0 /kT N(mI ) = N0e ~ . (5) FIG. 2. The population in each state depends on the magnetic When a population of protons are at thermal equilib- field strength, with a larger population in the lower energy, rium in the presence of a magnetic field, the lower en- spin up state. ergy, spin up state will have a higher population than the higher energy spin down state by Eq. 5 above because they have different mI . The unequally occupied states’ B. Classical Interpretation populations is also dependent on the magnitude of B0, as illustrated in Fig. 2. Thus, the populations of each state are time independent. However, it can be shown by As discussed above, in thermal equilibrium within a time-dependent perturbation theory that, by applying a magnetic field, a population of protons will have an un- perturbing magnetic field B1 rotating at frequency equal equal distribution of spin states, with a higher popula- 3 tion of protons being in the spin up state than the spin applied, the probabilies to be in each state are known. down state, as demonstrated by Eq. 4. This leads to a Two important pulse lengths are π/2 and π because at net magnetization in the proton population when placed these lengths of time, the population in the two states in a static magnetic field, where a net magnetic align- are equal and inverted, respectively. Thus, applying a ment parallel to the magnetic field, in the +z direction, π/2 pulse (applying H1 for an amount of time equal to is achieved. π/2 of the cycle) will cause the two populations to shift Classically, nuclei with nonzero spin can be considered from the equilibrium distribution of higher population in to be a small rotating bar magnet with magnetic moment spin up state to having equal populations in both states. µ and angular momentum vector I. If a constant mag- Similarly, applying a π pulse will cause the two popula- netic field B0 is applied along the z axis, B0 will apply a tions to be inverted, which, in a system in equilibrium, torque on the magnetic moment, leading to I precessing means a shift from spin up state having a higher popula- around the z axis with an angular frequency ω0 given by tion to the spin down state having a higher population. The actual length of time for a given H1 at frequency ω to achieve a π/2 and a π pulse are, respectively, ω0 = γB0, (6) π which is also known as the Larmor frequency (which is 2 π t π = and tπ = . (8) the same as Eq. 3, which describes the frequency re- 2 ω ω quired for transition between adjacent sublevels, but derived from two different phenomenon), with the precession being known as the Larmor precession. D. T1 and Spin-Lattice Relaxation When a magnetic field B1 orthogonal to and much weaker than the static magnetic field rotating around When a sample of nuclei are examined in absence of the z axis at an angular frequency ω is used, the nu- any magnetic field, the population of each state should clear magnetic dipole is torqued. When ω = ω0, a pro- be equal because the two states are degenerate. The ductive, non-cancelling torque is applied to the nuclear magnetization of a sample is given by magnetic dipole, increasing the angle θ between the precessing magnetic moment’s spin axis and B0. Since the energy of the nucleus in a magnetic field is Mz = (N1 − N2)µ, (9)

where N1 and N2 are the population in each of the states. Thus in the absence of a magnetic field, the magnetiza- E = −µ · B0 = −γ~IB0 cos θ, (7) tion should be zero. However, when placed in a static this change in θ due to B1 causes a change in the energy magnetic field, magnetization of the sample occurs be- of the nucleus, which is analagous to the transitions because the two states are now different in energy, which, tween sublevels explained in Sec. II A, the section about as predicted by Eq. 4, will lead to a higher population the quantum interpretation. Furthermore, if the system in the lower energy state. Since N1 6= N2, there is a net is observed from a rotating frame of reference around magnetization. However, this net magnetization when the z axis at an angular frequency of ω0, I will appear the sample is placed in a static magnetic field is not in- to precess around B1 with a frequency of ω = γB1 due stantaneous; a finite time is required for the population to Larmor precession. If B1 is administered for just the to reach equilibrium within B0. This magnetization pro- right duration of time (known as a π/2 pulse because it is cess can be described by a quarter of the cycle), the dipole is torqued just enough to begin to precessing in the xy plane, an event which dMz M0 − Mz can be observed if there is a coil receiver aligned with = − , (10) dt T the x axis. This precession in the xy plane can be seen 1 as a superposition of both spin up and spin down states where M0 is the magnetization as time goes to infinity and is analagous to the equally populated states in the and T1 is the characteristic time scale for reaching M0. quantum discussion above. Thus, T1 yields an idea of the amount of time it takes to reach magnetization from an unmagnetized sample. T1 is unique for each sample and typically ranges from C. π/2 and π Pulses microseconds to seconds. T1 is also known as the spin- lattice relaxation time because when going from equal The previous two subsections have asserted that the populations in both states to higher population in the probability to be in each state (spin up or spin down) lower energy state, energy is released from the nuclei to is time independent when nuclei are in a static magnetic the surrounding nuclei within the same molecule, which field B0. However, when a perturbing magnetic field H1 is is also known as the lattice. The time it takes for this applied, the probability to be in each state becomes time energy exchange from the nuclei to the lattice contributes dependent. Thus, for a given length of time that H1 is to why magnetization is not instantaneous. The study of 4 what causes different samples to have different T1 is a major topic in magnetic resonance.

E. T2 and Spin-Spin Relaxation

Also of interest is the amount of time it takes to reach equilibrium on the transverse plane. That is, how long would it take B0 to reduce magnetization from being only in one direction in the xy plane to being in every direc- FIG. 3. Block diagram of the apparatus used to study T1 and tion in the xy plane, and thus, equilibrium. One could T2. The pulse programmer sets how long a radiofrequency imagine if a π/2 pulse was administered after a sample is applied to the sample inside the static magnetic ﬁeld. A has been in thermal equilibrium within B0, the magne- receiver receives a signal from the sample and displays it on tization would then move from being on +z axis to +x the oscilloscope. axis. The question, then, is what is the time scale that it takes to cause magnetization in the +x axis to decay. This process is governed by the diﬀerential equations

dM M dM M x = − x and y = − y , (11) dt T2 dt T2 where Mx and My are the x and y components of the transverse magnetization and T2 is the characteristic relaxation time for spin-spin relaxation [2]. The solutions to these equations are

− t − t Mx(t) = M0e T2 and My(t) = M0e T2 . (12)

This is known as spin-spin relaxation because the decay to equilibrium is based off of the spins of nuclei in neighboring molecules. Because all nuclei in the sample have varying spin, the spin of the nuclei in neighboring molecules produce magnetic fields that affect the spin of FIG. 4. The sample chamber of the NMR specrometer used. nuclei in other molecules. These varying spins interact The main components are the receiver coil, the transmitter and causes decay to equilibrium to occur at a faster rate coil, and the two permenant magnets (not shown, but are coming in and out of the page). than simply due to spin-lattice interactions. As such, T2 values are generally smaller than T1 values for a given sample. shown, but going in and out of the page) are all orthogonal to each other. The two permenant magnets are coax- ial, with one having the south pole facing the sample and III. EXPERIMENTAL DESIGN the other having the north pole facing the sample. There are wires connecting the sample chamber to the control A. Experimental Apparatus panel to allow the control of the transmitted radiowaves and recording of the sample response. The receiver coil The apparatus used was a TeachSpin PS1-A NMR is wrapped around the sample tube while the transmit- spectrometer. The apparatus consists of a sample cham- ter coils are perpendicular to the axis of the tube, with ber, a control console, and an oscilliscope. A block di- coils sandwiching the sample tube. The sample tube was agram of the apparatus is shown in Fig. 3. The oscil- filled with at least 2 mL of sample, with the samples loscope is clearly labeled and the sample chamber is the being glycerol and mineral oil. box between the labeled permenant magnets. All the The control console is divided into three modules: the other components shown in the block diagram constitute 15 MHz receiver, the pulse programmer, and the com- the control console. pound oscillator, amplifier, and mixer. The receiver is The sample chamber is made of a small holder to place connected to the receiver coil from the sample chamber the sample vial in, a receiver coil, a transmitter coil, and to receive and amplify the radio frequency induced EMF two permenant magnets. As seen in Fig. 4, the receiver from the sample. The pulse programer is used to generate coil, transmitter coil, and two permenant magnets (not the pulses used in the experiments. It is able to generate 5 pulses ranging from 1 to 30 µs with the option of setting been inverted, with the spin down state having the higher the delay time between two pulses to be between 10 µs population. Then, approximately 2 ms later, a second to 9.99 s. It is also possible to select repitition times for pulse should be given for a length of time where the re- the pulse sequences from 1 ms to 10 s and the number ceiver signal is maximized, which is known as the π/2 of secondary pulses can be chosen from 0 to 99. The os- pulse. The receiver signal must be maximized because cillator is a tunable 15 MHz oscillator that can be tuned this informs the experimenter that the magnetization is by 1,000 Hz or 10 Hz. now pointing along the axis of the receiver coil. The re- The oscilloscope is connected to the mixer and the receiver signal should only have one peak and should look ceiver to allow recording and visualization of the voltage like the first peak in Fig. 5. This is known as a free in- output from both. It is through the oscilloscope that duction decay (FID) and represents the decaying of the most observations in this experiment were made. Data magnetization in the +x direction back into the +z di- was collected by reading from the oscilloscope. rection. The height of each FID should be measured and Before beginning the experiments the following default recorded as the time delay between the π pulse and the setup should be used: place the sample vial into the sam- π/2 pulse is increased in 1 ms increments. The height ple chamber, turn on both the oscilloscope and the con- should decrease to zero and then increase again until it trol console, the oscilloscope time scale set to 2.5 ms/box, levels off. the voltage scale for both mixer and receiver tuned to around 2.5 V/box and 1.5 V/box respectively, the receiver ”Tuning” knob tuned to maximize the receiver C. Spin Echo Experiment to Determine T2 reading on the oscilloscope, pulse programmer mode set to internal, repetition time set to 100 ms at 100%, synch To determine the time it takes for a transverse mag- A, A and B pulses both on, and the number of B pulses netization (in the +x axis) to dephase into no net trans- set to 1. The frequency on the oscillator is then tuned to verse magnetization, the first step required is to create around 15.53516 MHz for mineral oil and 15.53383 MHz a magnetization in the +x axis with a π/2 pulse, which for glycerol. However, the underlying goal of tuning the causes the population in both spin up and spin down frequency is to tune it such that the mixer signal is a states to be equal. While theoretically, the width of the single peak that decays into a flat line. FID that is measured after the pulse should yield the time required to dephase, because magnets all have spacial inhomogeneity, this is not true. This inhomogeneity B. Inversion Recovery Experiment to Determine T1 causes spins in different spacial positions to have a different Larmor frequency, which causes dephasing to occur To determine how long it takes for a nonequilibrium more quickly than the ideal case. Thus, after the FID, a magnetization to decay back to equilibrium, it would π pulse must be administered. This causes the direction be ideal to have a receiver along the +z axis, which is of the spins to flip 180 degrees. This essentially can- the equilibribum position. However, because the receiver cels out the spacial inhomogeneity by having the spins coils are not set up to receive in this axis, the magnetiza- that are dephasing faster, which have traveled farther, to tion should first be perturbed to place it in the nonequi- rephase at the same pace, thus recovering the net mag- libirum −z axis. Then, at different time intervals, the netization and yielding an echo of the FID as it rephases magnetization should be probed by perturbing it again and dephases again. Multiples of these π pulses can be to place it in the x axis, which allows the receiver coil to administered at same time delays apart. The height of determine the magnetization magnitude at a given time these echos can be recorded and a curve can be fitted to after a nonequilibrium position. From this data, T1 can the data to yield T by using Eq. 12. then be determined by fitting the data to a curve of the 2 This experiment requires two pulses, a π/2 pulse to form of magnetize along the +x axis, into the receiver coils, followed by a π pulse after the FID to obtain an echo of −t/T1 the FID. To ensure complete magnetization along the Mz(t) = M0(1 − 2e ), (13) +x axis, the first pulse should be set to a length of time which is derived from Eq. 10 with the initial condition such that the receiver signal, showing an FID (labeled of Mz(0) = −M0, where M0 is the equilibrium magneti- in Fig. 5), is maximized. Then, as soon as the FID has zation. decayed back to background noise, a second pulse with This experiment requires two pulses, a π pulse to invert a time of approximately double that of the first, should the magnetization from +z axis to −z axis, and a π/2 be applied, with the effect of creating an echo (labeled pulse to align the magnetization into the receiver coils in Fig. 5) peak. As the delay time is increased, the echo (+x axis). Thus, the first pulse should be set to being peak should decrease in height until no echo is seen. The a length of time where the receiver signal is minimized, echo peaks are recorded, along with the time delay be- since the −z axis is perpendicular to the receiver coils tween the beginning of the first pulse to the peak of the and thus will contribute no signal. This length pulse is echo (which is equivalent to twice the time delay between known as the π pulse and the spin population has now the first pulse and the second pulse). 6

10 1.4

FID 1.2 5

1 0

0.8 Echo Measurement −5 y = 8.61 * ( 1 − 2 * exp ( − t / 0.0403 ) ) 0.6 pi/2 Pulse Arbitrary Magnetization Magnitude Arbitrary Magnetization Echo 0.4 −10 pi Pulse 0 0.02 0.04 0.06 0.08 0.1 pi Pulse Echo Time from pi−pulse to pi/2−pulse (s) pi Pulse 0.2 FIG. 6. Plot of the glycerol inversion recovery experiment 0 0 100 200 300 400 500 600 700 800 900 data where T1 was determined from the x-intercept of the Arbitrary Time graph.

FIG. 5. Plot of a pulse sequence of π/2 followed by multiple 10 π pulses. This ﬁgure clearly demonstrates the FID and the echos used to determine T2. 5

IV. RESULTS 0

−5 For all T1 data, after collection, a plot was made of Measurement time delay from π-pulse to π/2-pulse in seconds versus y = 9.031 * ( 1 − 2 * exp ( − t / 0.02035 ) ) Arbitrary Magnetization Magnitude −10 arbitrary magnetization magnitude, which are shown in 0 0.02 0.04 0.06 0.08 0.1 Fig. 6 and Fig. 7 for glycerol and mineral oil respec- Time from pi−pulse to pi/2−pulse (s) tively. When making this plot, all data before the mag- nitization reaches 0 is plotted as negative because only FIG. 7. Plot of the mineral oil inversion recovery experiment absolute values were recorded. This plot was then fit to a data where T1 was determined from the x-intercept of the trend line based on Eq. 13, from which T1 was obtained. graph. Trendlines were also made for the upper bound and lower bound uncertainties that occured while determining peak heights. Another method that was used to solve for T1 T2 = 19.7 ms with an uncertainty of 1.0 ms. The oscil- takes advantage of the relation lator frequencies used for T2 for glycerol and mineral oil were 15.53453 and 15.53563 MHz respectively. The T2 uncertainties were calculated by obtaining trendlines of tn T1 = , (14) the upper and lower bound uncertainties and extracting ln(2) the T2 of the uncertainty trendlines and subtracting it from the T2 obtained from the measured values. where tn is the time delay from π-pulse to π/2-pulse, in Table I summarizes the results of our experiment. seconds, where the arbitrary magnetization magnitude is zero. Through these methods, it was discovered that, for glycerol, T1 = 41.4 ms with an uncertainty of 1.9 TABLE I. Summary of the T1 and T2 values obtained through ms. For mineral oil, it was discovered that T1 = 20.6 ms this experiment with an uncertainty of 1.0 ms. The oscillator frequencies Sample T1(ms) T2(ms) used for T1 for glycerol and mineral oil were 15.53383 and Glycerol 41.4 ± 1.9 47.6 ± 4.3 15.53516 MHz respectively. T1 was calculated by averag- Mineral Oil 20.6 ± 1.0 19.7 ± 1.0 ing the T1 obtained from both the trendline and Eq. 14. T1 uncertainty was obtained by chosing the largest of the uncertainties determined from each method. For all the T2 data, a plot was made of the time delay between the π/2-pulse and the echo, in seconds, versus V. DISCUSSION arbitrary magnetization magnitude, which is shown in Fig. 8 and Fig. 9 for glycerol and mineral oil respec- It is intriguing that for both glycerol and mineral oil, tively. A best fit line based on Eq. 12 was fit to the data T2 was found to be equal to T1 within uncertainty, con- and T2 was obtained from the trendline equation. From tradicting theory, which asserts that T1 is greater than this, we concluded that, for glycerol, T2 = 47.6 ms with T2. While the sources for experimental error could not an uncertainty of 4.3 ms. For mineral oil, we found that be found for why this is true, we believe that perhaps 7

3.5 Further experiments can be conducted with other samples. More interestingly, while tuning the oscillator for a 3 Measurement y = 3.329 * exp ( − t / 0.04762 ) resonant frequency, we found multiple resonant frequencies that worked for the samples we used. A meaningful 2.5 next step could be to determine T1 and T2 for these fre- 2 quencies also and compare the T1 and T2 values within the same molecule. 1.5 Arbitrary Magnetization Magnitude 1 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Time delay between pi/2−pulse and echo (s) 6

5 FIG. 8. Plot of the glyerol spin echo experiment data from Measurement which a trendline has been ﬁtted. The equation for the trend- 4 y = 6.627 * exp ( − t / 0.01968 ) line was used to determine T2. 3

2 one area of improvement would be to try to reduce the 1 uncertainty, which could be the underlying problem for Arbitrary Magnetization Magnitude 0 this disconnect. However, for glycerol, even reducing un- 0 0.01 0.02 0.03 0.04 0.05 0.06 certainty still places T2 greater than T1. Time delay between pi/2−pulse and echo (s) While our T2 was found to be greater than T1, which is unexpected, our data for mineral oil are within the FIG. 9. Plot of the mineral oil spin echo experiment data documented range for T2 and outside of the documented from which a trendline has been ﬁtted. The equation for the range for T1, which range from 6-60 ms and 30-150 ms trendline was used to determine T2. respectively [3]. Our data for glycerine for both T1 and VI. ACKNOWLEDGEMENTS T2 do not agree with documented data, which claim that the values are 87.5 and 58.0 ms respectively [4]. However, as neither of these sources were from peer reviewed jour- We want to thank Dr. Gregory Severn for his time, nals, the data presented can only be used for comparison patience, and help in troubleshooting and teaching the purposes. process and knowledge required to operate this lab.

[1] Melissinos, AC. Napolitano, J. Experiments in Modern [3] Wagner, EP. Understanding Precessional Frequency, Spin- Physics. Academic Pres, California, 2nd Edition, 2008. Lattice and Spin-Spin Interactions in Pulsed Nuclear Mag- [2] USD Physics Faculty. NMR Spin Echo Lab. TeachSpin, netic Resonance Spectroscopy. University of Pittsburgh 2009. Chem 1430 Manual, 2012. [4] Gibbs, ML. Nuclear Magnetic Resonance. Georgia Insti- tute of Technology Physics Lab Report, 2007.